Publications of Jian-Guo Liu    :chronological  alphabetical  combined  by tags listing:

%% Books   
@book{fds165493,
   Title = {Multi-scale phenomena in complex fluids, Modeling, Analysis
             and Numerical Simulations},
   Publisher = {World Scientific},
   Editor = {T. Hou and C. Liu and J.-G. Liu},
   Year = {2009},
   ISBN = {978-981-4273-25-1},
   Key = {fds165493}
}

@book{fds165494,
   Title = {Hyperbolic Problems: Theory, Numerics and Applications,
             volume I: Plenary & Invited Talks; volume II: Contributed
             Talks},
   Volume = {67},
   Series = {Proceedings of Symposia in Applied Mathematics},
   Publisher = {American Mathematical Society},
   Editor = {E. Tadmor and J.-G. Liu and A.E. Tzavaras},
   Year = {2009},
   ISBN = {978-0-8218-4728-2},
   Key = {fds165494}
}

@book{fds70657,
   Title = {Dynamics in Models of Coarsening, Coagulation, Condensation
             and Quantization},
   Publisher = {World Scientific},
   Editor = {W. Bao and J.-G. Liu},
   Year = {2007},
   ISBN = {9789812708502},
   Key = {fds70657}
}


%% Papers Published   
@article{fds374862,
   Author = {Feng, Y and Li, L and Liu, JG and Xu, X},
   Title = {EXISTENCE OF WEAK SOLUTIONS TO p-NAVIER-STOKES
             EQUATIONS},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {29},
   Number = {4},
   Pages = {1868-1890},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2024},
   Month = {April},
   url = {http://dx.doi.org/10.3934/dcdsb.2023159},
   Abstract = {We study the existence of weak solutions to the
             p-Navier-Stokes equations with a symmetric p-Laplacian on
             bounded domains. We construct a particular Schauder basis in
             W01, p(Ω) with divergence free constraint and prove
             existence of weak solutions using the Galerkin approximation
             via this basis. Meanwhile, in the proof, we establish a
             chain rule for the Lp integral of the weak solutions, which
             fixes a gap in our previous work. The equality of energy
             dissipation is also established for the weak solutions
             considered.},
   Doi = {10.3934/dcdsb.2023159},
   Key = {fds374862}
}

@article{fds375395,
   Author = {Stevens, JB and Riley, BA and Je, J and Gao, Y and Wang, C and Mowery, YM and Brizel, DM and Yin, F-F and Liu, J-G and Lafata, KJ},
   Title = {Radiomics on spatial-temporal manifolds via Fokker-Planck
             dynamics.},
   Journal = {Med Phys},
   Year = {2024},
   Month = {January},
   url = {http://dx.doi.org/10.1002/mp.16905},
   Abstract = {BACKGROUND: Delta radiomics is a high-throughput
             computational technique used to describe quantitative
             changes in serial, time-series imaging by considering the
             relative change in radiomic features of images extracted at
             two distinct time points. Recent work has demonstrated a
             lack of prognostic signal of radiomic features extracted
             using this technique. We hypothesize that this lack of
             signal is due to the fundamental assumptions made when
             extracting features via delta radiomics, and that other
             methods should be investigated. PURPOSE: The purpose of this
             work was to show a proof-of-concept of a new radiomics
             paradigm for sparse, time-series imaging data, where
             features are extracted from a spatial-temporal manifold
             modeling the time evolution between images, and to assess
             the prognostic value on patients with oropharyngeal cancer
             (OPC). METHODS: To accomplish this, we developed an
             algorithm to mathematically describe the relationship
             between two images acquired at time t = 0 $t = 0$ and t > 0
             $t > 0$ . These images serve as boundary conditions of a
             partial differential equation describing the transition from
             one image to the other. To solve this equation, we propagate
             the position and momentum of each voxel according to
             Fokker-Planck dynamics (i.e., a technique common in
             statistical mechanics). This transformation is driven by an
             underlying potential force uniquely determined by the
             equilibrium image. The solution generates a spatial-temporal
             manifold (3 spatial dimensions + time) from which we define
             dynamic radiomic features. First, our approach was
             numerically verified by stochastically sampling dynamic
             Gaussian processes of monotonically decreasing noise. The
             transformation from high to low noise was compared between
             our Fokker-Planck estimation and simulated ground-truth. To
             demonstrate feasibility and clinical impact, we applied our
             approach to 18 F-FDG-PET images to estimate early metabolic
             response of patients (n = 57) undergoing definitive
             (chemo)radiation for OPC. Images were acquired pre-treatment
             and 2-weeks intra-treatment (after 20 Gy). Dynamic radiomic
             features capturing changes in texture and morphology were
             then extracted. Patients were partitioned into two groups
             based on similar dynamic radiomic feature expression via
             k-means clustering and compared by Kaplan-Meier analyses
             with log-rank tests (p < 0.05). These results were
             compared to conventional delta radiomics to test the added
             value of our approach. RESULTS: Numerical results confirmed
             our technique can recover image noise characteristics given
             sparse input data as boundary conditions. Our technique was
             able to model tumor shrinkage and metabolic response. While
             no delta radiomics features proved prognostic, Kaplan-Meier
             analyses identified nine significant dynamic radiomic
             features. The most significant feature was
             Gray-Level-Size-Zone-Matrix gray-level variance
             (p = 0.011), which demonstrated prognostic improvement
             over its corresponding delta radiomic feature (p = 0.722).
             CONCLUSIONS: We developed, verified, and demonstrated the
             prognostic value of a novel, physics-based radiomics
             approach over conventional delta radiomics via data
             assimilation of quantitative imaging and differential
             equations.},
   Doi = {10.1002/mp.16905},
   Key = {fds375395}
}

@article{fds374859,
   Author = {Gao, Y and Liu, J-G},
   Title = {A Selection Principle for Weak KAM Solutions via
             Freidlin–Wentzell Large Deviation Principle of Invariant
             Measures},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {55},
   Number = {6},
   Pages = {6457-6495},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2023},
   Month = {December},
   url = {http://dx.doi.org/10.1137/22m1519717},
   Doi = {10.1137/22m1519717},
   Key = {fds374859}
}

@article{fds374860,
   Author = {Gao, Y and Liu, J-G},
   Title = {Large Deviation Principle and Thermodynamic Limit of
             Chemical Master Equation via Nonlinear Semigroup},
   Journal = {Multiscale Modeling & Simulation},
   Volume = {21},
   Number = {4},
   Pages = {1534-1569},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2023},
   Month = {December},
   url = {http://dx.doi.org/10.1137/22m1505633},
   Doi = {10.1137/22m1505633},
   Key = {fds374860}
}

@article{fds373536,
   Author = {Qi, D and Liu, J-G},
   Title = {High-order moment closure models with random batch method
             for efficient computation of multiscale turbulent
             systems.},
   Journal = {Chaos (Woodbury, N.Y.)},
   Volume = {33},
   Number = {10},
   Pages = {103133},
   Year = {2023},
   Month = {October},
   url = {http://dx.doi.org/10.1063/5.0160057},
   Abstract = {We propose a high-order stochastic-statistical moment
             closure model for efficient ensemble prediction of
             leading-order statistical moments and probability density
             functions in multiscale complex turbulent systems. The
             statistical moment equations are closed by a precise
             calibration of the high-order feedbacks using ensemble
             solutions of the consistent stochastic equations, suitable
             for modeling complex phenomena including non-Gaussian
             statistics and extreme events. To address challenges
             associated with closely coupled spatiotemporal scales in
             turbulent states and expensive large ensemble simulation for
             high-dimensional systems, we introduce efficient
             computational strategies using the random batch method
             (RBM). This approach significantly reduces the required
             ensemble size while accurately capturing essential
             high-order structures. Only a small batch of small-scale
             fluctuation modes is used for each time update of the
             samples, and exact convergence to the full model statistics
             is ensured through frequent resampling of the batches during
             time evolution. Furthermore, we develop a reduced-order
             model to handle systems with really high dimensions by
             linking the large number of small-scale fluctuation modes to
             ensemble samples of dominant leading modes. The
             effectiveness of the proposed models is validated by
             numerical experiments on the one-layer and two-layer Lorenz
             '96 systems, which exhibit representative chaotic features
             and various statistical regimes. The full and reduced-order
             RBM models demonstrate uniformly high skill in capturing the
             time evolution of crucial leading-order statistics,
             non-Gaussian probability distributions, while achieving
             significantly lower computational cost compared to direct
             Monte-Carlo approaches. The models provide effective tools
             for a wide range of real-world applications in prediction,
             uncertainty quantification, and data assimilation.},
   Doi = {10.1063/5.0160057},
   Key = {fds373536}
}

@article{fds368760,
   Author = {Wang, Y and Li, X and Konanur, M and Konkel, B and Seyferth, E and Brajer,
             N and Liu, J-G and Bashir, MR and Lafata, KJ},
   Title = {Towards optimal deep fusion of imaging and clinical data via
             a model-based description of fusion quality.},
   Journal = {Med Phys},
   Volume = {50},
   Number = {6},
   Pages = {3526-3537},
   Year = {2023},
   Month = {June},
   url = {http://dx.doi.org/10.1002/mp.16181},
   Abstract = {BACKGROUND: Due to intrinsic differences in data formatting,
             data structure, and underlying semantic information, the
             integration of imaging data with clinical data can be
             non-trivial. Optimal integration requires robust data
             fusion, that is, the process of integrating multiple data
             sources to produce more useful information than captured by
             individual data sources. Here, we introduce the concept of
             fusion quality for deep learning problems involving imaging
             and clinical data. We first provide a general theoretical
             framework and numerical validation of our technique. To
             demonstrate real-world applicability, we then apply our
             technique to optimize the fusion of CT imaging and hepatic
             blood markers to estimate portal venous hypertension, which
             is linked to prognosis in patients with cirrhosis of the
             liver. PURPOSE: To develop a measurement method of optimal
             data fusion quality deep learning problems utilizing both
             imaging data and clinical data. METHODS: Our approach is
             based on modeling the fully connected layer (FCL) of a
             convolutional neural network (CNN) as a potential function,
             whose distribution takes the form of the classical Gibbs
             measure. The features of the FCL are then modeled as random
             variables governed by state functions, which are interpreted
             as the different data sources to be fused. The probability
             density of each source, relative to the probability density
             of the FCL, represents a quantitative measure of
             source-bias. To minimize this source-bias and optimize CNN
             performance, we implement a vector-growing encoding scheme
             called positional encoding, where low-dimensional clinical
             data are transcribed into a rich feature space that
             complements high-dimensional imaging features. We first
             provide a numerical validation of our approach based on
             simulated Gaussian processes. We then applied our approach
             to patient data, where we optimized the fusion of CT images
             with blood markers to predict portal venous hypertension in
             patients with cirrhosis of the liver. This patient study was
             based on a modified ResNet-152 model that incorporates both
             images and blood markers as input. These two data sources
             were processed in parallel, fused into a single FCL, and
             optimized based on our fusion quality framework. RESULTS:
             Numerical validation of our approach confirmed that the
             probability density function of a fused feature space
             converges to a source-specific probability density function
             when source data are improperly fused. Our numerical results
             demonstrate that this phenomenon can be quantified as a
             measure of fusion quality. On patient data, the fused model
             consisting of both imaging data and positionally encoded
             blood markers at the theoretically optimal fusion quality
             metric achieved an AUC of 0.74 and an accuracy of 0.71. This
             model was statistically better than the imaging-only model
             (AUC = 0.60; accuracy = 0.62), the blood marker-only model
             (AUC = 0.58; accuracy = 0.60), and a variety of purposely
             sub-optimized fusion models (AUC = 0.61-0.70; accuracy =
             0.58-0.69). CONCLUSIONS: We introduced the concept of data
             fusion quality for multi-source deep learning problems
             involving both imaging and clinical data. We provided a
             theoretical framework, numerical validation, and real-world
             application in abdominal radiology. Our data suggests that
             CT imaging and hepatic blood markers provide complementary
             diagnostic information when appropriately
             fused.},
   Doi = {10.1002/mp.16181},
   Key = {fds368760}
}

@article{fds366912,
   Author = {Dou, X and Liu, JG and Zhou, Z},
   Title = {A TUMOR GROWTH MODEL WITH AUTOPHAGY: THE
             REACTION-(CROSS-)DIFFUSION SYSTEM AND ITS FREE BOUNDARY
             LIMIT},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {28},
   Number = {3},
   Pages = {1964-1992},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2023},
   Month = {March},
   url = {http://dx.doi.org/10.3934/dcdsb.2022154},
   Abstract = {In this paper, we propose a tumor growth model to
             incorporate and investigate the spatial effects of
             autophagy. The cells are classified into two phases: normal
             cells and autophagic cells, whose dynamics are also coupled
             with the nutrients. First, we construct a
             reaction-(cross-)diffusion system describing the evolution
             of cell densities, where the drift is determined by the
             negative gradient of the joint pressure, and the reaction
             terms manifest the unique mechanism of autophagy. Next, in
             the incompressible limit, such a cell density model
             naturally connects to a free boundary system, describing the
             geometric motion of the tumor region. Analyzing the free
             boundary model in a special case, we show that the ratio of
             the two phases of cells exponentially converges to a
             “well-mixed” limit. Within this “well-mixed” limit,
             we obtain an analytical solution of the free boundary system
             which indicates the exponential growth of the tumor size in
             the presence of autophagy in contrast to the linear growth
             without it. Numerical simulations are also provided to
             illustrate the analytical properties and to explore more
             scenarios.},
   Doi = {10.3934/dcdsb.2022154},
   Key = {fds366912}
}

@article{fds369041,
   Author = {Gao, Y and Li, T and Li, X and Liu, JG},
   Title = {TRANSITION PATH THEORY FOR LANGEVIN DYNAMICS ON MANIFOLDS:
             OPTIMAL CONTROL AND DATA-DRIVEN SOLVER},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {21},
   Number = {1},
   Pages = {1-33},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2023},
   Month = {March},
   url = {http://dx.doi.org/10.1137/21M1437883},
   Abstract = {We present a data-driven point of view for rare events,
             which represent conformational transitions in biochemical
             reactions modeled by overdamped Langevin dynamics on
             manifolds in high dimensions. We first reinterpret the
             transition state theory and the transition path theory from
             the optimal control viewpoint. Given a point cloud probing
             the manifold, we construct a discrete Markov chain with a
             Q-matrix computed from an approximated Voronoi tesselation
             via the point cloud. We use this Q-matrix to compute a
             discrete committor function whose level set automatically
             orders the point cloud. Then based on the committor
             function, an optimally controlled random walk on point
             clouds is constructed and utilized to efficiently sample
             transition paths, which become an almost sure event in O(1)
             time instead of a rare event in the original reaction
             dynamics. To compute the mean transition path efficiently, a
             local averaging algorithm based on the optimally controlled
             random walk is developed, which adapts the finite
             temperature string method to the controlled Monte Carlo
             samples. Numerical examples on sphere/torus including a
             conformational transition for the alanine dipeptide in
             vacuum are conducted to illustrate the data-driven solver
             for the transition path theory on point clouds. The mean
             transition path obtained via the controlled Monte Carlo
             simulations highly coincides with the computed dominant
             transition path in the transition path theory.},
   Doi = {10.1137/21M1437883},
   Key = {fds369041}
}

@article{fds369849,
   Author = {Qi, D and Liu, J-G},
   Title = {A random batch method for efficient ensemble forecasts of
             multiscale turbulent systems.},
   Journal = {Chaos (Woodbury, N.Y.)},
   Volume = {33},
   Number = {2},
   Pages = {023113},
   Year = {2023},
   Month = {February},
   url = {http://dx.doi.org/10.1063/5.0129127},
   Abstract = {A new efficient ensemble prediction strategy is developed
             for a multiscale turbulent model framework with emphasis on
             the nonlinear interactions between large and small-scale
             variables. The high computational cost in running large
             ensemble simulations of high-dimensional equations is
             effectively avoided by adopting a random batch decomposition
             of the wide spectrum of the fluctuation states, which is a
             characteristic feature of the multiscale turbulent systems.
             The time update of each ensemble sample is then only subject
             to a small portion of the small-scale fluctuation modes in
             one batch, while the true model dynamics with multiscale
             coupling is respected by frequent random resampling of the
             batches at each time updating step. We investigate both
             theoretical and numerical properties of the proposed method.
             First, the convergence of statistical errors in the random
             batch model approximation is shown rigorously independent of
             the sample size and full dimension of the system. Next, the
             forecast skill of the computational algorithm is tested on
             two representative models of turbulent flows exhibiting many
             key statistical phenomena with a direct link to realistic
             turbulent systems. The random batch method displays robust
             performance in capturing a series of crucial statistical
             features with general interests, including highly
             non-Gaussian fat-tailed probability distributions and
             intermittent bursts of instability, while requires a much
             lower computational cost than the direct ensemble approach.
             The efficient random batch method also facilitates the
             development of new strategies in uncertainty quantification
             and data assimilation for a wide variety of general complex
             turbulent systems in science and engineering.},
   Doi = {10.1063/5.0129127},
   Key = {fds369849}
}

@article{fds367493,
   Author = {Gao, Y and Liu, JG and Wu, N},
   Title = {Data-driven efficient solvers for Langevin dynamics on
             manifold in high dimensions},
   Journal = {Applied and Computational Harmonic Analysis},
   Volume = {62},
   Pages = {261-309},
   Year = {2023},
   Month = {January},
   url = {http://dx.doi.org/10.1016/j.acha.2022.09.003},
   Abstract = {We study the Langevin dynamics of a physical system with
             manifold structure M⊂Rp based on collected sample points
             {xi}i=1n⊂M that probe the unknown manifold M. Through the
             diffusion map, we first learn the reaction coordinates
             {yi}i=1n⊂N corresponding to {xi}i=1n, where N is a
             manifold diffeomorphic to M and isometrically embedded in
             Rℓ with ℓ≪p. The induced Langevin dynamics on N in
             terms of the reaction coordinates captures the slow time
             scale dynamics such as conformational changes in biochemical
             reactions. To construct an efficient and stable
             approximation for the Langevin dynamics on N, we leverage
             the corresponding Fokker-Planck equation on the manifold N
             in terms of the reaction coordinates y. We propose an
             implementable, unconditionally stable, data-driven finite
             volume scheme for this Fokker-Planck equation, which
             automatically incorporates the manifold structure of N.
             Furthermore, we provide a weighted L2 convergence analysis
             of the finite volume scheme to the Fokker-Planck equation on
             N. The proposed finite volume scheme leads to a Markov chain
             on {yi}i=1n with an approximated transition probability and
             jump rate between the nearest neighbor points. After an
             unconditionally stable explicit time discretization, the
             data-driven finite volume scheme gives an approximated
             Markov process for the Langevin dynamics on N and the
             approximated Markov process enjoys detailed balance,
             ergodicity, and other good properties.},
   Doi = {10.1016/j.acha.2022.09.003},
   Key = {fds367493}
}

@article{fds370086,
   Author = {Liu, JG and Tang, Y and Zhao, Y},
   Title = {ON THE EQUILIBRIUM OF THE POISSON-NERNST-PLANCK-BIKERMANN
             MODEL EQUIPPING WITH THE STERIC AND CORRELATION
             EFFECTS},
   Journal = {Communications in Mathematical Sciences},
   Volume = {21},
   Number = {2},
   Pages = {485-515},
   Year = {2023},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2023.v21.n2.a8},
   Abstract = {The Poisson-Nernst-Planck-Bikermann (PNPB) model, in which
             the ions and water molecules are treated as different
             species with non-uniform sizes and valences with
             interstitial voids, can describe the steric and correlation
             effects in ionic solution neglected by the
             Poisson-Nernst-Planck and Poisson-Boltzmann theories with
             point charge assumption. In the PNPB model, the electric
             potential is governed by the fourth-order Poisson-Bikermann
             (4PBik) equation instead of the Poisson equation so that it
             can describe the correlation effect. Moreover, the steric
             potential is included in the ionic and water fluxes as well
             as the equilibrium Fermi-like distributions which
             characterizes the steric effect quantitatively. In this
             work, we analyze the self-adjointness and the kernel of the
             fourth-order operator of the 4PBik equation. Also, we show
             the positivity of the void volume function and the convexity
             of the free energy. Following these properties, the
             well-posedness of the PNPB model in equilibrium is given.
             Furthermore, because the PNPB model has an energy dissipated
             structure, we adopt a finite volume scheme which preserves
             the energy dissipated property at the semi-discrete level.
             Various numerical investigations are given to show the
             parameter dependence of the steric effect to the steady
             state},
   Doi = {10.4310/CMS.2023.v21.n2.a8},
   Key = {fds370086}
}

@article{fds372916,
   Author = {Gao, Y and Liu, JG},
   Title = {Random Walk Approximation for Irreversible Drift-Diffusion
             Process on Manifold: Ergodicity, Unconditional Stability and
             Convergence},
   Journal = {Communications in Computational Physics},
   Volume = {34},
   Number = {1},
   Pages = {132-172},
   Year = {2023},
   Month = {January},
   url = {http://dx.doi.org/10.4208/cicp.OA-2023-0021},
   Abstract = {Irreversible drift-diffusion processes are very common in
             biochemical reactions. They have a non-equilibrium
             stationary state (invariant measure) which does not satisfy
             detailed balance. For the corresponding Fokker-Planck
             equation on a closed manifold, using Voronoi tessellation,
             we propose two upwind finite volume schemes with or without
             the information of the invariant measure. Both schemes
             possess stochastic Q-matrix structures and can be decomposed
             as a gradient flow part and a Hamiltonian flow part,
             enabling us to prove unconditional stability, ergodicity and
             error estimates. Based on the two upwind schemes, several
             numerical examples – including sampling accelerated by a
             mixture flow, image transformations and simulations for
             stochastic model of chaotic system – are conducted. These
             two structure-preserving schemes also give a natural random
             walk approximation for a generic irreversible
             drift-diffusion process on a manifold. This makes them
             suitable for adapting to manifold-related computations that
             arise from high-dimensional molecular dynamics
             simulations.},
   Doi = {10.4208/cicp.OA-2023-0021},
   Key = {fds372916}
}

@article{fds374861,
   Author = {Gao, Y and Liu, J-G and Li, W},
   Title = {Master equations for finite state mean field games with
             nonlinear activations},
   Journal = {Discrete and Continuous Dynamical Systems -
             B},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2023},
   url = {http://dx.doi.org/10.3934/dcdsb.2023204},
   Doi = {10.3934/dcdsb.2023204},
   Key = {fds374861}
}

@article{fds373606,
   Author = {Gao, Y and Liu, J-G},
   Title = {Stochastic Chemical Reaction Systems in Biology},
   Journal = {SIAM REVIEW},
   Volume = {65},
   Number = {2},
   Pages = {593-+},
   Year = {2023},
   Key = {fds373606}
}

@article{fds366136,
   Author = {Gao, Y and Liu, JG},
   Title = {Revisit of Macroscopic Dynamics for Some Non-equilibrium
             Chemical Reactions from a Hamiltonian Viewpoint},
   Journal = {Journal of Statistical Physics},
   Volume = {189},
   Number = {2},
   Publisher = {Springer Science and Business Media LLC},
   Year = {2022},
   Month = {November},
   url = {http://dx.doi.org/10.1007/s10955-022-02985-5},
   Abstract = {Most biochemical reactions in living cells are open systems
             interacting with environment through chemostats to exchange
             both energy and materials. At a mesoscopic scale, the number
             of each species in those biochemical reactions can be
             modeled by a random time-changed Poisson processes. To
             characterize macroscopic behaviors in the large number
             limit, the law of large numbers in the path space determines
             a mean-field limit nonlinear reaction rate equation
             describing the dynamics of the concentration of species,
             while the WKB expansion for the chemical master equation
             yields a Hamilton–Jacobi equation and the Legendre
             transform of the corresponding Hamiltonian gives the good
             rate function (action functional) in the large deviation
             principle. In this paper, we decompose a general macroscopic
             reaction rate equation into a conservative part and a
             dissipative part in terms of the stationary solution to the
             Hamilton–Jacobi equation. This stationary solution is used
             to determine the energy landscape and thermodynamics for
             general chemical reactions, which particularly maintains a
             positive entropy production rate at a non-equilibrium steady
             state. The associated energy dissipation law at both the
             mesoscopic and macroscopic levels is proved together with a
             passage from the mesoscopic to macroscopic one. A non-convex
             energy landscape emerges from the convex mesoscopic relative
             entropy functional in the large number limit, which picks up
             the non-equilibrium features. The existence of this
             stationary solution is ensured by the optimal control
             representation at an undetermined time horizon for the weak
             KAM solution to the stationary Hamilton–Jacobi equation.
             Furthermore, we use a symmetric Hamiltonian to study a class
             of non-equilibrium enzyme reactions, which leads to
             nonconvex energy landscape due to flux grouping degeneracy
             and reduces the conservative–dissipative decomposition to
             an Onsager-type strong gradient flow. This symmetric
             Hamiltonian implies that the transition paths between
             multiple steady states (rare events in biochemical
             reactions) is a modified time reversed least action path
             with associated path affinities and energy barriers. We
             illustrate this idea through a bistable catalysis reaction
             and compute the energy barrier for the transition path
             connecting two steady states via its energy
             landscape.},
   Doi = {10.1007/s10955-022-02985-5},
   Key = {fds366136}
}

@article{fds367494,
   Author = {Craig, K and Liu, JG and Lu, J and Marzuola, JL and Wang,
             L},
   Title = {A proximal-gradient algorithm for crystal surface
             evolution},
   Journal = {Numerische Mathematik},
   Volume = {152},
   Number = {3},
   Pages = {631-662},
   Year = {2022},
   Month = {November},
   url = {http://dx.doi.org/10.1007/s00211-022-01320-0},
   Abstract = {As a counterpoint to recent numerical methods for crystal
             surface evolution, which agree well with microscopic
             dynamics but suffer from significant stiffness that prevents
             simulation on fine spatial grids, we develop a new numerical
             method based on the macroscopic partial differential
             equation, leveraging its formal structure as the gradient
             flow of the total variation energy, with respect to a
             weighted H- 1 norm. This gradient flow structure relates to
             several metric space gradient flows of recent interest,
             including 2-Wasserstein flows and their generalizations to
             nonlinear mobilities. We develop a novel semi-implicit time
             discretization of the gradient flow, inspired by the
             classical minimizing movements scheme (known as the JKO
             scheme in the 2-Wasserstein case). We then use a primal dual
             hybrid gradient (PDHG) method to compute each element of the
             semi-implicit scheme. In one dimension, we prove convergence
             of the PDHG method to the semi-implicit scheme, under
             general integrability assumptions on the mobility and its
             reciprocal. Finally, by taking finite difference
             approximations of our PDHG method, we arrive at a fully
             discrete numerical algorithm, with iterations that converge
             at a rate independent of the spatial discretization: in
             particular, the convergence properties do not deteriorate as
             we refine our spatial grid. We close with several numerical
             examples illustrating the properties of our method,
             including facet formation at local maxima, pinning at local
             minima, and convergence as the spatial and temporal
             discretizations are refined.},
   Doi = {10.1007/s00211-022-01320-0},
   Key = {fds367494}
}

@article{fds364962,
   Author = {Li, L and Liu, JG and Tang, Y},
   Title = {Some Random Batch Particle Methods for the
             Poisson-Nernst-Planck and Poisson-Boltzmann
             Equations},
   Journal = {Communications in Computational Physics},
   Volume = {32},
   Number = {1},
   Pages = {41-82},
   Publisher = {Global Science Press},
   Year = {2022},
   Month = {July},
   url = {http://dx.doi.org/10.4208/cicp.OA-2021-0159},
   Abstract = {We consider in this paper random batch interacting particle
             methods for solving the Poisson-Nernst-Planck (PNP)
             equations, and thus the Poisson-Boltzmann (PB) equation as
             the equilibrium, in the external unbounded domain. To
             justify the simulation in a truncated domain, an error
             estimate of the truncation is proved in the symmetric cases
             for the PB equation. Then, the random batch interacting
             particle methods are introduced which are O(N) per time
             step. The particle methods can not only be considered as a
             numerical method for solving the PNP and PB equations, but
             also can be used as a direct simulation approach for the
             dynamics of the charged particles in solution. The particle
             methods are preferable due to their simplicity and
             adaptivity to complicated geometry, and may be interesting
             in describing the dynamics of the physical process.
             Moreover, it is feasible to incorporate more physical
             effects and interactions in the particle methods and to
             describe phenomena beyond the scope of the mean-field
             equations.},
   Doi = {10.4208/cicp.OA-2021-0159},
   Key = {fds364962}
}

@article{fds361926,
   Author = {Degond, P and Frouvelle, A and Liu, JG},
   Title = {FROM KINETIC TO FLUID MODELS OF LIQUID CRYSTALS BY THE
             MOMENT METHOD},
   Journal = {Kinetic and Related Models},
   Volume = {15},
   Number = {3},
   Pages = {417-465},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2022},
   Month = {June},
   url = {http://dx.doi.org/10.3934/krm.2021047},
   Abstract = {This paper deals with the convergence of the
             Doi-Navier-Stokes model of liquid crystals to the
             Ericksen-Leslie model in the limit of the Deborah number
             tending to zero. While the literature has investigated this
             problem by means of the Hilbert expansion method, we develop
             the moment method, i.e. a method that exploits conservation
             relations obeyed by the collision operator. These are
             non-classical conservation relations which are associated
             with a new concept, that of Generalized Collision Invariant
             (GCI). In this paper, we develop the GCI concept and relate
             it to geometrical and analytical structures of the collision
             operator. Then, the derivation of the limit model using the
             GCI is performed in an arbitrary number of spatial
             dimensions and with non-constant and non-uniform polymer
             density. This non-uniformity generates new terms in the
             Ericksen-Leslie model},
   Doi = {10.3934/krm.2021047},
   Key = {fds361926}
}

@article{fds363138,
   Author = {Liu, JG and Wang, Z and Zhang, Y and Zhou, Z},
   Title = {RIGOROUS JUSTIFICATION OF THE FOKKER-PLANCK EQUATIONS OF
             NEURAL NETWORKS BASED ON AN ITERATION PERSPECTIVE},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {54},
   Number = {1},
   Pages = {1270-1312},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2022},
   Month = {January},
   url = {http://dx.doi.org/10.1137/20M1338368},
   Abstract = {In this work, the primary goal is to establish a rigorous
             connection between the Fokker-Planck equation of neural
             networks and its microscopic model: the diffusion-jump
             stochastic process that captures the mean-field behavior of
             collections of neurons in the integrate-and-fire model. The
             proof is based on a novel iteration scheme: with an
             auxiliary random variable counting the firing events, both
             the density function of the stochastic process and the
             solution of the PDE problem admit series representations,
             and thus the difficulty in verifying the link between the
             density function and the PDE solution in each subproblem is
             greatly mitigated. The iteration approach provides a generic
             framework for integrating the probability approach with PDE
             techniques, with which we prove that the density function of
             the diffusion-jump stochastic process is indeed the
             classical solution of the Fokker-Planck equation with a
             unique flux-shift structure.},
   Doi = {10.1137/20M1338368},
   Key = {fds363138}
}

@article{fds359966,
   Author = {Liu, JG and Zhang, Z},
   Title = {EXISTENCE of GLOBAL WEAK SOLUTIONS of p-NAVIER-STOKES
             EQUATIONS},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {27},
   Number = {1},
   Pages = {469-486},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2022},
   Month = {January},
   url = {http://dx.doi.org/10.3934/dcdsb.2021051},
   Abstract = {This paper investigates the global existence of weak
             solutions for the incompressible p-Navier-Stokes equations
             in Rd (2 ≤ d ≤ p). The pNavier-Stokes equations are
             obtained by adding viscosity term to the p-Euler equations.
             The diffusion added is represented by the p-Laplacian of
             velocity and the p-Euler equations are derived as the
             Euler-Lagrange equations for the action represented by the
             Benamou-Brenier characterization of Wasserstein-p distances
             with constraint density to be characteristic
             functions.},
   Doi = {10.3934/dcdsb.2021051},
   Key = {fds359966}
}

@article{fds363681,
   Author = {Gao, Y and Liu, JG},
   Title = {PROJECTION METHOD FOR DROPLET DYNAMICS ON GROOVE-TEXTURED
             SURFACE WITH MERGING AND SPLITTING},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {44},
   Number = {2},
   Pages = {B310-B338},
   Year = {2022},
   Month = {January},
   url = {http://dx.doi.org/10.1137/20M1338563},
   Abstract = {The geometric motion of small droplets placed on an
             impermeable textured substrate is mainly driven by the
             capillary effect, the competition among surface tensions of
             three phases at the moving contact lines, and the
             impermeable substrate obstacle. After introducing an
             infinite dimensional manifold with an admissible tangent
             space on the boundary of the manifold, by Onsager's
             principle for an obstacle problem, we derive the associated
             parabolic variational inequalities. These variational
             inequalities can be used to compute the contact line
             dynamics with unavoidable merging and splitting of droplets
             due to the impermeable obstacle. To efficiently solve the
             parabolic variational inequality, we propose an
             unconditional stable explicit boundary updating scheme
             coupled with a projection method. The explicit boundary
             updating efficiently decouples the computation of the motion
             by mean curvature of the capillary surface and the moving
             contact lines. Meanwhile, the projection step efficiently
             splits the difficulties brought by the obstacle and the
             motion by mean curvature of the capillary surface.
             Furthermore, we prove the unconditional stability of the
             scheme and present an accuracy check. Convergence of the
             proposed scheme is also proved using a nonlinear
             Trotter-Kato product formula under the pinning contact line
             assumption. After incorporating the phase transition
             information at splitting points, several challenging
             examples including splitting and merging of droplets are
             demonstrated.},
   Doi = {10.1137/20M1338563},
   Key = {fds363681}
}

@article{fds363930,
   Author = {Li, L and Liu, JG and Liu, Z and Yang, Y and Zhou, Z},
   Title = {On Energy Stable Runge-Kutta Methods for the Water Wave
             Equation and its Simplified Non-Local Hyperbolic
             Model},
   Journal = {Communications in Computational Physics},
   Volume = {32},
   Number = {1},
   Pages = {222-258},
   Publisher = {Global Science Press},
   Year = {2022},
   Month = {January},
   url = {http://dx.doi.org/10.4208/cicp.OA-2021-0049},
   Abstract = {Although interest in numerical approximations of the water
             wave equation grows in recent years, the lack of rigorous
             analysis of its time discretization inhibits the design of
             more efficient algorithms. In practice of water wave
             simulations, the tradeoff between efficiency and stability
             has been a challenging problem. Thus to shed light on the
             stability condition for simulations of water waves, we focus
             on a model simplified from the water wave equation of
             infinite depth. This model preserves two main properties of
             the water wave equation: non-locality and hyperbolicity. For
             the constant coefficient case, we conduct systematic
             stability studies of the fully discrete approximation of
             such systems with the Fourier spectral approximation in
             space and general Runge-Kutta methods in time. As a result,
             an optimal time discretization strategy is provided in the
             form of a modified CFL condition, i.e. ∆t = O(√∆x).
             Meanwhile, the energy stable property is established for
             certain explicit Runge-Kutta methods. This CFL condition
             solves the problem of efficiency and stability: it allows
             numerical schemes to stay stable while resolves oscillations
             at the lowest requirement, which only produces acceptable
             computational load. In the variable coefficient case, the
             convergence of the semi-discrete approximation of it is
             presented, which naturally connects to the water wave
             equation. Analogue of these results for the water wave
             equation of finite depth is also discussed. To validate
             these theoretic observation, extensive numerical tests have
             been performed to verify the stability conditions.
             Simulations of the simplified hyperbolic model in the high
             frequency regime and the water wave equation are also
             provided.},
   Doi = {10.4208/cicp.OA-2021-0049},
   Key = {fds363930}
}

@article{fds359964,
   Author = {Gao, Y and Liu, JG},
   Title = {Surfactant-dependent contact line dynamics and droplet
             spreading on textured substrates: Derivations and
             computations},
   Journal = {Physica D: Nonlinear Phenomena},
   Volume = {428},
   Year = {2021},
   Month = {December},
   url = {http://dx.doi.org/10.1016/j.physd.2021.133067},
   Abstract = {We study spreading of a droplet, with insoluble surfactant
             covering its capillary surface, on a textured substrate. In
             this process, the surfactant-dependent surface tension
             dominates the behaviors of the whole dynamics, particularly
             the moving contact lines. This allows us to derive the full
             dynamics of the droplets laid by the insoluble surfactant:
             (i) the moving contact lines, (ii) the evolution of the
             capillary surface, (iii) the surfactant dynamics on this
             moving surface with a boundary condition at the contact
             lines and (iv) the incompressible viscous fluids inside the
             droplet. Our derivations base on Onsager's principle with
             Rayleigh dissipation functionals for either the viscous flow
             inside droplets or the motion by mean curvature of the
             capillary surface. We also prove the Rayleigh dissipation
             functional for viscous flow case is stronger than the one
             for the motion by mean curvature. After incorporating the
             textured substrate profile, we design a numerical scheme
             based on unconditionally stable explicit boundary updates
             and moving grids, which enable efficient computations for
             many challenging examples showing significant impacts of the
             surfactant to the deformation of droplets.},
   Doi = {10.1016/j.physd.2021.133067},
   Key = {fds359964}
}

@article{fds365497,
   Author = {Liu, J-G and Wang, Z and Xie, Y and Zhang, Y and Zhou,
             Z},
   Title = {Investigating the integrate and fire model as the limit of a
             random discharge model: a stochastic analysis
             perspective},
   Journal = {Mathematical Neuroscience and Applications},
   Volume = {Volume 1},
   Publisher = {Centre pour la Communication Scientifique Directe
             (CCSD)},
   Year = {2021},
   Month = {November},
   url = {http://dx.doi.org/10.46298/mna.7203},
   Abstract = {<jats:p>In the mean field integrate-and-fire model, the
             dynamics of a typical neuron within a large network is
             modeled as a diffusion-jump stochastic process whose jump
             takes place once the voltage reaches a threshold. In this
             work, the main goal is to establish the convergence
             relationship between the regularized process and the
             original one where in the regularized process, the jump
             mechanism is replaced by a Poisson dynamic, and jump
             intensity within the classically forbidden domain goes to
             infinity as the regularization parameter vanishes. On the
             macroscopic level, the Fokker-Planck equation for the
             process with random discharges (i.e. Poisson jumps) are
             defined on the whole space, while the equation for the limit
             process is on the half space. However, with the iteration
             scheme, the difficulty due to the domain differences has
             been greatly mitigated and the convergence for the
             stochastic process and the firing rates can be established.
             Moreover, we find a polynomial-order convergence for the
             distribution by a re-normalization argument in probability
             theory. Finally, by numerical experiments, we quantitatively
             explore the rate and the asymptotic behavior of the
             convergence for both linear and nonlinear
             models.</jats:p>},
   Doi = {10.46298/mna.7203},
   Key = {fds365497}
}

@article{fds359965,
   Author = {Gao, Y and Liu, JG and Liu, Z},
   Title = {Existence and rigidity of the vectorial peierls-nabarro
             model for dislocations in high dimensions},
   Journal = {Nonlinearity},
   Volume = {34},
   Number = {11},
   Pages = {7778-7828},
   Year = {2021},
   Month = {November},
   url = {http://dx.doi.org/10.1088/1361-6544/ac24e3},
   Abstract = {We focus on the existence and rigidity problems of the
             vectorial Peierls- Nabarro (PN) model for dislocations.
             Under the assumption that the misfit potential on the slip
             plane only depends on the shear displacement along the
             Burgers vector, a reduced non-local scalar Ginzburg-Landau
             equation with an anisotropic positive (if Poisson ratio
             belongs to (-1/2, 1/3)) singular kernel is derived on the
             slip plane. We first prove that minimizers of the PN energy
             for this reduced scalar problem exist. Starting from H1/2
             regularity, we prove that these minimizers are smooth 1D
             profiles only depending on the shear direction,
             monotonically and uniformly converge to two stable states at
             far fields in the direction of the Burgers vector. Then a De
             Giorgi-type conjecture of singlevariable symmetry for both
             minimizers and layer solutions is established. As a direct
             corollary, minimizers and layer solutions are unique up to
             translations. The proof of this De Giorgi-type conjecture
             relies on a delicate spectral analysis which is especially
             powerful for nonlocal pseudo-differential operatorswith
             strong maximal principle. All these results hold in any
             dimension since we work on the domain periodic in the
             transverse directions of the slip plane. The physical
             interpretation of this rigidity result is that the
             equilibrium dislocation on the slip plane only admits shear
             displacements and is a strictly monotonic 1D profile
             provided exclusive dependence of the misfit potential on the
             shear displacement.},
   Doi = {10.1088/1361-6544/ac24e3},
   Key = {fds359965}
}

@article{fds356793,
   Author = {Lafata, KJ and Chang, Y and Wang, C and Mowery, YM and Vergalasova, I and Niedzwiecki, D and Yoo, DS and Liu, J-G and Brizel, DM and Yin,
             F-F},
   Title = {Intrinsic radiomic expression patterns after 20 Gy
             demonstrate early metabolic response of oropharyngeal
             cancers.},
   Journal = {Med Phys},
   Volume = {48},
   Number = {7},
   Pages = {3767-3777},
   Year = {2021},
   Month = {July},
   url = {http://dx.doi.org/10.1002/mp.14926},
   Abstract = {PURPOSE: This study investigated the prognostic potential of
             intra-treatment PET radiomics data in patients undergoing
             definitive (chemo) radiation therapy for oropharyngeal
             cancer (OPC) on a prospective clinical trial. We
             hypothesized that the radiomic expression of OPC tumors
             after 20 Gy is associated with recurrence-free survival
             (RFS). MATERIALS AND METHODS: Sixty-four patients undergoing
             definitive (chemo)radiation for OPC were prospectively
             enrolled on an IRB-approved study. Investigational 18
             F-FDG-PET/CT images were acquired prior to treatment and
             2 weeks (20 Gy) into a seven-week course of therapy.
             Fifty-five quantitative radiomic features were extracted
             from the primary tumor as potential biomarkers of early
             metabolic response. An unsupervised data clustering
             algorithm was used to partition patients into clusters based
             only on their radiomic expression. Clustering results were
             naïvely compared to residual disease and/or subsequent
             recurrence and used to derive Kaplan-Meier estimators of
             RFS. To test whether radiomic expression provides prognostic
             value beyond conventional clinical features associated with
             head and neck cancer, multivariable Cox proportional hazards
             modeling was used to adjust radiomic clusters for T and N
             stage, HPV status, and change in tumor volume. RESULTS:
             While pre-treatment radiomics were not prognostic,
             intra-treatment radiomic expression was intrinsically
             associated with both residual/recurrent disease
             (P = 0.0256, χ 2 test) and RFS (HR = 7.53, 95%
             CI = 2.54-22.3; P = 0.0201). On univariate Cox analysis,
             radiomic cluster was associated with RFS (unadjusted
             HR = 2.70; 95% CI = 1.26-5.76; P = 0.0104) and
             maintained significance after adjustment for T, N staging,
             HPV status, and change in tumor volume after 20 Gy
             (adjusted HR = 2.69; 95% CI = 1.03-7.04; P = 0.0442).
             The particular radiomic characteristics associated with
             outcomes suggest that metabolic spatial heterogeneity after
             20 Gy portends complete and durable therapeutic response.
             This finding is independent of baseline metabolic imaging
             characteristics and clinical features of head and neck
             cancer, thus providing prognostic advantages over existing
             approaches. CONCLUSIONS: Our data illustrate the prognostic
             value of intra-treatment metabolic image interrogation,
             which may potentially guide adaptive therapy strategies for
             OPC patients and serve as a blueprint for other disease
             sites. The quality of our study was strengthened by its
             prospective image acquisition protocol, homogenous patient
             cohort, relatively long patient follow-up times, and
             unsupervised clustering formalism that is less prone to
             hyper-parameter tuning and over-fitting compared to
             supervised learning.},
   Doi = {10.1002/mp.14926},
   Key = {fds356793}
}

@article{fds355717,
   Author = {Hu, J and Liu, JG and Xie, Y and Zhou, Z},
   Title = {A structure preserving numerical scheme for Fokker-Planck
             equations of neuron networks: Numerical analysis and
             exploration},
   Journal = {Journal of Computational Physics},
   Volume = {433},
   Year = {2021},
   Month = {May},
   url = {http://dx.doi.org/10.1016/j.jcp.2021.110195},
   Abstract = {In this work, we are concerned with the Fokker-Planck
             equations associated with the Nonlinear Noisy Leaky
             Integrate-and-Fire model for neuron networks. Due to the
             jump mechanism at the microscopic level, such Fokker-Planck
             equations are endowed with an unconventional structure:
             transporting the boundary flux to a specific interior point.
             While the equations exhibit diversified solutions from
             various numerical observations, the properties of solutions
             are not yet completely understood, and by far there has been
             no rigorous numerical analysis work concerning such models.
             We propose a conservative and conditionally positivity
             preserving scheme for these Fokker-Planck equations, and we
             show that in the linear case, the semi-discrete scheme
             satisfies the discrete relative entropy estimate, which
             essentially matches the only known long time asymptotic
             solution property. We also provide extensive numerical tests
             to verify the scheme properties, and carry out several sets
             of numerical experiments, including finite-time blowup,
             convergence to equilibrium and capturing time-period
             solutions of the variant models.},
   Doi = {10.1016/j.jcp.2021.110195},
   Key = {fds355717}
}

@article{fds358861,
   Author = {Liu, JG and Wang, J and Zhao, Y and Zhou, Z},
   Title = {Field model for complex ionic fluids: Analytical properties
             and numerical investigation},
   Journal = {Communications in Computational Physics},
   Volume = {30},
   Number = {3},
   Pages = {874-902},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.4208/CICP.OA-2019-0223},
   Abstract = {In this paper, we consider the field model for complex ionic
             fluids with an energy variational structure, and analyze the
             well-posedness to this model with regularized kernels.
             Furthermore, we deduce the estimate of the maximal density
             function to quantify the finite size effect. On the
             numerical side, we adopt a finite volume scheme to the field
             model, which satisfies the following properties:
             positivity-preserving, mass conservation and energy
             dissipation. Besides, series of numerical experiments are
             provided to demonstrate the properties of the steady state
             and the finite size effect by showing the equilibrium
             profiles with different values of the parameter in the
             kernel.},
   Doi = {10.4208/CICP.OA-2019-0223},
   Key = {fds358861}
}

@article{fds359347,
   Author = {Liu, JG and Xu, X},
   Title = {Existence and incompressible limit of a tissue growth model
             with autophagy},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {53},
   Number = {5},
   Pages = {5215-5242},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.1137/21M1405253},
   Abstract = {In this paper we study a cross-diffusion system whose
             coefficient matrix is non-symmetric and degenerate. The
             system arises in the study of tissue growth with autophagy.
             The existence of a weak solution is established. We also
             investigate the limiting behavior of solutions as the
             pressure gets stiff. The so-called incompressible limit is a
             free boundary problem of Hele-Shaw type. Our key new
             discovery is that the usual energy estimate still holds as
             long as the time variable stays away from
             0.},
   Doi = {10.1137/21M1405253},
   Key = {fds359347}
}

@article{fds354038,
   Author = {Li, Q and Liu, JG and Shu, R},
   Title = {Sensitivity analysis of burgers' equation with
             shocks},
   Journal = {SIAM-ASA Journal on Uncertainty Quantification},
   Volume = {8},
   Number = {4},
   Pages = {1493-1521},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.1137/18M1211763},
   Abstract = {The generalized polynomial chaos (gPC) method has been
             extensively used in uncertainty quantification problems
             where equations contain random variables. For gPC to achieve
             high accuracy, PDE solutions need to have high regularity in
             the random space, but this is what hyperbolic type problems
             cannot provide. We provide a counterargument in this paper
             and show that even though the solution profile develops
             singularities in the random space, which destroys the
             spectral accuracy of gPC, the physical quantities (such as
             the shock emergence time, the shock location, and the shock
             strength) are all smooth functions of the uncertainties
             coming from both initial data and the wave speed. With
             proper shifting, the solution's polynomial interpolation
             approximates the real solution accurately, and the error
             decays as the order of the polynomial increases. Therefore
             this work provides a new perspective to “quantify
             uncertainties” and significantly improves the accuracy of
             the gPC method with a slight reformulation. We use the
             Burgers' equation as an example for thorough analysis, and
             the analysis could be extended to general conservation laws
             with convex fluxes.},
   Doi = {10.1137/18M1211763},
   Key = {fds354038}
}

@article{fds356794,
   Author = {Gao, Y and Liu, JG},
   Title = {Gradient flow formulation and second order numerical method
             for motion by mean curvature and contact line dynamics on
             rough surface},
   Journal = {Interfaces and Free Boundaries},
   Volume = {23},
   Number = {1},
   Pages = {103-158},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.4171/ifb/451},
   Abstract = {We study the dynamics of a droplet moving on an inclined
             rough surface in the absence of inertial and viscous stress
             effects. In this case, the dynamics of the droplet is a
             purely geometric motion in terms of the wetting domain and
             the capillary surface. Using a single graph representation,
             we interpret this geometric motion as a gradient flow on a
             manifold. We propose unconditionally stable first/second
             order numerical schemes to simulate this geometric motion of
             the droplet, which is described using motion by mean
             curvature coupled with moving contact lines. The schemes are
             based on (i) explicit moving boundaries, which decouple the
             dynamic updates of the contact lines and the capillary
             surface, (ii) an arbitrary Lagrangian-Eulerian method on
             moving grids and (iii) a predictor-corrector method with a
             nonlinear elliptic solver up to second order accuracy. For
             the case of quasi-static dynamics with continuous spatial
             variable in the numerical schemes, we prove the stability
             and convergence of the first/second order numerical schemes.
             To demonstrate the accuracy and long-time validation of the
             proposed schemes, several challenging computational examples
             - including breathing droplets, droplets on inhomogeneous
             rough surfaces and quasi-static Kelvin pendant droplets -
             are constructed and compared with exact solutions to
             quasi-static dynamics obtained by desingularized
             differential-algebraic system of equations
             (DAEs).},
   Doi = {10.4171/ifb/451},
   Key = {fds356794}
}

@article{fds365496,
   Author = {Gao, Y and Katsevich, AE and Liu, JG and Lu, J and Marzuola,
             JL},
   Title = {ANALYSIS OF A FOURTH-ORDER EXPONENTIAL PDE ARISING FROM A
             CRYSTAL SURFACE JUMP PROCESS WITH METROPOLIS-TYPE TRANSITION
             RATES},
   Journal = {Pure and Applied Analysis},
   Volume = {3},
   Number = {4},
   Pages = {595-612},
   Publisher = {Mathematical Sciences Publishers},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.2140/paa.2021.3.595},
   Abstract = {We analytically and numerically study a fourth-order PDE
             modeling rough crystal surface diffusion on the macroscopic
             level. We discuss existence of solutions globally in time
             and long-time dynamics for the PDE model. The PDE,
             originally derived by Katsevich is the continuum limit of a
             microscopic model of the surface dynamics, given by a Markov
             jump process with Metropolis-type transition rates. We
             outline the convergence argument, which depends on a
             simplifying assumption on the local equilibrium measure that
             is valid in the high-temperature regime. We provide
             numerical evidence for the convergence of the microscopic
             model to the PDE in this regime.},
   Doi = {10.2140/paa.2021.3.595},
   Key = {fds365496}
}

@article{fds366656,
   Author = {Liu, JG and Tang, M and Wang, L and Zhou, Z},
   Title = {Toward understanding the boundary propagation speeds in
             tumor growth models},
   Journal = {SIAM Journal on Applied Mathematics},
   Volume = {81},
   Number = {3},
   Pages = {1052-1076},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.1137/19M1296665},
   Abstract = {At the continuous level, we consider two types of tumor
             growth models: the cell density model, based on the fluid
             mechanical construction, is more favorable for scientific
             interpretation and numerical simulations, and the free
             boundary model, as the incompressible limit of the former,
             is more tractable when investigating the boundary
             propagation. In this work, we aim to investigate the
             boundary propagation speeds in those models based on
             asymptotic analysis of the free boundary model and efficient
             numerical simulations of the cell density model. We derive,
             for the first time, some analytical solutions for the free
             boundary model with pressure jumps across the tumor boundary
             in multidimensions with finite tumor sizes. We further show
             that in the large radius limit, the analytical solutions to
             the free boundary model in one and multiple spatial
             dimensions converge to traveling wave solutions. The
             convergence rate in the propagation speeds are algebraic in
             multidimensions as opposed to the exponential convergence in
             one dimension. We also propose an accurate front capturing
             numerical scheme for the cell density model, and extensive
             numerical tests are provided to illustrate the analytical
             findings.},
   Doi = {10.1137/19M1296665},
   Key = {fds366656}
}

@article{fds358862,
   Author = {Gao, Y and Jin, G and Liu, J-G},
   Title = {Inbetweening auto-animation via Fokker-Planck dynamics and
             thresholding},
   Journal = {Inverse Problems & Imaging},
   Volume = {15},
   Number = {5},
   Pages = {843-843},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2021},
   url = {http://dx.doi.org/10.3934/ipi.2021016},
   Abstract = {<jats:p xml:lang="fr">&lt;p style='text-indent:20px;'&gt;We
             propose an equilibrium-driven deformation algorithm (EDDA)
             to simulate the inbetweening transformations starting from
             an initial image to an equilibrium image, which covers
             images varying from a greyscale type to a colorful type on
             planes or manifolds. The algorithm is based on the
             Fokker-Planck dynamics on manifold, which automatically
             incorporates the manifold structure suggested by dataset and
             satisfies positivity, unconditional stability, mass
             conservation law and exponentially convergence. The
             thresholding scheme is adapted for the sharp interface
             dynamics and is used to achieve the finite time convergence.
             Using EDDA, three challenging examples, (I) facial aging
             process, (II) coronavirus disease 2019 (COVID-19) pneumonia
             invading/fading process, and (III) continental evolution
             process are computed efficiently.&lt;/p&gt;</jats:p>},
   Doi = {10.3934/ipi.2021016},
   Key = {fds358862}
}

@article{fds352860,
   Author = {Huang, H and Liu, JG and Pickl, P},
   Title = {On the Mean-Field Limit for the Vlasov–Poisson–Fokker–Planck
             System},
   Journal = {Journal of Statistical Physics},
   Volume = {181},
   Number = {5},
   Pages = {1915-1965},
   Year = {2020},
   Month = {December},
   url = {http://dx.doi.org/10.1007/s10955-020-02648-3},
   Abstract = {We rigorously justify the mean-field limit of an N-particle
             system subject to Brownian motions and interacting through
             the Newtonian potential in R3. Our result leads to a
             derivation of the Vlasov–Poisson–Fokker–Planck (VPFP)
             equations from the regularized microscopic N-particle
             system. More precisely, we show that the maximal distance
             between the exact microscopic trajectories and the
             mean-field trajectories is bounded by N-13+ε (163≤ε<136)
             with a blob size of N-δ (13≤δ<1954-2ε3) up to a
             probability of 1 - N-α for any α> 0. Moreover, we prove
             the convergence rate between the empirical measure
             associated to the regularized particle system and the
             solution of the VPFP equations. The technical novelty of
             this paper is that our estimates rely on the randomness
             coming from the initial data and from the Brownian
             motions.},
   Doi = {10.1007/s10955-020-02648-3},
   Key = {fds352860}
}

@article{fds351005,
   Author = {Gao, Y and Liu, JG and Lu, J and Marzuola, JL},
   Title = {Analysis of a continuum theory for broken bond crystal
             surface models with evaporation and deposition
             effects},
   Journal = {Nonlinearity},
   Volume = {33},
   Number = {8},
   Pages = {3816-3845},
   Year = {2020},
   Month = {August},
   url = {http://dx.doi.org/10.1088/1361-6544/ab853d},
   Abstract = {We study a 4th order degenerate parabolic PDE model in
             one-dimension with a 2nd order correction modeling the
             evolution of a crystal surface under the influence of both
             thermal fluctuations and evaporation/deposition effects.
             First, we provide a non-rigorous derivation of the PDE from
             an atomistic model using variations on kinetic Monte Carlo
             rates proposed by the last author with Weare [Marzuola J L
             and Weare J 2013 Phys. Rev. E 88 032403]. Then, we prove the
             existence of a global in time weak solution for the PDE by
             regularizing the equation in a way that allows us to apply
             the tools of Bernis-Friedman [Bernis F and Friedman A 1990
             J. Differ. Equ. 83 179-206]. The methods developed here can
             be applied to a large number of 4th order degenerate PDE
             models. In an appendix, we also discuss the global smooth
             solution with small data in the Weiner algebra framework
             following recent developments using tools of the second
             author with Robert Strain [Liu J G and Strain R M 2019
             Interfaces Free Boundaries 21 51-86].},
   Doi = {10.1088/1361-6544/ab853d},
   Key = {fds351005}
}

@article{fds366657,
   Author = {Gao, Y and Liu, JG},
   Title = {Large Time Behavior, Bi-Hamiltonian Structure, and Kinetic
             Formulation for a Complex Burgers Equation},
   Journal = {Quarterly of Applied Mathematics},
   Volume = {79},
   Number = {1},
   Pages = {120-123},
   Year = {2020},
   Month = {May},
   url = {http://dx.doi.org/10.1090/QAM/1573},
   Abstract = {We prove the existence and uniqueness of positive analytical
             solutions with positive initial data to the mean field
             equation (the Dyson equation) of the Dyson Brownian motion
             through the complex Burgers equation with a force term on
             the upper half complex plane. These solutions converge to a
             steady state given by Wigner's semicircle law. A unique
             global weak solution with nonnegative initial data to the
             Dyson equation is obtained, and some explicit solutions are
             given by Wigner's semicircle laws. We also construct a
             bi-Hamiltonian structure for the system of real and
             imaginary components of the complex Burgers equation
             (coupled Burgers system). We establish a kinetic formulation
             for the coupled Burgers system and prove the existence and
             uniqueness of entropy solutions. The coupled Burgers system
             in Lagrangian variable naturally leads to two interacting
             particle systems, the Fermi–Pasta–Ulam–Tsingou model
             with nearest-neighbor interactions, and the Calogero–Moser
             model. These two particle systems yield the same Lagrangian
             dynamics in the continuum limit.},
   Doi = {10.1090/QAM/1573},
   Key = {fds366657}
}

@article{fds356029,
   Author = {Jin, S and Li, L and Liu, JG},
   Title = {Convergence of the random batch method for interacting
             particles with disparate species and weights},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {59},
   Number = {2},
   Pages = {746-768},
   Year = {2020},
   Month = {March},
   url = {http://dx.doi.org/10.1137/20M1327641},
   Abstract = {We consider in this work the convergence of the random batch
             method proposed in our previous work [Jin et al., J. Comput.
             Phys., 400(2020), 108877] for interacting particles to the
             case of disparate species and weights. We show that the
             strong error is of O(√ τ) while the weak error is of
             O(τ) where τ is the time step between two random divisions
             of batches. Both types of convergence are uniform in N, the
             number of particles. The proof of strong convergence follows
             closely the proof in [Jin et al., J. Comput. Phys.,
             400(2020), 108877] for indistinguishable particles, but
             there are still some differences: Since there is no
             exchangeability now, we have to use a certain weighted
             average of the errors; some refined auxiliary lemmas have to
             be proved compared with our previous work. To show that the
             weak convergence of empirical measure is uniform in N,
             certain sharp estimates for the derivatives of the backward
             equations have been used. The weak convergence analysis is
             also illustrating for the convergence of the Random Batch
             Method for N-body Liouville equations.},
   Doi = {10.1137/20M1327641},
   Key = {fds356029}
}

@article{fds347984,
   Author = {Jin, S and Li, L and Liu, JG},
   Title = {Random Batch Methods (RBM) for interacting particle
             systems},
   Journal = {Journal of Computational Physics},
   Volume = {400},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.1016/j.jcp.2019.108877},
   Abstract = {We develop Random Batch Methods for interacting particle
             systems with large number of particles. These methods use
             small but random batches for particle interactions, thus the
             computational cost is reduced from O(N2) per time step to
             O(N), for a system with N particles with binary
             interactions. On one hand, these methods are efficient
             Asymptotic-Preserving schemes for the underlying particle
             systems, allowing N-independent time steps and also capture,
             in the N→∞ limit, the solution of the mean field limit
             which are nonlinear Fokker-Planck equations; on the other
             hand, the stochastic processes generated by the algorithms
             can also be regarded as new models for the underlying
             problems. For one of the methods, we give a particle number
             independent error estimate under some special interactions.
             Then, we apply these methods to some representative problems
             in mathematics, physics, social and data sciences, including
             the Dyson Brownian motion from random matrix theory,
             Thomson's problem, distribution of wealth, opinion dynamics
             and clustering. Numerical results show that the methods can
             capture both the transient solutions and the global
             equilibrium in these problems.},
   Doi = {10.1016/j.jcp.2019.108877},
   Key = {fds347984}
}

@article{fds350324,
   Author = {Feng, Y and Gao, T and Li, L and Liu, JG and Lu, Y},
   Title = {Uniform-in-time weak error analysis for stochastic gradient
             descent algorithms via diffusion approximation},
   Journal = {Communications in Mathematical Sciences},
   Volume = {18},
   Number = {1},
   Pages = {163-188},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2020.v18.n1.a7},
   Abstract = {Diffusion approximation provides weak approximation for
             stochastic gradient descent algorithms in a finite time
             horizon. In this paper, we introduce new tools motivated by
             the backward error analysis of numerical stochastic
             differential equations into the theoretical framework of
             diffusion approximation, extending the validity of the weak
             approximation from finite to infinite time horizon. The new
             techniques developed in this paper enable us to characterize
             the asymptotic behavior of constant-step-size SGD algorithms
             near a local minimum around which the objective functions
             are locally strongly convex, a goal previously unreachable
             within the diffusion approximation framework. Our analysis
             builds upon a truncated formal power expansion of the
             solution of a Kolmogorov equation arising from diffusion
             approximation, where the main technical ingredient is
             uniform-in-time bounds controlling the long-term behavior of
             the expansion coefficient functions near the local minimum.
             We expect these new techniques to bring new understanding of
             the behaviors of SGD near local minimum and greatly expand
             the range of applicability of diffusion approximation to
             cover wider and deeper aspects of stochastic optimization
             algorithms in data science.},
   Doi = {10.4310/CMS.2020.v18.n1.a7},
   Key = {fds350324}
}

@article{fds350325,
   Author = {Degond, P and Engel, M and Liu, JG and Pego, RL},
   Title = {A markov jump process modelling animal group size
             statistics},
   Journal = {Communications in Mathematical Sciences},
   Volume = {18},
   Number = {1},
   Pages = {55-89},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2020.v18.n1.a3},
   Abstract = {We translate a coagulation-fragmentation model, describing
             the dynamics of animal group size distributions, into a
             model for the population distribution and associate the
             nonlinear evolution equation with a Markov jump process of a
             type introduced in classic work of H. McKean. In particular
             this formalizes a model suggested by [H.-S. Niwa, J. Theo.
             Biol., 224:451(457, 2003] with simple coagulation and
             fragmentation rates. Based on the jump process, we develop a
             numerical scheme that allows us to approximate the
             equilibrium for the Niwa model, validated by comparison to
             analytical results by [Degond et al., J. Nonlinear Sci.,
             27(2):379(424, 2017], and study the population and size
             distributions for more complicated rates. Furthermore, the
             simulations are used to describe statistical properties of
             the underlying jump process. We additionally discuss the
             relation of the jump process to models expressed in
             stochastic differential equations and demonstrate that such
             a connection is justified in the case of nearest-neighbour
             interactions, as opposed to global interactions as in the
             Niwa model.},
   Doi = {10.4310/CMS.2020.v18.n1.a3},
   Key = {fds350325}
}

@article{fds350326,
   Author = {Li, L and Li, Y and Liu, JG and Liu, Z and Lu, J},
   Title = {A stochastic version of stein variational gradient descent
             for efficient sampling},
   Journal = {Communications in Applied Mathematics and Computational
             Science},
   Volume = {15},
   Number = {1},
   Pages = {37-63},
   Publisher = {Mathematical Sciences Publishers},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.2140/camcos.2020.15.37},
   Abstract = {We propose in this work RBM-SVGD, a stochastic version of
             the Stein variational gradient descent (SVGD) method for
             efficiently sampling from a given probability measure, which
             is thus useful for Bayesian inference. The method is to
             apply the random batch method (RBM) for interacting particle
             systems proposed by Jin et al. to the interacting particle
             systems in SVGD. While keeping the behaviors of SVGD, it
             reduces the computational cost, especially when the
             interacting kernel has long range. We prove that the one
             marginal distribution of the particles generated by this
             method converges to the one marginal of the interacting
             particle systems under Wasserstein-2 distance on fixed time
             interval T0; T U. Numerical examples verify the efficiency
             of this new version of SVGD.},
   Doi = {10.2140/camcos.2020.15.37},
   Key = {fds350326}
}

@article{fds351006,
   Author = {Li, L and Liu, JG},
   Title = {Large time behaviors of upwind schemes and B-schemes for
             fokker-planck equations on R by jump processes},
   Journal = {Mathematics of Computation},
   Volume = {89},
   Number = {325},
   Pages = {2283-2320},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.1090/mcom/3516},
   Abstract = {We revisit some standard schemes, including upwind schemes
             and some B-schemes, for linear conservation laws from the
             viewpoint of jump processes, allowing the study of them
             using probabilistic tools. For Fokker-Planck equations on R,
             in the case of weak confinement, we show that the numerical
             solutions converge to some stationary distributions. In the
             case of strong confinement, using a discrete Poincare
             inequality, we prove that the O(h) numeric error under ℓ1
             norm is uniform in time, and establish the uniform
             exponential convergence to the steady states. Compared with
             the traditional results of exponential convergence of these
             schemes, our result is in the whole space without boundary.
             We also establish similar results on the torus for which the
             stationary solution of the scheme does not have detailed
             balance. This work could motivate better understanding of
             numerical analysis for conservation laws, especially
             parabolic conservation laws, in unbounded
             domains.},
   Doi = {10.1090/mcom/3516},
   Key = {fds351006}
}

@article{fds354037,
   Author = {Gao, Y and Liu, JG and Luo, T and Xiang, Y},
   Title = {Revisit of the peierls-nabarro model for edge dislocations
             in Hilbert space},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {22},
   Number = {11},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.3934/dcdsb.2020224},
   Abstract = {In this paper, we revisit the mathematical validation of the
             Peierls–Nabarro (PN) models, which are multiscale models
             of dislocations that incorporate the detailed dislocation
             core structure. We focus on the static and dynamic PN models
             of an edge dislocation in Hilbert space. In a PN model, the
             total energy includes the elastic energy in the two
             half-space continua and a nonlinear potential energy, which
             is always infinite, across the slip plane. We revisit the
             relationship between the PN model in the full space and the
             reduced problem on the slip plane in terms of both governing
             equations and energy variations. The shear displacement jump
             is determined only by the reduced problem on the slip plane
             while the displacement fields in the two half spaces are
             determined by linear elasticity. We establish the existence
             and sharp regularities of classical solutions in Hilbert
             space. For both the reduced problem and the full PN model,
             we prove that a static solution is a global minimizer in a
             perturbed sense. We also show that there is a unique
             classical, global in time solution of the dynamic PN
             model.},
   Doi = {10.3934/dcdsb.2020224},
   Key = {fds354037}
}

@article{fds354040,
   Author = {LIU, JG and WANG, J},
   Title = {GLOBAL EXISTENCE FOR NERNST-PLANCK-NAVIER-STOKES SYSTEM IN
             RN},
   Journal = {Communications in Mathematical Sciences},
   Volume = {18},
   Number = {6},
   Pages = {1743-1754},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2020.v18.n6.a9},
   Abstract = {. In this note, we study the Nernst-Planck-Navier-Stokes
             system for the transport and diffusion of ions in
             electrolyte solutions. The key feature is to establish three
             energy-dissipation equalities. As their direct consequence,
             we obtain global existence for two-ionic species case in Rn,
             n ≥ 2, and multi-ionic species case in Rn, n =
             2,3.},
   Doi = {10.4310/CMS.2020.v18.n6.a9},
   Key = {fds354040}
}

@article{fds354041,
   Author = {LIU, JIANGUO and XU, X},
   Title = {A CLASS OF FUNCTIONAL INEQUALITIES AND THEIR APPLICATIONS TO
             FOURTH-ORDER NONLINEAR PARABOLIC EQUATIONS},
   Journal = {Communications in Mathematical Sciences},
   Volume = {18},
   Number = {7},
   Pages = {1911-1948},
   Publisher = {International Press of Boston},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2020.V18.N7.A5},
   Abstract = {We study a class of fourth-order nonlinear parabolic
             equations which include the thinfilm equation and the
             quantum drift-diffusion model as special cases. We
             investigate these equations by first developing functional
             inequalities of the type [Fourmula presented] which seem to
             be of interest in their own right.},
   Doi = {10.4310/CMS.2020.V18.N7.A5},
   Key = {fds354041}
}

@article{fds354039,
   Author = {Gao, Y and Liu, J-G},
   Title = {Long time behavior of dynamic solution to Peierls–Nabarro
             dislocation model},
   Journal = {Methods and Applications of Analysis},
   Volume = {27},
   Number = {2},
   Pages = {161-198},
   Publisher = {International Press of Boston},
   Year = {2020},
   url = {http://dx.doi.org/10.4310/maa.2020.v27.n2.a4},
   Doi = {10.4310/maa.2020.v27.n2.a4},
   Key = {fds354039}
}

@article{fds354042,
   Author = {Gao, Y and Liu, J-G},
   Title = {A note on parametric Bayesian inference via gradient
             flows},
   Journal = {Annals of Mathematical Sciences and Applications},
   Volume = {5},
   Number = {2},
   Pages = {261-282},
   Publisher = {International Press of Boston},
   Year = {2020},
   url = {http://dx.doi.org/10.4310/amsa.2020.v5.n2.a3},
   Doi = {10.4310/amsa.2020.v5.n2.a3},
   Key = {fds354042}
}

@article{fds366913,
   Author = {Gao, Y and Liu, J-G},
   Title = {LONG TIME BEHAVIOR OF DYNAMIC SOLUTION TO PEIERLS-NABARRO
             DISLOCATION MODEL},
   Journal = {METHODS AND APPLICATIONS OF ANALYSIS},
   Volume = {27},
   Number = {2},
   Pages = {161-197},
   Year = {2020},
   Key = {fds366913}
}

@article{fds347985,
   Author = {Li, L and Liu, JG and Yu, P},
   Title = {On the mean field limit for Brownian particles with Coulomb
             interaction in 3D},
   Journal = {Journal of Mathematical Physics},
   Volume = {60},
   Number = {11},
   Year = {2019},
   Month = {November},
   url = {http://dx.doi.org/10.1063/1.5114854},
   Abstract = {In this paper, we consider the mean field limit of Brownian
             particles with Coulomb repulsion in 3D space using
             compactness. Using a symmetrization technique, we are able
             to control the singularity and prove that the limit measure
             almost surely is a weak solution to the limiting nonlinear
             Fokker-Planck equation. Moreover, by proving that the energy
             almost surely is bounded by the initial energy, we improve
             the regularity of the weak solutions. By a natural
             assumption, we also establish the weak-strong uniqueness
             principle, which is closely related to the propagation of
             chaos.},
   Doi = {10.1063/1.5114854},
   Key = {fds347985}
}

@article{fds347986,
   Author = {Liu, JG and Pego, RL},
   Title = {On Local Singularities in Ideal Potential Flows with Free
             Surface},
   Journal = {Chinese Annals of Mathematics. Series B},
   Volume = {40},
   Number = {6},
   Pages = {925-948},
   Year = {2019},
   Month = {November},
   url = {http://dx.doi.org/10.1007/s11401-019-0167-z},
   Abstract = {Despite important advances in the mathematical analysis of
             the Euler equations for water waves, especially over the
             last two decades, it is not yet known whether local
             singularities can develop from smooth data in well-posed
             initial value problems. For ideal free-surface flow with
             zero surface tension and gravity, the authors review
             existing works that describe “splash singularities”,
             singular hyperbolic solutions related to jet formation and
             “flip-through”, and a recent construction of a singular
             free surface by Zubarev and Karabut that however involves
             unbounded negative pressure. The authors illustrate some of
             these phenomena with numerical computations of 2D flow based
             upon a conformal mapping formulation. Numerical tests with a
             different kind of initial data suggest the possibility that
             corner singularities may form in an unstable way from
             specially prepared initial data.},
   Doi = {10.1007/s11401-019-0167-z},
   Key = {fds347986}
}

@article{fds347987,
   Author = {Liu, JG and Pego, RL and Pu, Y},
   Title = {Well-posedness and derivative blow-up for a dispersionless
             regularized shallow water system},
   Journal = {Nonlinearity},
   Volume = {32},
   Number = {11},
   Pages = {4346-4376},
   Year = {2019},
   Month = {October},
   url = {http://dx.doi.org/10.1088/1361-6544/ab2cf1},
   Abstract = {We study local-time well-posedness and breakdown for
             solutions of regularized Saint-Venant equations (regularized
             classical shallow water equations) recently introduced by
             Clamond and Dutykh. The system is linearly non-dispersive,
             and smooth solutions conserve an H 1-equivalent energy. No
             shock discontinuities can occur, but the system is known to
             admit weakly singular shock-profile solutions that dissipate
             energy. We identify a class of small-energy smooth solutions
             that develop singularities in the first derivatives in
             finite time.},
   Doi = {10.1088/1361-6544/ab2cf1},
   Key = {fds347987}
}

@article{fds347988,
   Author = {Liu, JG and Pego, RL and Slepčev, D},
   Title = {Least action principles for incompressible flows and
             geodesics between shapes},
   Journal = {Calculus of Variations and Partial Differential
             Equations},
   Volume = {58},
   Number = {5},
   Year = {2019},
   Month = {October},
   url = {http://dx.doi.org/10.1007/s00526-019-1636-7},
   Abstract = {As V. I. Arnold observed in the 1960s, the Euler equations
             of incompressible fluid flow correspond formally to geodesic
             equations in a group of volume-preserving diffeomorphisms.
             Working in an Eulerian framework, we study incompressible
             flows of shapes as critical paths for action (kinetic
             energy) along transport paths constrained to have
             characteristic-function densities. The formal geodesic
             equations for this problem are Euler equations for
             incompressible, inviscid potential flow of fluid with zero
             pressure and surface tension on the free boundary. The
             problem of minimizing this action exhibits an instability
             associated with microdroplet formation, with the following
             outcomes: any two shapes of equal volume can be
             approximately connected by an Euler spray—a countable
             superposition of ellipsoidal geodesics. The infimum of the
             action is the Wasserstein distance squared, and is almost
             never attained except in dimension 1. Every Wasserstein
             geodesic between bounded densities of compact support
             provides a solution of the (compressible) pressureless Euler
             system that is a weak limit of (incompressible) Euler
             sprays.},
   Doi = {10.1007/s00526-019-1636-7},
   Key = {fds347988}
}

@article{fds347989,
   Author = {Lafata, KJ and Zhou, Z and Liu, J-G and Hong, J and Kelsey, CR and Yin,
             F-F},
   Title = {An Exploratory Radiomics Approach to Quantifying Pulmonary
             Function in CT Images.},
   Journal = {Sci Rep},
   Volume = {9},
   Number = {1},
   Pages = {11509},
   Year = {2019},
   Month = {August},
   url = {http://dx.doi.org/10.1038/s41598-019-48023-5},
   Abstract = {Contemporary medical imaging is becoming increasingly more
             quantitative. The emerging field of radiomics is a leading
             example. By translating unstructured data (i.e., images)
             into structured data (i.e., imaging features), radiomics can
             potentially characterize clinically useful imaging
             phenotypes. In this paper, an exploratory radiomics approach
             is used to investigate the potential association between
             quantitative imaging features and pulmonary function in CT
             images. Thirty-nine radiomic features were extracted from
             the lungs of 64 patients as potential imaging biomarkers for
             pulmonary function. Collectively, these features capture the
             morphology of the lungs, as well as intensity variations,
             fine-texture, and coarse-texture of the pulmonary tissue.
             The extracted lung radiomics data was compared to
             conventional pulmonary function tests. In general, patients
             with larger lungs of homogeneous, low attenuating pulmonary
             tissue (as measured via radiomics) were found to be
             associated with poor spirometry performance and a lower
             diffusing capacity for carbon monoxide. Unsupervised dynamic
             data clustering revealed subsets of patients with similar
             lung radiomic patterns that were found to be associated with
             similar forced expiratory volume in one second (FEV1)
             measurements. This implies that patients with similar
             radiomic feature vectors also presented with comparable
             spirometry performance, and were separable by varying
             degrees of pulmonary function as measured by
             imaging.},
   Doi = {10.1038/s41598-019-48023-5},
   Key = {fds347989}
}

@article{fds347990,
   Author = {Liu, JG and Tang, M and Wang, L and Zhou, Z},
   Title = {Analysis and computation of some tumor growth models with
             nutrient: From cell density models to free boundary
             dynamics},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {24},
   Number = {7},
   Pages = {3011-3035},
   Year = {2019},
   Month = {July},
   url = {http://dx.doi.org/10.3934/dcdsb.2018297},
   Abstract = {In this paper, we study a tumor growth equation along with
             various models for the nutrient component, including a in
             vitro model and a in vivo model. At the cell density level,
             the spatial availability of the tumor density n is governed
             by the Darcy law via the pressure p(n) = n γ . For finite
             γ, we prove some a priori estimates of the tumor growth
             model, such as boundedness of the nutrient density, and
             non-negativity and growth estimate of the tumor density. As
             γ → ∞, the cell density models formally converge to
             Hele-Shaw flow models, which determine the free boundary
             dynamics of the tumor tissue in the incompressible limit. We
             derive several analytical solutions to the Hele-Shaw flow
             models, which serve as benchmark solutions to the geometric
             motion of tumor front propagation. Finally, we apply a
             conservative and positivity preserving numerical scheme to
             the cell density models, with numerical results verifying
             the link between cell density models and the free boundary
             dynamical models.},
   Doi = {10.3934/dcdsb.2018297},
   Key = {fds347990}
}

@article{fds347991,
   Author = {Zhan, Q and Zhuang, M and Zhou, Z and Liu, JG and Liu,
             QH},
   Title = {Complete-Q Model for Poro-Viscoelastic Media in Subsurface
             Sensing: Large-Scale Simulation with an Adaptive DG
             Algorithm},
   Journal = {IEEE Transactions on Geoscience and Remote
             Sensing},
   Volume = {57},
   Number = {7},
   Pages = {4591-4599},
   Publisher = {Institute of Electrical and Electronics Engineers
             (IEEE)},
   Year = {2019},
   Month = {July},
   url = {http://dx.doi.org/10.1109/TGRS.2019.2891691},
   Abstract = {In this paper, full mechanisms of dissipation and dispersion
             in poro-viscoelastic media are accurately simulated in time
             domain. Specifically, four Q values are first proposed to
             depict a poro-viscoelastic medium: two for the attenuation
             of the bulk and shear moduli in the solid skeleton, one for
             the bulk modulus in the pore fluid, and the other one for
             the solid-fluid coupling. By introducing several sets of
             auxiliary ordinary differential equations, the Q factors are
             efficiently incorporated in a high-order discontinuous
             Galerkin algorithm. Consequently, in the mathematical sense,
             the Riemann problem is exactly solved, with the same form as
             the inviscid poroelastic material counterpart; in the
             practical sense, our algorithm requires nearly negligible
             extra time cost, while keeping the governing equations
             almost unchanged. Parenthetically, an arbitrarily
             nonconformal-mesh technique, in terms of both h- and
             p-adaptivity, is implemented to realize the domain
             decomposition for a flexible algorithm. Furthermore, our
             algorithm is verified with an analytical solution for the
             half-space modeling. A validation with an independent
             numerical solver, and an application to a large-scale
             realistic complex topography modeling demonstrate the
             accuracy, efficiency, flexibility, and capability in
             realistic subsurface sensing.},
   Doi = {10.1109/TGRS.2019.2891691},
   Key = {fds347991}
}

@article{fds347992,
   Author = {Liu, JG and Niethammer, B and Pego, RL},
   Title = {Self-similar Spreading in a Merging-Splitting Model of
             Animal Group Size},
   Journal = {Journal of Statistical Physics},
   Volume = {175},
   Number = {6},
   Pages = {1311-1330},
   Year = {2019},
   Month = {June},
   url = {http://dx.doi.org/10.1007/s10955-019-02280-w},
   Abstract = {In a recent study of certain merging-splitting models of
             animal-group size (Degond et al. in J Nonlinear Sci
             27(2):379–424, 2017), it was shown that an initial size
             distribution with infinite first moment leads to convergence
             to zero in weak sense, corresponding to unbounded growth of
             group size. In the present paper we show that for any such
             initial distribution with a power-law tail, the solution
             approaches a self-similar spreading form. A one-parameter
             family of such self-similar solutions exists, with densities
             that are completely monotone, having power-law behavior in
             both small and large size regimes, with different
             exponents.},
   Doi = {10.1007/s10955-019-02280-w},
   Key = {fds347992}
}

@article{fds341508,
   Author = {Liu, JG and Lu, J and Margetis, D and Marzuola, JL},
   Title = {Asymmetry in crystal facet dynamics of homoepitaxy by a
             continuum model},
   Journal = {Physica D: Nonlinear Phenomena},
   Volume = {393},
   Pages = {54-67},
   Year = {2019},
   Month = {June},
   url = {http://dx.doi.org/10.1016/j.physd.2019.01.004},
   Abstract = {In the absence of external material deposition, crystal
             surfaces usually relax to become flat by decreasing their
             free energy. We study analytically an asymmetry in the
             relaxation of macroscopic plateaus, facets, of a periodic
             surface corrugation in 1+1 dimensions via a continuum model
             below the roughening transition temperature. The model
             invokes a continuum evolution law expressed by a highly
             degenerate parabolic partial differential equation (PDE) for
             surface diffusion, which is related to the nonlinear
             gradient flow of a convex, singular surface free energy with
             a certain exponential mobility in homoepitaxy. This
             evolution law is motivated both by an atomistic broken-bond
             model and a mesoscale model for crystal steps. By
             constructing an explicit solution to this PDE, we
             demonstrate the lack of symmetry in the evolution of top and
             bottom facets in periodic surface profiles. Our explicit,
             analytical solution is compared to numerical simulations of
             the continuum law via a regularized surface free
             energy.},
   Doi = {10.1016/j.physd.2019.01.004},
   Key = {fds341508}
}

@article{fds347993,
   Author = {Gao, Y and Li, L and Liu, JG},
   Title = {Patched peakon weak solutions of the modified Camassa–Holm
             equation},
   Journal = {Physica D: Nonlinear Phenomena},
   Volume = {390},
   Pages = {15-35},
   Year = {2019},
   Month = {March},
   url = {http://dx.doi.org/10.1016/j.physd.2018.10.005},
   Abstract = {In this paper, we study traveling wave solutions and peakon
             weak solutions of the modified Camassa–Holm (mCH) equation
             with dispersive term 2kux for k∈R. We study traveling wave
             solutions through a Hamiltonian system obtained from the mCH
             equation by using a nonlinear transformation. The typical
             traveling wave solutions given by this Hamiltonian system
             are unbounded or multi-valued. We provide a method, called
             patching technic, to truncate these traveling wave solutions
             and patch different segments to obtain patched bounded
             single-valued peakon weak solutions which satisfy jump
             conditions at peakons. Then, we study some special peakon
             weak solutions constructed by the fundamental solution of
             the Helmholtz operator 1−∂xx, which can also be obtained
             by the patching technic. At last, we study some length and
             total signed area preserving closed planar curve flows that
             can be described by the mCH equation when k=1, for which we
             give a Hamiltonian structure and use the patched periodic
             peakon weak solutions to investigate loops with
             peakons.},
   Doi = {10.1016/j.physd.2018.10.005},
   Key = {fds347993}
}

@article{fds340536,
   Author = {Lafata, KJ and Hong, JC and Geng, R and Ackerson, BG and Liu, J-G and Zhou,
             Z and Torok, J and Kelsey, CR and Yin, F-F},
   Title = {Association of pre-treatment radiomic features with lung
             cancer recurrence following stereotactic body radiation
             therapy.},
   Journal = {Phys Med Biol},
   Volume = {64},
   Number = {2},
   Pages = {025007},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1088/1361-6560/aaf5a5},
   Abstract = {The purpose of this work was to investigate the potential
             relationship between radiomic features extracted from
             pre-treatment x-ray CT images and clinical outcomes
             following stereotactic body radiation therapy (SBRT) for
             non-small-cell lung cancer (NSCLC). Seventy patients who
             received SBRT for stage-1 NSCLC were retrospectively
             identified. The tumor was contoured on pre-treatment
             free-breathing CT images, from which 43 quantitative
             radiomic features were extracted to collectively capture
             tumor morphology, intensity, fine-texture, and
             coarse-texture. Treatment failure was defined based on
             cancer recurrence, local cancer recurrence, and non-local
             cancer recurrence following SBRT. The univariate association
             between each radiomic feature and each clinical endpoint was
             analyzed using Welch's t-test, and p-values were corrected
             for multiple hypothesis testing. Multivariate associations
             were based on regularized logistic regression with a
             singular value decomposition to reduce the dimensionality of
             the radiomics data. Two features demonstrated a
             statistically significant association with local failure:
             Homogeneity2 (p  =  0.022) and Long-Run-High-Gray-Level-Emphasis
             (p  =  0.048). These results indicate that
             relatively dense tumors with a homogenous coarse texture
             might be linked to higher rates of local recurrence.
             Multivariable logistic regression models produced maximum
             [Formula: see text] values of [Formula: see text], and
             [Formula: see text], for the recurrence, local recurrence,
             and non-local recurrence endpoints, respectively. The
             CT-based radiomic features used in this study may be more
             associated with local failure than non-local failure
             following SBRT for stage I NSCLC. This finding is supported
             by both univariate and multivariate analyses.},
   Doi = {10.1088/1361-6560/aaf5a5},
   Key = {fds340536}
}

@article{fds340920,
   Author = {Huang, H and Liu, JG and Lu, J},
   Title = {Learning interacting particle systems: Diffusion parameter
             estimation for aggregation equations},
   Journal = {Mathematical Models and Methods in Applied
             Sciences},
   Volume = {29},
   Number = {1},
   Pages = {1-29},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1142/S0218202519500015},
   Abstract = {In this paper, we study the parameter estimation of
             interacting particle systems subject to the Newtonian
             aggregation and Brownian diffusion. Specifically, we
             construct an estimator with partial observed data to
             approximate the diffusion parameter , and the estimation
             error is achieved. Furthermore, we extend this result to
             general aggregation equations with a bounded Lipschitz
             interaction field.},
   Doi = {10.1142/S0218202519500015},
   Key = {fds340920}
}

@article{fds347998,
   Author = {Frouvelle, A and Liu, JG},
   Title = {Long-Time Dynamics for a Simple Aggregation Equation on the
             Sphere},
   Journal = {Springer Proceedings in Mathematics and Statistics},
   Volume = {282},
   Pages = {457-479},
   Year = {2019},
   Month = {January},
   ISBN = {9783030150952},
   url = {http://dx.doi.org/10.1007/978-3-030-15096-9_16},
   Abstract = {We give a complete study of the asymptotic behavior of a
             simple model of alignment of unit vectors, both at the level
             of particles, which corresponds to a system of coupled
             differential equations, and at the continuum level, under
             the form of an aggregation equation on the sphere. We prove
             unconditional convergence towards an aligned asymptotic
             state. In the cases of the differential system and of
             symmetric initial data for the partial differential
             equation, we provide precise rates of convergence.},
   Doi = {10.1007/978-3-030-15096-9_16},
   Key = {fds347998}
}

@article{fds348010,
   Author = {Gao, Y and Liu, JG and Lu, XY},
   Title = {Gradient flow approach to an exponential thin film equation:
             Global existence and latent singularity},
   Journal = {ESAIM - Control, Optimisation and Calculus of
             Variations},
   Volume = {25},
   Pages = {49-49},
   Publisher = {E D P SCIENCES},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1051/cocv/2018037},
   Abstract = {In this work, we study a fourth order exponential equation,
             ut = Δe-Δu derived from thin film growth on crystal
             surface in multiple space dimensions. We use the gradient
             flow method in metric space to characterize the latent
             singularity in global strong solution, which is intrinsic
             due to high degeneration. We define a suitable functional,
             which reveals where the singularity happens, and then prove
             the variational inequality solution under very weak
             assumptions for initial data. Moreover, the existence of
             global strong solution is established with regular initial
             data.},
   Doi = {10.1051/cocv/2018037},
   Key = {fds348010}
}

@article{fds347997,
   Author = {De Hoop and MV and Liu, JG and Markowich, PA and Ussembayev,
             NS},
   Title = {Plane-wave analysis of a hyperbolic system of equations with
             relaxation in ℝd},
   Journal = {Communications in Mathematical Sciences},
   Volume = {17},
   Number = {1},
   Pages = {61-79},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.4310/cms.2019.v17.n1.a3},
   Abstract = {We consider a multi-dimensional scalar wave equation with
             memory corresponding to the viscoelastic material described
             by a generalized Zener model. We deduce that this relaxation
             system is an example of a non-strictly hyperbolic system
             satisfying Majda's block structure condition. Wellposedness
             of the associated Cauchy problem is established by showing
             that the symbol of the spatial derivatives is uniformly
             diagonalizable with real eigenvalues. A long-time stability
             result is obtained by plane-wave analysis when the memory
             term allows for dissipation of energy.},
   Doi = {10.4310/cms.2019.v17.n1.a3},
   Key = {fds347997}
}

@article{fds347999,
   Author = {Liu, A and Liu, JG and Lu, Y},
   Title = {On the rate of convergence of empirical measure in
             ∞-Wasserstein distance for unbounded density
             function},
   Journal = {Quarterly of Applied Mathematics},
   Volume = {77},
   Number = {4},
   Pages = {811-829},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1090/qam/1541},
   Abstract = {We consider a sequence of identical independently
             distributed random samples from an absolutely continuous
             probability measure in one dimension with unbounded density.
             We establish a new rate of convergence of the
             ∞-Wasserstein distance between the empirical measure of
             the samples and the true distribution, which extends the
             previous convergence result by Trillos and Slepčev to the
             case that the true distribution has an unbounded
             density.},
   Doi = {10.1090/qam/1541},
   Key = {fds347999}
}

@article{fds348000,
   Author = {Li, L and Liu, JG},
   Title = {A discretization of Caputo derivatives with application to
             time fractional SDEs and gradient flows},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {57},
   Number = {5},
   Pages = {2095-2120},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1137/19M123854X},
   Abstract = {We consider a discretization of Caputo derivatives resulted
             from deconvolving a scheme for the corresponding Volterra
             integral. Properties of this discretization, including signs
             of the coefficients, comparison principles, and stability of
             the corresponding implicit schemes, are proved by its
             linkage to Volterra integrals with completely monotone
             kernels. We then apply the backward scheme corresponding to
             this discretization to two time fractional dissipative
             problems, and these implicit schemes are helpful for the
             analysis of the corresponding problems. In particular, we
             show that the overdamped generalized Langevin equation with
             fractional noise has a unique limiting measure for strongly
             convex potentials and we establish the convergence of
             numerical solutions to the strong solutions of time
             fractional gradient flows. The proposed scheme and schemes
             derived using the same philosophy can be useful for many
             other applications as well.},
   Doi = {10.1137/19M123854X},
   Key = {fds348000}
}

@article{fds347994,
   Author = {Zhan, Q and Zhuang, M and Fang, Y and Liu, J-G and Liu,
             QH},
   Title = {Green's function for anisotropic dispersive poroelastic
             media based on the Radon transform and eigenvector
             diagonalization.},
   Journal = {Proceedings. Mathematical, physical, and engineering
             sciences},
   Volume = {475},
   Number = {2221},
   Pages = {20180610},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1098/rspa.2018.0610},
   Abstract = {A compact Green's function for general dispersive
             anisotropic poroelastic media in a full-frequency regime is
             presented for the first time. First, starting in a frequency
             domain, the anisotropic dispersion is exactly incorporated
             into the constitutive relationship, thus avoiding fractional
             derivatives in a time domain. Then, based on the Radon
             transform, the original three-dimensional differential
             equation is effectively reduced to a one-dimensional system
             in space. Furthermore, inspired by the strategy adopted in
             the characteristic analysis of hyperbolic equations, the
             eigenvector diagonalization method is applied to decouple
             the one-dimensional vector problem into several independent
             scalar equations. Consequently, the fundamental solutions
             are easily obtained. A further derivation shows that Green's
             function can be decomposed into circumferential and
             spherical integrals, corresponding to static and transient
             responses, respectively. The procedures shown in this study
             are also compatible with other pertinent multi-physics
             coupling problems, such as piezoelectric,
             magneto-electro-elastic and thermo-elastic materials.
             Finally, the verifications and validations with existing
             analytical solutions and numerical solvers corroborate the
             correctness of the proposed Green's function.},
   Doi = {10.1098/rspa.2018.0610},
   Key = {fds347994}
}

@article{fds347995,
   Author = {Liu, JG and Strain, RM},
   Title = {Global stability for solutions to the exponential PDE
             describing epitaxial growth},
   Journal = {Interfaces and Free Boundaries},
   Volume = {21},
   Number = {1},
   Pages = {61-86},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.4171/IFB/417},
   Abstract = {In this paper we prove the global existence, uniqueness,
             optimal large time decay rates, and uniform gain of
             analyticity for the exponential PDE ht D eh in the whole
             space Rdx . We assume the initial data is of medium size in
             the Wiener algebra A.Rd /; we use the initial condition h0 2
             A.Rd / which is scale-invariant with respect to the
             invariant scaling of the exponential PDE. This exponential
             PDE was derived in [18] and more recently in
             [22].},
   Doi = {10.4171/IFB/417},
   Key = {fds347995}
}

@article{fds347996,
   Author = {Lafata, K and Zhou, Z and Liu, JG and Yin, FF},
   Title = {Data clustering based on Langevin annealing with a
             self-consistent potential},
   Journal = {Quarterly of Applied Mathematics},
   Volume = {77},
   Number = {3},
   Pages = {591-613},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1090/qam/1521},
   Abstract = {This paper introduces a novel data clustering algorithm
             based on Langevin dynamics, where the associated potential
             is constructed directly from the data. To introduce a
             self-consistent potential, we adopt the potential model from
             the established Quantum Clustering method. The first step is
             to use a radial basis function to construct a density
             distribution from the data. A potential function is then
             constructed such that this density distribution is the
             ground state solution to the time-independent Schrödinger
             equation. The second step is to use this potential function
             with the Langevin dynamics at subcritical temperature to
             avoid ergodicity. The Langevin equations take a classical
             Gibbs distribution as the invariant measure, where the peaks
             of the distribution coincide with minima of the potential
             surface. The time dynamics of individual data points lead to
             different metastable states, which are interpreted as
             cluster centers. Clustering is therefore achieved when
             subsets of the data aggregate-as a result of the Langevin
             dynamics for a moderate period of time-in the neighborhood
             of a particular potential minimum. While the data points are
             pushed towards potential minima by the potential gradient,
             Brownian motion allows them to effectively tunnel through
             local potential barriers and escape saddle points into
             locations of the potential surface otherwise forbidden. The
             algorithm's feasibility is first established based on
             several illustrating examples and theoretical analyses,
             followed by a stricter evaluation using a standard benchmark
             dataset.},
   Doi = {10.1090/qam/1521},
   Key = {fds347996}
}

@article{fds348011,
   Author = {Hu, W and Li, CJ and Li, L and Liu, J-G},
   Title = {On the diffusion approximation of nonconvex stochastic
             gradient descent},
   Journal = {Annals of Mathematical Sciences and Applications},
   Volume = {4},
   Number = {1},
   Pages = {3-32},
   Publisher = {International Press of Boston},
   Year = {2019},
   url = {http://dx.doi.org/10.4310/amsa.2019.v4.n1.a1},
   Doi = {10.4310/amsa.2019.v4.n1.a1},
   Key = {fds348011}
}

@article{fds366914,
   Author = {Liu, J-G and Yang, R},
   Title = {PROPAGATION OF CHAOS FOR THE KELLER-SEGEL EQUATION WITH A
             LOGARITHMIC CUT-OFF},
   Journal = {METHODS AND APPLICATIONS OF ANALYSIS},
   Volume = {26},
   Number = {4},
   Pages = {319-348},
   Year = {2019},
   Key = {fds366914}
}

@article{fds338528,
   Author = {Gao, Y and Ji, H and Liu, JG and Witelski, TP},
   Title = {A vicinal surface model for epitaxial growth with
             logarithmic free energy},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {23},
   Number = {10},
   Pages = {4433-4453},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2018},
   Month = {December},
   url = {http://dx.doi.org/10.3934/dcdsb.2018170},
   Abstract = {We study a continuum model for solid films that arises from
             the modeling of one-dimensional step flows on a vicinal
             surface in the attachment-detachment-limited regime. The
             resulting nonlinear partial differential equation, ut =
             -u2(u3 + au)hhhh, gives the evolution for the surface slope
             u as a function of the local height h in a monotone step
             train. Subject to periodic boundary conditions and positive
             initial conditions, we prove the existence, uniqueness and
             positivity of global strong solutions to this PDE using two
             Lyapunov energy functions. The long time behavior of u
             converging to a constant that only depends on the initial
             data is also investigated both analytically and
             numerically.},
   Doi = {10.3934/dcdsb.2018170},
   Key = {fds338528}
}

@article{fds340760,
   Author = {Feng, Y and Li, L and Liu, JG and Xu, X},
   Title = {Continuous and discrete one dimensional autonomous
             fractional odes},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {23},
   Number = {8},
   Pages = {3109-3135},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2018},
   Month = {October},
   url = {http://dx.doi.org/10.3934/dcdsb.2017210},
   Abstract = {In this paper, we study 1D autonomous fractional ODEs D c
             γu = f(u); 0 < γ < 1, where u : [0;∞) → R is the
             unknown function and D c is the generalized Caputo
             derivative introduced by Li and Liu ( arXiv:1612.05103).
             Based on the existence and uniqueness theorem and regularity
             results in previous work, we show the monotonicity of
             solutions to the autonomous fractional ODEs and several
             versions of comparison principles. We also perform a
             detailed discussion of the asymptotic behavior for f(u) =
             Aup. In particular, based on an Osgood type blow-up
             criteria, we find relatively sharp bounds of the blow-up
             time in the case A > 0; p > 1. These bounds indicate that as
             the memory effect becomes stronger ( → 0), if the initial
             value is big, the blow-up time tends to zero while if the
             initial value is small, the blow-up time tends to infiinity.
             In the case A < 0; p > 1, we show that the solution decays
             to zero more slowly compared with the usual derivative.
             Lastly, we show several comparison principles and Gronwall
             inequalities for discretized equations, and perform some
             numerical simulations to confirm our analysis.},
   Doi = {10.3934/dcdsb.2017210},
   Key = {fds340760}
}

@article{fds335603,
   Author = {Feng, Y and Li, L and Liu, JG and Xu, X},
   Title = {A note on one-dimensional time fractional
             ODEs},
   Journal = {Applied Mathematics Letters},
   Volume = {83},
   Pages = {87-94},
   Publisher = {Elsevier BV},
   Year = {2018},
   Month = {September},
   url = {http://dx.doi.org/10.1016/j.aml.2018.03.015},
   Abstract = {In this note, we prove or re-prove several important results
             regarding one dimensional time fractional ODEs following our
             previous work Feng et al. [15]. Here we use the definition
             of Caputo derivative proposed in Li and Liu (2017) [5,7]
             based on a convolution group. In particular, we establish
             generalized comparison principles consistent with the new
             definition of Caputo derivatives. In addition, we establish
             the full asymptotic behaviors of the solutions for
             Dcγu=Aup. Lastly, we provide a simplified proof for the
             strict monotonicity and stability in initial values for the
             time fractional differential equations with weak
             assumptions.},
   Doi = {10.1016/j.aml.2018.03.015},
   Key = {fds335603}
}

@article{fds335604,
   Author = {Li, L and Liu, JG and Wang, L},
   Title = {Cauchy problems for Keller–Segel type time–space
             fractional diffusion equation},
   Journal = {Journal of Differential Equations},
   Volume = {265},
   Number = {3},
   Pages = {1044-1096},
   Publisher = {Elsevier BV},
   Year = {2018},
   Month = {August},
   url = {http://dx.doi.org/10.1016/j.jde.2018.03.025},
   Abstract = {This paper investigates Cauchy problems for nonlinear
             fractional time–space generalized Keller–Segel equation
             Dtβ0cρ+(−△)[Formula presented]ρ+∇⋅(ρB(ρ))=0,
             where Caputo derivative Dtβ0cρ models memory effects in
             time, fractional Laplacian (−△)[Formula presented]ρ
             represents Lévy diffusion and B(ρ)=−sn,γ∫Rn[Formula
             presented]ρ(y)dy is the Riesz potential with a singular
             kernel which takes into account the long rang interaction.
             We first establish Lr−Lq estimates and weighted estimates
             of the fundamental solutions (P(x,t),Y(x,t)) (or
             equivalently, the solution operators (Sαβ(t),Tαβ(t))).
             Then, we prove the existence and uniqueness of the mild
             solutions when initial data are in Lp spaces, or the
             weighted spaces. Similar to Keller–Segel equations, if the
             initial data are small in critical space Lpc(Rn)
             (pc=[Formula presented]), we construct the global existence.
             Furthermore, we prove the L1 integrability and integral
             preservation when the initial data are in L1(Rn)∩Lp(Rn) or
             L1(Rn)∩Lpc(Rn). Finally, some important properties of the
             mild solutions including the nonnegativity preservation,
             mass conservation and blowup behaviors are
             established.},
   Doi = {10.1016/j.jde.2018.03.025},
   Key = {fds335604}
}

@article{fds340537,
   Author = {Gao, Y and Liu, JG},
   Title = {The modified camassa-holm equation in lagrangian
             coordinates},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {23},
   Number = {6},
   Pages = {2545-2592},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2018},
   Month = {August},
   url = {http://dx.doi.org/10.3934/dcdsb.2018067},
   Abstract = {In this paper, we study the modified Camassa-Holm (mCH)
             equation in Lagrangian coordinates. For some initial data
             m0, we show that classical solutions to this equation blow
             up in finite time Tmax. Before Tmax, existence and
             uniqueness of classical solutions are established. Lifespan
             for classical solutions is obtained: Tmax ≥||m0||L∞
             ||m0||L1 . And there is a unique solution 1 X(ξ, t) to the
             Lagrange dynamics which is a strictly monotonic function of
             ξ for any t ∈ [0, Tmax): Xξ(·, t) > 0. As t approaching
             Tmax, we prove that the classical solution m(·, t) in
             Eulerian coordinates has a unique limit m(·, Tmax) in Radon
             measure space and there is a point ξ0 such that Xξ(ξ0,
             Tmax) = 0 which means Tmax is an onset time of collisions of
             characteristics. We also show that in some cases peakons are
             formed at Tmax. After Tmax, we regularize the Lagrange
             dynamics to prove global existence of weak solutions m in
             Radon measure space.},
   Doi = {10.3934/dcdsb.2018067},
   Key = {fds340537}
}

@article{fds335605,
   Author = {Liu, JG and Tang, M and Wang, L and Zhou, Z},
   Title = {An accurate front capturing scheme for tumor growth models
             with a free boundary limit},
   Journal = {Journal of Computational Physics},
   Volume = {364},
   Pages = {73-94},
   Publisher = {Elsevier BV},
   Year = {2018},
   Month = {July},
   url = {http://dx.doi.org/10.1016/j.jcp.2018.03.013},
   Abstract = {We consider a class of tumor growth models under the
             combined effects of density-dependent pressure and cell
             multiplication, with a free boundary model as its singular
             limit when the pressure-density relationship becomes highly
             nonlinear. In particular, the constitutive law connecting
             pressure p and density ρ is p(ρ)=[Formula
             presented]ρm−1, and when m≫1, the cell density ρ may
             evolve its support according to a pressure-driven geometric
             motion with sharp interface along its boundary. The
             nonlinearity and degeneracy in the diffusion bring great
             challenges in numerical simulations. Prior to the present
             paper, there is lack of standard mechanism to numerically
             capture the front propagation speed as m≫1. In this paper,
             we develop a numerical scheme based on a novel
             prediction-correction reformulation that can accurately
             approximate the front propagation even when the nonlinearity
             is extremely strong. We show that the semi-discrete scheme
             naturally connects to the free boundary limit equation as
             m→∞. With proper spatial discretization, the fully
             discrete scheme has improved stability, preserves
             positivity, and can be implemented without nonlinear
             solvers. Finally, extensive numerical examples in both one
             and two dimensions are provided to verify the claimed
             properties in various applications.},
   Doi = {10.1016/j.jcp.2018.03.013},
   Key = {fds335605}
}

@article{fds335606,
   Author = {Chen, K and Li, Q and Liu, JG},
   Title = {Online learning in optical tomography: A stochastic
             approach},
   Journal = {Inverse Problems},
   Volume = {34},
   Number = {7},
   Pages = {075010-075010},
   Publisher = {IOP Publishing},
   Year = {2018},
   Month = {May},
   url = {http://dx.doi.org/10.1088/1361-6420/aac220},
   Abstract = {We study the inverse problem of radiative transfer equation
             (RTE) using stochastic gradient descent method (SGD) in this
             paper. Mathematically, optical tomography amounts to
             recovering the optical parameters in RTE using the
             incoming-outgoing pair of light intensity. We formulate it
             as a PDE-constraint optimization problem, where the mismatch
             of computed and measured outgoing data is minimized with
             same initial data and RTE constraint. The memory and
             computation cost it requires, however, is typically
             prohibitive, especially in high dimensional space. Smart
             iterative solvers that only use partial information in each
             step is called for thereafter. Stochastic gradient descent
             method is an online learning algorithm that randomly selects
             data for minimizing the mismatch. It requires minimum memory
             and computation, and advances fast, therefore perfectly
             serves the purpose. In this paper we formulate the problem,
             in both nonlinear and its linearized setting, apply SGD
             algorithm and analyze the convergence performance.},
   Doi = {10.1088/1361-6420/aac220},
   Key = {fds335606}
}

@article{fds333565,
   Author = {Liu, JG and Xu, X},
   Title = {Partial regularity of weak solutions to a PDE system with
             cubic nonlinearity},
   Journal = {Journal of Differential Equations},
   Volume = {264},
   Number = {8},
   Pages = {5489-5526},
   Publisher = {ACADEMIC PRESS INC ELSEVIER SCIENCE},
   Year = {2018},
   Month = {April},
   url = {http://dx.doi.org/10.1016/j.jde.2018.01.001},
   Abstract = {In this paper we investigate regularity properties of weak
             solutions to a PDE system that arises in the study of
             biological transport networks. The system consists of a
             possibly singular elliptic equation for the scalar pressure
             of the underlying biological network coupled to a diffusion
             equation for the conductance vector of the network. There
             are several different types of nonlinearities in the system.
             Of particular mathematical interest is a term that is a
             polynomial function of solutions and their partial
             derivatives and this polynomial function has degree three.
             That is, the system contains a cubic nonlinearity. Only weak
             solutions to the system have been shown to exist. The
             regularity theory for the system remains fundamentally
             incomplete. In particular, it is not known whether or not
             weak solutions develop singularities. In this paper we
             obtain a partial regularity theorem, which gives an estimate
             for the parabolic Hausdorff dimension of the set of possible
             singular points.},
   Doi = {10.1016/j.jde.2018.01.001},
   Key = {fds333565}
}

@article{fds333566,
   Author = {Li, L and Liu, JG},
   Title = {p-Euler equations and p-Navier–Stokes equations},
   Journal = {Journal of Differential Equations},
   Volume = {264},
   Number = {7},
   Pages = {4707-4748},
   Publisher = {Elsevier BV},
   Year = {2018},
   Month = {April},
   url = {http://dx.doi.org/10.1016/j.jde.2017.12.023},
   Abstract = {We propose in this work new systems of equations which we
             call p-Euler equations and p-Navier–Stokes equations.
             p-Euler equations are derived as the Euler–Lagrange
             equations for the action represented by the
             Benamou–Brenier characterization of Wasserstein-p
             distances, with incompressibility constraint. p-Euler
             equations have similar structures with the usual Euler
             equations but the ‘momentum’ is the signed (p−1)-th
             power of the velocity. In the 2D case, the p-Euler equations
             have streamfunction-vorticity formulation, where the
             vorticity is given by the p-Laplacian of the streamfunction.
             By adding diffusion presented by γ-Laplacian of the
             velocity, we obtain what we call p-Navier–Stokes
             equations. If γ=p, the a priori energy estimates for the
             velocity and momentum have dual symmetries. Using these
             energy estimates and a time-shift estimate, we show the
             global existence of weak solutions for the p-Navier–Stokes
             equations in Rd for γ=p and p≥d≥2 through a compactness
             criterion.},
   Doi = {10.1016/j.jde.2017.12.023},
   Key = {fds333566}
}

@article{fds335607,
   Author = {Gao, Y and Liu, JG and Lu, XY and Xu, X},
   Title = {Maximal monotone operator theory and its applications to
             thin film equation in epitaxial growth on vicinal
             surface},
   Journal = {Calculus of Variations and Partial Differential
             Equations},
   Volume = {57},
   Number = {2},
   Publisher = {Springer Nature},
   Year = {2018},
   Month = {April},
   url = {http://dx.doi.org/10.1007/s00526-018-1326-x},
   Abstract = {In this work we consider (Formula presented.) which is
             derived from a thin film equation for epitaxial growth on
             vicinal surface. We formulate the problem as the gradient
             flow of a suitably-defined convex functional in a
             non-reflexive space. Then by restricting it to a Hilbert
             space and proving the uniqueness of its sub-differential, we
             can apply the classical maximal monotone operator theory.
             The mathematical difficulty is due to the fact that whh can
             appear as a positive Radon measure. We prove the existence
             of a global strong solution with hidden singularity. In
             particular, (1) holds almost everywhere when whh is replaced
             by its absolutely continuous part.},
   Doi = {10.1007/s00526-018-1326-x},
   Key = {fds335607}
}

@article{fds338622,
   Author = {Feng, Y and Li, L and Liu, JG},
   Title = {Semigroups of stochastic gradient descent and online
             principal component analysis: Properties and diffusion
             approximations},
   Journal = {Communications in Mathematical Sciences},
   Volume = {16},
   Number = {3},
   Pages = {777-789},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.4310/cms.2018.v16.n3.a8},
   Abstract = {We study the Markov semigroups for two important algorithms
             from machine learning: stochastic gradient descent (SGD) and
             online principal component analysis (PCA). We investigate
             the effects of small jumps on the properties of the
             semigroups. Properties including regularity preserving, L∞
             contraction are discussed. These semigroups are the dual of
             the semigroups for evolution of probability, while the
             latter are L1 contracting and positivity preserving. Using
             these properties, we show that stochastic differential
             equations (SDEs) in Rd (on the sphere Sd-1) can be used to
             approximate SGD (online PCA) weakly. These SDEs may be used
             to provide some insights of the behaviors of these
             algorithms.},
   Doi = {10.4310/cms.2018.v16.n3.a8},
   Key = {fds338622}
}

@article{fds338623,
   Author = {Li, L and Liu, JG},
   Title = {Some compactness criteria for weak solutions of time
             fractional pdes},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {50},
   Number = {4},
   Pages = {3963-3995},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1137/17M1145549},
   Abstract = {The Aubin-Lions lemma and its variants play crucial roles
             for the existence of weak solutions of nonlinear
             evolutionary PDEs. In this paper, we aim to develop some
             compactness criteria that are analogies of the Aubin-Lions
             lemma for the existence of weak solutions to time fractional
             PDEs. We first define the weak Caputo derivatives of order
             γ ϵ (0; 1) for functions valued in general Banach spaces,
             consistent with the traditional definition if the space is
             Rd and functions are absolutely continuous. Based on a
             Volterra-type integral form, we establish some time
             regularity estimates of the functions provided that the weak
             Caputo derivatives are in certain spaces. The compactness
             criteria are then established using the time regularity
             estimates. The existence of weak solutions for a special
             case of time fractional compressible Navier-Stokes equations
             with constant density and time fractional Keller-Segel
             equations in R2 are then proved as model problems. This work
             provides a framework for studying weak solutions of
             nonlinear time fractional PDEs.},
   Doi = {10.1137/17M1145549},
   Key = {fds338623}
}

@article{fds335608,
   Author = {Gao, Y and Li, L and Liu, JG},
   Title = {A dispersive regularization for the modified camassa–holm
             equation},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {50},
   Number = {3},
   Pages = {2807-2838},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1137/17M1132756},
   Abstract = {In this paper, we present a dispersive regularization
             approach to construct a global N-peakon weak solution to the
             modified Camassa–Holm equation (mCH) in one dimension. In
             particular, we perform a double mollification for the system
             of ODEs describing trajectories of N-peakon solutions and
             obtain N smoothed peakons without collisions. Though the
             smoothed peakons do not give a solution to the mCH equation,
             the weak consistency allows us to take the smoothing
             parameter to zero and the limiting function is a global
             N-peakon weak solution. The trajectories of the peakons in
             the constructed solution are globally Lipschitz continuous
             and do not cross each other. When N = 2, the solution is a
             sticky peakon weak solution. At last, using the N-peakon
             solutions and through a mean field limit process, we obtain
             global weak solutions for general initial data m0 in Radon
             measure space.},
   Doi = {10.1137/17M1132756},
   Key = {fds335608}
}

@article{fds335609,
   Author = {Li, L and Liu, JG},
   Title = {A generalized definition of caputo derivatives and its
             application to fractional odes},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {50},
   Number = {3},
   Pages = {2867-2900},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1137/17M1160318},
   Abstract = {We propose a generalized definition of Caputo derivatives
             from t = 0 of order \gamma \in (0, 1) using a convolution
             group, and we build a convenient framework for studying
             initial value problems of general nonlinear time fractional
             differential equations. Our strategy is to define a modified
             Riemann-Liouville fractional calculus which agrees with the
             traditional Riemann-Liouville definition for t > 0 but
             includes some singularities at t = 0 so that the group
             property holds. Then, making use of this fractional
             calculus, we introduce the generalized definition of Caputo
             derivatives. The new definition is consistent with various
             definitions in the literature while revealing the underlying
             group structure. The underlying group property makes many
             properties of Caputo derivatives natural. In particular, it
             allows us to deconvolve the fractional differential
             equations to integral equations with completely monotone
             kernels, which then enables us to prove the general
             comparison principle with the most general conditions. This
             then allows for a priori energy estimates of fractional
             PDEs. Since the new definition is valid for locally
             integrable functions that can blow up in finite time, it
             provides a framework for solutions to fractional ODEs and
             fractional PDEs. Many fundamental results for fractional
             ODEs are revisited within this framework under very weak
             conditions.},
   Doi = {10.1137/17M1160318},
   Key = {fds335609}
}

@article{fds333567,
   Author = {Li, L and Liu, JG},
   Title = {A note on deconvolution with completely monotone sequences
             and discrete fractional calculus},
   Journal = {Quarterly of Applied Mathematics},
   Volume = {76},
   Number = {1},
   Pages = {189-198},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1090/qam/1479},
   Abstract = {We study in this work convolution groups generated by
             completely monotone sequences related to the ubiquitous
             time-delay memory effect in physics and engineering. In the
             first part, we give an accurate description of the
             convolution inverse of a completely monotone sequence and
             show that the deconvolution with a completely monotone
             kernel is stable. In the second part, we study a discrete
             fractional calculus defined by the convolution group
             generated by the completely monotone sequence c(1) = (1, 1,
             1,..), and show the consistency with time-continuous
             Riemann-Liouville calculus, which may be suitable for
             modeling memory kernels in discrete time
             series.},
   Doi = {10.1090/qam/1479},
   Key = {fds333567}
}

@article{fds333568,
   Author = {Coquel, F and Jin, S and Liu, JG and Wang, L},
   Title = {Entropic sub-cell shock capturing schemes via Jin-Xin
             relaxation and glimm front sampling for scalar conservation
             laws},
   Journal = {Mathematics of Computation},
   Volume = {87},
   Number = {311},
   Pages = {1083-1126},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1090/mcom/3253},
   Abstract = {We introduce a sub-cell shock capturing method for scalar
             conservation laws built upon the Jin-Xin relaxation
             framework. Here, sub-cell shock capturing is achieved using
             the original defect measure correction technique. The
             proposed method exactly restores entropy shock solutions of
             the exact Riemann problem and, moreover, it produces
             monotone and entropy satisfying approximate self-similar
             solutions. These solutions are then sampled using Glimm's
             random choice method to advance in time. The resulting
             scheme combines the simplicity of the Jin-Xin relaxation
             method with the resolution of the Glimm's scheme to achieve
             the sharp (no smearing) capturing of discontinuities. The
             benefit of using defect measure corrections over usual
             sub-cell shock capturing methods is that the scheme can be
             easily made entropy satisfying with respect to infinitely
             many entropy pairs. Consequently, under a classical CFL
             condition, the method is proved to converge to the unique
             entropy weak solution of the Cauchy problem for general
             non-linear flux functions. Numerical results show that the
             proposed method indeed captures shocks-including interacting
             shocks-sharply without any smearing.},
   Doi = {10.1090/mcom/3253},
   Key = {fds333568}
}

@article{fds333569,
   Author = {Liu, JG and Wang, L and Zhou, Z},
   Title = {Positivity-preserving and asymptotic preserving method for
             2D Keller-Segal equations},
   Journal = {Mathematics of Computation},
   Volume = {87},
   Number = {311},
   Pages = {1165-1189},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1090/mcom/3250},
   Abstract = {We propose a semi-discrete scheme for 2D Keller-Segel
             equations based on a symmetrization reformation, which is
             equivalent to the convex splitting method and is free of any
             nonlinear solver. We show that, this new scheme is stable as
             long as the initial condition does not exceed certain
             threshold, and it asymptotically preserves the quasi-static
             limit in the transient regime. Furthermore, we show that the
             fully discrete scheme is conservative and positivity
             preserving, which makes it ideal for simulations. The
             analogical schemes for the radial symmetric cases and the
             subcritical degenerate cases are also presented and
             analyzed. With extensive numerical tests, we verify the
             claimed properties of the methods and demonstrate their
             superiority in various challenging applications.},
   Doi = {10.1090/mcom/3250},
   Key = {fds333569}
}

@article{fds348001,
   Author = {Jin, S and Liu, J-G and Ma, Z},
   Title = {Uniform spectral convergence of the stochastic Galerkin
             method for the linear transport equations with random inputs
             in diffusive regime and a micro–macro decomposition-based
             asymptotic-preserving method},
   Journal = {Research in the Mathematical Sciences},
   Volume = {4},
   Number = {1},
   Publisher = {Springer Science and Business Media LLC},
   Year = {2017},
   Month = {December},
   url = {http://dx.doi.org/10.1186/s40687-017-0105-1},
   Doi = {10.1186/s40687-017-0105-1},
   Key = {fds348001}
}

@article{fds329519,
   Author = {Li, L and Liu, JG and Lu, J},
   Title = {Fractional Stochastic Differential Equations Satisfying
             Fluctuation-Dissipation Theorem},
   Journal = {Journal of Statistical Physics},
   Volume = {169},
   Number = {2},
   Pages = {316-339},
   Publisher = {Springer Nature America, Inc},
   Year = {2017},
   Month = {October},
   url = {http://dx.doi.org/10.1007/s10955-017-1866-z},
   Abstract = {We propose in this work a fractional stochastic differential
             equation (FSDE) model consistent with the over-damped limit
             of the generalized Langevin equation model. As a result of
             the ‘fluctuation-dissipation theorem’, the differential
             equations driven by fractional Brownian noise to model
             memory effects should be paired with Caputo derivatives, and
             this FSDE model should be understood in an integral form. We
             establish the existence of strong solutions for such
             equations and discuss the ergodicity and convergence to
             Gibbs measure. In the linear forcing regime, we show
             rigorously the algebraic convergence to Gibbs measure when
             the ‘fluctuation-dissipation theorem’ is satisfied, and
             this verifies that satisfying ‘fluctuation-dissipation
             theorem’ indeed leads to the correct physical behavior. We
             further discuss possible approaches to analyze the
             ergodicity and convergence to Gibbs measure in the nonlinear
             forcing regime, while leave the rigorous analysis for future
             works. The FSDE model proposed is suitable for systems in
             contact with heat bath with power-law kernel and
             subdiffusion behaviors.},
   Doi = {10.1007/s10955-017-1866-z},
   Key = {fds329519}
}

@article{fds329520,
   Author = {Liu, JG and Ma, Z and Zhou, Z},
   Title = {Explicit and Implicit TVD Schemes for Conservation Laws with
             Caputo Derivatives},
   Journal = {Journal of Scientific Computing},
   Volume = {72},
   Number = {1},
   Pages = {291-313},
   Publisher = {Springer Nature},
   Year = {2017},
   Month = {July},
   url = {http://dx.doi.org/10.1007/s10915-017-0356-4},
   Abstract = {In this paper, we investigate numerical approximations of
             the scalar conservation law with the Caputo derivative,
             which introduces the memory effect. We construct the first
             order and the second order explicit upwind schemes for such
             equations, which are shown to be conditionally ℓ1
             contracting and TVD. However, the Caputo derivative leads to
             the modified CFL-type stability condition, (Δ t) α= O(Δ
             x) , where α∈ (0 , 1 ] is the fractional exponent in the
             derivative. When α is small, such strong constraint makes
             the numerical implementation extremely impractical. We have
             then proposed the implicit upwind scheme to overcome this
             issue, which is proved to be unconditionally ℓ1
             contracting and TVD. Various numerical tests are presented
             to validate the properties of the methods and provide more
             numerical evidence in interpreting the memory effect in
             conservation laws.},
   Doi = {10.1007/s10915-017-0356-4},
   Key = {fds329520}
}

@article{fds329521,
   Author = {Gao, Y and Ji, H and Liu, JG and Witelski, TP},
   Title = {Global existence of solutions to a tear film model with
             locally elevated evaporation rates},
   Journal = {Physica D: Nonlinear Phenomena},
   Volume = {350},
   Pages = {13-25},
   Publisher = {Elsevier BV},
   Year = {2017},
   Month = {July},
   url = {http://dx.doi.org/10.1016/j.physd.2017.03.005},
   Abstract = {Motivated by a model proposed by Peng et al. (2014) for
             break-up of tear films on human eyes, we study the dynamics
             of a generalized thin film model. The governing equations
             form a fourth-order coupled system of nonlinear parabolic
             PDEs for the film thickness and salt concentration subject
             to non-conservative effects representing evaporation. We
             analytically prove the global existence of solutions to this
             model with mobility exponents in several different ranges
             and present numerical simulations that are in agreement with
             the analytic results. We also numerically capture other
             interesting dynamics of the model, including finite-time
             rupture–shock phenomenon due to the instabilities caused
             by locally elevated evaporation rates, convergence to
             equilibrium and infinite-time thinning.},
   Doi = {10.1016/j.physd.2017.03.005},
   Key = {fds329521}
}

@article{fds329522,
   Author = {Gao, Y and Liu, JG and Lu, J},
   Title = {Continuum Limit of a Mesoscopic Model with Elasticity of
             Step Motion on Vicinal Surfaces},
   Journal = {Journal of Nonlinear Science},
   Volume = {27},
   Number = {3},
   Pages = {873-926},
   Publisher = {Springer Nature},
   Year = {2017},
   Month = {June},
   url = {http://dx.doi.org/10.1007/s00332-016-9354-1},
   Abstract = {This work considers the rigorous derivation of continuum
             models of step motion starting from a mesoscopic
             Burton–Cabrera–Frank-type model following the Xiang’s
             work (Xiang in SIAM J Appl Math 63(1):241–258, 2002). We
             prove that as the lattice parameter goes to zero, for a
             finite time interval, a modified discrete model converges to
             the strong solution of the limiting PDE with first-order
             convergence rate.},
   Doi = {10.1007/s00332-016-9354-1},
   Key = {fds329522}
}

@article{fds325701,
   Author = {Liu, JG and Wang, J},
   Title = {Global existence for a thin film equation with subcritical
             mass},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {22},
   Number = {4},
   Pages = {1461-1492},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2017},
   Month = {June},
   url = {http://dx.doi.org/10.3934/dcdsb.2017070},
   Abstract = {In this paper, we study existence of global entropy weak
             solutions to a critical-case unstable thin film equation in
             one-dimensional case ht + x(hn xxxh) + x(hn+2xh) = 0; where
             n 1. There exists a critical mass Mc = 2 p 6 3 found by
             Witelski et al. (2004 Euro. J. of Appl. Math. 15, 223-256)
             for n = 1. We obtain global existence of a non-negative
             entropy weak solution if initial mass is less than Mc. For n
             4, entropy weak solutions are positive and unique. For n =
             1, a finite time blow-up occurs for solutions with initial
             mass larger than Mc. For the Cauchy problem with n = 1 and
             initial mass less than Mc, we show that at least one of the
             following long-time behavior holds: the second moment goes
             to infinity as the time goes to infinity or h(tk) 0 in L1(R)
             for some subsequence tk 1.},
   Doi = {10.3934/dcdsb.2017070},
   Key = {fds325701}
}

@article{fds325700,
   Author = {Degond, P and Liu, JG and Pego, RL},
   Title = {Coagulation–Fragmentation Model for Animal Group-Size
             Statistics},
   Journal = {Journal of Nonlinear Science},
   Volume = {27},
   Number = {2},
   Pages = {379-424},
   Publisher = {Springer Nature},
   Year = {2017},
   Month = {April},
   url = {http://dx.doi.org/10.1007/s00332-016-9336-3},
   Abstract = {We study coagulation–fragmentation equations inspired by a
             simple model proposed in fisheries science to explain data
             for the size distribution of schools of pelagic fish.
             Although the equations lack detailed balance and admit no
             H-theorem, we are able to develop a rather complete
             description of equilibrium profiles and large-time behavior,
             based on recent developments in complex function theory for
             Bernstein and Pick functions. In the large-population
             continuum limit, a scaling-invariant regime is reached in
             which all equilibria are determined by a single scaling
             profile. This universal profile exhibits power-law behavior
             crossing over from exponent -23 for small size to -32 for
             large size, with an exponential cutoff.},
   Doi = {10.1007/s00332-016-9336-3},
   Key = {fds325700}
}

@article{fds329169,
   Author = {Cong, W and Liu, JG},
   Title = {Uniform L boundedness for a degenerate
             parabolic-parabolic Keller-Segel model},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {22},
   Number = {2},
   Pages = {307-338},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2017},
   Month = {March},
   url = {http://dx.doi.org/10.3934/dcdsb.2017015},
   Abstract = {This paper investigates the existence of a uniform in time
             L∞ bounded weak entropy solution for the quasilinear
             parabolic-parabolic KellerSegel model with the supercritical
             diffusion exponent 0 < m < 2 - 2/d in the multi-dimensional
             space ℝd under the condition that the L d(2-m)/2 norm of
             initial data is smaller than a universal constant. Moreover,
             the weak entropy solution u(x,t) satisfies mass conservation
             when m > 1-2/d. We also prove the local existence of weak
             entropy solutions and a blow-up criterion for general L1 ∩
             L∞ initial data.},
   Doi = {10.3934/dcdsb.2017015},
   Key = {fds329169}
}

@article{fds329524,
   Author = {Gao, Y and Liu, JG and Lu, J},
   Title = {Weak solution of a continuum model for vicinal surface in
             the attachment-detachment-limited regime},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {49},
   Number = {3},
   Pages = {1705-1731},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1094543},
   Abstract = {We study in this work a continuum model derived from a
             one-dimensional attachmentdetachment-limited type step flow
             on a vicinal surface, ut = -u2(u3)hhhh, where u, considered
             as a function of step height h, is the step slope of the
             surface. We formulate a notion of a weak solution to this
             continuum model and prove the existence of a global weak
             solution, which is positive almost everywhere. We also study
             the long time behavior of the weak solution and prove it
             converges to a constant solution as time goes to infinity.
             The space-time Hölder continuity of the weak solution is
             also discussed as a byproduct.},
   Doi = {10.1137/16M1094543},
   Key = {fds329524}
}

@article{fds331396,
   Author = {Liu, JG and Wang, J},
   Title = {A generalized Sz. Nagy inequality in higher dimensions and
             the critical thin film equation},
   Journal = {Nonlinearity},
   Volume = {30},
   Number = {1},
   Pages = {35-60},
   Publisher = {IOP Publishing},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1088/0951-7715/30/1/35},
   Abstract = {In this paper, we provide an alternative proof for the
             classical Sz. Nagy inequality in one dimension by a
             variational method and generalize it to higher dimensions d
             ≥ 1 J(h): = (∫ℝd|h|dx)a-1 ∫ℝd |∇h|2 dx/(∫ℝd
             |h|m+1 dx)a+1/m+1 ≥ β0, where m > 0 for d = 1, 2, 0 < m <
             d+2/d-2 for d ≥ 3, and a = d+2(m+1)/md. The Euler-Lagrange
             equation for critical points of J(h) in the non-negative
             radial decreasing function space is given by a free boundary
             problem for a generalized Lane-Emden equation, which has a
             unique solution (denoted by hc) and the solution determines
             the best constant for the above generalized Sz. Nagy
             inequality. The connection between the critical mass Mc =
             ∫Rdbl; hc dx = 2√2π/3 for the thin-film equation and
             the best constant of the Sz. Nagy inequality in one
             dimension was first noted by Witelski et al (2004 Eur. J.
             Appl. Math. 15 223-56). For the following critical thin film
             equation in multi-dimension d ≥ 2 ht + ∇ · (h ∇
             Delta; h) + ∇ · (h ∇ hm) = 0, x ϵ ℝd, where m = 1 +
             2/d, the critical mass is also given by Mc:= ∫ℝd hc dx.
             A finite time blow-up occurs for solutions with the initial
             mass larger than Mc. On the other hand, if the initial mass
             is less than Mc and a global non-negative entropy weak
             solution exists, then the second moment goes to infinity as
             t → ∞ or h(·, tk) ⇀ 0 in L1(ℝd) for some
             subsequence tk → ∞. This shows that a part of the mass
             spreads to infinity.},
   Doi = {10.1088/0951-7715/30/1/35},
   Key = {fds331396}
}

@article{fds329523,
   Author = {Huang, H and Liu, JG},
   Title = {Discrete-in-time random particle blob method for the
             Keller-Segel equation and convergence analysis},
   Journal = {Communications in Mathematical Sciences},
   Volume = {15},
   Number = {7},
   Pages = {1821-1842},
   Publisher = {International Press of Boston},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2017.v15.n7.a2},
   Abstract = {We establish an error estimate of a discrete-in-time random
             particle blob method for the Keller{Segel (KS) equation in
             ℝd (d≥2). With a blob size ε=N-1/d(d+1) log(N), we
             prove the convergence rate between the solution to the KS
             equation and the empirical measure of the random particle
             method under L2 norm in probability, where N is the number
             of the particles.},
   Doi = {10.4310/CMS.2017.v15.n7.a2},
   Key = {fds329523}
}

@article{fds330537,
   Author = {Degond, P and Herty, M and Liu, JG},
   Title = {Meanfield games and model predictive control},
   Journal = {Communications in Mathematical Sciences},
   Volume = {15},
   Number = {5},
   Pages = {1403-1422},
   Publisher = {International Press of Boston},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2017.v15.n5.a9},
   Abstract = {Mean-field games are games with a continuum of players that
             incorporate the timedimension through a control-theoretic
             approach. Recently, simpler approaches relying on the
             Best-Reply Strategy have been proposed. They assume that the
             agents navigate their strategies towards their goal by
             taking the direction of steepest descent of their cost
             function (i.e. the opposite of the utility function). In
             this paper, we explore the link between Mean-Field Games and
             the Best Reply Strategy approach. This is done by
             introducing a Model Predictive Control framework, which
             consists of setting the Mean-Field Game over a short time
             interval which recedes as time moves on. We show that the
             Model Predictive Control offers a compromise between a
             possibly unrealistic Mean-Field Game approach and the
             sub-optimal Best-Reply Strategy.},
   Doi = {10.4310/CMS.2017.v15.n5.a9},
   Key = {fds330537}
}

@article{fds323838,
   Author = {Degond, P and Liu, JG and Merino-Aceituno, S and Tardiveau,
             T},
   Title = {Continuum dynamics of the intention field under weakly
             cohesive social interaction},
   Journal = {Mathematical Models and Methods in Applied
             Sciences},
   Volume = {27},
   Number = {1},
   Pages = {159-182},
   Publisher = {World Scientific Pub Co Pte Lt},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1142/S021820251740005X},
   Abstract = {We investigate the long-Time dynamics of an opinion
             formation model inspired by a work by Borghesi, Bouchaud and
             Jensen. First, we derive a Fokker-Planck-Type equation under
             the assumption that interactions between individuals produce
             little consensus of opinion (grazing collision
             approximation). Second, we study conditions under which the
             Fokker-Planck equation has non-Trivial equilibria and derive
             the macroscopic limit (corresponding to the long-Time
             dynamics and spatially localized interactions) for the
             evolution of the mean opinion. Finally, we compare two
             different types of interaction rates: The original one given
             in the work of Borghesi, Bouchaud and Jensen (symmetric
             binary interactions) and one inspired from works by Motsch
             and Tadmor (non-symmetric binary interactions). We show that
             the first case leads to a conservative model for the density
             of the mean opinion whereas the second case leads to a
             non-conservative equation. We also show that the speed at
             which consensus is reached asymptotically for these two
             rates has fairly different density dependence.},
   Doi = {10.1142/S021820251740005X},
   Key = {fds323838}
}

@article{fds332012,
   Author = {Liu, JG and Yang, R},
   Title = {A random particle blob method for the keller-segel equation
             and convergence analysis},
   Journal = {Mathematics of Computation},
   Volume = {86},
   Number = {304},
   Pages = {725-745},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1090/mcom/3118},
   Abstract = {In this paper, we introduce a random particle blob method
             for the Keller-Segel equation (with dimension d ≥ 2) and
             establish a rigorous convergence analysis.},
   Doi = {10.1090/mcom/3118},
   Key = {fds332012}
}

@article{fds329525,
   Author = {Gao, Y and Liu, JG},
   Title = {Global convergence of a sticky particle method for the
             modified Camassa-Holm equation},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {49},
   Number = {2},
   Pages = {1267-1294},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1102069},
   Abstract = {In this paper, we prove convergence of a sticky particle
             method for the modified Camassa-Holm equation (mCH) with
             cubic nonlinearity in one dimension. As a byproduct, we
             prove global existence of weak solutions u with regularity:
             u and ux are space-time BV functions. The total variation of
             m(•, t) = u(•, t) - uxx(•, t) is bounded by the total
             variation of the initial data m0. We also obtain
             W1,1(ℝ)-stability of weak solutions when solutions are in
             L∞ (0, ∞; W1,2(ℝ)). (Notice that peakon weak solutions
             are not in W1,2(ℝ).) Finally, we provide some examples of
             nonuniqueness of peakon weak solutions to the mCH
             equation.},
   Doi = {10.1137/16M1102069},
   Key = {fds329525}
}

@article{fds330536,
   Author = {Liu, JG and Xu, X},
   Title = {Analytical validation of a continuum model for the evolution
             of a crystal surface in multiple space dimensions},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {49},
   Number = {3},
   Pages = {2220-2245},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1098474},
   Abstract = {In this paper we are concerned with the existence of a weak
             solution to the initial boundary value problem for the
             equation ∂u/∂t = Δ(Δu)-3. This problem arises in the
             mathematical modeling of the evolution of a crystal surface.
             Existence of a weak solution u with Δu ≥ 0 is obtained
             via a suitable substitution. Our investigations reveal the
             close connection between this problem and the equation
             ∂tρ+ρ2Δ2ρ3 = 0, another crystal surface model first
             proposed by H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare
             [Phys. D, 240 (2011), pp. 1771-1784].},
   Doi = {10.1137/16M1098474},
   Key = {fds330536}
}

@article{fds327636,
   Author = {Huang, H and Liu, JG},
   Title = {Error estimate of a random particle blob method for the
             Keller-Segel equation},
   Journal = {Mathematics of Computation},
   Volume = {86},
   Number = {308},
   Pages = {2719-2744},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1090/mcom/3174},
   Abstract = {We establish an optimal error estimate for a random particle
             blob method for the Keller-Segel equation in ℝd (d ≥ 2).
             With a blob size ε = hκ (1/2 < κ < 1), we prove a rate h|
             ln h| of convergence in ℓhp (p > d/1-κ) norm up to a
             probability 1-hC| ln h|, where h is the initial grid
             size.},
   Doi = {10.1090/mcom/3174},
   Key = {fds327636}
}

@article{fds323245,
   Author = {Huang, H and Liu, JG},
   Title = {Error estimates of the aggregation-diffusion splitting
             algorithms for the Keller-Segel equations},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {21},
   Number = {10},
   Pages = {3463-3478},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2016},
   Month = {December},
   url = {http://dx.doi.org/10.3934/dcdsb.2016107},
   Abstract = {In this paper, we discuss error estimates associated with
             three different aggregation-diffusion splitting schemes for
             the Keller-Segel equations. We start with one algorithm
             based on the Trotter product formula, and we show that the
             convergence rate is CΔt, where Δt is the time-step size.
             Secondly, we prove the convergence rate CΔt2 for the
             Strang's splitting. Lastly, we study a splitting scheme with
             the linear transport approximation, and prove the
             convergence rate CΔt.},
   Doi = {10.3934/dcdsb.2016107},
   Key = {fds323245}
}

@article{fds348494,
   Author = {Liu, J-G and Yang, R},
   Title = {Propagation of chaos for large Brownian particle system with
             Coulomb interaction},
   Journal = {Research in the Mathematical Sciences},
   Volume = {3},
   Number = {1},
   Publisher = {Springer Science and Business Media LLC},
   Year = {2016},
   Month = {December},
   url = {http://dx.doi.org/10.1186/s40687-016-0086-5},
   Doi = {10.1186/s40687-016-0086-5},
   Key = {fds348494}
}

@article{fds318453,
   Author = {Huang, H and Liu, JG},
   Title = {A note on Monge-Ampère Keller-Segel equation},
   Journal = {Applied Mathematics Letters},
   Volume = {61},
   Pages = {26-34},
   Publisher = {Elsevier BV},
   Year = {2016},
   Month = {November},
   url = {http://dx.doi.org/10.1016/j.aml.2016.05.003},
   Abstract = {This note studies the Monge-Ampère Keller-Segel equation in
             a periodic domain Td(d≥2), a fully nonlinear modification
             of the Keller-Segel equation where the Monge-Ampère
             equation det(I+2v)=u+1 substitutes for the usual Poisson
             equation Δv=u. The existence of global weak solutions is
             obtained for this modified equation. Moreover, we prove the
             regularity in L∞(0,T;L∞W1,1+γ(Td)) for some
             γ>0.},
   Doi = {10.1016/j.aml.2016.05.003},
   Key = {fds318453}
}

@article{fds320551,
   Author = {Liu, JG and Wang, J},
   Title = {A Note on L∞-Bound and Uniqueness to a Degenerate
             Keller-Segel Model},
   Journal = {Acta Applicandae Mathematicae},
   Volume = {142},
   Number = {1},
   Pages = {173-188},
   Publisher = {Springer Nature},
   Year = {2016},
   Month = {April},
   ISSN = {0167-8019},
   url = {http://dx.doi.org/10.1007/s10440-015-0022-5},
   Abstract = {In this note we establish the uniform (Formula presented.)
             -bound for the weak solutions to a degenerate Keller-Segel
             equation with the diffusion exponent (Formula presented.)
             under a sharp condition on the initial data for the global
             existence. As a consequence, the uniqueness of the weak
             solutions is also proved.},
   Doi = {10.1007/s10440-015-0022-5},
   Key = {fds320551}
}

@article{fds315797,
   Author = {Herschlag, G and Liu, JG and Layton, AT},
   Title = {Fluid extraction across pumping and permeable walls in the
             viscous limit},
   Journal = {Physics of Fluids},
   Volume = {28},
   Number = {4},
   Pages = {041902-041902},
   Publisher = {AIP Publishing},
   Year = {2016},
   Month = {April},
   ISSN = {1070-6631},
   url = {http://dx.doi.org/10.1063/1.4946005},
   Abstract = {In biological transport mechanisms such as insect
             respiration and renal filtration, fluid travels along a
             leaky channel allowing material exchange with systems
             exterior to the channel. The channels in these systems may
             undergo peristaltic pumping which is thought to enhance the
             material exchange. To date, little analytic work has been
             done to study the effect of pumping on material extraction
             across the channel walls. In this paper, we examine a fluid
             extraction model in which fluid flowing through a leaky
             channel is exchanged with fluid in a reservoir. The channel
             walls are allowed to contract and expand uniformly,
             simulating a pumping mechanism. In order to efficiently
             determine solutions of the model, we derive a formal power
             series solution for the Stokes equations in a finite channel
             with uniformly contracting/expanding permeable walls. This
             flow has been well studied in the case in which the normal
             velocity at the channel walls is proportional to the wall
             velocity. In contrast we do not assume flow that is
             proportional to the wall velocity, but flow that is driven
             by hydrostatic pressure, and we use Darcy's law to close our
             system for normal wall velocity. We incorporate our flow
             solution into a model that tracks the material pressure
             exterior to the channel. We use this model to examine flux
             across the channel-reservoir barrier and demonstrate that
             pumping can either enhance or impede fluid extraction across
             channel walls. We find that associated with each set of
             physical flow and pumping parameters, there are optimal
             reservoir conditions that maximize the amount of material
             flowing from the channel into the reservoir.},
   Doi = {10.1063/1.4946005},
   Key = {fds315797}
}

@article{fds333570,
   Author = {Liu, J-G and Wang, J},
   Title = {Refined hyper-contractivity and uniqueness for the
             Keller–Segel equations},
   Journal = {Applied Mathematics Letters},
   Volume = {52},
   Pages = {212-219},
   Publisher = {Elsevier BV},
   Year = {2016},
   Month = {February},
   url = {http://dx.doi.org/10.1016/j.aml.2015.09.001},
   Doi = {10.1016/j.aml.2015.09.001},
   Key = {fds333570}
}

@article{fds329526,
   Author = {Chen, J and Liu, JG and Zhou, Z},
   Title = {On a Schrödinger-Landau-Lifshitz system: Variational
             structure and numerical methods},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {14},
   Number = {4},
   Pages = {1463-1487},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2016},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M106947X},
   Abstract = {From a variational perspective, we derive a series of
             magnetization and quantum spin current systems coupled via
             an "s-d" potential term, including the Schrödinger-Landau-Lifshitz-
             Maxwell system, the Pauli-Landau-Lifshitz system, and the
             Schrödinger-Landau-Lifshitz system with successive
             simplifications. For the latter two systems, we propose
             using the time splitting spectral method for the quantum
             spin current and the Gauss-Seidel projection method for the
             magnetization. Accuracy of the time splitting spectral
             method applied to the Pauli equation is analyzed and
             verified by numerous examples. Moreover, behaviors of the
             Schrödinger-Landau- Lifshitz system in different "s-d"
             coupling regimes are explored numerically.},
   Doi = {10.1137/16M106947X},
   Key = {fds329526}
}

@article{fds318454,
   Author = {Huang, H and Liu, JG},
   Title = {Well-posedness for the keller-segel equation with fractional
             laplacian and the theory of propagation of
             chaos},
   Journal = {Kinetic and Related Models},
   Volume = {9},
   Number = {4},
   Pages = {715-748},
   Year = {2016},
   Month = {January},
   url = {http://dx.doi.org/10.3934/krm.2016013},
   Abstract = {This paper investigates the generalized Keller-Segel (KS)
             system with a nonlocal diffusion term -ν(-Δ) α/2 ρ (1 <
             α < 2). Firstly, the global existence of weak solutions is
             proved for the initial density ρ0 ∈ L1∩L d/α (ℝd) (d
             ≥ 2) with [norm of matrix]ρ0[norm of matrix] d/α < K,
             where K is a universal constant only depending on d, α, ν.
             Moreover, the conservation of mass holds true and the weak
             solution satisfies some hyper-contractive and decay
             estimates in Lr for any 1 < r < ∞. Secondly, for the more
             general initial data ρ0 ∈ L1 ∩ L2(ℝd) (d = 2, 3), the
             local existence is obtained. Thirdly, for ρ0 ∈ L1 (ℝd;
             (1 + |x|)dx ∩ L∞(ℝd)( d ≥ 2) with [norm of
             matrix]ρ0[norm of matrix]d/α < K, we prove the uniqueness
             and stability of weak solutions under Wasserstein metric
             through the method of associating the KS equation with a
             self-consistent stochastic process driven by the
             rotationally invariant α-stable Lévy process Lα(t). Also,
             we prove the weak solution is L1 bounded uniformly in time.
             Lastly, we consider the N-particle interacting system with
             the Lévy process Lα(t) and the Newtonian potential
             aggregation and prove that the expectation of collision time
             between particles is below a universal constant if the
             moment ∫ℝd |x| γρ0dx for some 1 < γ < α is below a
             universal constant K γ and ν is also below a universal
             constant. Meanwhile, we prove the propagation of chaos as N
             → ∞ for the interacting particle system with a cut-off
             parameter ε ~ (ln N)-1/d, and show that the mean field
             limit equation is exactly the generalized KS
             equation.},
   Doi = {10.3934/krm.2016013},
   Key = {fds318454}
}

@article{fds318455,
   Author = {Cong, W and Liu, JG},
   Title = {A degenerate p-laplacian keller-segel model},
   Journal = {Kinetic and Related Models},
   Volume = {9},
   Number = {4},
   Pages = {687-714},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2016},
   Month = {January},
   url = {http://dx.doi.org/10.3934/krm.2016012},
   Abstract = {This paper investigates the existence of a uniform in time
             L∞ bounded weak solution for the p-Laplacian Keller-Segel
             system with the supercritical diffusion exponent 1 < p <
             3d/d+1 in the multi-dimensional space ℝd under the
             condition that the L d(3-p)/p norm of initial data is
             smaller than a universal constant. We also prove the local
             existence of weak solutions and a blow-up criterion for
             general L1 ∩L∞ initial data.},
   Doi = {10.3934/krm.2016012},
   Key = {fds318455}
}

@article{fds323246,
   Author = {Liu, JG and Xu, X},
   Title = {Existence theorems for a multidimensional crystal surface
             model},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {48},
   Number = {6},
   Pages = {3667-3687},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2016},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1059400},
   Abstract = {In this paper we study the existence assertion of the
             initial boundary value problem for the equation @u/@t =
             Δe-Δu. This problem arises in the mathematical description
             of the evolution of crystal surfaces. Our investigations
             reveal that the exponent in the equation can have a singular
             part in the sense of the Lebesgue decomposition theorem, and
             the exponential nonlinearity somehow "cancels" it out. The
             net result is that we obtain a solution u that satisfies the
             equation and the initial boundary conditions in the almost
             everywhere (a.e.) sense.},
   Doi = {10.1137/16M1059400},
   Key = {fds323246}
}

@article{fds320552,
   Author = {Liu, JG and Pego, RL},
   Title = {On generating functions of hausdorff moment
             sequences},
   Journal = {Transactions of the American Mathematical
             Society},
   Volume = {368},
   Number = {12},
   Pages = {8499-8518},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2016},
   Month = {January},
   url = {http://dx.doi.org/10.1090/tran/6618},
   Abstract = {The class of generating functions for completely monotone
             sequences (moments of finite positive measures on [0, 1])
             has an elegant characterization as the class of Pick
             functions analytic and positive on (−∞, 1). We establish
             this and another such characterization and develop a variety
             of consequences. In particular, we characterize generating
             functions for moments of convex and concave probability
             distribution functions on [0, 1]. Also we provide a simple
             analytic proof that for any real p and r with p > 0, the
             Fuss-Catalan or Raney numbers (Formula Presented) are the
             moments of a probability distribution on some interval [0,
             τ] if and only if p ≥ 1 and p ≥ r ≥ 0. The same
             statement holds for the binomial coefficients (Formula
             Presented).},
   Doi = {10.1090/tran/6618},
   Key = {fds320552}
}

@article{fds320553,
   Author = {Liu, JG and Zhang, Y},
   Title = {Convergence of diffusion-drift many particle systems in
             probability under a sobolev norm},
   Journal = {Springer Proceedings in Mathematics and Statistics},
   Volume = {162},
   Series = {Proceedings of Particle Systems and Partial Differential
             Equations - III},
   Pages = {195-223},
   Publisher = {Springer International Publishing},
   Year = {2016},
   Month = {January},
   ISBN = {9783319321424},
   url = {http://dx.doi.org/10.1007/978-3-319-32144-8_10},
   Abstract = {In this paperwedevelop a newmartingale method to showthe
             convergence of the regularized empirical measure of many
             particle systems in probability under a Sobolev norm to the
             corresponding mean field PDE. Our method works well for the
             simple case of Fokker Planck equation and we can estimate a
             lower bound of the rate of convergence. This method can be
             generalized to more complicated systems with
             interactions.},
   Doi = {10.1007/978-3-319-32144-8_10},
   Key = {fds320553}
}

@article{fds362424,
   Author = {Duan, Y and Liu, J-G},
   Title = {Error estimate of the particle method for the
             $b$-equation},
   Journal = {Methods and Applications of Analysis},
   Volume = {23},
   Number = {2},
   Pages = {119-154},
   Publisher = {International Press of Boston},
   Year = {2016},
   url = {http://dx.doi.org/10.4310/maa.2016.v23.n2.a1},
   Doi = {10.4310/maa.2016.v23.n2.a1},
   Key = {fds362424}
}

@article{fds362425,
   Author = {Liu, J-G and Zhang, Y},
   Title = {Convergence of stochastic interacting particle systems in
             probability under a Sobolev norm},
   Journal = {Annals of Mathematical Sciences and Applications},
   Volume = {1},
   Number = {2},
   Pages = {251-299},
   Publisher = {International Press of Boston},
   Year = {2016},
   url = {http://dx.doi.org/10.4310/amsa.2016.v1.n2.a1},
   Doi = {10.4310/amsa.2016.v1.n2.a1},
   Key = {fds362425}
}

@article{fds341422,
   Author = {Degond, P and Frouvelle, A and Liu, JG},
   Title = {Phase Transitions, Hysteresis, and Hyperbolicity for
             Self-Organized Alignment Dynamics},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {216},
   Number = {1},
   Pages = {63-115},
   Year = {2015},
   Month = {April},
   url = {http://dx.doi.org/10.1007/s00205-014-0800-7},
   Abstract = {We provide a complete and rigorous description of phase
             transitions for kinetic models of self-propelled particles
             interacting through alignment. These models exhibit a
             competition between alignment and noise. Both the alignment
             frequency and noise intensity depend on a measure of the
             local alignment. We show that, in the spatially homogeneous
             case, the phase transition features (number and nature of
             equilibria, stability, convergence rate, phase diagram,
             hysteresis) are totally encoded in how the ratio between the
             alignment and noise intensities depend on the local
             alignment. In the spatially inhomogeneous case, we derive
             the macroscopic models associated to the stable equilibria
             and classify their hyperbolicity according to the same
             function.},
   Doi = {10.1007/s00205-014-0800-7},
   Key = {fds341422}
}

@article{fds246842,
   Author = {Xue, Y and Wang, C and Liu, JG},
   Title = {Simple Finite Element Numerical Simulation of Incompressible
             Flow Over Non-rectangular Domains and the Super-Convergence
             Analysis},
   Journal = {Journal of Scientific Computing},
   Volume = {65},
   Number = {3},
   Pages = {1189-1216},
   Publisher = {Springer Nature},
   Year = {2015},
   Month = {March},
   ISSN = {0885-7474},
   url = {http://dx.doi.org/10.1007/s10915-015-0005-8},
   Abstract = {In this paper, we apply a simple finite element numerical
             scheme, proposed in an earlier work (Liu in Math Comput
             70(234):579–593, 2000), to perform a high resolution
             numerical simulation of incompressible flow over an
             irregular domain and analyze its boundary layer separation.
             Compared with many classical finite element fluid solvers,
             this numerical method avoids a Stokes solver, and only two
             Poisson-like equations need to be solved at each time
             step/stage. In addition, its combination with the fully
             explicit fourth order Runge–Kutta (RK4) time
             discretization enables us to compute high Reynolds number
             flow in a very efficient way. As an application of this
             robust numerical solver, the dynamical mechanism of the
             boundary layer separation for a triangular cavity flow with
             Reynolds numbers $$Re=10^4$$Re=104 and $$Re=10^5$$Re=105,
             including the precise values of bifurcation location and
             critical time, are reported in this paper. In addition, we
             provide a super-convergence analysis for the simple finite
             element numerical scheme, using linear elements over a
             uniform triangulation with right triangles.},
   Doi = {10.1007/s10915-015-0005-8},
   Key = {fds246842}
}

@article{fds246843,
   Author = {Lu, J and Liu, JG and Margetis, D},
   Title = {Emergence of step flow from an atomistic scheme of epitaxial
             growth in 1+1 dimensions},
   Journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter
             Physics},
   Volume = {91},
   Number = {3},
   Pages = {032403},
   Year = {2015},
   Month = {March},
   ISSN = {1539-3755},
   url = {http://dx.doi.org/10.1103/PhysRevE.91.032403},
   Abstract = {The Burton-Cabrera-Frank (BCF) model for the flow of line
             defects (steps) on crystal surfaces has offered useful
             insights into nanostructure evolution. This model has rested
             on phenomenological grounds. Our goal is to show via scaling
             arguments the emergence of the BCF theory for noninteracting
             steps from a stochastic atomistic scheme of a kinetic
             restricted solid-on-solid model in one spatial dimension.
             Our main assumptions are: adsorbed atoms (adatoms) form a
             dilute system, and elastic effects of the crystal lattice
             are absent. The step edge is treated as a front that
             propagates via probabilistic rules for atom attachment and
             detachment at the step. We formally derive a quasistatic
             step flow description by averaging out the stochastic scheme
             when terrace diffusion, adatom desorption, and deposition
             from above are present.},
   Doi = {10.1103/PhysRevE.91.032403},
   Key = {fds246843}
}

@article{fds300222,
   Author = {Chertock, A and Liu, JG and Pendleton, T},
   Title = {Elastic collisions among peakon solutions for the
             Camassa-Holm equation},
   Journal = {Applied Numerical Mathematics},
   Volume = {93},
   Pages = {30-46},
   Publisher = {Elsevier BV},
   Year = {2015},
   Month = {January},
   ISSN = {0168-9274},
   url = {http://dx.doi.org/10.1016/j.apnum.2014.01.001},
   Abstract = {The purpose of this paper is to study the dynamics of the
             interaction among a special class of solutions of the
             one-dimensional Camassa-Holm equation. The equation yields
             soliton solutions whose identity is preserved through
             nonlinear interactions. These solutions are characterized by
             a discontinuity at the peak in the wave shape and are thus
             called peakon solutions. We apply a particle method to the
             Camassa-Holm equation and show that the nonlinear
             interaction among the peakon solutions resembles an elastic
             collision, i.e., the total energy and momentum of the system
             before the peakon interaction is equal to the total energy
             and momentum of the system after the collision. From this
             result, we provide several numerical illustrations which
             support the analytical study, as well as showcase the merits
             of using a particle method to simulate solutions to the
             Camassa-Holm equation under a wide class of initial
             data.},
   Doi = {10.1016/j.apnum.2014.01.001},
   Key = {fds300222}
}

@article{fds313338,
   Author = {Herschlag, G and Liu, JG and Layton, AT},
   Title = {An exact solution for stokes flow in a channel with
             arbitrarily large wall permeability},
   Journal = {SIAM Journal on Applied Mathematics},
   Volume = {75},
   Number = {5},
   Pages = {2246-2267},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2015},
   Month = {January},
   ISSN = {0036-1399},
   url = {http://dx.doi.org/10.1137/140995854},
   Abstract = {We derive an exact solution for Stokes flow in a channel
             with permeable walls. At the channel walls, the normal
             component of the fluid velocity is described by Darcy's law,
             and the tangential component of the fluid velocity is
             described by the no slip condition. The pressure exterior to
             the channel is assumed to be constant. Although this problem
             has been well studied, typical studies assume that the
             permeability of the wall is small relative to other
             nondimensional parameters; this work relaxes this assumption
             and explores a regime in parameter space that has not yet
             been well studied. A consequence of this relaxation is that
             transverse velocity is no longer necessarily small when
             compared with the axial velocity. We use our result to
             explore how existing asymptotic theories break down in the
             limit of large permeability for channels of small
             length.},
   Doi = {10.1137/140995854},
   Key = {fds313338}
}

@article{fds365498,
   Author = {Degond, P and Frouvelle, A and Liu, J-G and Motsch, S and Navoret,
             L},
   Title = {Macroscopic models of collective motion and
             self-organization},
   Journal = {Séminaire Laurent Schwartz — EDP et applications},
   Volume = {2012 - 2013},
   Pages = {1-27},
   Publisher = {Cellule MathDoc/CEDRAM},
   Year = {2014},
   Month = {November},
   url = {http://dx.doi.org/10.5802/slsedp.32},
   Doi = {10.5802/slsedp.32},
   Key = {fds365498}
}

@article{fds246846,
   Author = {Degond, P and Liu, J-G and Ringhofer, C},
   Title = {Evolution of wealth in a non-conservative economy driven by
             local Nash equilibria.},
   Journal = {Philosophical transactions. Series A, Mathematical,
             physical, and engineering sciences},
   Volume = {372},
   Number = {2028},
   Pages = {20130394},
   Publisher = {The Royal Society},
   Year = {2014},
   Month = {November},
   ISSN = {1364-503X},
   url = {http://dx.doi.org/10.1098/rsta.2013.0394},
   Abstract = {We develop a model for the evolution of wealth in a
             non-conservative economic environment, extending a theory
             developed in Degond et al. (2014 J. Stat. Phys. 154, 751-780
             (doi:10.1007/s10955-013-0888-4)). The model considers a
             system of rational agents interacting in a game-theoretical
             framework. This evolution drives the dynamics of the agents
             in both wealth and economic configuration variables. The
             cost function is chosen to represent a risk-averse strategy
             of each agent. That is, the agent is more likely to interact
             with the market, the more predictable the market, and
             therefore the smaller its individual risk. This yields a
             kinetic equation for an effective single particle agent
             density with a Nash equilibrium serving as the local
             thermodynamic equilibrium. We consider a regime of scale
             separation where the large-scale dynamics is given by a
             hydrodynamic closure with this local equilibrium. A class of
             generalized collision invariants is developed to overcome
             the difficulty of the non-conservative property in the
             hydrodynamic closure derivation of the large-scale dynamics
             for the evolution of wealth distribution. The result is a
             system of gas dynamics-type equations for the density and
             average wealth of the agents on large scales. We recover the
             inverse Gamma distribution, which has been previously
             considered in the literature, as a local equilibrium for
             particular choices of the cost function.},
   Doi = {10.1098/rsta.2013.0394},
   Key = {fds246846}
}

@article{fds246848,
   Author = {Coquel, F and Jin, S and Liu, JG and Wang, L},
   Title = {Well-Posedness and Singular Limit of a Semilinear Hyperbolic
             Relaxation System with a Two-Scale Discontinuous Relaxation
             Rate},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {214},
   Number = {3},
   Pages = {1051-1084},
   Year = {2014},
   Month = {October},
   ISSN = {0003-9527},
   url = {http://dx.doi.org/10.1007/s00205-014-0773-6},
   Abstract = {Nonlinear hyperbolic systems with relaxations may encounter
             different scales of relaxation time, which is a prototype
             multiscale phenomenon that arises in many applications. In
             such a problem the relaxation time is of O(1) in part of the
             domain and very small in the remaining domain in which the
             solution can be approximated by the zero relaxation limit
             which can be solved numerically much more efficiently. For
             the Jin–Xin relaxation system in such a two-scale setting,
             we establish its wellposedness and singular limit as the
             (smaller) relaxation time goes to zero. The limit is a
             multiscale coupling problem which couples the original
             Jin–Xin system on the domain when the relaxation time is
             O(1) with its relaxation limit in the other domain through
             interface conditions which can be derived by matched
             interface layer analysis.As a result, we also establish the
             well-posedness and regularity (such as boundedness in sup
             norm with bounded total variation and L1-contraction) of the
             coupling problem, thus providing a rigorous mathematical
             foundation, in the general nonlinear setting, to the
             multiscale domain decomposition method for this two-scale
             problem originally proposed in Jin et al. in Math. Comp. 82,
             749–779, 2013.},
   Doi = {10.1007/s00205-014-0773-6},
   Key = {fds246848}
}

@article{fds246857,
   Author = {Johnston, H and Wang, C and Liu, JG},
   Title = {A Local Pressure Boundary Condition Spectral Collocation
             Scheme for the Three-Dimensional Navier–Stokes
             Equations},
   Journal = {Journal of Scientific Computing},
   Volume = {60},
   Number = {3},
   Pages = {612-626},
   Publisher = {Springer Nature},
   Year = {2014},
   Month = {September},
   ISSN = {0885-7474},
   url = {http://dx.doi.org/10.1007/s10915-013-9808-7},
   Abstract = {A spectral collocation scheme for the three-dimensional
             incompressible (u,p) formulation of the Navier–Stokes
             equations, in domains Ω with a non-periodic boundary
             condition, is described. The key feature is the high order
             approximation, by means of a local Hermite interpolant, of a
             Neumann boundary condition for use in the numerical solution
             of the pressure Poisson system. The time updates of the
             velocity u and pressure p are decoupled as a result of
             treating the pressure gradient in the momentum equation
             explicitly in time. The pressure update is computed from a
             pressure Poisson equation. Extension of the overall
             methodology to the Boussinesq system is also described. The
             uncoupling of the pressure and velocity time updates results
             in a highly efficient scheme that is simple to implement and
             well suited for simulating moderate to high Reynolds and
             Rayleigh number flows. Accuracy checks are presented, along
             with simulations of the lid-driven cavity flow and a
             differentially heated cavity flow, to demonstrate the scheme
             produces accurate three-dimensional results at a reasonable
             computational cost.},
   Doi = {10.1007/s10915-013-9808-7},
   Key = {fds246857}
}

@article{fds246862,
   Author = {Duan, Y and Liu, JG},
   Title = {Convergence analysis of the vortex blob method for the
             b-equation},
   Journal = {Discrete and Continuous Dynamical Systems- Series
             A},
   Volume = {34},
   Number = {5},
   Pages = {1995-2011},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2014},
   Month = {May},
   ISSN = {1078-0947},
   url = {http://dx.doi.org/10.3934/dcds.2014.34.1995},
   Abstract = {In this paper, we prove the convergence of the vortex blob
             method for a family of nonlinear evolutionary partial
             differential equations (PDEs), the so-called b-equation.
             This kind of PDEs, including the Camassa-Holm equation and
             the Degasperis-Procesi equation, has many applications in
             diverse scientific fields. Our convergence analysis also
             provides a proof for the existence of the global weak
             solution to the b-equation when the initial data is a
             nonnegative Radon measure with compact support.},
   Doi = {10.3934/dcds.2014.34.1995},
   Key = {fds246862}
}

@article{fds246858,
   Author = {Bian, S and Liu, JG and Zou, C},
   Title = {Ultra-contractivity for keller-segel model with diffusion
             exponent m > 1-2/d},
   Journal = {Kinetic and Related Models},
   Volume = {7},
   Number = {1},
   Pages = {9-28},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2014},
   Month = {March},
   ISSN = {1937-5093},
   url = {http://dx.doi.org/10.3934/krm.2014.7.9},
   Abstract = {This paper establishes the hyper-contractivity in L∞(ℝd)
             (it's known as ultra-contractivity) for the
             multi-dimensional Keller-Segel systems with the diffusion
             exponent m > 1-2/d. The results show that for the super-
             critical and critical case 1-2/d < m ≤ 2-2/d, if
             ∥U0∥d(2-m)/2 < Cd, m where Cd, m is a universal
             constant, then for any t > 0 ∥u(.,t)∥L∞(ℝd) is
             bounded and decays as t goes to infinity. For the
             subcritical case m > 2-2/d, the solution u(.,t)∈
             L∞(ℝd) with any initial data U0 ∈ L1+(ℝd) for any
             positive time.},
   Doi = {10.3934/krm.2014.7.9},
   Key = {fds246858}
}

@article{fds246849,
   Author = {Degond, P and Herty, M and Liu, JG},
   Title = {Flow on sweeping networks},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {12},
   Number = {2},
   Pages = {538-565},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2014},
   Month = {January},
   ISSN = {1540-3459},
   url = {http://dx.doi.org/10.1137/130927061},
   Abstract = {We introduce a cellular automaton model coupled with a
             transport equation for flows on graphs. The direction of the
             flow is described by a switching process where the switching
             probability dynamically changes according to the value of
             the transported quantity in the neighboring cells. A
             motivation is pedestrian dynamics during panic situations in
             a small corridor where the propagation of people in a part
             of the corridor can be either left- or right-going. Under
             the assumptions of propagation of chaos and mean-field
             limit, we derive a master equation and the corresponding
             mean-field kinetic and macroscopic models. Steady-states are
             computed and analyzed and exhibit the possibility of
             multiple metastable states and hysteresis. © 2014 Society
             for Industrial and Applied Mathematics.},
   Doi = {10.1137/130927061},
   Key = {fds246849}
}

@article{fds246851,
   Author = {Chen, X and Li, X and Liu, JG},
   Title = {Existence and uniqueness of global weak solution to a
             kinetic model for the sedimentation of rod-like
             particles},
   Journal = {Communications in Mathematical Sciences},
   Volume = {12},
   Number = {8},
   Pages = {1579-1601},
   Publisher = {International Press of Boston},
   Year = {2014},
   Month = {January},
   ISSN = {1539-6746},
   url = {http://dx.doi.org/10.4310/CMS.2014.v12.n8.a10},
   Abstract = {We investigate a kinetic model for the sedimentation of
             dilute suspensions of rod-like particles under gravity,
             deduced by Helzel, Otto, and Tzavaras (2011), which couples
             the impressible (Navier-)Stokes equation with the
             Fokker-Planck equation. With a no-flux boundary condition
             for the distribution function, we establish the existence
             and uniqueness of a global weak solution to the two
             dimensional model involving the Stokes equation. ©
             2014.},
   Doi = {10.4310/CMS.2014.v12.n8.a10},
   Key = {fds246851}
}

@article{fds333571,
   Author = {Degond, P and Frouvelle, A and Liu, J-G},
   Title = {A NOTE ON PHASE TRANSITIONS FOR THE SMOLUCHOWSKI EQUATION
             WITH DIPOLAR POTENTIAL},
   Journal = {HYPERBOLIC PROBLEMS: THEORY, NUMERICS, APPLICATIONS},
   Volume = {8},
   Pages = {179-192},
   Booktitle = {Proceedings of the Fourteenth International Conference on
             Hyperbolic Problems: Theory, Numerics and
             Application},
   Publisher = {AMER INST MATHEMATICAL SCIENCES-AIMS},
   Editor = {Ancona, F and Bressan, A and Marcati, P and Marson,
             A},
   Year = {2014},
   Month = {January},
   Key = {fds333571}
}

@article{fds337236,
   Author = {Chae, D and Degond, P and Liu, JG},
   Title = {Well-posedness for hall-magnetohydrodynamics},
   Journal = {Annales de l'Institut Henri Poincare (C) Analyse Non
             Lineaire},
   Volume = {31},
   Number = {3},
   Pages = {555-565},
   Year = {2014},
   Month = {January},
   url = {http://dx.doi.org/10.1016/j.anihpc.2013.04.006},
   Abstract = {We prove local existence of smooth solutions for large data
             and global smooth solutions for small data to the
             incompressible, resistive, viscous or inviscid Hall-MHD
             model. We also show a Liouville theorem for the stationary
             solutions. © 2013 Elsevier Masson SAS. All rights
             reserved.},
   Doi = {10.1016/j.anihpc.2013.04.006},
   Key = {fds337236}
}

@article{fds246856,
   Author = {Goudon, T and Jin, S and Liu, J-G and Yan, B},
   Title = {Asymptotic-preserving schemes for kinetic-fluid modeling of
             disperse two-phase flows with variable fluid
             density},
   Journal = {International Journal for Numerical Methods in
             Fluids},
   Volume = {75},
   Number = {2},
   Pages = {81-102},
   Publisher = {WILEY},
   Year = {2014},
   ISSN = {0271-2091},
   url = {http://dx.doi.org/10.1002/fld.3885},
   Abstract = {We are concerned with a coupled system describing the
             interaction between suspended particles and a dense fluid.
             The particles are modeled by a kinetic equation of
             Vlasov-Fokker-Planck type, and the fluid is described by the
             incompressible Navier-Stokes system, with variable density.
             The systems are coupled through drag forces. High friction
             regimes lead to a purely hydrodynamic description of the
             mixture. We design first and second order
             asymptotic-preserving schemes suited to such regimes. We
             extend the method introduced in [Goudon T, Jin S, Liu JG,
             Yan B. Journal of Computational Physics 2013; 246:145-164]
             to the case of variable density in compressible flow. We
             check the accuracy and the asymptotic-preserving property
             numerically. We set up a few numerical experiments to
             demonstrate the ability of the scheme in capturing intricate
             interactions between the two phases on a wide range of
             physical parameters and geometric situations. © 2014 John
             Wiley & Sons, Ltd.},
   Doi = {10.1002/fld.3885},
   Key = {fds246856}
}

@article{fds246866,
   Author = {Bian, S and Liu, JG},
   Title = {Dynamic and Steady States for Multi-Dimensional Keller-Segel
             Model with Diffusion Exponent m > 0},
   Journal = {Communications in Mathematical Physics},
   Volume = {323},
   Number = {3},
   Pages = {1017-1070},
   Publisher = {Springer Nature},
   Year = {2013},
   Month = {November},
   ISSN = {0010-3616},
   url = {http://dx.doi.org/10.1007/s00220-013-1777-z},
   Abstract = {This paper investigates infinite-time spreading and
             finite-time blow-up for the Keller-Segel system. For 0 < m
             ≤ 2 - 2/d, the L p space for both dynamic and steady
             solutions are detected with (Formula presented.). Firstly,
             the global existence of the weak solution is proved for
             small initial data in L p. Moreover, when m > 1 - 2/d, the
             weak solution preserves mass and satisfies the
             hyper-contractive estimates in L q for any p < q < ∞.
             Furthermore, for slow diffusion 1 < m ≤ 2 - 2/d, this weak
             solution is also a weak entropy solution which blows up at
             finite time provided by the initial negative free energy.
             For m > 2 - 2/d, the hyper-contractive estimates are also
             obtained. Finally, we focus on the L p norm of the steady
             solutions, it is shown that the energy critical exponent m =
             2d/(d + 2) is the critical exponent separating finite L p
             norm and infinite L p norm for the steady state solutions.
             © 2013 Springer-Verlag Berlin Heidelberg.},
   Doi = {10.1007/s00220-013-1777-z},
   Key = {fds246866}
}

@article{fds246864,
   Author = {Chen, X and Liu, JG},
   Title = {Analysis of polymeric flow models and related compactness
             theorems in weighted spaces},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {45},
   Number = {3},
   Pages = {1179-1215},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2013},
   Month = {October},
   ISSN = {0036-1410},
   url = {http://dx.doi.org/10.1137/120887850},
   Abstract = {We studied coupled systems of the Fokker-Planck equation and
             the Navier-Stokes equation modeling the Hookean and the
             finitely extensible nonlinear elastic (FENE)-type polymeric
             flows. We proved the continuous embedding and compact
             embedding theorems in weighted spaces that naturally arise
             from related entropy estimates. These embedding estimates
             are shown to be sharp. For the Hookean polymeric system with
             a center-of-mass diffusion and a superlinear spring
             potential, we proved the existence of a global weak
             solution. Moreover, we were able to tackle the FENE model
             with L2 initial data for the polymer density instead of the
             L∞ counterpart in the literature. © 2013 Society for
             Industrial and Applied Mathematics.},
   Doi = {10.1137/120887850},
   Key = {fds246864}
}

@article{fds246869,
   Author = {Goudon, T and Jin, S and Liu, JG and Yan, B},
   Title = {Asymptotic-preserving schemes for kinetic-fluid modeling of
             disperse two-phase flows},
   Journal = {Journal of Computational Physics},
   Volume = {246},
   Pages = {145-164},
   Publisher = {Elsevier BV},
   Year = {2013},
   Month = {August},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1016/j.jcp.2013.03.038},
   Abstract = {We consider a system coupling the incompressible
             Navier-Stokes equations to the Vlasov-Fokker-Planck
             equation. Such a problem arises in the description of
             particulate flows. We design a numerical scheme to simulate
             the behavior of the system. This scheme is
             asymptotic-preserving, thus efficient in both the kinetic
             and hydrodynamic regimes. It has a numerical stability
             condition controlled by the non-stiff convection operator,
             with an implicit treatment of the stiff drag term and the
             Fokker-Planck operator. Yet, consistent to a standard
             asymptotic-preserving Fokker-Planck solver or an
             incompressible Navier-Stokes solver, only the
             conjugate-gradient method and fast Poisson and Helmholtz
             solvers are needed. Numerical experiments are presented to
             demonstrate the accuracy and asymptotic behavior of the
             scheme, with several interesting applications. © 2013
             Elsevier Inc.},
   Doi = {10.1016/j.jcp.2013.03.038},
   Key = {fds246869}
}

@article{fds246870,
   Author = {Chen, X and Liu, JG},
   Title = {Global weak entropy solution to Doi-Saintillan-Shelley model
             for active and passive rod-like and ellipsoidal particle
             suspensions},
   Journal = {Journal of Differential Equations},
   Volume = {254},
   Number = {7},
   Pages = {2764-2802},
   Publisher = {Elsevier BV},
   Year = {2013},
   Month = {April},
   ISSN = {0022-0396},
   url = {http://dx.doi.org/10.1016/j.jde.2013.01.005},
   Abstract = {We prove the existence of the global weak entropy solution
             to the Doi-Saintillan-Shelley model for active and passive
             rod-like particle suspensions, which couples a Fokker-Planck
             equation with the incompressible Navier-Stokes or Stokes
             equation, under the no-flux boundary conditions,
             L2(Ω;L1(Sd-1)) initial data, and finite initial entropy for
             the particle distribution function in two and three
             dimensions. Furthermore, for the model with the Stokes
             equation, we obtain the global L2(Ω×Sd-1) weak solution in
             two and three dimensions and the uniqueness in two
             dimension. © 2013 Elsevier Inc..},
   Doi = {10.1016/j.jde.2013.01.005},
   Key = {fds246870}
}

@article{fds246861,
   Author = {Huang, YL and Liu, JG and Wang, WC},
   Title = {A generalized mac scheme on curvilinear domains},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {35},
   Number = {5},
   Pages = {B953-B986},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2013},
   Month = {January},
   ISSN = {1064-8275},
   url = {http://dx.doi.org/10.1137/120875508},
   Abstract = {We propose a simple finite difference scheme for
             Navier-Stokes equations in primitive formulation on
             curvilinear domains. With proper boundary treatment and
             interplay between covariant and contravariant components,
             the spatial discretization admits exact Hodge decomposition
             and energy identity. As a result, the pressure can be
             decoupled from the momentum equation with explicit time
             stepping. No artificial pressure boundary condition is
             needed. In addition, it can be shown that this spatially
             compatible discretization leads to uniform inf-sup
             condition, which plays a crucial role in the pressure
             approximation of both dynamic and steady state calculations.
             Numerical experiments demonstrate the robustness and
             efficiency of our scheme. Copyright © by SIAM. Unauthorized
             reproduction of this article is prohibited.},
   Doi = {10.1137/120875508},
   Key = {fds246861}
}

@article{fds246859,
   Author = {Degond, P and Liu, J-G and Ringhofer, C},
   Title = {Evolution of the Distribution of Wealth in an Economic
             Environment Driven by Local Nash Equilibria},
   Journal = {Journal of Statistical Physics},
   Volume = {154},
   Number = {3},
   Pages = {1-30},
   Publisher = {Springer Nature},
   Year = {2013},
   ISSN = {0022-4715},
   url = {http://dx.doi.org/10.1007/s10955-013-0888-4},
   Abstract = {We present and analyze a model for the evolution of the
             wealth distribution within a heterogeneous economic
             environment. The model considers a system of rational agents
             interacting in a game theoretical framework, through fairly
             general assumptions on the cost function. This evolution
             drives the dynamic of the agents in both wealth and economic
             configuration variables. We consider a regime of scale
             separation where the large scale dynamics is given by a
             hydrodynamic closure with a Nash equilibrium serving as the
             local thermodynamic equilibrium. The result is a system of
             gas dynamics-type equations for the density and average
             wealth of the agents on large scales. We recover the inverse
             gamma distribution as an equilibrium in the particular case
             of quadratic cost functions which has been previously
             considered in the literature. © 2013 Springer
             Science+Business Media New York.},
   Doi = {10.1007/s10955-013-0888-4},
   Key = {fds246859}
}

@article{fds246860,
   Author = {Chen, X and Jüngel, A and Liu, J-G},
   Title = {A Note on Aubin-Lions-Dubinskiǐ Lemmas},
   Journal = {Acta Applicandae Mathematicae},
   Volume = {133},
   Number = {1},
   Pages = {1-11},
   Year = {2013},
   ISSN = {0167-8019},
   url = {http://dx.doi.org/10.1007/s10440-013-9858-8},
   Abstract = {Strong compactness results for families of functions in
             seminormed nonnegative cones in the spirit of the
             Aubin-Lions-Dubinskiǐ lemma are proven, refining some
             recent results in the literature. The first theorem sharpens
             slightly a result of Dubinskiǐ (in Mat. Sb.
             67(109):609-642, 1965) for seminormed cones. The second
             theorem applies to piecewise constant functions in time and
             sharpens slightly the results of Dreher and Jüngel (in
             Nonlinear Anal. 75:3072-3077, 2012) and Chen and Liu (in
             Appl. Math. Lett. 25:2252-2257, 2012). An application is
             given, which is useful in the study of porous-medium or
             fast-diffusion type equations. © 2013 Springer
             Science+Business Media.},
   Doi = {10.1007/s10440-013-9858-8},
   Key = {fds246860}
}

@article{fds246863,
   Author = {Degond, P and Liu, J-G and Ringhofer, C},
   Title = {Large-Scale Dynamics of Mean-Field Games Driven by Local
             Nash Equilibria},
   Journal = {Journal of Nonlinear Science},
   Volume = {24},
   Number = {1},
   Pages = {1-23},
   Year = {2013},
   ISSN = {0938-8974},
   url = {http://dx.doi.org/10.1007/s00332-013-9185-2},
   Abstract = {We introduce a new mean field kinetic model for systems of
             rational agents interacting in a game-theoretical framework.
             This model is inspired from noncooperative anonymous games
             with a continuum of players and Mean-Field Games. The large
             time behavior of the system is given by a macroscopic
             closure with a Nash equilibrium serving as the local
             thermodynamic equilibrium. An application of the presented
             theory to a social model (herding behavior) is discussed. ©
             Springer Science+Business Media New York
             2013.},
   Doi = {10.1007/s00332-013-9185-2},
   Key = {fds246863}
}

@article{fds246867,
   Author = {Chae, D and Degond, P and Liu, J-G},
   Title = {Well-posedness for Hall-magnetohydrodynamics},
   Journal = {Annales de l'Institut Henri Poincare. Annales: Analyse Non
             Lineaire/Nonlinear Analysis},
   Volume = {31},
   Number = {3},
   Pages = {555-565},
   Publisher = {Elsevier BV},
   Year = {2013},
   ISSN = {0294-1449},
   url = {http://dx.doi.org/10.1016/j.anihpc.2013.04.006},
   Abstract = {We prove local existence of smooth solutions for large data
             and global smooth solutions for small data to the
             incompressible, resistive, viscous or inviscid Hall-MHD
             model. We also show a Liouville theorem for the stationary
             solutions. © 2013 Elsevier Masson SAS. All rights
             reserved.},
   Doi = {10.1016/j.anihpc.2013.04.006},
   Key = {fds246867}
}

@article{fds246896,
   Author = {Jin, S and Liu, JG and Wang, L},
   Title = {A domain decomposition method for semilinear hyperbolic
             systems with two-scale relaxations},
   Journal = {Math. Comp.},
   Volume = {82},
   Number = {282},
   Pages = {749-779},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2013},
   url = {http://dx.doi.org/10.1090/S0025-5718-2012-02643-3},
   Abstract = {We present a domain decomposition method on a semilinear
             hyperbolic system with multiple relaxation times. In the
             region where the relaxation time is small, an asymptotic
             equilibrium equation can be used for computational
             efficiency. An interface condition based on the sign of the
             characteristic speed at the interface is provided to couple
             the two systems in a domain decomposition setting. A
             rigorous analysis, based on the Laplace Transform, on the L2
             error estimate is presented for the linear case, which shows
             how the error of the domain decomposition method depends on
             the smaller relaxation time, and the boundary and interface
             layer effects. The given convergence rate is optimal. We
             present a numerical implementation of this domain
             decomposition method, and give some numerical results in
             order to study the performance of this method. © 2012
             American Mathematical Society.},
   Doi = {10.1090/S0025-5718-2012-02643-3},
   Key = {fds246896}
}

@article{fds362426,
   Author = {Degond, P and Liu, J-G and Motsch, S and Panferov,
             V},
   Title = {Hydrodynamic models of self-organized dynamics: Derivation
             and existence theory},
   Journal = {Methods and Applications of Analysis},
   Volume = {20},
   Number = {2},
   Pages = {89-114},
   Publisher = {International Press of Boston},
   Year = {2013},
   url = {http://dx.doi.org/10.4310/maa.2013.v20.n2.a1},
   Doi = {10.4310/maa.2013.v20.n2.a1},
   Key = {fds362426}
}

@article{fds220112,
   Author = {A. Chertock and J.-G. Liu and T. Pendleton},
   Title = {Convergence analysis of the particle method for the
             Camassa-Holm equation},
   Pages = {365-373},
   Booktitle = {Proceedings of the 13th International Conference on
             ``Hyperbolic Problems: Theory, Numerics and
             Applications"},
   Publisher = {Higher Education Press},
   Address = {Beijing},
   Year = {2012},
   Key = {fds220112}
}

@article{fds246887,
   Author = {Chae, D and Liu, JG},
   Title = {Blow-up, zero alpha limit and the Liouville type theorem for
             the Euler-Poincare equations},
   Journal = {Comm. Math. Phy.,},
   Volume = {314},
   Number = {3},
   Pages = {671-687},
   Publisher = {Springer Nature},
   Year = {2012},
   ISSN = {0010-3616},
   url = {http://dx.doi.org/10.1007/s00220-012-1534-8},
   Abstract = {In this paper we study the Euler-Poincaré equations in ℝ
             N. We prove local existence of weak solutions in W 2,p(ℝ
             N),p>N, and local existence of unique classical solutions in
             H k(ℝ N),k> N/2+3, as well as a blow-up criterion. For the
             zero dispersion equation (α = 0) we prove a finite time
             blow-up of the classical solution. We also prove that as the
             dispersion parameter vanishes, the weak solution converges
             to a solution of the zero dispersion equation with sharp
             rate as α → 0, provided that the limiting solution
             belongs to C([0,T); H k(ℝ N)) with k > N/2 + 3. For the
             stationary weak solutions of the Euler-Poincaré equations
             we prove a Liouville type theorem. Namely, for α > 0 any
             weak solution u ∈ H 1(ℝ N) is u=0; for α= 0 any weak
             solution u ∈ L 2(ℝ N) is u=0. © 2012
             Springer-Verlag.},
   Doi = {10.1007/s00220-012-1534-8},
   Key = {fds246887}
}

@article{fds246888,
   Author = {Chen, X and Liu, JG},
   Title = {Two Nonlinear Compactness Theorems in L^p(0,T;B)},
   Journal = {Appl. Math. Lett.},
   Volume = {25},
   Number = {12},
   Pages = {2252-2257},
   Publisher = {Elsevier BV},
   Year = {2012},
   ISSN = {0893-9659},
   url = {http://dx.doi.org/10.1016/j.aml.2012.06.012},
   Abstract = {We establish two nonlinear compactness theorems in Lp(0,T;B)
             with hypothesis on time translations, which are nonlinear
             counterparts of two results by Simon (1987) [1]. The first
             theorem sharpens a result by Maitre (2003) [10] and is
             important in the study of doubly nonlinear ellipticparabolic
             equations. Based on this theorem, we then obtain a time
             translation counterpart of a result by Dubinskiǐ (1965)
             [5], which is supposed to be useful in the study of some
             nonlinear kinetic equations (e.g. the FENE-type beadspring
             chains model). © 2012 Elsevier Ltd. All rights
             reserved.},
   Doi = {10.1016/j.aml.2012.06.012},
   Key = {fds246888}
}

@article{fds246889,
   Author = {Chen, L and Liu, JG and Wang, J},
   Title = {Multi-dimensional degenerate Keller-Segel system with
             critical diffusion exponent 2n/(n+2)},
   Journal = {SIAM J. Math Anal},
   Volume = {44},
   Number = {2},
   Pages = {1077-1102},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2012},
   ISSN = {0036-1410},
   url = {http://dx.doi.org/10.1137/110839102},
   Abstract = {This paper deals with a degenerate diffusion
             Patlak-Keller-Segel system in n = 3 dimension. The main
             difference between the current work and many other recent
             studies on the same model is that we study the diffusion
             exponent m = 2n/(n + 2), which is smaller than the usual
             exponent m* = 2-2/n used in other studies. With the exponent
             m = 2n/(n + 2), the associated free energy is conformal
             invariant, and there is a family of stationary solutions
             Uλ,x0 (x) = C(λ/ λ 2+|x-x0| 2 ) n+2/2 λ < 0, σ0 ? ℝn.
             For radially symmetric solutions, we prove that if the
             initial data are strictly below Uλ,0(x) for some λ, then
             the solution vanishes in L1 loc as tλ8; if the initial data
             are strictly above Uλ,0(x) for some λ, then the solution
             either blows up at a finite time or has a mass concentration
             at r = 0 as time goes to infinity. For general initial data,
             we prove that there is a global weak solution provided that
             the Lm norm of initial density is less than a universal
             constant, and the weak solution vanishes as time goes to
             infinity. We also prove a finite time blow-up of the
             solution if the Lm norm for initial data is larger than the
             Lm norm of Uλ,x0 (x), which is constant independent of λ
             and x0, and the free energy of initial data is smaller than
             that of Uλ,x0(x). © 2012 Society for Industrial and
             Applied Mathematics.},
   Doi = {10.1137/110839102},
   Key = {fds246889}
}

@article{fds246890,
   Author = {Frouvelle, A and Liu, JG},
   Title = {Dynamics in a kinetic model of oriented particles with phase
             transition},
   Journal = {SIAM J. Math Anal},
   Volume = {44},
   Number = {2},
   Pages = {791-826},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2012},
   ISSN = {0036-1410},
   url = {http://dx.doi.org/10.1137/110823912},
   Abstract = {Motivated by a phenomenon of phase transition in a model of
             alignment of selfpropelled particles, we obtain a kinetic
             mean-field equation which is nothing more than the
             Smoluchowski equation on the sphere with dipolar potential.
             In this self-contained article, using only basic tools, we
             analyze the dynamics of this equation in any dimension. We
             first prove global wellposedness of this equation, starting
             with an initial condition in any Sobolev space. We then
             compute all possible steady states. There is a threshold for
             the noise parameter: over this threshold, the only
             equilibrium is the uniform distribution, and under this
             threshold, the other equilibria are the Fisher-von Mises
             distributions with arbitrary direction and a concentration
             parameter determined by the intensity of the noise. For any
             initial condition, we give a rigorous proof of convergence
             of the solution to a steady state as time goes to infinity.
             In particular, when the noise is under the threshold and
             with nonzero initial mean velocity, the solution converges
             exponentially fast to a unique Fisher- von Mises
             distribution. We also found a new conservation relation,
             which can be viewed as a convex quadratic entropy when the
             noise is above the threshold. This provides a uniform
             exponential rate of convergence to the uniform distribution.
             At the threshold, we show algebraic decay to the uniform
             distribution. © 2012 Society for Industrial and Applied
             Mathematics.},
   Doi = {10.1137/110823912},
   Key = {fds246890}
}

@article{fds246891,
   Author = {Carrillo, J and Chen, L and Liu, JG and Wang, J},
   Title = {A note on the subcritical two dimensional Keller-Segel
             system},
   Journal = {Acta Applicanda Mathematicae},
   Volume = {119},
   Number = {1},
   Pages = {43-55},
   Publisher = {Springer Nature},
   Year = {2012},
   ISSN = {0167-8019},
   url = {http://dx.doi.org/10.1007/s10440-011-9660-4},
   Abstract = {The existence of solution for the 2D-Keller-Segel system in
             the subcritical case, i.e. when the initial mass is less
             than 8π, is reproved. Instead of using the entropy in the
             free energy and free energy dissipation, which was used in
             the proofs (Blanchet et al. in SIAM J. Numer. Anal.
             46:691-721, 2008; Electron. J. Differ. Equ. Conf. 44:32,
             2006 (electronic)), the potential energy term is fully
             utilized by adapting Delort's theory on 2D incompressible
             Euler equation (Delort in J. Am. Math. Soc. 4:553-386,
             1991). © 2011 Springer Science+Business Media
             B.V.},
   Doi = {10.1007/s10440-011-9660-4},
   Key = {fds246891}
}

@article{fds246892,
   Author = {Degond, P and Liu, JG},
   Title = {Hydrodynamics of self-alignment interactions with precession
             and derivation of the Landau-Lifschitz-Gilbert
             equation},
   Journal = {Math. Models Methods Appl. Sci.},
   Volume = {22},
   Number = {SUPPL.1},
   Pages = {1114001-1114018},
   Publisher = {World Scientific Pub Co Pte Lt},
   Year = {2012},
   ISSN = {0218-2025},
   url = {http://dx.doi.org/10.1142/S021820251140001X},
   Abstract = {We consider a kinetic model of self-propelled particles with
             alignment interaction and with precession about the
             alignment direction. We derive a hydrodynamic system for the
             local density and velocity orientation of the particles. The
             system consists of the conservative equation for the local
             density and a non-conservative equation for the orientation.
             First, we assume that the alignment interaction is purely
             local and derive a first-order system. However, we show that
             this system may lose its hyperbolicity. Under the assumption
             of weakly nonlocal interaction, we derive diffusive
             corrections to the first-order system which lead to the
             combination of a heat flow of the harmonic map and
             LandauLifschitzGilbert dynamics. In the particular case of
             zero self-propelling speed, the resulting model reduces to
             the phenomenological LandauLifschitzGilbert equations.
             Therefore the present theory provides a kinetic formulation
             of classical micromagnetization models and spin dynamics. ©
             2012 World Scientific Publishing Company.},
   Doi = {10.1142/S021820251140001X},
   Key = {fds246892}
}

@article{fds246893,
   Author = {Chertock, A and Liu, JG and Pendleton, T},
   Title = {Convergence of a particle method and global weak solutions
             of a family of evolutionary PDEs},
   Journal = {SIAM J. Numer. Anal.},
   Volume = {50},
   Number = {1},
   Pages = {1-21},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2012},
   ISSN = {0036-1429},
   url = {http://dx.doi.org/10.1137/110831386},
   Abstract = {The purpose of this paper is to provide global existence and
             uniqueness results for a family of fluid transport equations
             by establishing convergence results for the particle method
             applied to these equations. The considered family of PDEs is
             a collection of strongly nonlinear equations which yield
             traveling wave solutions and can be used to model a variety
             of flows in fluid dynamics. We apply a particle method to
             the studied evolutionary equations and provide a new
             self-contained method for proving its convergence. The
             latter is accomplished by using the concept of space-time
             bounded variation and the associated compactness properties.
             From this result, we prove the existence of a unique global
             weak solution in some special cases and obtain stronger
             regularity properties of the solution than previously
             established. © 2012 Society for Industrial and Applied
             Mathematics.},
   Doi = {10.1137/110831386},
   Key = {fds246893}
}

@article{fds246894,
   Author = {Haack, J and Jin, S and Liu, JG},
   Title = {An all-speed asymptotic-preserving method for the isentropic
             Euler and Navier-Stokes equations},
   Journal = {Commun. Comput. Phy.},
   Volume = {12},
   Number = {4},
   Pages = {955-980},
   Publisher = {Global Science Press},
   Year = {2012},
   ISSN = {1815-2406},
   url = {http://dx.doi.org/10.4208/cicp.250910.131011a},
   Abstract = {The computation of compressible flows becomes more
             challenging when the Mach number has different orders of
             magnitude. When the Mach number is of order one, modern
             shock capturing methods are able to capture shocks and other
             complex structures with high numerical resolutions. However,
             if the Mach number is small, the acoustic waves lead to
             stiffness in time and excessively large numerical viscosity,
             thus demanding much smaller time step and mesh size than
             normally needed for incompressible flow simulation. In this
             paper, we develop an all-speed asymptotic preserving (AP)
             numerical scheme for the compressible isentropic Euler and
             Navier-Stokes equations that is uniformly stable and
             accurate for all Mach numbers. Our idea is to split the
             system into two parts: one involves a slow, nonlinear and
             conservative hyperbolic system adequate for the use of
             modern shock capturing methods and the other a linear
             hyperbolic system which contains the stiff acoustic
             dynamics, to be solved implicitly. This implicit part is
             reformulated into a standard pressure Poisson projection
             system and thus possesses sufficient structure for efficient
             fast Fourier transform solution techniques. In the zero Mach
             number limit, the scheme automatically becomes a projection
             method-like incompressible solver. We present numerical
             results in one and two dimensions in both compressible and
             incompressible regimes. © 2012 Global-Science
             Press.},
   Doi = {10.4208/cicp.250910.131011a},
   Key = {fds246894}
}

@article{fds246895,
   Author = {Degond, P and Frouvell, A and Liu, JG},
   Title = {Macroscopic limits and phase transition in a system of
             self-propelled particles},
   Journal = {J Nonlinear Sci.},
   Volume = {23},
   Number = {3},
   Pages = {427-456},
   Publisher = {Springer Nature},
   Year = {2012},
   ISSN = {0938-8974},
   url = {http://dx.doi.org/10.1007/s00332-012-9157-y},
   Abstract = {We investigate systems of self-propelled particles with
             alignment interaction. Compared to previous work (Degond and
             Motsch, Math. Models Methods Appl. Sci. 18:1193-1215, 2008a;
             Frouvelle, Math. Models Methods Appl. Sci., 2012), the force
             acting on the particles is not normalized, and this
             modification gives rise to phase transitions from disordered
             states at low density to aligned states at high densities.
             This model is the space-inhomogeneous extension of
             (Frouvelle and Liu, Dynamics in a kinetic model of oriented
             particles with phase transition, 2012), in which the
             existence and stability of the equilibrium states were
             investigated. When the density is lower than a threshold
             value, the dynamics is described by a nonlinear diffusion
             equation. By contrast, when the density is larger than this
             threshold value, the dynamics is described by a similar
             hydrodynamic model for self-alignment interactions as
             derived in (Degond and Motsch, Math. Models Methods Appl.
             Sci. 18:1193-1215, 2008a; Frouvelle, Math. Models Methods
             Appl. Sci., 2012). However, the modified normalization of
             the force gives rise to different convection speeds, and the
             resulting model may lose its hyperbolicity in some regions
             of the state space. © 2012 Springer Science+Business Media
             New York.},
   Doi = {10.1007/s00332-012-9157-y},
   Key = {fds246895}
}

@article{fds246899,
   Author = {Zheng, W and Gao, H and Liu, JG and Zhang, Y and Ye, Q and Swank,
             C},
   Title = {General solution to gradient-induced transverse and
             longitudinal relaxation of spins undergoing restricted
             diffusion},
   Journal = {Physical Review A - Atomic, Molecular, and Optical
             Physics},
   Volume = {84},
   Number = {5},
   Pages = {053411-8},
   Publisher = {American Physical Society (APS)},
   Year = {2011},
   Month = {November},
   ISSN = {1050-2947},
   url = {http://dx.doi.org/10.1103/PhysRevA.84.053411},
   Abstract = {We develop an approach, by calculating the autocorrelation
             function of spins, to derive the magnetic field
             gradient-induced transverse (T2) relaxation of spins
             undergoing restricted diffusion. This approach is an
             extension to the method adopted by McGregor. McGregor's
             approach solves the problem only in the fast diffusion
             limit; however, our approach yields a single analytical
             solution suitable in all diffusion regimes, including the
             intermediate regime. This establishes a direct connection
             between the well-known slow diffusion result of Torrey and
             the fast diffusion result. We also perform free induction
             decay measurements on spin-exchange optically polarized 3He
             gas with different diffusion constants. The measured
             transverse relaxation profiles are compared with the theory
             and satisfactory agreement has been found throughout all
             diffusion regimes. In addition to the transverse relaxation,
             this approach is also applicable to solving the longitudinal
             relaxation (T 1) regardless of the diffusion limits. It
             turns out that the longitudinal relaxation in the slow
             diffusion limit differs by a factor of 2 from that in the
             fast diffusion limit. © 2011 American Physical
             Society.},
   Doi = {10.1103/PhysRevA.84.053411},
   Key = {fds246899}
}

@article{fds246897,
   Author = {Liu, JG and Lorz, A},
   Title = {A coupled chemotaxis-fluid model: Global
             existence},
   Journal = {Ann. I. H. Poincare, AN},
   Volume = {28},
   Number = {5},
   Pages = {643-652},
   Publisher = {Elsevier BV},
   Year = {2011},
   ISSN = {0294-1449},
   url = {http://dx.doi.org/10.1016/j.anihpc.2011.04.005},
   Abstract = {We consider a model arising from biology, consisting of
             chemotaxis equations coupled to viscous incompressible fluid
             equations through transport and external forcing. Global
             existence of solutions to the Cauchy problem is investigated
             under certain conditions. Precisely, for the
             chemotaxis-Navier- Stokes system in two space dimensions, we
             obtain global existence for large data. In three space
             dimensions, we prove global existence of weak solutions for
             the chemotaxis-Stokes system with nonlinear diffusion for
             the cell density.© 2011 Elsevier Masson SAS. All rights
             reserved.},
   Doi = {10.1016/j.anihpc.2011.04.005},
   Key = {fds246897}
}

@article{fds246898,
   Author = {Acheritogaray, M and Degond, P and Frouvelle, A and Liu,
             JG},
   Title = {Kinetic formulation and global existence for the
             Hall-Magneto-hydrodynamics system},
   Journal = {Kinetic and Related Models},
   Volume = {4},
   Number = {4},
   Pages = {901-918},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2011},
   ISSN = {1937-5093},
   url = {http://dx.doi.org/10.3934/krm.2011.4.901},
   Abstract = {This paper deals with the derivation and analysis of the the
             Hall Magneto-Hydrodynamic equations. We first provide a
             derivation of this system from a two-fluids Euler-Maxwell
             system for electrons and ions, through a set of scaling
             limits. We also propose a kinetic formulation for the
             Hall-MHD equa- tions which contains as fluid closure
             different variants of the Hall-MHD model. Then, we prove the
             existence of global weak solutions for the incompressible
             viscous resistive Hall-MHD model. We use the particular
             structure of the Hall term which has zero contribution to
             the energy identity. Finally, we discuss particular
             solutions in the form of axisymmetric purely swirling
             magnetic fields and propose some regularization of the Hall
             equation. © American Institute of Mathematical
             Sciences.},
   Doi = {10.3934/krm.2011.4.901},
   Key = {fds246898}
}

@article{fds246904,
   Author = {Huang, YL and Liu, JG and Wang, WC},
   Title = {An FFT based fast Poisson solver on spherical
             shells},
   Journal = {Commun. Comput. Phy.},
   Volume = {9},
   Number = {3},
   Pages = {649-667},
   Publisher = {Global Science Press},
   Year = {2011},
   ISSN = {1815-2406},
   url = {http://dx.doi.org/10.4208/cicp.060509.080609s},
   Abstract = {We present a fast Poisson solver on spherical shells. With a
             special change of variable, the radial part of the Laplacian
             transforms to a constant coefficient differ- ential
             operator. As a result, the Fast Fourier Transform can be
             applied to solve the Poisson equation with O(N^3 logN)
             operations. Numerical examples have confirmed the accuracy
             and robustness of the new scheme.},
   Doi = {10.4208/cicp.060509.080609s},
   Key = {fds246904}
}

@article{fds246900,
   Author = {Liu, JG and Liu, J and Pego, RL},
   Title = {Stable and accurate pressure approximation for unsteady
             incompressible viscous flow},
   Journal = {Journal of Computational Physics},
   Volume = {229},
   Number = {9},
   Pages = {3428-3453},
   Publisher = {Elsevier BV},
   Year = {2010},
   Month = {January},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1016/j.jcp.2010.01.010},
   Abstract = {How to properly specify boundary conditions for pressure is
             a longstanding problem for the incompressible Navier-Stokes
             equations with no-slip boundary conditions. An analytical
             resolution of this issue stems from a recently developed
             formula for the pressure in terms of the commutator of the
             Laplacian and Leray projection operators. Here we make use
             of this formula to (a) improve the accuracy of computing
             pressure in two kinds of existing time-discrete projection
             methods implicit in viscosity only, and (b) devise new
             higher-order accurate time-discrete projection methods that
             extend a slip-correction idea behind the well-known
             finite-difference scheme of Kim and Moin. We test these
             schemes for stability and accuracy using various
             combinations of C0 finite elements. For all three kinds of
             time discretization, one can obtain third-order accuracy for
             both pressure and velocity without a time-step stability
             restriction of diffusive type. Furthermore, two kinds of
             projection methods are found stable using piecewise-linear
             elements for both velocity and pressure. © 2010 Elsevier
             Inc.},
   Doi = {10.1016/j.jcp.2010.01.010},
   Key = {fds246900}
}

@article{fds304584,
   Author = {Liu, JG and Pego, RL},
   Title = {Stable discretization of magnetohydrodynamics in bounded
             domains},
   Journal = {Communications in Mathematical Sciences},
   Volume = {8},
   Number = {1},
   Pages = {235-251},
   Publisher = {International Press of Boston},
   Year = {2010},
   Month = {January},
   ISSN = {1539-6746},
   url = {http://dx.doi.org/10.4310/CMS.2010.v8.n1.a12},
   Abstract = {We study a semi-implicit time-difference scheme for
             magnetohydrodynamics of a viscous and resistive
             incompressible fluid in a bounded smooth domain with a
             perfectly conducting boundary. In the scheme, the velocity
             and magnetic fields are updated by solving simple Helmholtz
             equations. Pressure is treated explicitly in time, by
             solving Poisson equations corresponding to a recently
             de-veloped formula for the Navier-Stokes pressure involving
             the commutator of Laplacian and Leray projection operators.
             We prove stability of the time-difference scheme, and deduce
             a local-time well-posedness theorem for MHD dynamics
             extended to ignore the divergence-free constraint on
             velocity and magnetic fields. These fields are
             divergence-free for all later time if they are initially so.
             © 2010 International Press.},
   Doi = {10.4310/CMS.2010.v8.n1.a12},
   Key = {fds304584}
}

@article{fds246905,
   Author = {Liu, JG and Mieussens, L},
   Title = {Analysis of an asymptotic preserving scheme for linear
             kinetic equations in the diffusion limit},
   Journal = {SIAM J. Numer. Anal.},
   Volume = {48},
   Number = {4},
   Pages = {1474-1491},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2010},
   ISSN = {0036-1429},
   url = {http://hdl.handle.net/10161/4316 Duke open
             access},
   Abstract = {We present a mathematical analysis of the asymptotic
             preserving scheme proposed in [M. Lemou and L. Mieussens,
             SIAM J. Sci. Comput., 31 (2008), pp. 334–368] for linear
             transport equations in kinetic and diffusive regimes. We
             prove that the scheme is uniformly stable and accurate with
             respect to the mean free path of the particles. This
             property is satisfied under an explicitly given CFL
             condition. This condition tends to a parabolic CFL condition
             for small mean free paths and is close to a convection CFL
             condition for large mean free paths. Our analysis is based
             on very simple energy estimates.},
   Doi = {10.1137/090772770},
   Key = {fds246905}
}

@article{fds246928,
   Author = {Liu, JG and Pego, R},
   Title = {Stable discretization of magnetohydrodynamics in bounded
             domains},
   Journal = {Commun. Math. Sci.},
   Volume = {8},
   Number = {1},
   Pages = {234-251},
   Publisher = {INT PRESS BOSTON, INC},
   Year = {2010},
   ISSN = {1539-6746},
   Abstract = {We study a semi-implicit time-difference scheme for
             magnetohydrodynamics of a viscous and resistive
             incompressible fluid in a bounded smooth domain with a
             perfectly conducting boundary. In the scheme, the velocity
             and magnetic fields are updated by solving simple Helmholtz
             equations. Pressure is treated explicitly in time, by
             solving Poisson equations corresponding to a recently
             de-veloped formula for the Navier-Stokes pressure involving
             the commutator of Laplacian and Leray projection operators.
             We prove stability of the time-difference scheme, and deduce
             a local-time well-posedness theorem for MHD dynamics
             extended to ignore the divergence-free constraint on
             velocity and magnetic fields. These fields are
             divergence-free for all later time if they are initially so.
             © 2010 International Press.},
   Key = {fds246928}
}

@article{fds246943,
   Author = {Liu, JG and Liu, J and Pego, RL},
   Title = {Error estimates for finite-element Navier-Stokes solvers
             without standard Inf-Sup conditions},
   Journal = {Chinese Annals of Mathematics. Series B},
   Volume = {30},
   Number = {6},
   Pages = {743-768},
   Publisher = {Springer Nature},
   Year = {2009},
   Month = {December},
   ISSN = {0252-9599},
   url = {http://dx.doi.org/10.1007/s11401-009-0116-3},
   Abstract = {The authors establish error estimates for recently developed
             finite-element methods for incompressible viscous flow in
             domains with no-slip boundary conditions. The methods arise
             by discretization of a well-posed extended Navier-Stokes
             dynamics for which pressure is determined from current
             velocity and force fields. The methods use C1 elements for
             velocity and C0 elements for pressure. A stability estimate
             is proved for a related finite-element projection method
             close to classical time-splitting methods of Orszag,
             Israeli, DeVille and Karniadakis. © Editorial Office of CAM
             and Springer-Verlag Berlin Heidelberg 2009.},
   Doi = {10.1007/s11401-009-0116-3},
   Key = {fds246943}
}

@article{fds246944,
   Author = {Liu, JG and Wang, WC},
   Title = {Characterization and regularity for axisymmetric solenoidal
             vector elds with application to Navier-Stokes
             equation},
   Journal = {SIAM J. Math. Anal.},
   Volume = {41},
   Number = {5},
   Pages = {1825-1850},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2009},
   ISSN = {0036-1410},
   url = {http://dx.doi.org/10.1137/080739744},
   Abstract = {We consider the vorticity-stream formulation of axisymmetric
             incompressible flows and its equivalence with the primitive
             formulation. It is shown that, to characterize the
             regularity of a divergence free axisymmetric vector field in
             terms of the swirling components, an extra set of pole
             conditions is necessary to give a full description of the
             regu larity. In addition, smooth solutions up to the axis of
             rotation give rise to smooth solutions of primitive
             formulation in the case of the Navier-Stokes equation, but
             not the Euler equation. We also establish a proper weak
             formulation and show its equivalence to Leray's formulation.
             © 2009 Society for Industrial and Applied
             Mathematics.},
   Doi = {10.1137/080739744},
   Key = {fds246944}
}

@article{fds246945,
   Author = {Ha, SY and Liu, JG},
   Title = {A simple proof of the Cucker-Smale flocking dynamics and
             mean-field limit},
   Journal = {Commun. Math. Sci.},
   Volume = {7},
   Number = {2},
   Pages = {297-325},
   Publisher = {International Press of Boston},
   Year = {2009},
   ISSN = {1539-6746},
   url = {http://dx.doi.org/10.4310/CMS.2009.v7.n2.a2},
   Abstract = {We present a simple proof on the formation of flocking to
             the Cucker-Smale system based on the explicit construction
             of a Lyapunov functional. Our results also provide a unified
             condition on the initial states in which the exponential
             convergence to flocking state will occur. For large particle
             systems, we give a rigorous justification for the mean-field
             limit from the many particle Cucker-Smale system to the
             Vlasov equation with flocking dissipation as the number of
             particles goes to infinity. © 2009 International
             Press.},
   Doi = {10.4310/CMS.2009.v7.n2.a2},
   Key = {fds246945}
}

@article{fds246946,
   Author = {Degond, P and Liu, JG and Vignal, MH},
   Title = {Analysis of an asymptotic preserving scheme for the
             Euler-Poisson system in the quasineutral
             limit},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {46},
   Number = {3},
   Pages = {1298-1322},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2008},
   Month = {November},
   ISSN = {0036-1429},
   url = {http://dx.doi.org/10.1137/070690584},
   Keywords = {stiffness • Debye length • electron plasma period
             • Burgers-Poisson • sheath problem •
             Klein-Gordon},
   Abstract = {In a previous work [P. Crispel, P. Degond, and M.-H. Vignal,
             J. Comput. Phys., 223 (2007), pp. 208-234], a new numerical
             discretization of the Euler-Poisson system was proposed.
             This scheme is "asymptotic preserving" in the quasineutral
             limit (i.e., when the Debye length ε tends to zero), which
             means that it becomes consistent with the limit model when
             ε → 0. In the present work, we show that the stability
             domain of the present scheme is independent of ε. This
             stability analysis is performed on the Fourier transformed
             (with respect to the space variable) linearized system. We
             show that the stability property is more robust when a
             space-decentered scheme is used (which brings in some
             numerical dissipation) rather than a space-centered scheme.
             The linearization is first performed about a zero mean
             velocity and then about a nonzero mean velocity. At the
             various stages of the analysis, our scheme is compared with
             more classical schemes and its improved stability property
             is outlined. The analysis of a fully discrete (in space and
             time) version of the scheme is also given. Finally, some
             considerations about a model nonlinear problem, the
             Burgers-Poisson problem, are also discussed. © 2008 Society
             for Industrial and Applied Mathematics.},
   Doi = {10.1137/070690584},
   Key = {fds246946}
}

@article{fds246948,
   Author = {Lu, X and Lin, P and Liu, JG},
   Title = {Analysis of a sequential regularization method for the
             unsteady Navier-Stokes equations},
   Journal = {Mathematics of Computation},
   Volume = {77},
   Number = {263},
   Pages = {1467-1494},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2008},
   Month = {July},
   ISSN = {0025-5718},
   url = {http://dx.doi.org/10.1090/S0025-5718-08-02087-5},
   Keywords = {Navier-Stokes equations • iterative penalty method
             • implicit parabolic PDE • error estimates •
             constrained dynamical system • stabilization
             method},
   Abstract = {The incompressibility constraint makes Navier-Stokes
             equations difficult. A reformulation to a better posed
             problem is needed before solving it numerically. The
             sequential regularization method (SRM) is a reformulation
             which combines the penalty method with a stabilization
             method in the context of constrained dynamical systems and
             has the benefit of both methods. In the paper, we study the
             existence and uniqueness for the solution of the SRM and
             provide a simple proof of the convergence of the solution of
             the SRM to the solution of the Navier-Stokes equations. We
             also give error estimates for the time discretized SRM
             formulation. ©2008 American Mathematical
             Society.},
   Doi = {10.1090/S0025-5718-08-02087-5},
   Key = {fds246948}
}

@article{fds246941,
   Author = {Lin, P and Liu, JG and Lu, X},
   Title = {Long time numerical solution of the Navier-Stokes equations
             based on a sequential regularization formulation},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {31},
   Number = {1},
   Pages = {398-419},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2008},
   Month = {January},
   ISSN = {1064-8275},
   url = {http://dx.doi.org/10.1137/060673722},
   Abstract = {The sequential regularization method is a reformulation of
             the unsteady Navier-Stokes equations from the viewpoint of
             constrained dynamical systems or the approximate
             Helmholtz-Hodge projection. In this paper we study the long
             time behavior of the sequential regularization formulation.
             We give a uniform-in-time estimate between the solution of
             the reformulated system and that of the Navier-Stokes
             equations. We also conduct an error analysis for the
             temporal discrete system and show that the error bound is
             independent of time. A couple of long time flow examples are
             computed to demonstrate this method. © 2008 Society for
             Industrial and Applied Mathematics.},
   Doi = {10.1137/060673722},
   Key = {fds246941}
}

@article{fds246942,
   Author = {Liu, JG and Wang, C},
   Title = {A fourth order numerical method for the primtive equations
             formulated in mean vorticity},
   Journal = {Communications in Computational Physics},
   Volume = {4},
   Number = {1},
   Pages = {26-55},
   Year = {2008},
   Month = {January},
   ISSN = {1815-2406},
   Abstract = {A fourth-order finite difference method is proposed and
             studied for the primitive equations (PEs) of large-scale
             atmospheric and oceanic flow based on mean vorticity
             formulation. Since the vertical average of the horizontal
             velocity field is divergence-free, we can introduce mean
             vorticity and mean stream function which are connected by a
             2-D Poisson equation. As a result, the PEs can be
             reformulated such that the prognostic equation for the
             horizontal velocity is replaced by evolutionary equations
             for the mean vorticity field and the vertical derivative of
             the horizontal velocity. The mean vorticity equation is
             approximated by a compact difference scheme due to the
             difficulty of the mean vorticity boundary condition, while
             fourth-order long-stencil approximations are utilized to
             deal with transport type equations for computational
             convenience. The numerical values for the total velocity
             field (both horizontal and vertical) are statically
             determined by a discrete realization of a differential
             equation at each fixed horizontal point. The method is
             highly efficient and is capable of producing highly resolved
             solutions at a reasonable computational cost. The full
             fourth-order accuracy is checked by an example of the
             reformulated PEs with force terms. Additionally, numerical
             results of a large-scale oceanic circulation are presented.
             © 2008 Global-Science Press.},
   Key = {fds246942}
}

@article{fds246940,
   Author = {Hsia, CH and Liu, JG and Wang, C},
   Title = {Structural stability and bifurcation for 2D incompressible
             ows with symmetry},
   Journal = {Meth. Appl. Anal.},
   Volume = {15},
   Pages = {495-512},
   Year = {2008},
   Key = {fds246940}
}

@article{fds246949,
   Author = {Antman, SS and Liu, JG},
   Title = {Basic themes and pretty problems of nonlinear solid
             mechanics},
   Journal = {Milan Journal of Mathematics},
   Volume = {75},
   Number = {1},
   Pages = {135-176},
   Publisher = {Springer Nature},
   Year = {2007},
   Month = {December},
   ISSN = {1424-9286},
   url = {http://dx.doi.org/10.1007/s00032-007-0068-6},
   Keywords = {Nonlinear solid mechanics • radial motions •
             existence • multiplicity • blowup • inverse
             problems • quasistaticity • control •
             invariant artificial viscosity and shock
             structure},
   Abstract = {The first part of this paper describes some important
             underlying themes in the mathematical theory of continuum
             mechanics that are distinct from formulating and analyzing
             governing equations. The main part of this paper is devoted
             to a survey of some concrete, conceptually simple, pretty
             problems that help illuminate the underlying themes. The
             paper concludes with a discussion of the crucial role of
             invariant constitutive equations in computation. © 2007
             Birkhaueser.},
   Doi = {10.1007/s00032-007-0068-6},
   Key = {fds246949}
}

@article{fds246958,
   Author = {Moore, J and Cheng, Z and Hao, J and Guo, G and Liu, J-G and Lin, C and Yu,
             LL},
   Title = {Effects of solid-state yeast treatment on the antioxidant
             properties and protein and fiber compositions of common hard
             wheat bran.},
   Journal = {Journal of agricultural and food chemistry},
   Volume = {55},
   Number = {25},
   Pages = {10173-10182},
   Year = {2007},
   Month = {December},
   ISSN = {0021-8561},
   url = {http://dx.doi.org/10.1021/jf071590o},
   Abstract = {The bran fraction of wheat grain is known to contain
             significant quantities of bioactive components. This study
             evaluated the potential of solid-state yeast fermentation to
             improve the health beneficial properties of wheat bran,
             including extractable antioxidant properties, protein
             contents, and soluble and insoluble fiber compositions.
             Three commercial food grade yeast preparations were
             evaluated in the study along with the effects of yeast dose,
             treatment time, and their interaction with the beneficial
             components. Solid-state yeast treatments were able to
             significantly increase releasable antioxidant properties
             ranging from 28 to 65, from 0 to 20, from 13 to 19, from 0
             to 25, from 50 to 100, and from 3 to 333% for scavenging
             capacities against peroxyl (ORAC), ABTS cation, DPPH and
             hydroxyl radicals, total phenolic contents (TPC), and
             phenolic acids, respectively. Yeast treatment increased
             protein content 11-12% but did not significantly alter the
             fiber composition of wheat bran. Effects of solid-state
             yeast treatment on both ORAC and TPC of wheat bran were
             altered by yeast dose, treatment time, and their
             interaction. Results suggest that solid-state yeast
             treatment may be a commercially viable postharvest procedure
             for improving the health beneficial properties of wheat bran
             and other wheat-based food ingredients.},
   Doi = {10.1021/jf071590o},
   Key = {fds246958}
}

@article{fds246880,
   Author = {Liu, JG and Liu, J and Pego, RL},
   Title = {Stability and convergence of efficient Navier-Stokes solvers
             via a commutator estimate},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {60},
   Number = {10},
   Pages = {1443-1487},
   Publisher = {WILEY},
   Year = {2007},
   Month = {October},
   ISSN = {0010-3640},
   url = {http://dx.doi.org/10.1002/cpa.20178},
   Abstract = {For strong solutions of the incompressible Navier-Stokes
             equations in bounded domains with velocity specified at the
             boundary, we establish the unconditional stability and
             convergence of discretization schemes that decouple the
             updates of pressure and velocity through explicit time
             stepping for pressure. These schemes require no solution of
             stationary Stokes systems, nor any compatibility between
             velocity and pressure spaces to ensure an inf-sup condition,
             and are representative of a class of highly efficient
             computational methods that have recently emerged. The proofs
             are simple, based upon a new, sharp estimate for the
             commutator of the Laplacian and Helmholtz projection
             operators. This allows us to treat an unconstrained
             formulation of the Navier-Stokes equations as a perturbed
             diffusion equation. ©2006 Wiley Periodicals,
             Inc.},
   Doi = {10.1002/cpa.20178},
   Key = {fds246880}
}

@article{fds139011,
   Author = {J.-G. Liu and Jie Liu and R. Pego},
   Title = {Estimates on the Stokes pressure by partitioning the energy
             of harmonic functions},
   Pages = {251--270},
   Booktitle = {Kyoto Conference on the Navier-Stokes equations and their
             Applications},
   Publisher = {Kyoto Univ.},
   Editor = {Y. Giga and H. Kozono and H. Okamoto and Y. Shibta},
   Year = {2007},
   Abstract = {We show that in a tubular domain with sufficiently small
             width, the normal and tangential gradients of a harmonic
             function have almost the same L2 norm. This estimate yields
             a sharp estimate of the pressure in terms of the viscosity
             term in the Navier-Stokes equation with no-slip boundary
             condition. By consequence, one can analyze the Navier-
             Stokes equations simply as a perturbed vector diffusion
             equation instead of as a perturbed Stokes system. As an
             application, we describe a rather easy approach to establish
             a new isomorphism theorem for the non-homogeneous Stokes
             system.},
   Key = {fds139011}
}

@article{fds246903,
   Author = {Liu, JG and Liu, J and Pego, R},
   Title = {Stability and convergence of efficient Navier-Stokes solvers
             via a commutator estimate via a commutator
             estimate},
   Journal = {Comm. Pure Appl. Math.},
   Volume = {60},
   Pages = {1443-1487},
   Year = {2007},
   Key = {fds246903}
}

@article{fds246947,
   Author = {Degond, P and Jin, S and Liu, JG},
   Title = {Mach-number uniform asymptotic- preserving Gauge schemes for
             compressible flows},
   Journal = {Bulletin of the Institute of Mathematics Academia Sinica
             (New Series)},
   Volume = {2},
   Pages = {851-892},
   Year = {2007},
   Keywords = {Mach number uniform method • Euler equations •
             Navier-Stokes equations • Asymptotic Preserving schemes
             • gauge schemes • compressible fluids •
             Low-Mach number limit • macro-micro decomposition
             • semi-implicit scheme • Euler-Poisson system
             • Navier-Stokes-Poisson system},
   Abstract = {We present novel algorithms for compressible flows that are
             efficient for all Mach numbers. The approach is based on
             several ingredients: semi-implicit schemes, the gauge
             decomposition of the velocity field and a second order
             formulation of the density equation (in the isentropic case)
             and of the energy equation (in the full Navier-Stokes case).
             Additionally, we show that our approach corresponds to a
             micro-macro decomposition of the model, where the macro
             field corresponds to the incompressible component satisfying
             a perturbed low Mach number limit equation and the micro
             field is the potential component of the velocity. Finally,
             we also use the conservative variables in order to obtain a
             proper conservative formulation of the equations when the
             Mach number is order unity. We successively consider the
             isentropic case, the full Navier-Stokes case, and the
             isentropic Navier-Stokes-Poisson case. In this work, we only
             concentrate on the question of the time discretization and
             show that the proposed method leads to Asymptotic Preserving
             schemes for compressible flows in the low Mach number
             limit.},
   Key = {fds246947}
}

@article{fds246960,
   Author = {Liu, JG and Wang, WC},
   Title = {Convergence analysis of the energy and helicity preserving
             scheme for axisymmetric flows},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {44},
   Number = {6},
   Pages = {2456-2480},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2006},
   Month = {December},
   ISSN = {0036-1429},
   url = {http://dx.doi.org/10.1137/050639314},
   Abstract = {We give an error estimate for the energy and helicity
             preserving scheme (EHPS) in second order finite difference
             setting on axisymmetric incompressible flows with swirling
             velocity. This is accomplished by a weighted energy
             estimate, along with careful and nonstandard local
             truncation error analysis near the geometric singularity and
             a far field decay estimate for the stream function. A key
             ingredient in our a priori estimate is the permutation
             identities associated with the Jacobians, which are also a
             unique feature that distinguishes EHPS from standard finite
             difference schemes. © 2006 Society for Industrial and
             Applied Mathematics.},
   Doi = {10.1137/050639314},
   Key = {fds246960}
}

@article{fds246901,
   Author = {Degond, P and Liu, JG and Mieussens, L},
   Title = {Macroscopic fluid models with localized kinetic upscaling
             effects},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {5},
   Number = {3},
   Pages = {940-979},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2006},
   Month = {September},
   ISSN = {1540-3459},
   url = {http://dx.doi.org/10.1137/060651574},
   Keywords = {Kinetic-Fluid coupling, Kinetic equation, Hydrodynamic
             approximation, Diffusion approximation},
   Abstract = {This paper presents a general methodology to design
             macroscopic fluid models that take into account localized
             kinetic upscaling effects. The fluid models are solved in
             the whole domain together with a localized kinetic upscaling
             that corrects the fluid model wherever it is necessary. This
             upscaling is obtained by solving a kinetic equation on the
             nonequilibrium part of the distribution function. This
             equation is solved only locally and is related to the fluid
             equation through a downscaling effect. The method does not
             need to find an interface condition as do usual domain
             decomposition methods to match fluid and kinetic
             representations. We show our approach applies to problems
             that have a hydrodynamic time scale as well as to problems
             with diffusion time scale. Simple numerical schemes are
             proposed to discretize our models, and several numerical
             examples are used to validate the method. © 2006 Society
             for Industrial and Applied Mathematics.},
   Doi = {10.1137/060651574},
   Key = {fds246901}
}

@article{fds246957,
   Author = {Moore, J and Liu, J-G and Zhou, K and Yu, LL},
   Title = {Effects of genotype and environment on the antioxidant
             properties of hard winter wheat bran.},
   Journal = {Journal of agricultural and food chemistry},
   Volume = {54},
   Number = {15},
   Pages = {5313-5322},
   Year = {2006},
   Month = {July},
   ISSN = {0021-8561},
   url = {http://dx.doi.org/10.1021/jf060381l},
   Abstract = {Recent consumer interest in controlling and preventing
             chronic diseases through improved diet has promoted research
             on the bioactive components of agricultural products. Wheat
             is an important agricultural and dietary commodity worldwide
             with known antioxidant properties concentrated mostly in the
             bran fraction. The objective of this study was to determine
             the relative contributions of genotype (G) and growing
             environment (E) to hard winter wheat bran antioxidant
             properties, as well as correlations of these properties to
             growing conditions. Bran samples of 20 hard winter wheat
             varieties grown in two locations were examined for their
             free radical scavenging capacities against DPPH, ABTS
             cation, peroxyl (ORAC), and superoxide anion radicals and
             chelating properties, as well as their total phenolics and
             phenolic acid compositions. Results showed significant
             differences for all antioxidant properties tested and
             multiple significant correlations between these properties.
             A factorial designed analysis of variance for these data and
             pooled previously published data showed similar results for
             four of the six antioxidant properties, indicating that G
             effects were considerably larger than E effects for
             chelating capacity and DPPH radical scavenging properties,
             whereas E was much stronger than G for ABTS cation radical
             scavenging capacity and total phenolics, although small
             interaction effects (GxE) were significant for all
             antioxidant properties analyzed. Results also showed
             significant correlations between temperature stress or solar
             radiation and some antioxidant properties. These results
             indicate that each antioxidant property of hard winter wheat
             bran is influenced differently by genotype and growing
             conditions.},
   Doi = {10.1021/jf060381l},
   Key = {fds246957}
}

@article{fds139013,
   Author = {J.-G. Liu and Jie Liu and R. Pego},
   Title = {On incompressible Navier-Stokes dynamics: a new approach for
             analysis and computation},
   Pages = {29--44},
   Booktitle = {Proceedings of the Tenth International Conference on
             Hyperbolic Problems},
   Publisher = {Yokohama Publishers, Inc.},
   Editor = {F. Asakura and etc},
   Year = {2006},
   Key = {fds139013}
}

@article{fds246964,
   Author = {Liu, JG and Samelson, R and Wang, C},
   Title = {Global weak solution of planetary geostrophic equations with
             inviscid geostrophic balance},
   Journal = {Applicable Analysis},
   Volume = {85},
   Number = {6-7},
   Pages = {593-605},
   Year = {2006},
   url = {http://dx.doi.org/10.1080/00036810500328299},
   Abstract = {A reformulation of the planetary geostrophic equations
             (PGEs) with the inviscid balance equation is proposed and
             the existence of global weak solutions is established,
             provided that the mechanical force satisfies an integral
             constraint. There is only one prognostic equation for the
             temperature field, and the velocity field is statically
             determined by the planetary geostrophic balance combined
             with the incompressibility condition. Furthermore, the
             velocity profile can be accurately represented as a function
             of the temperature gradient. In particular, the vertical
             velocity depends only on the first-order derivative of the
             temperature. As a result, the bound for the L∞ (0, t 1 ; L
             2 ) ∩ L 2 (0, t 1 ; H 1 ) norm of the temperature field is
             sufficient to show the existence of the weak solution. ©
             2006, Taylor & Francis Group, LLC.},
   Doi = {10.1080/00036810500328299},
   Key = {fds246964}
}

@article{fds246902,
   Author = {Liu, JG and Wang, WC},
   Title = {Energy and helicity preserving schemes for hydro- and
             magnetohydro-dynamics flows with symmetry},
   Journal = {Journal of Computational Physics},
   Volume = {200},
   Number = {1},
   Pages = {8-33},
   Publisher = {Elsevier BV},
   Year = {2004},
   Month = {October},
   url = {http://dx.doi.org/10.1016/j.jcp.2004.03.005},
   Abstract = {We propose a class of simple and efficient numerical scheme
             for incompressible fluid equations with coordinate symmetry.
             By introducing a generalized vorticity-stream formulation,
             the divergence free constraints are automatically satisfied.
             In addition, with explicit treatment of the nonlinear terms
             and local vorticity boundary condition, the Navier-Stokes
             (MHD, respectively) equation essentially decouples into 2
             (4, respectively) scalar equation and thus the scheme is
             very efficient. Moreover, with proper discretization of the
             nonlinear terms, the scheme preserves both energy and
             helicity identities numerically. This is achieved by
             recasting the nonlinear terms (convection, vorticity
             stretching, geometric source, Lorentz force and
             electro-motive force) in terms of Jacobians. This
             conservative property is valid even in the presence of the
             pole singularity for axisymmetric flows. The exact
             conservation of energy and helicity has effectively
             eliminated excessive numerical viscosity. Numerical examples
             have demonstrated both accuracy and efficiency of the
             scheme. Finally, local mesh refinement near the boundary can
             also be easily incorporated into the scheme without extra
             cost. © 2004 Elsevier Inc. All rights reserved.},
   Doi = {10.1016/j.jcp.2004.03.005},
   Key = {fds246902}
}

@article{fds246963,
   Author = {Ghil, M and Liu, JG and Wang, C and Wang, S},
   Title = {Boundary-layer separation and adverse pressure gradient for
             2-D viscous incompressible flow},
   Journal = {Physica D: Nonlinear Phenomena},
   Volume = {197},
   Number = {1-2},
   Pages = {149-173},
   Publisher = {Elsevier BV},
   Year = {2004},
   Month = {October},
   ISSN = {0167-2789},
   url = {http://dx.doi.org/10.1016/j.physd.2004.06.012},
   Abstract = {We study the detailed process of bifurcation in the flow's
             topological structure for a two-dimensional (2-D)
             incompressible flow subject to no-slip boundary conditions
             and its connection with boundary-layer separation. The
             boundary-layer separation theory of M. Ghil, T. Ma and S.
             Wang, based on the structural-bifurcation concept, is
             translated into vorticity form. The vorticily formulation of
             the theory shows that structural bifurcation occurs whenever
             a degenerate singular point for the vorticity appears on the
             boundary; this singular point is characterized by nonzero
             tangential second-order derivative and nonzero time
             derivative of the vorticity. Furthermore, we prove the
             presence of an adverse pressure gradient at the critical
             point, due to reversal in the direction of the pressure
             force with respect to the basic shear flow at this point. A
             numerical example of 2-D driven-cavity flow, governed by the
             Navier Stokes equations, is presented; boundary-layer
             separation occurs, the bifurcation criterion is satisfied,
             and an adverse pressure gradient is shown to be present. ©
             2004 Elsevier B.V. All rights reserved.},
   Doi = {10.1016/j.physd.2004.06.012},
   Key = {fds246963}
}

@article{fds304585,
   Author = {Li, B and Liu, JG},
   Title = {Epitaxial growth without slope selection: Energetics,
             coarsening, and dynamic scaling},
   Journal = {Journal of Nonlinear Science},
   Volume = {14},
   Number = {5},
   Pages = {429-451},
   Publisher = {Springer Nature},
   Year = {2004},
   Month = {October},
   ISSN = {0938-8974},
   url = {http://dx.doi.org/10.1007/s00332-004-0634-9},
   Abstract = {We study a continuum model for epitaxial growth of thin
             films in which the slope of mound structure of film surface
             increases. This model is a diffusion equation for the
             surface height profile h which is assumed to satisfy the
             periodic boundary condition. The equation happens to possess
             a Liapunov or "free-energy" functional. This functional
             consists of the term |Δ h| 2, which represents the surface
             diffusion, and-log (1 + |∇ h| 2), which describes the
             effect of kinetic asymmetry in the adatom
             attachment-detachment. We first prove for large time t that
             the interface width-the standard deviation of the height
             profile-is bounded above by O(t 1/2), the averaged gradient
             is bounded above by O(t 1/4), and the averaged energy is
             bounded below by O(-log t). We then consider a small
             coefficient ε 2 of |Δ h| 2 with ε = 1/L and L the linear
             size of the underlying system, and study the energy
             asymptotics in the large system limit ε → 0. We show that
             global minimizers of the free-energy functional exist for
             each ε > 0, the L 2-norm of the gradient of any global
             minimizer scales as O(1/ε), and the global minimum energy
             scales as O( log ε). The existence of global energy
             minimizers and a scaling argument are used to construct a
             sequence of equilibrium solutions with different
             wavelengths. Finally, we apply our minimum energy estimates
             to derive bounds in terms of the linear system size L for
             the saturation interface width and the corresponding
             saturation time. © 2005 Springer.},
   Doi = {10.1007/s00332-004-0634-9},
   Key = {fds304585}
}

@article{fds246962,
   Author = {Johnston, H and Liu, JG},
   Title = {Accurate, stable and efficient Navier-Stokes solvers based
             on explicit treatment of the pressure term},
   Journal = {Journal of Computational Physics},
   Volume = {199},
   Number = {1},
   Pages = {221-259},
   Publisher = {Elsevier BV},
   Year = {2004},
   Month = {September},
   url = {http://dx.doi.org/10.1016/j.jcp.2004.02.009},
   Abstract = {We present numerical schemes for the incompressible
             Navier-Stokes equations based on a primitive variable
             formulation in which the incompressibility constraint has
             been replaced by a pressure Poisson equation. The pressure
             is treated explicitly in time, completely decoupling the
             computation of the momentum and kinematic equations. The
             result is a class of extremely efficient Navier-Stokes
             solvers. Full time accuracy is achieved for all flow
             variables. The key to the schemes is a Neumann boundary
             condition for the pressure Poisson equation which enforces
             the incompressibility condition for the velocity field.
             Irrespective of explicit or implicit time discretization of
             the viscous term in the momentum equation the explicit time
             discretization of the pressure term does not affect the time
             step constraint. Indeed, we prove unconditional stability of
             the new formulation for the Stokes equation with explicit
             treatment of the pressure term and first or second order
             implicit treatment of the viscous term. Systematic numerical
             experiments for the full Navier-Stokes equations indicate
             that a second order implicit time discretization of the
             viscous term, with the pressure and convective terms treated
             explicitly, is stable under the standard CFL condition.
             Additionally, various numerical examples are presented,
             including both implicit and explicit time discretizations,
             using spectral and finite difference spatial
             discretizations, demonstrating the accuracy, flexibility and
             efficiency of this class of schemes. In particular, a
             Galerkin formulation is presented requiring only C0 elements
             to implement. © 2004 Elsevier Inc. All rights
             reserved.},
   Doi = {10.1016/j.jcp.2004.02.009},
   Key = {fds246962}
}

@article{fds246956,
   Author = {Wang, C and Liu, JG and Johnston, H},
   Title = {Analysis of a fourth order finite difference method for the
             incompressible Boussinesq equations},
   Journal = {Numerische Mathematik},
   Volume = {97},
   Number = {3},
   Pages = {555-594},
   Publisher = {Springer Nature},
   Year = {2004},
   Month = {May},
   url = {http://dx.doi.org/10.1007/s00211-003-0508-3},
   Abstract = {The convergence of a fourth order finite difference method
             for the 2-D unsteady, viscous incompressible Boussinesq
             equations, based on the vorticity-stream function
             formulation, is established in this article. A compact
             fourth order scheme is used to discretize the momentum
             equation, and long-stencil fourth order operators are
             applied to discretize the temperature transport equation. A
             local vorticity boundary condition is used to enforce the
             no-slip boundary condition for the velocity. One-sided
             extrapolation is used near the boundary, dependent on the
             type of boundary condition for the temperature, to prescribe
             the temperature at "ghost" points lying outside of the
             computational domain. Theoretical results of the stability
             and accuracy of the method are also provided. In numerical
             experiments the method has been shown to be capable of
             producing highly resolved solutions at a reasonable
             computational cost.},
   Doi = {10.1007/s00211-003-0508-3},
   Key = {fds246956}
}

@article{fds246954,
   Author = {Lin, HE and Liu, JG and Xu, WQ},
   Title = {Effects of small viscosity and far field boundary conditions
             for hyperbolic systems},
   Journal = {Communications on Pure and Applied Analysis},
   Volume = {3},
   Number = {2},
   Pages = {267-290},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2004},
   Month = {January},
   ISSN = {1534-0392},
   url = {http://dx.doi.org/10.3934/cpaa.2004.3.267},
   Abstract = {In this paper we study the effects of small viscosity term
             and the far-field boundary conditions for systems of
             convection-diffusion equations in the zero viscosity limit.
             The far-field boundary conditions are classified and the
             corresponding solution structures are analyzed. It is
             confirmed that the Neumann type of far-field boundary
             condition is preferred. On the other hand, we also identify
             a class of improperly coupled boundary conditions which lead
             to catastrophic reflection waves dominating the inlet in the
             zero viscosity limit. The analysis is performed on the
             linearized convection-diffusion model which well describes
             the behavior at the far field for many physical and
             engineering systems such as fluid dynamical equations and
             electro-magnetic equations. The results obtained here should
             provide some theoretical guidance for designing effective
             far field boundary conditions.},
   Doi = {10.3934/cpaa.2004.3.267},
   Key = {fds246954}
}

@article{fds246955,
   Author = {Liu, JG and Xu, WQ},
   Title = {Far field boundary condition for convection diffusion
             equation at zero viscosity limit},
   Journal = {Quarterly of Applied Mathematics},
   Volume = {62},
   Number = {1},
   Pages = {27-52},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2004},
   Month = {January},
   url = {http://dx.doi.org/10.1090/qam/2032571},
   Abstract = {In this paper, we give a systematic study of the boundary
             layer behavior for linear convection-diffusion equation in
             the zero viscosity limit. We analyze the boundary layer
             structures in the viscous solution and derive the boundary
             condition satisfied by the viscosity limit as a solution of
             the inviscid equation. The results confirm that the Neumann
             type of far-field boundary condition is preferred in the
             outlet and characteristic boundary dondition. Under some
             appropriate regularity and compatibility conditions on the
             initial and boundary data, we obtain optimal error estimates
             between the full viscous solution and the inviscid solution
             with suitable boundary layer corrections. These results hold
             in arbitrary space dimensions and similar statements also
             hold for the strip problem This model well describes the
             behavior at the far-field for many physical and engineering
             systems such as fluid dynamical equation and
             electro-magnetic equation. The results obtained here should
             provide some theoretical guidance for designing effective
             far-field boundary conditions.},
   Doi = {10.1090/qam/2032571},
   Key = {fds246955}
}

@article{fds304583,
   Author = {Liu, JG and Wang, C},
   Title = {High order finite difference methods for unsteady
             incompressible flows in multi-connected domains},
   Journal = {Computers and Fluids},
   Volume = {33},
   Number = {2},
   Pages = {223-255},
   Publisher = {Elsevier BV},
   Year = {2004},
   Month = {January},
   url = {http://dx.doi.org/10.1016/S0045-7930(03)00037-9},
   Abstract = {Using the vorticity and stream function variables is an
             effective way to compute 2-D incompressible flow due to the
             facts that the incompressibility constraint for the velocity
             is automatically satisfied, the pressure variable is
             eliminated, and high order schemes can be efficiently
             implemented. However, a difficulty arises in a
             multi-connected computational domain in determining the
             constants for the stream function on the boundary of the
             "holes". This is an especially challenging task for the
             calculation of unsteady flows, since these constants vary
             with time to reflect the total fluxes of the flow in each
             sub-channel. In this paper, we propose an efficient method
             in a finite difference setting to solve this problem and
             present some numerical experiments, including an accuracy
             check of a Taylor vortex-type flow, flow past a
             non-symmetric square, and flow in a heat exchanger. © 2003
             Elsevier Ltd. All rights reserved.},
   Doi = {10.1016/S0045-7930(03)00037-9},
   Key = {fds304583}
}

@article{fds246959,
   Author = {Li, B and Liu, JG},
   Title = {Eptaxial growth without slope selection: energetics,
             coarsening, and dynamic scaling},
   Journal = {J. Nonlinear Sci.},
   Volume = {14},
   Number = {5},
   Pages = {429-451},
   Year = {2004},
   ISSN = {0938-8974},
   url = {http://dx.doi.org/10.1007/s00332-004-0634-9},
   Abstract = {We study a continuum model for epitaxial growth of thin
             films in which the slope of mound structure of film surface
             increases. This model is a diffusion equation for the
             surface height profile h which is assumed to satisfy the
             periodic boundary condition. The equation happens to possess
             a Liapunov or "free-energy" functional. This functional
             consists of the term |Δ h| 2, which represents the
             surface diffusion, and-log (1 + |∇ h| 2), which
             describes the effect of kinetic asymmetry in the adatom
             attachment-detachment. We first prove for large time t that
             the interface width-the standard deviation of the height
             profile-is bounded above by O(t 1/2), the averaged gradient
             is bounded above by O(t 1/4), and the averaged energy is
             bounded below by O(-log t). We then consider a small
             coefficient ε 2 of |Δ h| 2 with ε = 1/L and L the
             linear size of the underlying system, and study the energy
             asymptotics in the large system limit ε → 0. We
             show that global minimizers of the free-energy functional
             exist for each ε &gt; 0, the L 2-norm of the gradient of
             any global minimizer scales as O(1/ε), and the global
             minimum energy scales as O( log ε). The existence of
             global energy minimizers and a scaling argument are used to
             construct a sequence of equilibrium solutions with different
             wavelengths. Finally, we apply our minimum energy estimates
             to derive bounds in terms of the linear system size L for
             the saturation interface width and the corresponding
             saturation time. © 2005 Springer.},
   Doi = {10.1007/s00332-004-0634-9},
   Key = {fds246959}
}

@article{fds246965,
   Author = {Liu, JG and Wang, C},
   Title = {High order finite difference method for unsteady
             incompressible flow on multi-connected domain in
             vorticity-stream function formulation},
   Journal = {Computer and Fluids},
   Volume = {33},
   Number = {2},
   Pages = {223-255},
   Year = {2004},
   url = {http://dx.doi.org/10.1016/S0045-7930(03)00037-9},
   Abstract = {Using the vorticity and stream function variables is an
             effective way to compute 2-D incompressible flow due to the
             facts that the incompressibility constraint for the velocity
             is automatically satisfied, the pressure variable is
             eliminated, and high order schemes can be efficiently
             implemented. However, a difficulty arises in a
             multi-connected computational domain in determining the
             constants for the stream function on the boundary of the
             "holes". This is an especially challenging task for the
             calculation of unsteady flows, since these constants vary
             with time to reflect the total fluxes of the flow in each
             sub-channel. In this paper, we propose an efficient method
             in a finite difference setting to solve this problem and
             present some numerical experiments, including an accuracy
             check of a Taylor vortex-type flow, flow past a
             non-symmetric square, and flow in a heat exchanger. ©
             2003 Elsevier Ltd. All rights reserved.},
   Doi = {10.1016/S0045-7930(03)00037-9},
   Key = {fds246965}
}

@article{fds246953,
   Author = {Duraisamy, K and Baeder, JD and Liu, JG},
   Title = {Concepts and Application of Time-Limiters to High Resolution
             Schemes},
   Journal = {Journal of Scientific Computing},
   Volume = {19},
   Number = {1-3},
   Pages = {139-162},
   Year = {2003},
   Month = {December},
   ISSN = {0885-7474},
   url = {http://dx.doi.org/10.1023/A:1025395707090},
   Abstract = {A new class of implicit high-order non-oscillatory time
             integration schemes is introduced in a method-of-lines
             framework. These schemes can be used in conjunction with an
             appropriate spatial discretization scheme for the numerical
             solution of time dependent conservation equations. The main
             concept behind these schemes is that the order of accuracy
             in time is dropped locally in regions where the time
             evolution of the solution is not smooth. By doing this, an
             attempt is made at locally satisfying monotonicity
             conditions, while maintaining a high order of accuracy in
             most of the solution domain. When a linear high order time
             integration scheme is used along with a high order spatial
             discretization, enforcement of monotonicity imposes severe
             time-step restrictions. We propose to apply limiters to
             these time-integration schemes, thus making them non-linear.
             When these new schemes are used with high order spatial
             discretizations, solutions remain non-oscillatory for much
             larger time-steps as compared to linear time integration
             schemes. Numerical results obtained on scalar conservation
             equations and systems of conservation equations are highly
             promising.},
   Doi = {10.1023/A:1025395707090},
   Key = {fds246953}
}

@article{fds246966,
   Author = {Li, B and Liu, JG},
   Title = {Thin film epitaxy with or without slope selection},
   Journal = {European Journal of Applied Mathematics},
   Volume = {14},
   Number = {6},
   Pages = {713-743},
   Publisher = {Cambridge University Press (CUP)},
   Year = {2003},
   Month = {December},
   url = {http://dx.doi.org/10.1017/S095679250300528X},
   Abstract = {Two nonlinear diffusion equations for thin film epitaxy,
             with or without slope selection, are studied in this work.
             The nonlinearity models the Ehrlich-Schwoebel effect - the
             kinetic asymmetry in attachment and detachment of adatoms to
             and from terrace boundaries. Both perturbation analysis and
             numerical simulation are presented to show that such an
             atomistic effect is the origin of a nonlinear morphological
             instability, in a rough-smooth-rough pattern, that has been
             experimentally observed as transient in an early stage of
             epitaxial growth on rough surfaces. Initial-boundary-value
             problems for both equations are proven to be well-posed, and
             the solution regularity is also obtained. Galerkin spectral
             approximations are studied to provide both a priori bounds
             for proving the well-posedness and numerical schemes for
             simulation. Numerical results are presented to confirm part
             of the analysis and to explore the difference between the
             two models on coarsening dynamics.},
   Doi = {10.1017/S095679250300528X},
   Key = {fds246966}
}

@article{fds246968,
   Author = {Liu, JG and Wang, C and Johnston, H},
   Title = {A Fourth Order Scheme for Incompressible Boussinesq
             Equations},
   Journal = {Journal of Scientific Computing},
   Volume = {18},
   Number = {2},
   Pages = {253-285},
   Year = {2003},
   Month = {April},
   ISSN = {0885-7474},
   url = {http://dx.doi.org/10.1023/A:1021168924020},
   Abstract = {A fourth order finite difference method is presented for the
             2D unsteady viscous incompressible Boussinesq equations in
             vorticity-stream function formulation. The method is
             especially suitable for moderate to large Reynolds number
             flows. The momentum equation is discretized by a compact
             fourth order scheme with the no-slip boundary condition
             enforced using a local vorticity boundary condition. Fourth
             order long-stencil discretizations are used for the
             temperature transport equation with one-sided extrapolation
             applied near the boundary. The time stepping scheme for both
             equations is classical fourth order Runge-Kutta. The method
             is highly efficient. The main computation consists of the
             solution of two Poisson-like equations at each Runge-Kutta
             time stage for which standard FFT based fast Poisson solvers
             are used. An example of Lorenz flow is presented, in which
             the full fourth order accuracy is checked. The numerical
             simulation of a strong shear flow induced by a temperature
             jump, is resolved by two perfectly matching resolutions.
             Additionally, we present benchmark quality simulations of a
             differentially-heated cavity problem. This flow was the
             focus of a special session at the first MIT conference on
             Computational Fluid and Solid Mechanics in June
             2001.},
   Doi = {10.1023/A:1021168924020},
   Key = {fds246968}
}

@article{fds246951,
   Author = {Wang, C and Liu, JG},
   Title = {Positivity property of second-order flux-splitting schemes
             for the compressible Euler equations},
   Journal = {Discrete and Continuous Dynamical Systems - Series
             B},
   Volume = {3},
   Number = {2},
   Pages = {201-228},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2003},
   Month = {January},
   url = {http://dx.doi.org/10.3934/dcdsb.2003.3.201},
   Abstract = {A class of upwind flux splitting methods in the Euler
             equations of compressible flow is considered in this paper.
             Using the property that Euler flux F(U) is a homogeneous
             function of degree one in U, we reformulate the splitting
             fluxes with F+ = A+U, F- = A -U, and the corresponding
             matrices are either symmetric or symmetrizable and keep only
             non-negative and non-positive eigenvalues. That leads to the
             conclusion that the first order schemes are positive in the
             sense of Lax-Liu [18], which implies that it is L2- stable
             in some suitable sense. Moreover, the second order scheme is
             a stable perturbation of the first order scheme, so that the
             positivity of the second order schemes is also established,
             under a CFL-like condition. In addition, these splitting
             methods preserve the positivity of density and
             energy.},
   Doi = {10.3934/dcdsb.2003.3.201},
   Key = {fds246951}
}

@article{fds246952,
   Author = {Chainais-Hillairet, C and Liu, JG and Peng, YJ},
   Title = {Finite volume scheme for multi-dimensional drift-diffusion
             equations and convergence analysis},
   Journal = {Mathematical Modelling and Numerical Analysis},
   Volume = {37},
   Number = {2},
   Pages = {319-338},
   Publisher = {E D P SCIENCES},
   Year = {2003},
   Month = {January},
   url = {http://dx.doi.org/10.1051/m2an:2003028},
   Abstract = {We introduce a finite volume scheme for multi-dimensional
             drift-diffusion equations. Such equations arise from the
             theory of semiconductors and are composed of two continuity
             equations coupled with a Poisson equation. In the case that
             the continuity equations are non degenerate, we prove the
             convergence of the scheme and then the existence of
             solutions to the problem. The key point of the proof relies
             on the construction of an approximate gradient of the
             electric potential which allows us to deal with coupled
             terms in the continuity equations. Finally, a numerical
             example is given to show the efficiency of the
             scheme.},
   Doi = {10.1051/m2an:2003028},
   Key = {fds246952}
}

@article{fds366915,
   Author = {Weinan, E and Liu, JG},
   Title = {ADDENDUM TO “GAUGE METHOD FOR VISCOUS INCOMPRESSIBLE
             FLOWS”*},
   Journal = {Communications in Mathematical Sciences},
   Volume = {1},
   Number = {4},
   Pages = {837-837},
   Year = {2003},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2003.v1.n4.a10},
   Abstract = {Gauge transformation is a well-known concept in physics and
             has been used as a computational tool also. In fluid
             dynamics, Buttke was the first to use it as a computational
             tool to design vortex methods [1], following earlier work of
             Oseledets and others [3]. An alternative formulation was
             found by Maddocks and Pego [2] using the impetus-striction
             variables. This formulation does not seem to have the
             problem of numerical instability at the linear level. These
             authors are mainly concerned with writing down the
             Hamiltonian formulation of Euler’s equation, whereas we
             are mainly concerned with using the gauge freedom to
             overcome the difficulties with boundary condition.},
   Doi = {10.4310/CMS.2003.v1.n4.a10},
   Key = {fds366915}
}

@article{fds246950,
   Author = {Wang, C and Liu, JG},
   Title = {Fourth order convergence of a compact difference solver for
             incompressible flow},
   Journal = {Commun. Appl. Anal.},
   Volume = {7},
   Pages = {171-191},
   Year = {2003},
   Key = {fds246950}
}

@article{fds246961,
   Author = {Weinan, E and Liu, JG},
   Title = {Gauge method for viscous incompressible flows},
   Journal = {Comm. Math. Sci.},
   Volume = {1},
   Pages = {317-332},
   Year = {2003},
   Key = {fds246961}
}

@article{fds246967,
   Author = {Chern, IL and Liu, JG and Wang, WC},
   Title = {Accurate evaluation of electrostatics for macromolecules in
             solution},
   Journal = {Methods and Applications of Analysis},
   Volume = {10},
   Pages = {309-328},
   Year = {2003},
   Key = {fds246967}
}

@article{fds246939,
   Author = {Johnston, H and Liu, JG},
   Title = {Finite difference schemes for incompressible flow based on
             local pressure boundary conditions},
   Journal = {Journal of Computational Physics},
   Volume = {180},
   Number = {1},
   Pages = {120-154},
   Publisher = {Elsevier BV},
   Year = {2002},
   Month = {July},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1006/jcph.2002.7079},
   Abstract = {In this paper we discuss the derivation and use of local
             pressure boundary conditions for finite difference schemes
             for the unsteady incompressible Navier-Stokes equations in
             the velocity-pressure formulation. Their use is especially
             well suited for the computation of moderate to large
             Reynolds number flows. We explore the similarities between
             the implementation and use of local pressure boundary
             conditions and local vorticity boundary conditions in the
             design of numerical schemes for incompressible flow in 2D.
             In their respective formulations, when these local numerical
             boundary conditions are coupled with a fully explicit
             convectively stable time stepping procedure, the resulting
             methods are, simple to implement and highly efficient.
             Unlike the vorticity formulation, the use of the local
             pressure boundary condition approach is readily applicable
             to 3D flows. The simplicity of the local pressure boundary
             condition approach and its easy adaptation to more general
             flow settings make the resulting scheme an attractive
             alternative to the more popular methods for solving the
             Navier-Stokes equations in the velocity-pressure
             formulation. We present numerical results of a second-order
             finite difference scheme on a nonstaggered grid using local
             pressure boundary conditions. Stability and accuracy of the
             scheme applied to Stokes flow is demonstrated using normal
             mode analysis. Also described is the extension of the method
             to variable density flows. © 2002 Elsevier Science
             (USA).},
   Doi = {10.1006/jcph.2002.7079},
   Key = {fds246939}
}

@article{fds246937,
   Author = {Wang, C and Liu, JG},
   Title = {Analysis of finite difference schemes for unsteady
             Navier-Stokes equations in vorticity formulation},
   Journal = {Numerische Mathematik},
   Volume = {91},
   Number = {3},
   Pages = {543-576},
   Year = {2002},
   Month = {May},
   url = {http://dx.doi.org/10.1007/s002110100311},
   Abstract = {In this paper, we provide stability and convergence analysis
             for a class of finite difference schemes for unsteady
             incompressible Navier-Stokes equations in vorticity-stream
             function formulation. The no-slip boundary condition for the
             velocity is converted into local vorticity boundary
             conditions. Thorn's formula, Wilkes' formula, or other local
             formulas in the earlier literature can be used in the second
             order method; while high order formulas, such as Briley's
             formula, can be used in the fourth order compact difference
             scheme proposed by E and Liu. The stability analysis of
             these long-stencil formulas cannot be directly derived from
             straightforward manipulations since more than one interior
             point is involved in the formula. The main idea of the
             stability analysis is to control local terms by global
             quantities via discrete elliptic regularity for stream
             function. We choose to analyze the second order scheme with
             Wilkes' formula in detail. In this case, we can avoid the
             complicated technique necessitated by the Strang-type high
             order expansions. As a consequence, our analysis results in
             almost optimal regularity assumption for the exact solution.
             The above methodology is very general. We also give a
             detailed analysis for the fourth order scheme using a 1-D
             Stokes model.},
   Doi = {10.1007/s002110100311},
   Key = {fds246937}
}

@article{fds246938,
   Author = {Weinan, E and Liu, JG},
   Title = {Projection method III: Spatial discretization on the
             staggered grid},
   Journal = {Mathematics of Computation},
   Volume = {71},
   Number = {237},
   Pages = {27-47},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2002},
   Month = {January},
   url = {http://dx.doi.org/10.1090/S0025-5718-01-01313-8},
   Abstract = {In E & Liu (SIAM J Numer. Anal., 1995), we studied
             convergence and the structure of the error for several
             projection methods when the spatial variable was kept
             continuous (we call this the semi-discrete case). In this
             paper, we address similar questions for the fully discrete
             case when the spatial variables are discretized using a
             staggered grid. We prove that the numerical solution in
             velocity has full accuracy up to the boundary, despite the
             fact that there are numerical boundary layers present in the
             semi-discrete solutions.},
   Doi = {10.1090/S0025-5718-01-01313-8},
   Key = {fds246938}
}

@article{fds246934,
   Author = {Liu, JG and Wang, WC},
   Title = {An energy-preserving MAC-Yee scheme for the incompressible
             MHD equation},
   Journal = {Journal of Computational Physics},
   Volume = {174},
   Number = {1},
   Pages = {12-37},
   Publisher = {Elsevier BV},
   Year = {2001},
   Month = {November},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1006/jcph.2001.6772},
   Abstract = {We propose a simple and efficient finite-difference method
             for the incompressible MHD equation. The numerical method
             combines the advantage of the MAC scheme for the
             Navier-Stokes equation and Yee's scheme for the Maxwell
             equation. In particular, the semi-discrete version of our
             scheme introduces no numerical dissipation and preserves the
             energy identity exactly. © 2001 Elsevier
             Science.},
   Doi = {10.1006/jcph.2001.6772},
   Key = {fds246934}
}

@article{fds304582,
   Author = {Liu, JG and Xin, Z},
   Title = {Convergence of the point vortex method for 2-D vortex
             sheet},
   Journal = {Mathematics of Computation},
   Volume = {70},
   Number = {234},
   Pages = {595-606},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2001},
   Month = {April},
   url = {http://dx.doi.org/10.1090/S0025-5718-00-01271-0},
   Abstract = {We give an elementary proof of the convergence of the point
             vortex method (PVM) to a classical weak solution for the
             two-dimensional incompressible Euler equations with initial
             vorticity being a finite Radon measure of distinguished sign
             and the initial velocity of locally bounded energy. This
             includes the important example of vortex sheets, which
             exhibits the classical Kelvin-Helmholtz instability. A
             surprise fact is that although the velocity fields generated
             by the point vortex method do not have bounded local kinetic
             energy, the limiting velocity field is shown to have a
             bounded local kinetic energy.},
   Doi = {10.1090/S0025-5718-00-01271-0},
   Key = {fds304582}
}

@article{fds246873,
   Author = {Liu, JG and Weinan, E},
   Title = {Simple finite element method in vorticity formulation for
             incompressible flows},
   Journal = {Mathematics of Computation},
   Volume = {70},
   Number = {234},
   Pages = {579-593},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2001},
   Month = {April},
   url = {http://dx.doi.org/10.1090/S0025-5718-00-01239-4},
   Abstract = {A very simple and efficient finite element method is
             introduced for two and three dimensional viscous
             incompressible flows using the vorticity formulation. This
             method relies on recasting the traditional finite element
             method in the spirit of the high order accurate finite
             difference methods introduced by the authors in another
             work. Optimal accuracy of arbitrary order can be achieved
             using standard finite element or spectral elements. The
             method is convectively stable and is particularly suited for
             moderate to high Reynolds number flows.},
   Doi = {10.1090/S0025-5718-00-01239-4},
   Key = {fds246873}
}

@article{fds246935,
   Author = {Liu, JG and Weinan, E},
   Title = {Simple finite element method in vorticity formulation for
             incompressible flow},
   Journal = {Math. Comp.},
   Volume = {69},
   Pages = {1385-1407},
   Year = {2001},
   Key = {fds246935}
}

@article{fds246936,
   Author = {Liu, JG and Xin, Z},
   Title = {Convergence of point vortex method for 2-D vortex
             sheet},
   Journal = {Math. Comp.},
   Volume = {70},
   Number = {234},
   Pages = {565-606},
   Year = {2001},
   url = {http://dx.doi.org/10.1090/S0025-5718-00-01271-0},
   Abstract = {We give an elementary proof of the convergence of the point
             vortex method (PVM) to a classical weak solution for the
             two-dimensional incompressible Euler equations with initial
             vorticity being a finite Radon measure of distinguished sign
             and the initial velocity of locally bounded energy. This
             includes the important example of vortex sheets, which
             exhibits the classical Kelvin-Helmholtz instability. A
             surprise fact is that although the velocity fields generated
             by the point vortex method do not have bounded local kinetic
             energy, the limiting velocity field is shown to have a
             bounded local kinetic energy.},
   Doi = {10.1090/S0025-5718-00-01271-0},
   Key = {fds246936}
}

@article{fds246933,
   Author = {Weinan, E and Liu, JG},
   Title = {Gauge finite element method for incompressible
             flows},
   Journal = {International Journal for Numerical Methods in
             Fluids},
   Volume = {34},
   Number = {8},
   Pages = {701-710},
   Publisher = {WILEY},
   Year = {2000},
   Month = {December},
   ISSN = {0271-2091},
   url = {http://dx.doi.org/10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B},
   Abstract = {A finite element method for computing viscous incompressible
             flows based on the gauge formulation introduced in [Weinan
             E. Liu J-G. Gauge method for viscous incompressible flows.
             Journal of Computational Physics (submitted)] is presented.
             This formulation replaces the pressure by a gauge variable.
             This new gauge variable is a numerical tool and differs from
             the standard gauge variable that arises from decomposing a
             compressible velocity field. It has the advantage that an
             additional boundary condition can be assigned to the gauge
             variable, thus eliminating the issue of a pressure boundary
             condition associated with the original primitive variable
             formulation. The computational task is then reduced to
             solving standard heat and Poisson equations, which are
             approximated by straightforward, piecewise linear (or
             higher-order) finite elements. This method can achieve
             high-order accuracy at a cost comparable with that of
             solving standard heat and Poisson equations. It is naturally
             adapted to complex geometry and it is much simpler than
             traditional finite elements methods for incompressible
             flows. Several numerical examples on both structured and
             unstructured grids are presented. Copyright © 2000 John
             Wiley & Sons, Ltd.},
   Doi = {10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B},
   Key = {fds246933}
}

@article{fds246931,
   Author = {Liu, JG and Shu, CW},
   Title = {A High-Order Discontinuous Galerkin Method for 2D
             Incompressible Flows},
   Journal = {Journal of Computational Physics},
   Volume = {160},
   Number = {2},
   Pages = {577-596},
   Publisher = {Elsevier BV},
   Year = {2000},
   Month = {May},
   url = {http://dx.doi.org/10.1006/jcph.2000.6475},
   Abstract = {In this paper we introduce a high-order discontinuous
             Galerkin method for two-dimensional incompressible flow in
             the vorticity stream-function formulation. The momentum
             equation is treated explicitly, utilizing the efficiency of
             the discontinuous Galerkin method. The stream function is
             obtained by a standard Poisson solver using continuous
             finite elements. There is a natural matching between these
             two finite element spaces, since the normal component of the
             velocity field is continuous across element boundaries. This
             allows for a correct upwinding gluing in the discontinuous
             Galerkin framework, while still maintaining total energy
             conservation with no numerical dissipation and total
             enstrophy stability. The method is efficient for inviscid or
             high Reynolds number flows. Optimal error estimates are
             proved and verified by numerical experiments. © 2000
             Academic Press.},
   Doi = {10.1006/jcph.2000.6475},
   Key = {fds246931}
}

@article{fds246930,
   Author = {Liu, JG and Xin, Z},
   Title = {Convergence of a Galerkin method for 2-D discontinuous Euler
             flows},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {53},
   Number = {6},
   Pages = {786-798},
   Publisher = {Wiley},
   Year = {2000},
   Month = {January},
   url = {http://dx.doi.org/10.1002/(SICI)1097-0312(200006)53:6<786::AID-CPA3>3.0.CO;2-Y},
   Abstract = {We prove the convergence of a discontinuous Galerkin method
             approximating the 2-D incompressible Euler equations with
             discontinuous initial vorticity: ω0 ∈ L2(Ω).
             Furthermore, when ω0 ∈ L∞(Ω), the whole sequence is
             shown to be strongly convergent. This is the first
             convergence result in numerical approximations of this
             general class of discontinuous flows. Some important flows
             such as vortex patches belong to this class. © 2000 John
             Wiley & Sons, Inc.},
   Doi = {10.1002/(SICI)1097-0312(200006)53:6<786::AID-CPA3>3.0.CO;2-Y},
   Key = {fds246930}
}

@article{fds246932,
   Author = {Wang, C and Liu, JG},
   Title = {Convergence of gauge method for incompressible
             flow},
   Journal = {Mathematics of Computation},
   Volume = {69},
   Number = {232},
   Pages = {1385-1407},
   Year = {2000},
   Month = {January},
   url = {http://dx.doi.org/10.1090/s0025-5718-00-01248-5},
   Abstract = {A new formulation, a gauge formulation of the incompressible
             Navier-Stokes equations in terms of an auxiliary field a and
             a gauge variable φ, u = a + ∇φ, was proposed recently by
             E and Liu. This paper provides a theoretical analysis of
             their formulation and verifies the computational advantages.
             We discuss the implicit gauge method, which uses backward
             Euler or Crank-Nicolson in time discretization. However, the
             boundary conditions for the auxiliary field a are
             implemented explicitly (vertical extrapolation). The
             resulting momentum equation is decoupled from the kinematic
             equation, and the computational cost is reduced to solving a
             standard heat and Poisson equation. Moreover, such explicit
             boundary conditions for the auxiliary field a will be shown
             to be unconditionally stable for Stokes equations. For the
             full nonlinear Navier-Stokes equations the time stepping
             constraint is reduced to the standard CFL constraint Δt/Δx
             ≤ C. We also prove first order convergence of the gauge
             method when we use MAC grids as our spatial discretization.
             The optimal error estimate for the velocity field is also
             obtained.},
   Doi = {10.1090/s0025-5718-00-01248-5},
   Key = {fds246932}
}

@article{fds246927,
   Author = {Lefloch, PG and Liu, JG},
   Title = {Generalized monotone schemes, discrete paths of extrema, and
             discrete entropy conditions},
   Journal = {Mathematics of Computation},
   Volume = {68},
   Number = {227},
   Pages = {1025-1055},
   Year = {1999},
   Month = {January},
   url = {http://dx.doi.org/10.1090/s0025-5718-99-01062-5},
   Abstract = {Solutions of conservation laws satisfy the monotonicity
             property: the number of local extrema is a non-increasing
             function of time, and local maximum/minimum values
             decrease/increase monotonically in time. This paper
             investigates this property from a numerical standpoint. We
             introduce a class of fully discrete in space and time, high
             order accurate, difference schemes, called generalized
             monotone schemes. Convergence toward the entropy solution is
             proven via a new technique of proof, assuming that the
             initial data has a finite number of extremum values only,
             and the flux-function is strictly convex. We define discrete
             paths of extrema by tracking local extremum values in the
             approximate solution. In the course of the analysis we
             establish the pointwise convergence of the trace of the
             solution along a path of extremum. As a corollary, we obtain
             a proof of convergence for a MUSCL-type scheme that is
             second order accurate away from sonic points and
             extrema.},
   Doi = {10.1090/s0025-5718-99-01062-5},
   Key = {fds246927}
}

@article{fds246929,
   Author = {Wang, ZJ and Liu, JG and Childress, S},
   Title = {Connection between corner vortices and shear layer
             instability in flow past an ellipse},
   Journal = {Physics of Fluids},
   Volume = {11},
   Number = {9},
   Pages = {2446-2448},
   Year = {1999},
   Month = {January},
   url = {http://dx.doi.org/10.1063/1.870108},
   Abstract = {We investigate, by numerical simulation, the shear layer
             instability associated with the outer layer of a spiral
             vortex formed behind an impulsively started thin ellipse.
             The unstable free shear layer undergoes a secondary
             instability. We connect this instability with the dynamics
             of corner vortices adjacent to the tip of the ellipse by
             observing that the typical turnover time of the corner
             vortex matches the period of the unstable mode in the shear
             layer. We suggest that the corner vortex acts as a signal
             generator, and produces periodic perturbation which triggers
             the instability. © 1999 American Institute of
             Physics.},
   Doi = {10.1063/1.870108},
   Key = {fds246929}
}

@article{fds246926,
   Author = {Choi, H and Liu, JG},
   Title = {The Reconstruction of Upwind Fluxes for Conservation Laws:
             Its Behavior in Dynamic and Steady State
             Calculations},
   Journal = {Journal of Computational Physics},
   Volume = {144},
   Number = {2},
   Pages = {237-256},
   Publisher = {Elsevier BV},
   Year = {1998},
   Month = {August},
   url = {http://dx.doi.org/10.1006/jcph.1998.5970},
   Abstract = {The Euler equation of compressible flows is solved by the
             finite volume method, where high order accuracy is achieved
             by the reconstruction of each component of upwind fluxes of
             a flux splitting using the biased averaging procedure.
             Compared to the solution reconstruction in Godunov-type
             methods, its implementation is simple and easy, and the
             computational complexity is relatively low. This approach is
             parameter-free and requires neither a Riemann solver nor
             field-by-field decomposition. The numerical results from
             both dynamic and steady state calculations demonstrate the
             accuracy and robustness of this approach. Some techniques
             for the acceleration of the convergence to the steady state
             are discussed, including multigrid and multistage
             Runge-Kutta time methods. © 1998 Academic
             Press.},
   Doi = {10.1006/jcph.1998.5970},
   Key = {fds246926}
}

@article{fds246925,
   Author = {Xu, E and Liu, JG},
   Title = {Pricing of mortgage-backed securities with option-adjusted
             spread},
   Journal = {Managerial Finance},
   Volume = {24},
   Pages = {94-109},
   Year = {1998},
   Key = {fds246925}
}

@article{fds246922,
   Author = {E, W and Liu, JG},
   Title = {Finite Difference Methods for 3D Viscous Incompressible
             Flows in the Vorticity-Vector Potential Formulation on
             Nonstaggered Grids},
   Journal = {Journal of Computational Physics},
   Volume = {138},
   Number = {1},
   Pages = {57-82},
   Publisher = {Elsevier BV},
   Year = {1997},
   Month = {November},
   url = {http://dx.doi.org/10.1006/jcph.1997.5815},
   Abstract = {Simple, efficient, and accurate finite difference methods
             are introduced for 3D unsteady viscous incompressible flows
             in the vorticity-vector potential formulation on
             nonstaggered grids. Two different types of methods are
             discussed. They differ in the implementation of the normal
             component of the vorticity boundary condition and
             consequently the enforcement of the divergence free
             condition for vorticity. Both second-order and fourth-order
             accurate schemes are developed. A detailed accuracy test is
             performed, revealing the structure of the error and the
             effect of how the convective terms are discretized near the
             boundary. The influence of the divergence free condition for
             vorticity to the overall accuracy is studied. Results on the
             cubic driven cavity flow at Reynolds number 500 and 3200 are
             shown and compared with that of the MAC scheme. © 1997
             Academic Press.},
   Doi = {10.1006/jcph.1997.5815},
   Key = {fds246922}
}

@article{fds246923,
   Author = {Chen, GQ and Liu, JG},
   Title = {Convergence of difference schemes with high resolution for
             conservation laws},
   Journal = {Mathematics of Computation},
   Volume = {66},
   Number = {219},
   Pages = {1027-1053},
   Year = {1997},
   Month = {January},
   url = {http://dx.doi.org/10.1090/s0025-5718-97-00859-4},
   Abstract = {We are concerned with the convergence of Lax-Weridroff type
             schemes with high resolution to the entropy solutions fo:
             conservation laws. These schemes include the original
             Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and
             its two step versions-the Richtrayer scheme and the
             MacCormack scheme. For the convex scalar conservation laws
             with algebraic growth flux functions, we prove the
             convergence of these schemes to the weak solutions
             satisfying appropriate entropy inequalities. The proof is
             based on detailed Lp estimates of the approximate solutions,
             H-1 compactness estimates of the corresponding entropy
             dissipation measures, and some compensated compactness
             frameworks. Then these techniques are generalized to study
             the convergence problem for the nonconvex scalar case and
             the hyperbolic systems of conservation laws.},
   Doi = {10.1090/s0025-5718-97-00859-4},
   Key = {fds246923}
}

@article{fds246924,
   Author = {Weinan, E and Liu, JG},
   Title = {Finite difference schemes for incompressible flows in the
             velocity - impulse density formulation},
   Journal = {Journal of Computational Physics},
   Volume = {130},
   Number = {1},
   Pages = {67-76},
   Publisher = {Elsevier BV},
   Year = {1997},
   Month = {January},
   url = {http://dx.doi.org/10.1006/jcph.1996.5537},
   Abstract = {We consider finite difference schemes based on the impulse
             density variable. We show that the original velocity -
             impulse density formulation of Oseledets is marginally
             ill-posed for the inviscid flow, and this has the
             consequence that some ordinarily stable numerical methods in
             other formulations become unstable in the velocity - impulse
             density formulation. We present numerical evidence of this
             instability. We then discuss the construction of stable
             finite difference schemes by requiring that at the numerical
             level the nonlinear terms be convertible to similar terms in
             the primitive variable formulation. Finally we give a
             simplified velocity - impulse density formulation which is
             free of these complications and yet retains the nice
             features of the original velocity - impulse density
             formulation with regard to the treatment of boundary. We
             present numerical results on this simplified formulation for
             the driven cavity flow on both the staggered and
             non-staggered grids. © 1997 Academic Press.},
   Doi = {10.1006/jcph.1996.5537},
   Key = {fds246924}
}

@article{fds246916,
   Author = {Weinan, E and Liu, JG},
   Title = {Vorticity boundary condition and related issues for finite
             difference schemes},
   Journal = {Journal of Computational Physics},
   Volume = {124},
   Number = {2},
   Pages = {368-382},
   Publisher = {Elsevier BV},
   Year = {1996},
   Month = {March},
   url = {http://dx.doi.org/10.1006/jcph.1996.0066},
   Abstract = {This paper discusses three basic issues related to the
             design of finite difference schemes for unsteady viscous
             incompressible flows using vorticity formulations: the
             boundary condition for vorticity, an efficient time-stepping
             procedure, and the relation between these schemes and the
             ones based on velocity-pressure formulation. We show that
             many of the newly developed global vorticity boundary
             conditions can actually be written as some local formulas
             derived earlier. We also show that if we couple a standard
             centered difference scheme with third-or fourth-order
             explicit Runge-Kutta methods, the resulting schemes have no
             cell Reynolds number constraints. For high Reynolds number
             flows, these schemes are stable under the CFL condition
             given by the convective terms. Finally, we show that the
             classical MAC scheme is the same as Thom's formula coupled
             with second-order centered differences in the interior, in
             the sense that one can define discrete vorticity in a
             natural way for the MAC scheme and get the same values as
             the ones computed from Thom's formula. We use this to derive
             an efficient fourth-order Runge-Kutta time discretization
             for the MAC scheme from the one for Thom's formula. We
             present numerical results for driven cavity flow at high
             Reynolds number (105). © 1996 Academic Press,
             Inc.},
   Doi = {10.1006/jcph.1996.0066},
   Key = {fds246916}
}

@article{fds246915,
   Author = {Jin, S and Liu, JG},
   Title = {The effects of numerical viscosities: I. Slowly moving
             shocks},
   Journal = {Journal of Computational Physics},
   Volume = {126},
   Number = {2},
   Pages = {373-389},
   Publisher = {Elsevier BV},
   Year = {1996},
   Month = {January},
   url = {http://dx.doi.org/10.1006/jcph.1996.0144},
   Abstract = {We begin a systematical study on the effect of numerical
             viscosities. In this paper we investigate the behavior of
             shock-capturing methods for slowly moving shocks. It is
             known that for slowly moving shocks even a first-order
             scheme, such as the Godunov or Roe type methods, will
             generate downstream oscillatory wave patterns that cannot be
             effectively damped by the dissipation of these first-order
             schemes. The purpose of this paper is to understand the
             formation and behavior of these downstream patterns. Our
             study shows that the downstream errors are generated by the
             unsteady nature of the viscous shock profiles and behave
             diffusively. The scenario is as follows. When solving the
             compressible Euler equations by shock capturing methods, the
             smeared density profile introduces a momentum spike at the
             shock location if the shock moves slowly. Downstream waves
             will necessarily emerge in order to balance the momentum
             mass carried by the spike for the momentum conservation.
             Although each family of waves decays in l∞ and l2 while
             they preserve the same mass, the perturbing nature of the
             viscous or spike profile is a constant source for the
             generation of new downstream waves, causing spurious
             solutions for all time. Higher order TVD or ENO type
             interpolations accentuate this problem. © 1996 Academic
             Press, Inc.},
   Doi = {10.1006/jcph.1996.0144},
   Key = {fds246915}
}

@article{fds246917,
   Author = {Weinan, E and Liu, JG},
   Title = {Essentially compact schemes for unsteady viscous
             incompressible flows},
   Journal = {Journal of Computational Physics},
   Volume = {126},
   Number = {1},
   Pages = {122-138},
   Publisher = {Elsevier BV},
   Year = {1996},
   Month = {January},
   url = {http://dx.doi.org/10.1006/jcph.1996.0125},
   Abstract = {A new fourth-order accurate finite difference scheme for the
             computation of unsteady viscous incompressible flows is
             introduced. The scheme is based on the vorticity-stream
             function formulation. It is essentially compact and has the
             nice features of a compact scheme with regard to the
             treatment of boundary conditions. It is also very efficient,
             at every time step or Runge-Kutta stage, only two
             Poisson-like equations have to be solved. The Poisson-like
             equations are amenable to standard fast Poisson solvers
             usually designed for second order schemes. Detailed
             comparison with the second-order scheme shows the clear
             superiority of this new fourth-order scheme in resolving
             both the boundary layers and the gross features of the flow.
             This efficient fourth-order scheme also made it possible to
             compute the driven cavity flow at Reynolds number 106 on a
             10242 grid at a reasonable cost. Fourth-order convergence is
             proved under mild regularity requirements. This is the first
             such result to our knowledge. © 1996 Academic Press,
             Inc.},
   Doi = {10.1006/jcph.1996.0125},
   Key = {fds246917}
}

@article{fds246918,
   Author = {Weinan, E and Liu, JG},
   Title = {Projection method II: Godunov-Ryabenki analysis},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {33},
   Number = {4},
   Pages = {1597-1621},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {1996},
   Month = {January},
   url = {http://dx.doi.org/10.1137/s003614299426450x},
   Abstract = {This is the second of a series of papers on the subject of
             projection methods for viscous incompressible flow
             calculations. The purpose of the present paper is to explain
             why the accuracy of the velocity approximation is not
             affected by (1) the numerical boundary layers in the
             approximation of pressure and the intermediate velocity
             field and (2) the noncommutativity of the projection
             operator and the laplacian. This is done by using a
             Godunov-Ryabenki type of analysis in a rigorous fashion. By
             doing so, we hope to be able to convey the message that
             normal mode analysis is basically sufficient for
             understanding the stability and accuracy of a
             finite-difference method for the Navier-Stokes equation even
             in the presence of boundaries. As an example, we analyze the
             second-order projection method based on pressure increment
             formulations used by van Kan and Bell, Colella, and Glaz.
             The leading order error term in this case is of O(Δt) and
             behaves as high frequency oscillations over the whole
             domain, compared with the O(Δt1/2) numerical boundary
             layers found in the second-order Kim-Moin
             method.},
   Doi = {10.1137/s003614299426450x},
   Key = {fds246918}
}

@article{fds246919,
   Author = {Levermore, CD and Liu, JG},
   Title = {Large oscillations arising in a dispersive numerical
             scheme},
   Journal = {Physica D: Nonlinear Phenomena},
   Volume = {99},
   Number = {2-3},
   Pages = {191-216},
   Publisher = {Elsevier BV},
   Year = {1996},
   Month = {January},
   url = {http://dx.doi.org/10.1016/S0167-2789(96)00157-1},
   Abstract = {We study the oscillatory behavior that arises in solutions
             of a dispersive numerical scheme for the Hopf equation
             whenever the classical solution of that equation develops a
             singularity. Modulation equations are derived that describe
             period-two oscillations so long as the solution of those
             equations takes values for which the equations are
             hyperbolic. However, those equations have an elliptic region
             that may be entered by its solutions in a unite time, after
             which the corresponding period-two oscillations are seen to
             break down. This kind of phenomenon has not been observed
             for integrable schemes. The generation and propagation of
             period-two oscillations are asymptotically analyzed and a
             matching formula is found for the transition between
             oscillatory and nonoscillatory regions. Modulation equations
             are also presented for period-three oscillations. Numerical
             experiments are carried out that illustrate our analysis. ©
             1996 Elsevier Science B.V. All rights reserved.},
   Doi = {10.1016/S0167-2789(96)00157-1},
   Key = {fds246919}
}

@article{fds246920,
   Author = {Liu, JG and Xin, Z},
   Title = {Kinetic and viscous boundary layers for broadwell
             equations},
   Journal = {Transport Theory and Statistical Physics},
   Volume = {25},
   Number = {3-5},
   Pages = {447-461},
   Publisher = {Informa UK Limited},
   Year = {1996},
   Month = {January},
   url = {http://dx.doi.org/10.1080/00411459608220713},
   Abstract = {In this paper, we investigate the boundary layer behavior of
             solutions to the one dimensional Broadwell model of the
             nonlinear Boltzmann equation for small mean free path. We
             consider the analogue of Maxwell's diffusive and the
             reflexive boundary conditions. It is found that even for
             such a simple model, there are boundary layers due to purely
             kinetic effects which cannot be detected by the
             corresponding Navier-Stokes system. It is also found
             numerically that a compressive boundary layer is not always
             stable in the sense that it may detach from the boundary and
             move into the interior of the gas as a shock
             layer.},
   Doi = {10.1080/00411459608220713},
   Key = {fds246920}
}

@article{fds246914,
   Author = {Jin, S and Liu, JG},
   Title = {Oscillations induced by numerical viscosities},
   Journal = {Mat. Contemp.},
   Volume = {10},
   Pages = {169-180},
   Year = {1996},
   Key = {fds246914}
}

@article{fds246921,
   Author = {Liu, JG and Xin, Z},
   Title = {Boundary layer behavior in the fluid-dynamic limit for a
             nonlinear model Boltzmann equation},
   Journal = {Arch. Rat. Mech. Anal.},
   Volume = {135},
   Number = {1},
   Pages = {61-105},
   Publisher = {Springer Nature},
   Year = {1996},
   url = {http://dx.doi.org/10.1007/BF02198435},
   Abstract = {In this paper, we study the fluid-dynamic limit for the
             one-dimensional Broadwell model of the nonlinear Boltzmann
             equation in the presence of boundaries. We consider an
             analogue of Maxwell's diffusive and reflective boundary
             conditions. The boundary layers can be classified as either
             compressive or expansive in terms of the associated
             characteristic fields. We show that both expansive and
             compressive boundary layers (before detachment) are
             nonlinearly stable and that the layer effects are localized
             so that the fluid dynamic approximation is valid away from
             the boundary. We also show that the same conclusion holds
             for short time without the structural conditions on the
             boundary layers. A rigorous estimate for the distance
             between the kinetic solution and the fluid-dynamic solution
             in terms of the mean-free path in the L∞ -norm is obtained
             provided that the interior fluid flow is smooth. The rate of
             convergence is optimal.},
   Doi = {10.1007/BF02198435},
   Key = {fds246921}
}

@article{fds362427,
   Author = {E, W and Liu, J-G},
   Title = {Finite difference schemes for incompressible flows in
             vorticity formulations},
   Journal = {ESAIM: Proceedings},
   Volume = {1},
   Pages = {181-195},
   Publisher = {EDP Sciences},
   Editor = {Gagnon, Y and Cottet, G-H and G., D and F., A and Meiburg,
             E},
   Year = {1996},
   url = {http://dx.doi.org/10.1051/proc:1996009},
   Doi = {10.1051/proc:1996009},
   Key = {fds362427}
}

@article{fds246912,
   Author = {Weinan, E and Liu, JG},
   Title = {Projection method I: convergence and numerical boundary
             layers},
   Journal = {SIAM J. Numer. Anal.},
   Volume = {32},
   Number = {4},
   Pages = {1017-1057},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {1995},
   url = {http://dx.doi.org/10.1137/0732047},
   Doi = {10.1137/0732047},
   Key = {fds246912}
}

@article{fds246913,
   Author = {Liu, JG and Xin, Z},
   Title = {Convergence of vortex methods for weak solutions to the 2-D
             Euler equations with vortex sheets data},
   Journal = {Comm. Pure Appl. Math.},
   Volume = {48},
   Number = {6},
   Pages = {611-628},
   Year = {1995},
   url = {http://dx.doi.org/10.1002/cpa.3160480603},
   Abstract = {We prove the convergence of vortex blob methods to classical
             weak solutions for the two‐dimensional incompressible
             Euler equations with initial data satisfying the conditions
             that the vorticity is a finite Radon measure of
             distinguished sign and the kinetic energy is locally
             bounded. This includes the important example of vortex
             sheets. The result is valid as long as the computational
             grid size h does not exceed the smoothing blob size ε,
             i.e., h/ε ≦ C.. ©1995 John Wiley & Sons, Inc. Copyright
             © 1995 Wiley Periodicals, Inc., A Wiley
             Company},
   Doi = {10.1002/cpa.3160480603},
   Key = {fds246913}
}

@article{fds246911,
   Author = {Jin, S and Liu, JG},
   Title = {Relaxation and diffusion enhanced dispersive
             waves},
   Journal = {Proceedings of The Royal Society of London, Series A:
             Mathematical and Physical Sciences},
   Volume = {446},
   Number = {1928},
   Pages = {555-563},
   Year = {1994},
   Month = {January},
   url = {http://dx.doi.org/10.1098/rspa.1994.0120},
   Abstract = {The development of shocks in nonlinear hyperbolic
             conservation laws may be regularized through either
             diffusion or relaxation. However, we have observed
             surprisingly that for some physical problems, when both of
             the smoothing factors diffusion and relaxation coexist,
             under appropriate asymptotic assumptions, the dispersive
             waves are enhanced. This phenomenon is studied
             asymptotically in the sense of the Chapman-Enskog expansion
             and demonstrated numerically.},
   Doi = {10.1098/rspa.1994.0120},
   Key = {fds246911}
}

@article{fds246910,
   Author = {Lefloch, P and Liu, JG},
   Title = {Discrete entropy and monotonicity criteria for hyperbolic
             conservation laws},
   Journal = {C.R. Acad. Sci. Paris.},
   Volume = {319},
   Number = {8},
   Pages = {881-886},
   Publisher = {ELSEVIER FRANCE-EDITIONS SCIENTIFIQUES MEDICALES
             ELSEVIER},
   Year = {1994},
   Key = {fds246910}
}

@article{fds359206,
   Author = {Chen, G-Q and Liu, J-G},
   Title = {Convergence of Second-Order Schemes for Isentropic Gas
             Dynamics},
   Journal = {Mathematics of Computation},
   Volume = {61},
   Number = {204},
   Pages = {607-607},
   Publisher = {JSTOR},
   Year = {1993},
   Month = {October},
   url = {http://dx.doi.org/10.2307/2153243},
   Doi = {10.2307/2153243},
   Key = {fds359206}
}

@article{fds246909,
   Author = {Liu, JG and Xin, Z},
   Title = {Nonlinear stability of discrete shocks for systems of
             conservation laws},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {125},
   Number = {3},
   Pages = {217-256},
   Publisher = {Springer Nature},
   Year = {1993},
   Month = {September},
   ISSN = {0003-9527},
   url = {http://dx.doi.org/10.1007/BF00383220},
   Abstract = {In this paper we study the asymptotic nonlinear stability of
             discrete shocks for the Lax-Friedrichs scheme for
             approximating general m×m systems of nonlinear hyperbolic
             conservation laws. It is shown that weak single discrete
             shocks for such a scheme are nonlinearly stable in the
             Lp-norm for all p ≧ 1, provided that the sums of the
             initial perturbations equal zero. These results should shed
             light on the convergence of the numerical solution
             constructed by the Lax-Friedrichs scheme for the
             single-shock solution of system of hyperbolic conservation
             laws. If the Riemann solution corresponding to the given
             far-field states is a superposition of m single shocks from
             each characteristic family, we show that the corresponding
             multiple discrete shocks are nonlinearly stable in Lp (P ≧
             2). These results are proved by using both a weighted
             estimate and a characteristic energy method based on the
             internal structures of the discrete shocks and the essential
             monotonicity of the Lax-Friedrichs scheme. © 1993
             Springer-Verlag.},
   Doi = {10.1007/BF00383220},
   Key = {fds246909}
}

@article{fds348002,
   Author = {Liu, J-G and Xin, Z},
   Title = {L 1 -Stability of Stationary Discrete Shocks},
   Journal = {Mathematics of Computation},
   Volume = {60},
   Number = {201},
   Pages = {233-233},
   Publisher = {JSTOR},
   Year = {1993},
   Month = {January},
   url = {http://dx.doi.org/10.2307/2153163},
   Doi = {10.2307/2153163},
   Key = {fds348002}
}

@article{fds246906,
   Author = {Chen, GQ and Liu, JG},
   Title = {Convergence of second-order schemes for isentropic gas
             dynamics},
   Journal = {Math. Comp.},
   Volume = {61},
   Number = {204},
   Pages = {607-629},
   Publisher = {AMER MATHEMATICAL SOC},
   Year = {1993},
   url = {http://dx.doi.org/10.2307/2153243},
   Abstract = {Convergence of a second-order shock-capturing scheme for the
             system of isentropic gas dynamics with L initial data is
             established by analyzing the entropy dissipation measures.
             This scheme is modified from the classical MUSCL scheme to
             treat the vacuum problem in gas fluids and to capture local
             entropy near shock waves. Convergence of this scheme for the
             piston problem is also discussed. © 1993 American
             Mathematical Society. ∞},
   Doi = {10.2307/2153243},
   Key = {fds246906}
}

@article{fds246907,
   Author = {Engquist, B and Liu, JG},
   Title = {Numerical methods for oscillatory solutions to hyperbolic
             problems},
   Journal = {Comm. Pure Appl. Math.},
   Volume = {46},
   Number = {10},
   Pages = {1327-1361},
   Publisher = {WILEY},
   Year = {1993},
   url = {http://dx.doi.org/10.1002/cpa.3160461003},
   Abstract = {Difference approximations of hyperbolic partial differential
             equations with highly oscillatory coefficients and initial
             values are studied. Analysis of strong and weak convergence
             is carried out in the practically interesting case when the
             discretization step sizes are essentially independent of the
             oscillatory wave lengths. © 1993 John Wiley & Sons, Inc.
             Copyright © 1993 Wiley Periodicals, Inc., A Wiley
             Company},
   Doi = {10.1002/cpa.3160461003},
   Key = {fds246907}
}

@article{fds246908,
   Author = {Liu, JG and Xin, Z},
   Title = {L1-stability of stationary discrete shocks},
   Journal = {Math. Comp.},
   Volume = {60},
   Number = {201},
   Pages = {233-244},
   Publisher = {American Mathematical Society (AMS)},
   Year = {1993},
   url = {http://dx.doi.org/10.1090/S0025-5718-1993-1159170-7},
   Abstract = {The nonlinear stability in the Lpnorm, p 1 , of stationary
             weak discrete shocks for the Lax-Friedrichs scheme
             approximating general m x m systems of nonlinear hyperbolic
             conservation laws is proved, provided that the summations of
             the initial perturbations equal zero. The result is proved
             by using both a weighted estimate and characteristic energy
             method based on the internal structures of the discrete
             shocks and the essential monotonicity of the Lax-Friedrichs
             scheme. © 1993 American Mathematical Society.},
   Doi = {10.1090/S0025-5718-1993-1159170-7},
   Key = {fds246908}
}


%% Papers Accepted   
@article{fds320739,
   Author = {P. Degond and J.-G. Liu and S. Merino-Aceituno and T.
             Tardiveau},
   Title = {Continuum dynamics of the intention field under weakly
             cohesive social interactions},
   Journal = {Math. Models Methods Appl. Sci.},
   Year = {2016},
   Key = {fds320739}
}

@article{fds320743,
   Author = {Y. Gao and J.-G. Liu and J. Lu},
   Title = {Continuum limit of a mesoscopic model of step motion on
             vicinal surfaces},
   Journal = {J. Nonlinear Science},
   Year = {2016},
   Key = {fds320743}
}