%% @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"><p style='text-indent:20px;'>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.</p></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{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} } @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} } @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} } @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} } @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} } @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 > 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} }