%% Books
@book{fds165493,
Title = {Multi-scale phenomena in complex fluids, Modeling, Analysis
and Numerical Simulations},
Publisher = {World Scientific},
Editor = {T. Hou and C. Liu and J.-G. Liu},
Year = {2009},
ISBN = {978-981-4273-25-1},
Key = {fds165493}
}
@book{fds165494,
Title = {Hyperbolic Problems: Theory, Numerics and Applications,
volume I: Plenary & Invited Talks; volume II: Contributed
Talks},
Volume = {67},
Series = {Proceedings of Symposia in Applied Mathematics},
Publisher = {American Mathematical Society},
Editor = {E. Tadmor and J.-G. Liu and A.E. Tzavaras},
Year = {2009},
ISBN = {978-0-8218-4728-2},
Key = {fds165494}
}
@book{fds70657,
Title = {Dynamics in Models of Coarsening, Coagulation, Condensation
and Quantization},
Publisher = {World Scientific},
Editor = {W. Bao and J.-G. Liu},
Year = {2007},
ISBN = {9789812708502},
Key = {fds70657}
}
%% Papers Published
@article{fds376894,
Author = {Cherepanov, V and Liu, JG and Qian, Z},
Title = {On the Dynamics of the Boundary Vorticity for Incompressible
Viscous Flows},
Journal = {Journal of Scientific Computing},
Volume = {99},
Number = {2},
Year = {2024},
Month = {May},
url = {http://dx.doi.org/10.1007/s10915-024-02498-1},
Abstract = {The dynamical equation of the boundary vorticity has been
obtained, which shows that the viscosity at a solid wall is
doubled as if the fluid became more viscous at the boundary.
For certain viscous flows the boundary vorticity can be
determined via the dynamical equation up to bounded errors
for all time, without the need of knowing the details of the
main stream flows. We then validate the dynamical equation
by carrying out stochastic direct numerical simulations
(i.e. the random vortex method for wall-bounded
incompressible viscous flows) by two different means of
updating the boundary vorticity, one using mollifiers of the
Biot–Savart singular integral kernel, another using the
dynamical equations.},
Doi = {10.1007/s10915-024-02498-1},
Key = {fds376894}
}
@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{fds376895,
Author = {Liu, J-G and Pego, RL},
Title = {A Simple Construction of Fat Cantor Sets},
Journal = {The American Mathematical Monthly},
Pages = {1-1},
Publisher = {Informa UK Limited},
Year = {2024},
Month = {March},
url = {http://dx.doi.org/10.1080/00029890.2024.2322909},
Doi = {10.1080/00029890.2024.2322909},
Key = {fds376895}
}
@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{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},
Volume = {29},
Number = {7},
Pages = {2837-2879},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2024},
url = {http://dx.doi.org/10.3934/dcdsb.2023204},
Doi = {10.3934/dcdsb.2023204},
Key = {fds374861}
}
@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{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{fds362424,
Author = {Duan, Y and Liu, J-G},
Title = {Error estimate of the particle method for the
$b$-equation},
Journal = {Methods and Applications of Analysis},
Volume = {23},
Number = {2},
Pages = {119-154},
Publisher = {International Press of Boston},
Year = {2016},
url = {http://dx.doi.org/10.4310/maa.2016.v23.n2.a1},
Doi = {10.4310/maa.2016.v23.n2.a1},
Key = {fds362424}
}
@article{fds362425,
Author = {Liu, J-G and Zhang, Y},
Title = {Convergence of stochastic interacting particle systems in
probability under a Sobolev norm},
Journal = {Annals of Mathematical Sciences and Applications},
Volume = {1},
Number = {2},
Pages = {251-299},
Publisher = {International Press of Boston},
Year = {2016},
url = {http://dx.doi.org/10.4310/amsa.2016.v1.n2.a1},
Doi = {10.4310/amsa.2016.v1.n2.a1},
Key = {fds362425}
}
@article{fds341422,
Author = {Degond, P and Frouvelle, A and Liu, JG},
Title = {Phase Transitions, Hysteresis, and Hyperbolicity for
Self-Organized Alignment Dynamics},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {216},
Number = {1},
Pages = {63-115},
Year = {2015},
Month = {April},
url = {http://dx.doi.org/10.1007/s00205-014-0800-7},
Abstract = {We provide a complete and rigorous description of phase
transitions for kinetic models of self-propelled particles
interacting through alignment. These models exhibit a
competition between alignment and noise. Both the alignment
frequency and noise intensity depend on a measure of the
local alignment. We show that, in the spatially homogeneous
case, the phase transition features (number and nature of
equilibria, stability, convergence rate, phase diagram,
hysteresis) are totally encoded in how the ratio between the
alignment and noise intensities depend on the local
alignment. In the spatially inhomogeneous case, we derive
the macroscopic models associated to the stable equilibria
and classify their hyperbolicity according to the same
function.},
Doi = {10.1007/s00205-014-0800-7},
Key = {fds341422}
}
@article{fds246842,
Author = {Xue, Y and Wang, C and Liu, JG},
Title = {Simple Finite Element Numerical Simulation of Incompressible
Flow Over Non-rectangular Domains and the Super-Convergence
Analysis},
Journal = {Journal of Scientific Computing},
Volume = {65},
Number = {3},
Pages = {1189-1216},
Publisher = {Springer Nature},
Year = {2015},
Month = {March},
ISSN = {0885-7474},
url = {http://dx.doi.org/10.1007/s10915-015-0005-8},
Abstract = {In this paper, we apply a simple finite element numerical
scheme, proposed in an earlier work (Liu in Math Comput
70(234):579–593, 2000), to perform a high resolution
numerical simulation of incompressible flow over an
irregular domain and analyze its boundary layer separation.
Compared with many classical finite element fluid solvers,
this numerical method avoids a Stokes solver, and only two
Poisson-like equations need to be solved at each time
step/stage. In addition, its combination with the fully
explicit fourth order Runge–Kutta (RK4) time
discretization enables us to compute high Reynolds number
flow in a very efficient way. As an application of this
robust numerical solver, the dynamical mechanism of the
boundary layer separation for a triangular cavity flow with
Reynolds numbers $$Re=10^4$$Re=104 and $$Re=10^5$$Re=105,
including the precise values of bifurcation location and
critical time, are reported in this paper. In addition, we
provide a super-convergence analysis for the simple finite
element numerical scheme, using linear elements over a
uniform triangulation with right triangles.},
Doi = {10.1007/s10915-015-0005-8},
Key = {fds246842}
}
@article{fds246843,
Author = {Lu, J and Liu, JG and Margetis, D},
Title = {Emergence of step flow from an atomistic scheme of epitaxial
growth in 1+1 dimensions},
Journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter
Physics},
Volume = {91},
Number = {3},
Pages = {032403},
Year = {2015},
Month = {March},
ISSN = {1539-3755},
url = {http://dx.doi.org/10.1103/PhysRevE.91.032403},
Abstract = {The Burton-Cabrera-Frank (BCF) model for the flow of line
defects (steps) on crystal surfaces has offered useful
insights into nanostructure evolution. This model has rested
on phenomenological grounds. Our goal is to show via scaling
arguments the emergence of the BCF theory for noninteracting
steps from a stochastic atomistic scheme of a kinetic
restricted solid-on-solid model in one spatial dimension.
Our main assumptions are: adsorbed atoms (adatoms) form a
dilute system, and elastic effects of the crystal lattice
are absent. The step edge is treated as a front that
propagates via probabilistic rules for atom attachment and
detachment at the step. We formally derive a quasistatic
step flow description by averaging out the stochastic scheme
when terrace diffusion, adatom desorption, and deposition
from above are present.},
Doi = {10.1103/PhysRevE.91.032403},
Key = {fds246843}
}
@article{fds300222,
Author = {Chertock, A and Liu, JG and Pendleton, T},
Title = {Elastic collisions among peakon solutions for the
Camassa-Holm equation},
Journal = {Applied Numerical Mathematics},
Volume = {93},
Pages = {30-46},
Publisher = {Elsevier BV},
Year = {2015},
Month = {January},
ISSN = {0168-9274},
url = {http://dx.doi.org/10.1016/j.apnum.2014.01.001},
Abstract = {The purpose of this paper is to study the dynamics of the
interaction among a special class of solutions of the
one-dimensional Camassa-Holm equation. The equation yields
soliton solutions whose identity is preserved through
nonlinear interactions. These solutions are characterized by
a discontinuity at the peak in the wave shape and are thus
called peakon solutions. We apply a particle method to the
Camassa-Holm equation and show that the nonlinear
interaction among the peakon solutions resembles an elastic
collision, i.e., the total energy and momentum of the system
before the peakon interaction is equal to the total energy
and momentum of the system after the collision. From this
result, we provide several numerical illustrations which
support the analytical study, as well as showcase the merits
of using a particle method to simulate solutions to the
Camassa-Holm equation under a wide class of initial
data.},
Doi = {10.1016/j.apnum.2014.01.001},
Key = {fds300222}
}
@article{fds313338,
Author = {Herschlag, G and Liu, JG and Layton, AT},
Title = {An exact solution for stokes flow in a channel with
arbitrarily large wall permeability},
Journal = {SIAM Journal on Applied Mathematics},
Volume = {75},
Number = {5},
Pages = {2246-2267},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2015},
Month = {January},
ISSN = {0036-1399},
url = {http://dx.doi.org/10.1137/140995854},
Abstract = {We derive an exact solution for Stokes flow in a channel
with permeable walls. At the channel walls, the normal
component of the fluid velocity is described by Darcy's law,
and the tangential component of the fluid velocity is
described by the no slip condition. The pressure exterior to
the channel is assumed to be constant. Although this problem
has been well studied, typical studies assume that the
permeability of the wall is small relative to other
nondimensional parameters; this work relaxes this assumption
and explores a regime in parameter space that has not yet
been well studied. A consequence of this relaxation is that
transverse velocity is no longer necessarily small when
compared with the axial velocity. We use our result to
explore how existing asymptotic theories break down in the
limit of large permeability for channels of small
length.},
Doi = {10.1137/140995854},
Key = {fds313338}
}
@article{fds365498,
Author = {Degond, P and Frouvelle, A and Liu, J-G and Motsch, S and Navoret,
L},
Title = {Macroscopic models of collective motion and
self-organization},
Journal = {Séminaire Laurent Schwartz — EDP et applications},
Volume = {2012 - 2013},
Pages = {1-27},
Publisher = {Cellule MathDoc/CEDRAM},
Year = {2014},
Month = {November},
url = {http://dx.doi.org/10.5802/slsedp.32},
Doi = {10.5802/slsedp.32},
Key = {fds365498}
}
@article{fds246846,
Author = {Degond, P and Liu, J-G and Ringhofer, C},
Title = {Evolution of wealth in a non-conservative economy driven by
local Nash equilibria.},
Journal = {Philosophical transactions. Series A, Mathematical,
physical, and engineering sciences},
Volume = {372},
Number = {2028},
Pages = {20130394},
Publisher = {The Royal Society},
Year = {2014},
Month = {November},
ISSN = {1364-503X},
url = {http://dx.doi.org/10.1098/rsta.2013.0394},
Abstract = {We develop a model for the evolution of wealth in a
non-conservative economic environment, extending a theory
developed in Degond et al. (2014 J. Stat. Phys. 154, 751-780
(doi:10.1007/s10955-013-0888-4)). The model considers a
system of rational agents interacting in a game-theoretical
framework. This evolution drives the dynamics of the agents
in both wealth and economic configuration variables. The
cost function is chosen to represent a risk-averse strategy
of each agent. That is, the agent is more likely to interact
with the market, the more predictable the market, and
therefore the smaller its individual risk. This yields a
kinetic equation for an effective single particle agent
density with a Nash equilibrium serving as the local
thermodynamic equilibrium. We consider a regime of scale
separation where the large-scale dynamics is given by a
hydrodynamic closure with this local equilibrium. A class of
generalized collision invariants is developed to overcome
the difficulty of the non-conservative property in the
hydrodynamic closure derivation of the large-scale dynamics
for the evolution of wealth distribution. The result is a
system of gas dynamics-type equations for the density and
average wealth of the agents on large scales. We recover the
inverse Gamma distribution, which has been previously
considered in the literature, as a local equilibrium for
particular choices of the cost function.},
Doi = {10.1098/rsta.2013.0394},
Key = {fds246846}
}
@article{fds246848,
Author = {Coquel, F and Jin, S and Liu, JG and Wang, L},
Title = {Well-Posedness and Singular Limit of a Semilinear Hyperbolic
Relaxation System with a Two-Scale Discontinuous Relaxation
Rate},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {214},
Number = {3},
Pages = {1051-1084},
Year = {2014},
Month = {October},
ISSN = {0003-9527},
url = {http://dx.doi.org/10.1007/s00205-014-0773-6},
Abstract = {Nonlinear hyperbolic systems with relaxations may encounter
different scales of relaxation time, which is a prototype
multiscale phenomenon that arises in many applications. In
such a problem the relaxation time is of O(1) in part of the
domain and very small in the remaining domain in which the
solution can be approximated by the zero relaxation limit
which can be solved numerically much more efficiently. For
the Jin–Xin relaxation system in such a two-scale setting,
we establish its wellposedness and singular limit as the
(smaller) relaxation time goes to zero. The limit is a
multiscale coupling problem which couples the original
Jin–Xin system on the domain when the relaxation time is
O(1) with its relaxation limit in the other domain through
interface conditions which can be derived by matched
interface layer analysis.As a result, we also establish the
well-posedness and regularity (such as boundedness in sup
norm with bounded total variation and L1-contraction) of the
coupling problem, thus providing a rigorous mathematical
foundation, in the general nonlinear setting, to the
multiscale domain decomposition method for this two-scale
problem originally proposed in Jin et al. in Math. Comp. 82,
749–779, 2013.},
Doi = {10.1007/s00205-014-0773-6},
Key = {fds246848}
}
@article{fds246857,
Author = {Johnston, H and Wang, C and Liu, JG},
Title = {A Local Pressure Boundary Condition Spectral Collocation
Scheme for the Three-Dimensional Navier–Stokes
Equations},
Journal = {Journal of Scientific Computing},
Volume = {60},
Number = {3},
Pages = {612-626},
Publisher = {Springer Nature},
Year = {2014},
Month = {September},
ISSN = {0885-7474},
url = {http://dx.doi.org/10.1007/s10915-013-9808-7},
Abstract = {A spectral collocation scheme for the three-dimensional
incompressible (u,p) formulation of the Navier–Stokes
equations, in domains Ω with a non-periodic boundary
condition, is described. The key feature is the high order
approximation, by means of a local Hermite interpolant, of a
Neumann boundary condition for use in the numerical solution
of the pressure Poisson system. The time updates of the
velocity u and pressure p are decoupled as a result of
treating the pressure gradient in the momentum equation
explicitly in time. The pressure update is computed from a
pressure Poisson equation. Extension of the overall
methodology to the Boussinesq system is also described. The
uncoupling of the pressure and velocity time updates results
in a highly efficient scheme that is simple to implement and
well suited for simulating moderate to high Reynolds and
Rayleigh number flows. Accuracy checks are presented, along
with simulations of the lid-driven cavity flow and a
differentially heated cavity flow, to demonstrate the scheme
produces accurate three-dimensional results at a reasonable
computational cost.},
Doi = {10.1007/s10915-013-9808-7},
Key = {fds246857}
}
@article{fds246862,
Author = {Duan, Y and Liu, JG},
Title = {Convergence analysis of the vortex blob method for the
b-equation},
Journal = {Discrete and Continuous Dynamical Systems- Series
A},
Volume = {34},
Number = {5},
Pages = {1995-2011},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2014},
Month = {May},
ISSN = {1078-0947},
url = {http://dx.doi.org/10.3934/dcds.2014.34.1995},
Abstract = {In this paper, we prove the convergence of the vortex blob
method for a family of nonlinear evolutionary partial
differential equations (PDEs), the so-called b-equation.
This kind of PDEs, including the Camassa-Holm equation and
the Degasperis-Procesi equation, has many applications in
diverse scientific fields. Our convergence analysis also
provides a proof for the existence of the global weak
solution to the b-equation when the initial data is a
nonnegative Radon measure with compact support.},
Doi = {10.3934/dcds.2014.34.1995},
Key = {fds246862}
}
@article{fds246858,
Author = {Bian, S and Liu, JG and Zou, C},
Title = {Ultra-contractivity for keller-segel model with diffusion
exponent m > 1-2/d},
Journal = {Kinetic and Related Models},
Volume = {7},
Number = {1},
Pages = {9-28},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2014},
Month = {March},
ISSN = {1937-5093},
url = {http://dx.doi.org/10.3934/krm.2014.7.9},
Abstract = {This paper establishes the hyper-contractivity in L∞(ℝd)
(it's known as ultra-contractivity) for the
multi-dimensional Keller-Segel systems with the diffusion
exponent m > 1-2/d. The results show that for the super-
critical and critical case 1-2/d < m ≤ 2-2/d, if
∥U0∥d(2-m)/2 < Cd, m where Cd, m is a universal
constant, then for any t > 0 ∥u(.,t)∥L∞(ℝd) is
bounded and decays as t goes to infinity. For the
subcritical case m > 2-2/d, the solution u(.,t)∈
L∞(ℝd) with any initial data U0 ∈ L1+(ℝd) for any
positive time.},
Doi = {10.3934/krm.2014.7.9},
Key = {fds246858}
}
@article{fds246849,
Author = {Degond, P and Herty, M and Liu, JG},
Title = {Flow on sweeping networks},
Journal = {Multiscale Modeling and Simulation},
Volume = {12},
Number = {2},
Pages = {538-565},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2014},
Month = {January},
ISSN = {1540-3459},
url = {http://dx.doi.org/10.1137/130927061},
Abstract = {We introduce a cellular automaton model coupled with a
transport equation for flows on graphs. The direction of the
flow is described by a switching process where the switching
probability dynamically changes according to the value of
the transported quantity in the neighboring cells. A
motivation is pedestrian dynamics during panic situations in
a small corridor where the propagation of people in a part
of the corridor can be either left- or right-going. Under
the assumptions of propagation of chaos and mean-field
limit, we derive a master equation and the corresponding
mean-field kinetic and macroscopic models. Steady-states are
computed and analyzed and exhibit the possibility of
multiple metastable states and hysteresis. © 2014 Society
for Industrial and Applied Mathematics.},
Doi = {10.1137/130927061},
Key = {fds246849}
}
@article{fds246851,
Author = {Chen, X and Li, X and Liu, JG},
Title = {Existence and uniqueness of global weak solution to a
kinetic model for the sedimentation of rod-like
particles},
Journal = {Communications in Mathematical Sciences},
Volume = {12},
Number = {8},
Pages = {1579-1601},
Publisher = {International Press of Boston},
Year = {2014},
Month = {January},
ISSN = {1539-6746},
url = {http://dx.doi.org/10.4310/CMS.2014.v12.n8.a10},
Abstract = {We investigate a kinetic model for the sedimentation of
dilute suspensions of rod-like particles under gravity,
deduced by Helzel, Otto, and Tzavaras (2011), which couples
the impressible (Navier-)Stokes equation with the
Fokker-Planck equation. With a no-flux boundary condition
for the distribution function, we establish the existence
and uniqueness of a global weak solution to the two
dimensional model involving the Stokes equation. ©
2014.},
Doi = {10.4310/CMS.2014.v12.n8.a10},
Key = {fds246851}
}
@article{fds333571,
Author = {Degond, P and Frouvelle, A and Liu, J-G},
Title = {A NOTE ON PHASE TRANSITIONS FOR THE SMOLUCHOWSKI EQUATION
WITH DIPOLAR POTENTIAL},
Journal = {HYPERBOLIC PROBLEMS: THEORY, NUMERICS, APPLICATIONS},
Volume = {8},
Pages = {179-192},
Booktitle = {Proceedings of the Fourteenth International Conference on
Hyperbolic Problems: Theory, Numerics and
Application},
Publisher = {AMER INST MATHEMATICAL SCIENCES-AIMS},
Editor = {Ancona, F and Bressan, A and Marcati, P and Marson,
A},
Year = {2014},
Month = {January},
Key = {fds333571}
}
@article{fds337236,
Author = {Chae, D and Degond, P and Liu, JG},
Title = {Well-posedness for hall-magnetohydrodynamics},
Journal = {Annales de l'Institut Henri Poincare (C) Analyse Non
Lineaire},
Volume = {31},
Number = {3},
Pages = {555-565},
Year = {2014},
Month = {January},
url = {http://dx.doi.org/10.1016/j.anihpc.2013.04.006},
Abstract = {We prove local existence of smooth solutions for large data
and global smooth solutions for small data to the
incompressible, resistive, viscous or inviscid Hall-MHD
model. We also show a Liouville theorem for the stationary
solutions. © 2013 Elsevier Masson SAS. All rights
reserved.},
Doi = {10.1016/j.anihpc.2013.04.006},
Key = {fds337236}
}
@article{fds246856,
Author = {Goudon, T and Jin, S and Liu, J-G and Yan, B},
Title = {Asymptotic-preserving schemes for kinetic-fluid modeling of
disperse two-phase flows with variable fluid
density},
Journal = {International Journal for Numerical Methods in
Fluids},
Volume = {75},
Number = {2},
Pages = {81-102},
Publisher = {WILEY},
Year = {2014},
ISSN = {0271-2091},
url = {http://dx.doi.org/10.1002/fld.3885},
Abstract = {We are concerned with a coupled system describing the
interaction between suspended particles and a dense fluid.
The particles are modeled by a kinetic equation of
Vlasov-Fokker-Planck type, and the fluid is described by the
incompressible Navier-Stokes system, with variable density.
The systems are coupled through drag forces. High friction
regimes lead to a purely hydrodynamic description of the
mixture. We design first and second order
asymptotic-preserving schemes suited to such regimes. We
extend the method introduced in [Goudon T, Jin S, Liu JG,
Yan B. Journal of Computational Physics 2013; 246:145-164]
to the case of variable density in compressible flow. We
check the accuracy and the asymptotic-preserving property
numerically. We set up a few numerical experiments to
demonstrate the ability of the scheme in capturing intricate
interactions between the two phases on a wide range of
physical parameters and geometric situations. © 2014 John
Wiley & Sons, Ltd.},
Doi = {10.1002/fld.3885},
Key = {fds246856}
}
@article{fds246866,
Author = {Bian, S and Liu, JG},
Title = {Dynamic and Steady States for Multi-Dimensional Keller-Segel
Model with Diffusion Exponent m > 0},
Journal = {Communications in Mathematical Physics},
Volume = {323},
Number = {3},
Pages = {1017-1070},
Publisher = {Springer Nature},
Year = {2013},
Month = {November},
ISSN = {0010-3616},
url = {http://dx.doi.org/10.1007/s00220-013-1777-z},
Abstract = {This paper investigates infinite-time spreading and
finite-time blow-up for the Keller-Segel system. For 0 < m
≤ 2 - 2/d, the L p space for both dynamic and steady
solutions are detected with (Formula presented.). Firstly,
the global existence of the weak solution is proved for
small initial data in L p. Moreover, when m > 1 - 2/d, the
weak solution preserves mass and satisfies the
hyper-contractive estimates in L q for any p < q < ∞.
Furthermore, for slow diffusion 1 < m ≤ 2 - 2/d, this weak
solution is also a weak entropy solution which blows up at
finite time provided by the initial negative free energy.
For m > 2 - 2/d, the hyper-contractive estimates are also
obtained. Finally, we focus on the L p norm of the steady
solutions, it is shown that the energy critical exponent m =
2d/(d + 2) is the critical exponent separating finite L p
norm and infinite L p norm for the steady state solutions.
© 2013 Springer-Verlag Berlin Heidelberg.},
Doi = {10.1007/s00220-013-1777-z},
Key = {fds246866}
}
@article{fds246864,
Author = {Chen, X and Liu, JG},
Title = {Analysis of polymeric flow models and related compactness
theorems in weighted spaces},
Journal = {SIAM Journal on Mathematical Analysis},
Volume = {45},
Number = {3},
Pages = {1179-1215},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2013},
Month = {October},
ISSN = {0036-1410},
url = {http://dx.doi.org/10.1137/120887850},
Abstract = {We studied coupled systems of the Fokker-Planck equation and
the Navier-Stokes equation modeling the Hookean and the
finitely extensible nonlinear elastic (FENE)-type polymeric
flows. We proved the continuous embedding and compact
embedding theorems in weighted spaces that naturally arise
from related entropy estimates. These embedding estimates
are shown to be sharp. For the Hookean polymeric system with
a center-of-mass diffusion and a superlinear spring
potential, we proved the existence of a global weak
solution. Moreover, we were able to tackle the FENE model
with L2 initial data for the polymer density instead of the
L∞ counterpart in the literature. © 2013 Society for
Industrial and Applied Mathematics.},
Doi = {10.1137/120887850},
Key = {fds246864}
}
@article{fds246869,
Author = {Goudon, T and Jin, S and Liu, JG and Yan, B},
Title = {Asymptotic-preserving schemes for kinetic-fluid modeling of
disperse two-phase flows},
Journal = {Journal of Computational Physics},
Volume = {246},
Pages = {145-164},
Publisher = {Elsevier BV},
Year = {2013},
Month = {August},
ISSN = {0021-9991},
url = {http://dx.doi.org/10.1016/j.jcp.2013.03.038},
Abstract = {We consider a system coupling the incompressible
Navier-Stokes equations to the Vlasov-Fokker-Planck
equation. Such a problem arises in the description of
particulate flows. We design a numerical scheme to simulate
the behavior of the system. This scheme is
asymptotic-preserving, thus efficient in both the kinetic
and hydrodynamic regimes. It has a numerical stability
condition controlled by the non-stiff convection operator,
with an implicit treatment of the stiff drag term and the
Fokker-Planck operator. Yet, consistent to a standard
asymptotic-preserving Fokker-Planck solver or an
incompressible Navier-Stokes solver, only the
conjugate-gradient method and fast Poisson and Helmholtz
solvers are needed. Numerical experiments are presented to
demonstrate the accuracy and asymptotic behavior of the
scheme, with several interesting applications. © 2013
Elsevier Inc.},
Doi = {10.1016/j.jcp.2013.03.038},
Key = {fds246869}
}
@article{fds246870,
Author = {Chen, X and Liu, JG},
Title = {Global weak entropy solution to Doi-Saintillan-Shelley model
for active and passive rod-like and ellipsoidal particle
suspensions},
Journal = {Journal of Differential Equations},
Volume = {254},
Number = {7},
Pages = {2764-2802},
Publisher = {Elsevier BV},
Year = {2013},
Month = {April},
ISSN = {0022-0396},
url = {http://dx.doi.org/10.1016/j.jde.2013.01.005},
Abstract = {We prove the existence of the global weak entropy solution
to the Doi-Saintillan-Shelley model for active and passive
rod-like particle suspensions, which couples a Fokker-Planck
equation with the incompressible Navier-Stokes or Stokes
equation, under the no-flux boundary conditions,
L2(Ω;L1(Sd-1)) initial data, and finite initial entropy for
the particle distribution function in two and three
dimensions. Furthermore, for the model with the Stokes
equation, we obtain the global L2(Ω×Sd-1) weak solution in
two and three dimensions and the uniqueness in two
dimension. © 2013 Elsevier Inc..},
Doi = {10.1016/j.jde.2013.01.005},
Key = {fds246870}
}
@article{fds246861,
Author = {Huang, YL and Liu, JG and Wang, WC},
Title = {A generalized mac scheme on curvilinear domains},
Journal = {SIAM Journal on Scientific Computing},
Volume = {35},
Number = {5},
Pages = {B953-B986},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2013},
Month = {January},
ISSN = {1064-8275},
url = {http://dx.doi.org/10.1137/120875508},
Abstract = {We propose a simple finite difference scheme for
Navier-Stokes equations in primitive formulation on
curvilinear domains. With proper boundary treatment and
interplay between covariant and contravariant components,
the spatial discretization admits exact Hodge decomposition
and energy identity. As a result, the pressure can be
decoupled from the momentum equation with explicit time
stepping. No artificial pressure boundary condition is
needed. In addition, it can be shown that this spatially
compatible discretization leads to uniform inf-sup
condition, which plays a crucial role in the pressure
approximation of both dynamic and steady state calculations.
Numerical experiments demonstrate the robustness and
efficiency of our scheme. Copyright © by SIAM. Unauthorized
reproduction of this article is prohibited.},
Doi = {10.1137/120875508},
Key = {fds246861}
}
@article{fds246859,
Author = {Degond, P and Liu, J-G and Ringhofer, C},
Title = {Evolution of the Distribution of Wealth in an Economic
Environment Driven by Local Nash Equilibria},
Journal = {Journal of Statistical Physics},
Volume = {154},
Number = {3},
Pages = {1-30},
Publisher = {Springer Nature},
Year = {2013},
ISSN = {0022-4715},
url = {http://dx.doi.org/10.1007/s10955-013-0888-4},
Abstract = {We present and analyze a model for the evolution of the
wealth distribution within a heterogeneous economic
environment. The model considers a system of rational agents
interacting in a game theoretical framework, through fairly
general assumptions on the cost function. This evolution
drives the dynamic of the agents in both wealth and economic
configuration variables. We consider a regime of scale
separation where the large scale dynamics is given by a
hydrodynamic closure with a Nash equilibrium serving as the
local thermodynamic equilibrium. The result is a system of
gas dynamics-type equations for the density and average
wealth of the agents on large scales. We recover the inverse
gamma distribution as an equilibrium in the particular case
of quadratic cost functions which has been previously
considered in the literature. © 2013 Springer
Science+Business Media New York.},
Doi = {10.1007/s10955-013-0888-4},
Key = {fds246859}
}
@article{fds246860,
Author = {Chen, X and Jüngel, A and Liu, J-G},
Title = {A Note on Aubin-Lions-Dubinskiǐ Lemmas},
Journal = {Acta Applicandae Mathematicae},
Volume = {133},
Number = {1},
Pages = {1-11},
Year = {2013},
ISSN = {0167-8019},
url = {http://dx.doi.org/10.1007/s10440-013-9858-8},
Abstract = {Strong compactness results for families of functions in
seminormed nonnegative cones in the spirit of the
Aubin-Lions-Dubinskiǐ lemma are proven, refining some
recent results in the literature. The first theorem sharpens
slightly a result of Dubinskiǐ (in Mat. Sb.
67(109):609-642, 1965) for seminormed cones. The second
theorem applies to piecewise constant functions in time and
sharpens slightly the results of Dreher and Jüngel (in
Nonlinear Anal. 75:3072-3077, 2012) and Chen and Liu (in
Appl. Math. Lett. 25:2252-2257, 2012). An application is
given, which is useful in the study of porous-medium or
fast-diffusion type equations. © 2013 Springer
Science+Business Media.},
Doi = {10.1007/s10440-013-9858-8},
Key = {fds246860}
}
@article{fds246863,
Author = {Degond, P and Liu, J-G and Ringhofer, C},
Title = {Large-Scale Dynamics of Mean-Field Games Driven by Local
Nash Equilibria},
Journal = {Journal of Nonlinear Science},
Volume = {24},
Number = {1},
Pages = {1-23},
Year = {2013},
ISSN = {0938-8974},
url = {http://dx.doi.org/10.1007/s00332-013-9185-2},
Abstract = {We introduce a new mean field kinetic model for systems of
rational agents interacting in a game-theoretical framework.
This model is inspired from noncooperative anonymous games
with a continuum of players and Mean-Field Games. The large
time behavior of the system is given by a macroscopic
closure with a Nash equilibrium serving as the local
thermodynamic equilibrium. An application of the presented
theory to a social model (herding behavior) is discussed. ©
Springer Science+Business Media New York
2013.},
Doi = {10.1007/s00332-013-9185-2},
Key = {fds246863}
}
@article{fds246867,
Author = {Chae, D and Degond, P and Liu, J-G},
Title = {Well-posedness for Hall-magnetohydrodynamics},
Journal = {Annales de l'Institut Henri Poincare. Annales: Analyse Non
Lineaire/Nonlinear Analysis},
Volume = {31},
Number = {3},
Pages = {555-565},
Publisher = {Elsevier BV},
Year = {2013},
ISSN = {0294-1449},
url = {http://dx.doi.org/10.1016/j.anihpc.2013.04.006},
Abstract = {We prove local existence of smooth solutions for large data
and global smooth solutions for small data to the
incompressible, resistive, viscous or inviscid Hall-MHD
model. We also show a Liouville theorem for the stationary
solutions. © 2013 Elsevier Masson SAS. All rights
reserved.},
Doi = {10.1016/j.anihpc.2013.04.006},
Key = {fds246867}
}
@article{fds246896,
Author = {Jin, S and Liu, JG and Wang, L},
Title = {A domain decomposition method for semilinear hyperbolic
systems with two-scale relaxations},
Journal = {Math. Comp.},
Volume = {82},
Number = {282},
Pages = {749-779},
Publisher = {American Mathematical Society (AMS)},
Year = {2013},
url = {http://dx.doi.org/10.1090/S0025-5718-2012-02643-3},
Abstract = {We present a domain decomposition method on a semilinear
hyperbolic system with multiple relaxation times. In the
region where the relaxation time is small, an asymptotic
equilibrium equation can be used for computational
efficiency. An interface condition based on the sign of the
characteristic speed at the interface is provided to couple
the two systems in a domain decomposition setting. A
rigorous analysis, based on the Laplace Transform, on the L2
error estimate is presented for the linear case, which shows
how the error of the domain decomposition method depends on
the smaller relaxation time, and the boundary and interface
layer effects. The given convergence rate is optimal. We
present a numerical implementation of this domain
decomposition method, and give some numerical results in
order to study the performance of this method. © 2012
American Mathematical Society.},
Doi = {10.1090/S0025-5718-2012-02643-3},
Key = {fds246896}
}
@article{fds362426,
Author = {Degond, P and Liu, J-G and Motsch, S and Panferov,
V},
Title = {Hydrodynamic models of self-organized dynamics: Derivation
and existence theory},
Journal = {Methods and Applications of Analysis},
Volume = {20},
Number = {2},
Pages = {89-114},
Publisher = {International Press of Boston},
Year = {2013},
url = {http://dx.doi.org/10.4310/maa.2013.v20.n2.a1},
Doi = {10.4310/maa.2013.v20.n2.a1},
Key = {fds362426}
}
@article{fds220112,
Author = {A. Chertock and J.-G. Liu and T. Pendleton},
Title = {Convergence analysis of the particle method for the
Camassa-Holm equation},
Pages = {365-373},
Booktitle = {Proceedings of the 13th International Conference on
``Hyperbolic Problems: Theory, Numerics and
Applications"},
Publisher = {Higher Education Press},
Address = {Beijing},
Year = {2012},
Key = {fds220112}
}
@article{fds246887,
Author = {Chae, D and Liu, JG},
Title = {Blow-up, zero alpha limit and the Liouville type theorem for
the Euler-Poincare equations},
Journal = {Comm. Math. Phy.,},
Volume = {314},
Number = {3},
Pages = {671-687},
Publisher = {Springer Nature},
Year = {2012},
ISSN = {0010-3616},
url = {http://dx.doi.org/10.1007/s00220-012-1534-8},
Abstract = {In this paper we study the Euler-Poincaré equations in ℝ
N. We prove local existence of weak solutions in W 2,p(ℝ
N),p>N, and local existence of unique classical solutions in
H k(ℝ N),k> N/2+3, as well as a blow-up criterion. For the
zero dispersion equation (α = 0) we prove a finite time
blow-up of the classical solution. We also prove that as the
dispersion parameter vanishes, the weak solution converges
to a solution of the zero dispersion equation with sharp
rate as α → 0, provided that the limiting solution
belongs to C([0,T); H k(ℝ N)) with k > N/2 + 3. For the
stationary weak solutions of the Euler-Poincaré equations
we prove a Liouville type theorem. Namely, for α > 0 any
weak solution u ∈ H 1(ℝ N) is u=0; for α= 0 any weak
solution u ∈ L 2(ℝ N) is u=0. © 2012
Springer-Verlag.},
Doi = {10.1007/s00220-012-1534-8},
Key = {fds246887}
}
@article{fds246888,
Author = {Chen, X and Liu, JG},
Title = {Two Nonlinear Compactness Theorems in L^p(0,T;B)},
Journal = {Appl. Math. Lett.},
Volume = {25},
Number = {12},
Pages = {2252-2257},
Publisher = {Elsevier BV},
Year = {2012},
ISSN = {0893-9659},
url = {http://dx.doi.org/10.1016/j.aml.2012.06.012},
Abstract = {We establish two nonlinear compactness theorems in Lp(0,T;B)
with hypothesis on time translations, which are nonlinear
counterparts of two results by Simon (1987) [1]. The first
theorem sharpens a result by Maitre (2003) [10] and is
important in the study of doubly nonlinear ellipticparabolic
equations. Based on this theorem, we then obtain a time
translation counterpart of a result by Dubinskiǐ (1965)
[5], which is supposed to be useful in the study of some
nonlinear kinetic equations (e.g. the FENE-type beadspring
chains model). © 2012 Elsevier Ltd. All rights
reserved.},
Doi = {10.1016/j.aml.2012.06.012},
Key = {fds246888}
}
@article{fds246889,
Author = {Chen, L and Liu, JG and Wang, J},
Title = {Multi-dimensional degenerate Keller-Segel system with
critical diffusion exponent 2n/(n+2)},
Journal = {SIAM J. Math Anal},
Volume = {44},
Number = {2},
Pages = {1077-1102},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2012},
ISSN = {0036-1410},
url = {http://dx.doi.org/10.1137/110839102},
Abstract = {This paper deals with a degenerate diffusion
Patlak-Keller-Segel system in n = 3 dimension. The main
difference between the current work and many other recent
studies on the same model is that we study the diffusion
exponent m = 2n/(n + 2), which is smaller than the usual
exponent m* = 2-2/n used in other studies. With the exponent
m = 2n/(n + 2), the associated free energy is conformal
invariant, and there is a family of stationary solutions
Uλ,x0 (x) = C(λ/ λ 2+|x-x0| 2 ) n+2/2 λ < 0, σ0 ? ℝn.
For radially symmetric solutions, we prove that if the
initial data are strictly below Uλ,0(x) for some λ, then
the solution vanishes in L1 loc as tλ8; if the initial data
are strictly above Uλ,0(x) for some λ, then the solution
either blows up at a finite time or has a mass concentration
at r = 0 as time goes to infinity. For general initial data,
we prove that there is a global weak solution provided that
the Lm norm of initial density is less than a universal
constant, and the weak solution vanishes as time goes to
infinity. We also prove a finite time blow-up of the
solution if the Lm norm for initial data is larger than the
Lm norm of Uλ,x0 (x), which is constant independent of λ
and x0, and the free energy of initial data is smaller than
that of Uλ,x0(x). © 2012 Society for Industrial and
Applied Mathematics.},
Doi = {10.1137/110839102},
Key = {fds246889}
}
@article{fds246890,
Author = {Frouvelle, A and Liu, JG},
Title = {Dynamics in a kinetic model of oriented particles with phase
transition},
Journal = {SIAM J. Math Anal},
Volume = {44},
Number = {2},
Pages = {791-826},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2012},
ISSN = {0036-1410},
url = {http://dx.doi.org/10.1137/110823912},
Abstract = {Motivated by a phenomenon of phase transition in a model of
alignment of selfpropelled particles, we obtain a kinetic
mean-field equation which is nothing more than the
Smoluchowski equation on the sphere with dipolar potential.
In this self-contained article, using only basic tools, we
analyze the dynamics of this equation in any dimension. We
first prove global wellposedness of this equation, starting
with an initial condition in any Sobolev space. We then
compute all possible steady states. There is a threshold for
the noise parameter: over this threshold, the only
equilibrium is the uniform distribution, and under this
threshold, the other equilibria are the Fisher-von Mises
distributions with arbitrary direction and a concentration
parameter determined by the intensity of the noise. For any
initial condition, we give a rigorous proof of convergence
of the solution to a steady state as time goes to infinity.
In particular, when the noise is under the threshold and
with nonzero initial mean velocity, the solution converges
exponentially fast to a unique Fisher- von Mises
distribution. We also found a new conservation relation,
which can be viewed as a convex quadratic entropy when the
noise is above the threshold. This provides a uniform
exponential rate of convergence to the uniform distribution.
At the threshold, we show algebraic decay to the uniform
distribution. © 2012 Society for Industrial and Applied
Mathematics.},
Doi = {10.1137/110823912},
Key = {fds246890}
}
@article{fds246891,
Author = {Carrillo, J and Chen, L and Liu, JG and Wang, J},
Title = {A note on the subcritical two dimensional Keller-Segel
system},
Journal = {Acta Applicanda Mathematicae},
Volume = {119},
Number = {1},
Pages = {43-55},
Publisher = {Springer Nature},
Year = {2012},
ISSN = {0167-8019},
url = {http://dx.doi.org/10.1007/s10440-011-9660-4},
Abstract = {The existence of solution for the 2D-Keller-Segel system in
the subcritical case, i.e. when the initial mass is less
than 8π, is reproved. Instead of using the entropy in the
free energy and free energy dissipation, which was used in
the proofs (Blanchet et al. in SIAM J. Numer. Anal.
46:691-721, 2008; Electron. J. Differ. Equ. Conf. 44:32,
2006 (electronic)), the potential energy term is fully
utilized by adapting Delort's theory on 2D incompressible
Euler equation (Delort in J. Am. Math. Soc. 4:553-386,
1991). © 2011 Springer Science+Business Media
B.V.},
Doi = {10.1007/s10440-011-9660-4},
Key = {fds246891}
}
@article{fds246892,
Author = {Degond, P and Liu, JG},
Title = {Hydrodynamics of self-alignment interactions with precession
and derivation of the Landau-Lifschitz-Gilbert
equation},
Journal = {Math. Models Methods Appl. Sci.},
Volume = {22},
Number = {SUPPL.1},
Pages = {1114001-1114018},
Publisher = {World Scientific Pub Co Pte Lt},
Year = {2012},
ISSN = {0218-2025},
url = {http://dx.doi.org/10.1142/S021820251140001X},
Abstract = {We consider a kinetic model of self-propelled particles with
alignment interaction and with precession about the
alignment direction. We derive a hydrodynamic system for the
local density and velocity orientation of the particles. The
system consists of the conservative equation for the local
density and a non-conservative equation for the orientation.
First, we assume that the alignment interaction is purely
local and derive a first-order system. However, we show that
this system may lose its hyperbolicity. Under the assumption
of weakly nonlocal interaction, we derive diffusive
corrections to the first-order system which lead to the
combination of a heat flow of the harmonic map and
LandauLifschitzGilbert dynamics. In the particular case of
zero self-propelling speed, the resulting model reduces to
the phenomenological LandauLifschitzGilbert equations.
Therefore the present theory provides a kinetic formulation
of classical micromagnetization models and spin dynamics. ©
2012 World Scientific Publishing Company.},
Doi = {10.1142/S021820251140001X},
Key = {fds246892}
}
@article{fds246893,
Author = {Chertock, A and Liu, JG and Pendleton, T},
Title = {Convergence of a particle method and global weak solutions
of a family of evolutionary PDEs},
Journal = {SIAM J. Numer. Anal.},
Volume = {50},
Number = {1},
Pages = {1-21},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2012},
ISSN = {0036-1429},
url = {http://dx.doi.org/10.1137/110831386},
Abstract = {The purpose of this paper is to provide global existence and
uniqueness results for a family of fluid transport equations
by establishing convergence results for the particle method
applied to these equations. The considered family of PDEs is
a collection of strongly nonlinear equations which yield
traveling wave solutions and can be used to model a variety
of flows in fluid dynamics. We apply a particle method to
the studied evolutionary equations and provide a new
self-contained method for proving its convergence. The
latter is accomplished by using the concept of space-time
bounded variation and the associated compactness properties.
From this result, we prove the existence of a unique global
weak solution in some special cases and obtain stronger
regularity properties of the solution than previously
established. © 2012 Society for Industrial and Applied
Mathematics.},
Doi = {10.1137/110831386},
Key = {fds246893}
}
@article{fds246894,
Author = {Haack, J and Jin, S and Liu, JG},
Title = {An all-speed asymptotic-preserving method for the isentropic
Euler and Navier-Stokes equations},
Journal = {Commun. Comput. Phy.},
Volume = {12},
Number = {4},
Pages = {955-980},
Publisher = {Global Science Press},
Year = {2012},
ISSN = {1815-2406},
url = {http://dx.doi.org/10.4208/cicp.250910.131011a},
Abstract = {The computation of compressible flows becomes more
challenging when the Mach number has different orders of
magnitude. When the Mach number is of order one, modern
shock capturing methods are able to capture shocks and other
complex structures with high numerical resolutions. However,
if the Mach number is small, the acoustic waves lead to
stiffness in time and excessively large numerical viscosity,
thus demanding much smaller time step and mesh size than
normally needed for incompressible flow simulation. In this
paper, we develop an all-speed asymptotic preserving (AP)
numerical scheme for the compressible isentropic Euler and
Navier-Stokes equations that is uniformly stable and
accurate for all Mach numbers. Our idea is to split the
system into two parts: one involves a slow, nonlinear and
conservative hyperbolic system adequate for the use of
modern shock capturing methods and the other a linear
hyperbolic system which contains the stiff acoustic
dynamics, to be solved implicitly. This implicit part is
reformulated into a standard pressure Poisson projection
system and thus possesses sufficient structure for efficient
fast Fourier transform solution techniques. In the zero Mach
number limit, the scheme automatically becomes a projection
method-like incompressible solver. We present numerical
results in one and two dimensions in both compressible and
incompressible regimes. © 2012 Global-Science
Press.},
Doi = {10.4208/cicp.250910.131011a},
Key = {fds246894}
}
@article{fds246895,
Author = {Degond, P and Frouvell, A and Liu, JG},
Title = {Macroscopic limits and phase transition in a system of
self-propelled particles},
Journal = {J Nonlinear Sci.},
Volume = {23},
Number = {3},
Pages = {427-456},
Publisher = {Springer Nature},
Year = {2012},
ISSN = {0938-8974},
url = {http://dx.doi.org/10.1007/s00332-012-9157-y},
Abstract = {We investigate systems of self-propelled particles with
alignment interaction. Compared to previous work (Degond and
Motsch, Math. Models Methods Appl. Sci. 18:1193-1215, 2008a;
Frouvelle, Math. Models Methods Appl. Sci., 2012), the force
acting on the particles is not normalized, and this
modification gives rise to phase transitions from disordered
states at low density to aligned states at high densities.
This model is the space-inhomogeneous extension of
(Frouvelle and Liu, Dynamics in a kinetic model of oriented
particles with phase transition, 2012), in which the
existence and stability of the equilibrium states were
investigated. When the density is lower than a threshold
value, the dynamics is described by a nonlinear diffusion
equation. By contrast, when the density is larger than this
threshold value, the dynamics is described by a similar
hydrodynamic model for self-alignment interactions as
derived in (Degond and Motsch, Math. Models Methods Appl.
Sci. 18:1193-1215, 2008a; Frouvelle, Math. Models Methods
Appl. Sci., 2012). However, the modified normalization of
the force gives rise to different convection speeds, and the
resulting model may lose its hyperbolicity in some regions
of the state space. © 2012 Springer Science+Business Media
New York.},
Doi = {10.1007/s00332-012-9157-y},
Key = {fds246895}
}
@article{fds246899,
Author = {Zheng, W and Gao, H and Liu, JG and Zhang, Y and Ye, Q and Swank,
C},
Title = {General solution to gradient-induced transverse and
longitudinal relaxation of spins undergoing restricted
diffusion},
Journal = {Physical Review A - Atomic, Molecular, and Optical
Physics},
Volume = {84},
Number = {5},
Pages = {053411-8},
Publisher = {American Physical Society (APS)},
Year = {2011},
Month = {November},
ISSN = {1050-2947},
url = {http://dx.doi.org/10.1103/PhysRevA.84.053411},
Abstract = {We develop an approach, by calculating the autocorrelation
function of spins, to derive the magnetic field
gradient-induced transverse (T2) relaxation of spins
undergoing restricted diffusion. This approach is an
extension to the method adopted by McGregor. McGregor's
approach solves the problem only in the fast diffusion
limit; however, our approach yields a single analytical
solution suitable in all diffusion regimes, including the
intermediate regime. This establishes a direct connection
between the well-known slow diffusion result of Torrey and
the fast diffusion result. We also perform free induction
decay measurements on spin-exchange optically polarized 3He
gas with different diffusion constants. The measured
transverse relaxation profiles are compared with the theory
and satisfactory agreement has been found throughout all
diffusion regimes. In addition to the transverse relaxation,
this approach is also applicable to solving the longitudinal
relaxation (T 1) regardless of the diffusion limits. It
turns out that the longitudinal relaxation in the slow
diffusion limit differs by a factor of 2 from that in the
fast diffusion limit. © 2011 American Physical
Society.},
Doi = {10.1103/PhysRevA.84.053411},
Key = {fds246899}
}
@article{fds246897,
Author = {Liu, JG and Lorz, A},
Title = {A coupled chemotaxis-fluid model: Global
existence},
Journal = {Ann. I. H. Poincare, AN},
Volume = {28},
Number = {5},
Pages = {643-652},
Publisher = {Elsevier BV},
Year = {2011},
ISSN = {0294-1449},
url = {http://dx.doi.org/10.1016/j.anihpc.2011.04.005},
Abstract = {We consider a model arising from biology, consisting of
chemotaxis equations coupled to viscous incompressible fluid
equations through transport and external forcing. Global
existence of solutions to the Cauchy problem is investigated
under certain conditions. Precisely, for the
chemotaxis-Navier- Stokes system in two space dimensions, we
obtain global existence for large data. In three space
dimensions, we prove global existence of weak solutions for
the chemotaxis-Stokes system with nonlinear diffusion for
the cell density.© 2011 Elsevier Masson SAS. All rights
reserved.},
Doi = {10.1016/j.anihpc.2011.04.005},
Key = {fds246897}
}
@article{fds246898,
Author = {Acheritogaray, M and Degond, P and Frouvelle, A and Liu,
JG},
Title = {Kinetic formulation and global existence for the
Hall-Magneto-hydrodynamics system},
Journal = {Kinetic and Related Models},
Volume = {4},
Number = {4},
Pages = {901-918},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2011},
ISSN = {1937-5093},
url = {http://dx.doi.org/10.3934/krm.2011.4.901},
Abstract = {This paper deals with the derivation and analysis of the the
Hall Magneto-Hydrodynamic equations. We first provide a
derivation of this system from a two-fluids Euler-Maxwell
system for electrons and ions, through a set of scaling
limits. We also propose a kinetic formulation for the
Hall-MHD equa- tions which contains as fluid closure
different variants of the Hall-MHD model. Then, we prove the
existence of global weak solutions for the incompressible
viscous resistive Hall-MHD model. We use the particular
structure of the Hall term which has zero contribution to
the energy identity. Finally, we discuss particular
solutions in the form of axisymmetric purely swirling
magnetic fields and propose some regularization of the Hall
equation. © American Institute of Mathematical
Sciences.},
Doi = {10.3934/krm.2011.4.901},
Key = {fds246898}
}
@article{fds246904,
Author = {Huang, YL and Liu, JG and Wang, WC},
Title = {An FFT based fast Poisson solver on spherical
shells},
Journal = {Commun. Comput. Phy.},
Volume = {9},
Number = {3},
Pages = {649-667},
Publisher = {Global Science Press},
Year = {2011},
ISSN = {1815-2406},
url = {http://dx.doi.org/10.4208/cicp.060509.080609s},
Abstract = {We present a fast Poisson solver on spherical shells. With a
special change of variable, the radial part of the Laplacian
transforms to a constant coefficient differ- ential
operator. As a result, the Fast Fourier Transform can be
applied to solve the Poisson equation with O(N^3 logN)
operations. Numerical examples have confirmed the accuracy
and robustness of the new scheme.},
Doi = {10.4208/cicp.060509.080609s},
Key = {fds246904}
}
@article{fds246900,
Author = {Liu, JG and Liu, J and Pego, RL},
Title = {Stable and accurate pressure approximation for unsteady
incompressible viscous flow},
Journal = {Journal of Computational Physics},
Volume = {229},
Number = {9},
Pages = {3428-3453},
Publisher = {Elsevier BV},
Year = {2010},
Month = {January},
ISSN = {0021-9991},
url = {http://dx.doi.org/10.1016/j.jcp.2010.01.010},
Abstract = {How to properly specify boundary conditions for pressure is
a longstanding problem for the incompressible Navier-Stokes
equations with no-slip boundary conditions. An analytical
resolution of this issue stems from a recently developed
formula for the pressure in terms of the commutator of the
Laplacian and Leray projection operators. Here we make use
of this formula to (a) improve the accuracy of computing
pressure in two kinds of existing time-discrete projection
methods implicit in viscosity only, and (b) devise new
higher-order accurate time-discrete projection methods that
extend a slip-correction idea behind the well-known
finite-difference scheme of Kim and Moin. We test these
schemes for stability and accuracy using various
combinations of C0 finite elements. For all three kinds of
time discretization, one can obtain third-order accuracy for
both pressure and velocity without a time-step stability
restriction of diffusive type. Furthermore, two kinds of
projection methods are found stable using piecewise-linear
elements for both velocity and pressure. © 2010 Elsevier
Inc.},
Doi = {10.1016/j.jcp.2010.01.010},
Key = {fds246900}
}
@article{fds304584,
Author = {Liu, JG and Pego, RL},
Title = {Stable discretization of magnetohydrodynamics in bounded
domains},
Journal = {Communications in Mathematical Sciences},
Volume = {8},
Number = {1},
Pages = {235-251},
Publisher = {International Press of Boston},
Year = {2010},
Month = {January},
ISSN = {1539-6746},
url = {http://dx.doi.org/10.4310/CMS.2010.v8.n1.a12},
Abstract = {We study a semi-implicit time-difference scheme for
magnetohydrodynamics of a viscous and resistive
incompressible fluid in a bounded smooth domain with a
perfectly conducting boundary. In the scheme, the velocity
and magnetic fields are updated by solving simple Helmholtz
equations. Pressure is treated explicitly in time, by
solving Poisson equations corresponding to a recently
de-veloped formula for the Navier-Stokes pressure involving
the commutator of Laplacian and Leray projection operators.
We prove stability of the time-difference scheme, and deduce
a local-time well-posedness theorem for MHD dynamics
extended to ignore the divergence-free constraint on
velocity and magnetic fields. These fields are
divergence-free for all later time if they are initially so.
© 2010 International Press.},
Doi = {10.4310/CMS.2010.v8.n1.a12},
Key = {fds304584}
}
@article{fds246905,
Author = {Liu, JG and Mieussens, L},
Title = {Analysis of an asymptotic preserving scheme for linear
kinetic equations in the diffusion limit},
Journal = {SIAM J. Numer. Anal.},
Volume = {48},
Number = {4},
Pages = {1474-1491},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2010},
ISSN = {0036-1429},
url = {http://hdl.handle.net/10161/4316 Duke open
access},
Abstract = {We present a mathematical analysis of the asymptotic
preserving scheme proposed in [M. Lemou and L. Mieussens,
SIAM J. Sci. Comput., 31 (2008), pp. 334–368] for linear
transport equations in kinetic and diffusive regimes. We
prove that the scheme is uniformly stable and accurate with
respect to the mean free path of the particles. This
property is satisfied under an explicitly given CFL
condition. This condition tends to a parabolic CFL condition
for small mean free paths and is close to a convection CFL
condition for large mean free paths. Our analysis is based
on very simple energy estimates.},
Doi = {10.1137/090772770},
Key = {fds246905}
}
@article{fds246928,
Author = {Liu, JG and Pego, R},
Title = {Stable discretization of magnetohydrodynamics in bounded
domains},
Journal = {Commun. Math. Sci.},
Volume = {8},
Number = {1},
Pages = {234-251},
Publisher = {INT PRESS BOSTON, INC},
Year = {2010},
ISSN = {1539-6746},
Abstract = {We study a semi-implicit time-difference scheme for
magnetohydrodynamics of a viscous and resistive
incompressible fluid in a bounded smooth domain with a
perfectly conducting boundary. In the scheme, the velocity
and magnetic fields are updated by solving simple Helmholtz
equations. Pressure is treated explicitly in time, by
solving Poisson equations corresponding to a recently
de-veloped formula for the Navier-Stokes pressure involving
the commutator of Laplacian and Leray projection operators.
We prove stability of the time-difference scheme, and deduce
a local-time well-posedness theorem for MHD dynamics
extended to ignore the divergence-free constraint on
velocity and magnetic fields. These fields are
divergence-free for all later time if they are initially so.
© 2010 International Press.},
Key = {fds246928}
}
@article{fds246943,
Author = {Liu, JG and Liu, J and Pego, RL},
Title = {Error estimates for finite-element Navier-Stokes solvers
without standard Inf-Sup conditions},
Journal = {Chinese Annals of Mathematics. Series B},
Volume = {30},
Number = {6},
Pages = {743-768},
Publisher = {Springer Nature},
Year = {2009},
Month = {December},
ISSN = {0252-9599},
url = {http://dx.doi.org/10.1007/s11401-009-0116-3},
Abstract = {The authors establish error estimates for recently developed
finite-element methods for incompressible viscous flow in
domains with no-slip boundary conditions. The methods arise
by discretization of a well-posed extended Navier-Stokes
dynamics for which pressure is determined from current
velocity and force fields. The methods use C1 elements for
velocity and C0 elements for pressure. A stability estimate
is proved for a related finite-element projection method
close to classical time-splitting methods of Orszag,
Israeli, DeVille and Karniadakis. © Editorial Office of CAM
and Springer-Verlag Berlin Heidelberg 2009.},
Doi = {10.1007/s11401-009-0116-3},
Key = {fds246943}
}
@article{fds246944,
Author = {Liu, JG and Wang, WC},
Title = {Characterization and regularity for axisymmetric solenoidal
vector elds with application to Navier-Stokes
equation},
Journal = {SIAM J. Math. Anal.},
Volume = {41},
Number = {5},
Pages = {1825-1850},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2009},
ISSN = {0036-1410},
url = {http://dx.doi.org/10.1137/080739744},
Abstract = {We consider the vorticity-stream formulation of axisymmetric
incompressible flows and its equivalence with the primitive
formulation. It is shown that, to characterize the
regularity of a divergence free axisymmetric vector field in
terms of the swirling components, an extra set of pole
conditions is necessary to give a full description of the
regu larity. In addition, smooth solutions up to the axis of
rotation give rise to smooth solutions of primitive
formulation in the case of the Navier-Stokes equation, but
not the Euler equation. We also establish a proper weak
formulation and show its equivalence to Leray's formulation.
© 2009 Society for Industrial and Applied
Mathematics.},
Doi = {10.1137/080739744},
Key = {fds246944}
}
@article{fds246945,
Author = {Ha, SY and Liu, JG},
Title = {A simple proof of the Cucker-Smale flocking dynamics and
mean-field limit},
Journal = {Commun. Math. Sci.},
Volume = {7},
Number = {2},
Pages = {297-325},
Publisher = {International Press of Boston},
Year = {2009},
ISSN = {1539-6746},
url = {http://dx.doi.org/10.4310/CMS.2009.v7.n2.a2},
Abstract = {We present a simple proof on the formation of flocking to
the Cucker-Smale system based on the explicit construction
of a Lyapunov functional. Our results also provide a unified
condition on the initial states in which the exponential
convergence to flocking state will occur. For large particle
systems, we give a rigorous justification for the mean-field
limit from the many particle Cucker-Smale system to the
Vlasov equation with flocking dissipation as the number of
particles goes to infinity. © 2009 International
Press.},
Doi = {10.4310/CMS.2009.v7.n2.a2},
Key = {fds246945}
}
@article{fds246946,
Author = {Degond, P and Liu, JG and Vignal, MH},
Title = {Analysis of an asymptotic preserving scheme for the
Euler-Poisson system in the quasineutral
limit},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {46},
Number = {3},
Pages = {1298-1322},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2008},
Month = {November},
ISSN = {0036-1429},
url = {http://dx.doi.org/10.1137/070690584},
Keywords = {stiffness • Debye length • electron plasma period
• Burgers-Poisson • sheath problem •
Klein-Gordon},
Abstract = {In a previous work [P. Crispel, P. Degond, and M.-H. Vignal,
J. Comput. Phys., 223 (2007), pp. 208-234], a new numerical
discretization of the Euler-Poisson system was proposed.
This scheme is "asymptotic preserving" in the quasineutral
limit (i.e., when the Debye length ε tends to zero), which
means that it becomes consistent with the limit model when
ε → 0. In the present work, we show that the stability
domain of the present scheme is independent of ε. This
stability analysis is performed on the Fourier transformed
(with respect to the space variable) linearized system. We
show that the stability property is more robust when a
space-decentered scheme is used (which brings in some
numerical dissipation) rather than a space-centered scheme.
The linearization is first performed about a zero mean
velocity and then about a nonzero mean velocity. At the
various stages of the analysis, our scheme is compared with
more classical schemes and its improved stability property
is outlined. The analysis of a fully discrete (in space and
time) version of the scheme is also given. Finally, some
considerations about a model nonlinear problem, the
Burgers-Poisson problem, are also discussed. © 2008 Society
for Industrial and Applied Mathematics.},
Doi = {10.1137/070690584},
Key = {fds246946}
}
@article{fds246948,
Author = {Lu, X and Lin, P and Liu, JG},
Title = {Analysis of a sequential regularization method for the
unsteady Navier-Stokes equations},
Journal = {Mathematics of Computation},
Volume = {77},
Number = {263},
Pages = {1467-1494},
Publisher = {American Mathematical Society (AMS)},
Year = {2008},
Month = {July},
ISSN = {0025-5718},
url = {http://dx.doi.org/10.1090/S0025-5718-08-02087-5},
Keywords = {Navier-Stokes equations • iterative penalty method
• implicit parabolic PDE • error estimates •
constrained dynamical system • stabilization
method},
Abstract = {The incompressibility constraint makes Navier-Stokes
equations difficult. A reformulation to a better posed
problem is needed before solving it numerically. The
sequential regularization method (SRM) is a reformulation
which combines the penalty method with a stabilization
method in the context of constrained dynamical systems and
has the benefit of both methods. In the paper, we study the
existence and uniqueness for the solution of the SRM and
provide a simple proof of the convergence of the solution of
the SRM to the solution of the Navier-Stokes equations. We
also give error estimates for the time discretized SRM
formulation. ©2008 American Mathematical
Society.},
Doi = {10.1090/S0025-5718-08-02087-5},
Key = {fds246948}
}
@article{fds246941,
Author = {Lin, P and Liu, JG and Lu, X},
Title = {Long time numerical solution of the Navier-Stokes equations
based on a sequential regularization formulation},
Journal = {SIAM Journal on Scientific Computing},
Volume = {31},
Number = {1},
Pages = {398-419},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2008},
Month = {January},
ISSN = {1064-8275},
url = {http://dx.doi.org/10.1137/060673722},
Abstract = {The sequential regularization method is a reformulation of
the unsteady Navier-Stokes equations from the viewpoint of
constrained dynamical systems or the approximate
Helmholtz-Hodge projection. In this paper we study the long
time behavior of the sequential regularization formulation.
We give a uniform-in-time estimate between the solution of
the reformulated system and that of the Navier-Stokes
equations. We also conduct an error analysis for the
temporal discrete system and show that the error bound is
independent of time. A couple of long time flow examples are
computed to demonstrate this method. © 2008 Society for
Industrial and Applied Mathematics.},
Doi = {10.1137/060673722},
Key = {fds246941}
}
@article{fds246942,
Author = {Liu, JG and Wang, C},
Title = {A fourth order numerical method for the primtive equations
formulated in mean vorticity},
Journal = {Communications in Computational Physics},
Volume = {4},
Number = {1},
Pages = {26-55},
Year = {2008},
Month = {January},
ISSN = {1815-2406},
Abstract = {A fourth-order finite difference method is proposed and
studied for the primitive equations (PEs) of large-scale
atmospheric and oceanic flow based on mean vorticity
formulation. Since the vertical average of the horizontal
velocity field is divergence-free, we can introduce mean
vorticity and mean stream function which are connected by a
2-D Poisson equation. As a result, the PEs can be
reformulated such that the prognostic equation for the
horizontal velocity is replaced by evolutionary equations
for the mean vorticity field and the vertical derivative of
the horizontal velocity. The mean vorticity equation is
approximated by a compact difference scheme due to the
difficulty of the mean vorticity boundary condition, while
fourth-order long-stencil approximations are utilized to
deal with transport type equations for computational
convenience. The numerical values for the total velocity
field (both horizontal and vertical) are statically
determined by a discrete realization of a differential
equation at each fixed horizontal point. The method is
highly efficient and is capable of producing highly resolved
solutions at a reasonable computational cost. The full
fourth-order accuracy is checked by an example of the
reformulated PEs with force terms. Additionally, numerical
results of a large-scale oceanic circulation are presented.
© 2008 Global-Science Press.},
Key = {fds246942}
}
@article{fds246940,
Author = {Hsia, CH and Liu, JG and Wang, C},
Title = {Structural stability and bifurcation for 2D incompressible
ows with symmetry},
Journal = {Meth. Appl. Anal.},
Volume = {15},
Pages = {495-512},
Year = {2008},
Key = {fds246940}
}
@article{fds246949,
Author = {Antman, SS and Liu, JG},
Title = {Basic themes and pretty problems of nonlinear solid
mechanics},
Journal = {Milan Journal of Mathematics},
Volume = {75},
Number = {1},
Pages = {135-176},
Publisher = {Springer Nature},
Year = {2007},
Month = {December},
ISSN = {1424-9286},
url = {http://dx.doi.org/10.1007/s00032-007-0068-6},
Keywords = {Nonlinear solid mechanics • radial motions •
existence • multiplicity • blowup • inverse
problems • quasistaticity • control •
invariant artificial viscosity and shock
structure},
Abstract = {The first part of this paper describes some important
underlying themes in the mathematical theory of continuum
mechanics that are distinct from formulating and analyzing
governing equations. The main part of this paper is devoted
to a survey of some concrete, conceptually simple, pretty
problems that help illuminate the underlying themes. The
paper concludes with a discussion of the crucial role of
invariant constitutive equations in computation. © 2007
Birkhaueser.},
Doi = {10.1007/s00032-007-0068-6},
Key = {fds246949}
}
@article{fds246958,
Author = {Moore, J and Cheng, Z and Hao, J and Guo, G and Liu, J-G and Lin, C and Yu,
LL},
Title = {Effects of solid-state yeast treatment on the antioxidant
properties and protein and fiber compositions of common hard
wheat bran.},
Journal = {Journal of agricultural and food chemistry},
Volume = {55},
Number = {25},
Pages = {10173-10182},
Year = {2007},
Month = {December},
ISSN = {0021-8561},
url = {http://dx.doi.org/10.1021/jf071590o},
Abstract = {The bran fraction of wheat grain is known to contain
significant quantities of bioactive components. This study
evaluated the potential of solid-state yeast fermentation to
improve the health beneficial properties of wheat bran,
including extractable antioxidant properties, protein
contents, and soluble and insoluble fiber compositions.
Three commercial food grade yeast preparations were
evaluated in the study along with the effects of yeast dose,
treatment time, and their interaction with the beneficial
components. Solid-state yeast treatments were able to
significantly increase releasable antioxidant properties
ranging from 28 to 65, from 0 to 20, from 13 to 19, from 0
to 25, from 50 to 100, and from 3 to 333% for scavenging
capacities against peroxyl (ORAC), ABTS cation, DPPH and
hydroxyl radicals, total phenolic contents (TPC), and
phenolic acids, respectively. Yeast treatment increased
protein content 11-12% but did not significantly alter the
fiber composition of wheat bran. Effects of solid-state
yeast treatment on both ORAC and TPC of wheat bran were
altered by yeast dose, treatment time, and their
interaction. Results suggest that solid-state yeast
treatment may be a commercially viable postharvest procedure
for improving the health beneficial properties of wheat bran
and other wheat-based food ingredients.},
Doi = {10.1021/jf071590o},
Key = {fds246958}
}
@article{fds246880,
Author = {Liu, JG and Liu, J and Pego, RL},
Title = {Stability and convergence of efficient Navier-Stokes solvers
via a commutator estimate},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {60},
Number = {10},
Pages = {1443-1487},
Publisher = {WILEY},
Year = {2007},
Month = {October},
ISSN = {0010-3640},
url = {http://dx.doi.org/10.1002/cpa.20178},
Abstract = {For strong solutions of the incompressible Navier-Stokes
equations in bounded domains with velocity specified at the
boundary, we establish the unconditional stability and
convergence of discretization schemes that decouple the
updates of pressure and velocity through explicit time
stepping for pressure. These schemes require no solution of
stationary Stokes systems, nor any compatibility between
velocity and pressure spaces to ensure an inf-sup condition,
and are representative of a class of highly efficient
computational methods that have recently emerged. The proofs
are simple, based upon a new, sharp estimate for the
commutator of the Laplacian and Helmholtz projection
operators. This allows us to treat an unconstrained
formulation of the Navier-Stokes equations as a perturbed
diffusion equation. ©2006 Wiley Periodicals,
Inc.},
Doi = {10.1002/cpa.20178},
Key = {fds246880}
}
@article{fds139011,
Author = {J.-G. Liu and Jie Liu and R. Pego},
Title = {Estimates on the Stokes pressure by partitioning the energy
of harmonic functions},
Pages = {251--270},
Booktitle = {Kyoto Conference on the Navier-Stokes equations and their
Applications},
Publisher = {Kyoto Univ.},
Editor = {Y. Giga and H. Kozono and H. Okamoto and Y. Shibta},
Year = {2007},
Abstract = {We show that in a tubular domain with sufficiently small
width, the normal and tangential gradients of a harmonic
function have almost the same L2 norm. This estimate yields
a sharp estimate of the pressure in terms of the viscosity
term in the Navier-Stokes equation with no-slip boundary
condition. By consequence, one can analyze the Navier-
Stokes equations simply as a perturbed vector diffusion
equation instead of as a perturbed Stokes system. As an
application, we describe a rather easy approach to establish
a new isomorphism theorem for the non-homogeneous Stokes
system.},
Key = {fds139011}
}
@article{fds246903,
Author = {Liu, JG and Liu, J and Pego, R},
Title = {Stability and convergence of efficient Navier-Stokes solvers
via a commutator estimate via a commutator
estimate},
Journal = {Comm. Pure Appl. Math.},
Volume = {60},
Pages = {1443-1487},
Year = {2007},
Key = {fds246903}
}
@article{fds246947,
Author = {Degond, P and Jin, S and Liu, JG},
Title = {Mach-number uniform asymptotic- preserving Gauge schemes for
compressible flows},
Journal = {Bulletin of the Institute of Mathematics Academia Sinica
(New Series)},
Volume = {2},
Pages = {851-892},
Year = {2007},
Keywords = {Mach number uniform method • Euler equations •
Navier-Stokes equations • Asymptotic Preserving schemes
• gauge schemes • compressible fluids •
Low-Mach number limit • macro-micro decomposition
• semi-implicit scheme • Euler-Poisson system
• Navier-Stokes-Poisson system},
Abstract = {We present novel algorithms for compressible flows that are
efficient for all Mach numbers. The approach is based on
several ingredients: semi-implicit schemes, the gauge
decomposition of the velocity field and a second order
formulation of the density equation (in the isentropic case)
and of the energy equation (in the full Navier-Stokes case).
Additionally, we show that our approach corresponds to a
micro-macro decomposition of the model, where the macro
field corresponds to the incompressible component satisfying
a perturbed low Mach number limit equation and the micro
field is the potential component of the velocity. Finally,
we also use the conservative variables in order to obtain a
proper conservative formulation of the equations when the
Mach number is order unity. We successively consider the
isentropic case, the full Navier-Stokes case, and the
isentropic Navier-Stokes-Poisson case. In this work, we only
concentrate on the question of the time discretization and
show that the proposed method leads to Asymptotic Preserving
schemes for compressible flows in the low Mach number
limit.},
Key = {fds246947}
}
@article{fds246960,
Author = {Liu, JG and Wang, WC},
Title = {Convergence analysis of the energy and helicity preserving
scheme for axisymmetric flows},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {44},
Number = {6},
Pages = {2456-2480},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2006},
Month = {December},
ISSN = {0036-1429},
url = {http://dx.doi.org/10.1137/050639314},
Abstract = {We give an error estimate for the energy and helicity
preserving scheme (EHPS) in second order finite difference
setting on axisymmetric incompressible flows with swirling
velocity. This is accomplished by a weighted energy
estimate, along with careful and nonstandard local
truncation error analysis near the geometric singularity and
a far field decay estimate for the stream function. A key
ingredient in our a priori estimate is the permutation
identities associated with the Jacobians, which are also a
unique feature that distinguishes EHPS from standard finite
difference schemes. © 2006 Society for Industrial and
Applied Mathematics.},
Doi = {10.1137/050639314},
Key = {fds246960}
}
@article{fds246901,
Author = {Degond, P and Liu, JG and Mieussens, L},
Title = {Macroscopic fluid models with localized kinetic upscaling
effects},
Journal = {Multiscale Modeling and Simulation},
Volume = {5},
Number = {3},
Pages = {940-979},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2006},
Month = {September},
ISSN = {1540-3459},
url = {http://dx.doi.org/10.1137/060651574},
Keywords = {Kinetic-Fluid coupling, Kinetic equation, Hydrodynamic
approximation, Diffusion approximation},
Abstract = {This paper presents a general methodology to design
macroscopic fluid models that take into account localized
kinetic upscaling effects. The fluid models are solved in
the whole domain together with a localized kinetic upscaling
that corrects the fluid model wherever it is necessary. This
upscaling is obtained by solving a kinetic equation on the
nonequilibrium part of the distribution function. This
equation is solved only locally and is related to the fluid
equation through a downscaling effect. The method does not
need to find an interface condition as do usual domain
decomposition methods to match fluid and kinetic
representations. We show our approach applies to problems
that have a hydrodynamic time scale as well as to problems
with diffusion time scale. Simple numerical schemes are
proposed to discretize our models, and several numerical
examples are used to validate the method. © 2006 Society
for Industrial and Applied Mathematics.},
Doi = {10.1137/060651574},
Key = {fds246901}
}
@article{fds246957,
Author = {Moore, J and Liu, J-G and Zhou, K and Yu, LL},
Title = {Effects of genotype and environment on the antioxidant
properties of hard winter wheat bran.},
Journal = {Journal of agricultural and food chemistry},
Volume = {54},
Number = {15},
Pages = {5313-5322},
Year = {2006},
Month = {July},
ISSN = {0021-8561},
url = {http://dx.doi.org/10.1021/jf060381l},
Abstract = {Recent consumer interest in controlling and preventing
chronic diseases through improved diet has promoted research
on the bioactive components of agricultural products. Wheat
is an important agricultural and dietary commodity worldwide
with known antioxidant properties concentrated mostly in the
bran fraction. The objective of this study was to determine
the relative contributions of genotype (G) and growing
environment (E) to hard winter wheat bran antioxidant
properties, as well as correlations of these properties to
growing conditions. Bran samples of 20 hard winter wheat
varieties grown in two locations were examined for their
free radical scavenging capacities against DPPH, ABTS
cation, peroxyl (ORAC), and superoxide anion radicals and
chelating properties, as well as their total phenolics and
phenolic acid compositions. Results showed significant
differences for all antioxidant properties tested and
multiple significant correlations between these properties.
A factorial designed analysis of variance for these data and
pooled previously published data showed similar results for
four of the six antioxidant properties, indicating that G
effects were considerably larger than E effects for
chelating capacity and DPPH radical scavenging properties,
whereas E was much stronger than G for ABTS cation radical
scavenging capacity and total phenolics, although small
interaction effects (GxE) were significant for all
antioxidant properties analyzed. Results also showed
significant correlations between temperature stress or solar
radiation and some antioxidant properties. These results
indicate that each antioxidant property of hard winter wheat
bran is influenced differently by genotype and growing
conditions.},
Doi = {10.1021/jf060381l},
Key = {fds246957}
}
@article{fds139013,
Author = {J.-G. Liu and Jie Liu and R. Pego},
Title = {On incompressible Navier-Stokes dynamics: a new approach for
analysis and computation},
Pages = {29--44},
Booktitle = {Proceedings of the Tenth International Conference on
Hyperbolic Problems},
Publisher = {Yokohama Publishers, Inc.},
Editor = {F. Asakura and etc},
Year = {2006},
Key = {fds139013}
}
@article{fds246964,
Author = {Liu, JG and Samelson, R and Wang, C},
Title = {Global weak solution of planetary geostrophic equations with
inviscid geostrophic balance},
Journal = {Applicable Analysis},
Volume = {85},
Number = {6-7},
Pages = {593-605},
Year = {2006},
url = {http://dx.doi.org/10.1080/00036810500328299},
Abstract = {A reformulation of the planetary geostrophic equations
(PGEs) with the inviscid balance equation is proposed and
the existence of global weak solutions is established,
provided that the mechanical force satisfies an integral
constraint. There is only one prognostic equation for the
temperature field, and the velocity field is statically
determined by the planetary geostrophic balance combined
with the incompressibility condition. Furthermore, the
velocity profile can be accurately represented as a function
of the temperature gradient. In particular, the vertical
velocity depends only on the first-order derivative of the
temperature. As a result, the bound for the L∞ (0, t 1 ; L
2 ) ∩ L 2 (0, t 1 ; H 1 ) norm of the temperature field is
sufficient to show the existence of the weak solution. ©
2006, Taylor & Francis Group, LLC.},
Doi = {10.1080/00036810500328299},
Key = {fds246964}
}
@article{fds246902,
Author = {Liu, JG and Wang, WC},
Title = {Energy and helicity preserving schemes for hydro- and
magnetohydro-dynamics flows with symmetry},
Journal = {Journal of Computational Physics},
Volume = {200},
Number = {1},
Pages = {8-33},
Publisher = {Elsevier BV},
Year = {2004},
Month = {October},
url = {http://dx.doi.org/10.1016/j.jcp.2004.03.005},
Abstract = {We propose a class of simple and efficient numerical scheme
for incompressible fluid equations with coordinate symmetry.
By introducing a generalized vorticity-stream formulation,
the divergence free constraints are automatically satisfied.
In addition, with explicit treatment of the nonlinear terms
and local vorticity boundary condition, the Navier-Stokes
(MHD, respectively) equation essentially decouples into 2
(4, respectively) scalar equation and thus the scheme is
very efficient. Moreover, with proper discretization of the
nonlinear terms, the scheme preserves both energy and
helicity identities numerically. This is achieved by
recasting the nonlinear terms (convection, vorticity
stretching, geometric source, Lorentz force and
electro-motive force) in terms of Jacobians. This
conservative property is valid even in the presence of the
pole singularity for axisymmetric flows. The exact
conservation of energy and helicity has effectively
eliminated excessive numerical viscosity. Numerical examples
have demonstrated both accuracy and efficiency of the
scheme. Finally, local mesh refinement near the boundary can
also be easily incorporated into the scheme without extra
cost. © 2004 Elsevier Inc. All rights reserved.},
Doi = {10.1016/j.jcp.2004.03.005},
Key = {fds246902}
}
@article{fds246963,
Author = {Ghil, M and Liu, JG and Wang, C and Wang, S},
Title = {Boundary-layer separation and adverse pressure gradient for
2-D viscous incompressible flow},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {197},
Number = {1-2},
Pages = {149-173},
Publisher = {Elsevier BV},
Year = {2004},
Month = {October},
ISSN = {0167-2789},
url = {http://dx.doi.org/10.1016/j.physd.2004.06.012},
Abstract = {We study the detailed process of bifurcation in the flow's
topological structure for a two-dimensional (2-D)
incompressible flow subject to no-slip boundary conditions
and its connection with boundary-layer separation. The
boundary-layer separation theory of M. Ghil, T. Ma and S.
Wang, based on the structural-bifurcation concept, is
translated into vorticity form. The vorticily formulation of
the theory shows that structural bifurcation occurs whenever
a degenerate singular point for the vorticity appears on the
boundary; this singular point is characterized by nonzero
tangential second-order derivative and nonzero time
derivative of the vorticity. Furthermore, we prove the
presence of an adverse pressure gradient at the critical
point, due to reversal in the direction of the pressure
force with respect to the basic shear flow at this point. A
numerical example of 2-D driven-cavity flow, governed by the
Navier Stokes equations, is presented; boundary-layer
separation occurs, the bifurcation criterion is satisfied,
and an adverse pressure gradient is shown to be present. ©
2004 Elsevier B.V. All rights reserved.},
Doi = {10.1016/j.physd.2004.06.012},
Key = {fds246963}
}
@article{fds304585,
Author = {Li, B and Liu, JG},
Title = {Epitaxial growth without slope selection: Energetics,
coarsening, and dynamic scaling},
Journal = {Journal of Nonlinear Science},
Volume = {14},
Number = {5},
Pages = {429-451},
Publisher = {Springer Nature},
Year = {2004},
Month = {October},
ISSN = {0938-8974},
url = {http://dx.doi.org/10.1007/s00332-004-0634-9},
Abstract = {We study a continuum model for epitaxial growth of thin
films in which the slope of mound structure of film surface
increases. This model is a diffusion equation for the
surface height profile h which is assumed to satisfy the
periodic boundary condition. The equation happens to possess
a Liapunov or "free-energy" functional. This functional
consists of the term |Δ h| 2, which represents the surface
diffusion, and-log (1 + |∇ h| 2), which describes the
effect of kinetic asymmetry in the adatom
attachment-detachment. We first prove for large time t that
the interface width-the standard deviation of the height
profile-is bounded above by O(t 1/2), the averaged gradient
is bounded above by O(t 1/4), and the averaged energy is
bounded below by O(-log t). We then consider a small
coefficient ε 2 of |Δ h| 2 with ε = 1/L and L the linear
size of the underlying system, and study the energy
asymptotics in the large system limit ε → 0. We show that
global minimizers of the free-energy functional exist for
each ε > 0, the L 2-norm of the gradient of any global
minimizer scales as O(1/ε), and the global minimum energy
scales as O( log ε). The existence of global energy
minimizers and a scaling argument are used to construct a
sequence of equilibrium solutions with different
wavelengths. Finally, we apply our minimum energy estimates
to derive bounds in terms of the linear system size L for
the saturation interface width and the corresponding
saturation time. © 2005 Springer.},
Doi = {10.1007/s00332-004-0634-9},
Key = {fds304585}
}
@article{fds246962,
Author = {Johnston, H and Liu, JG},
Title = {Accurate, stable and efficient Navier-Stokes solvers based
on explicit treatment of the pressure term},
Journal = {Journal of Computational Physics},
Volume = {199},
Number = {1},
Pages = {221-259},
Publisher = {Elsevier BV},
Year = {2004},
Month = {September},
url = {http://dx.doi.org/10.1016/j.jcp.2004.02.009},
Abstract = {We present numerical schemes for the incompressible
Navier-Stokes equations based on a primitive variable
formulation in which the incompressibility constraint has
been replaced by a pressure Poisson equation. The pressure
is treated explicitly in time, completely decoupling the
computation of the momentum and kinematic equations. The
result is a class of extremely efficient Navier-Stokes
solvers. Full time accuracy is achieved for all flow
variables. The key to the schemes is a Neumann boundary
condition for the pressure Poisson equation which enforces
the incompressibility condition for the velocity field.
Irrespective of explicit or implicit time discretization of
the viscous term in the momentum equation the explicit time
discretization of the pressure term does not affect the time
step constraint. Indeed, we prove unconditional stability of
the new formulation for the Stokes equation with explicit
treatment of the pressure term and first or second order
implicit treatment of the viscous term. Systematic numerical
experiments for the full Navier-Stokes equations indicate
that a second order implicit time discretization of the
viscous term, with the pressure and convective terms treated
explicitly, is stable under the standard CFL condition.
Additionally, various numerical examples are presented,
including both implicit and explicit time discretizations,
using spectral and finite difference spatial
discretizations, demonstrating the accuracy, flexibility and
efficiency of this class of schemes. In particular, a
Galerkin formulation is presented requiring only C0 elements
to implement. © 2004 Elsevier Inc. All rights
reserved.},
Doi = {10.1016/j.jcp.2004.02.009},
Key = {fds246962}
}
@article{fds246956,
Author = {Wang, C and Liu, JG and Johnston, H},
Title = {Analysis of a fourth order finite difference method for the
incompressible Boussinesq equations},
Journal = {Numerische Mathematik},
Volume = {97},
Number = {3},
Pages = {555-594},
Publisher = {Springer Nature},
Year = {2004},
Month = {May},
url = {http://dx.doi.org/10.1007/s00211-003-0508-3},
Abstract = {The convergence of a fourth order finite difference method
for the 2-D unsteady, viscous incompressible Boussinesq
equations, based on the vorticity-stream function
formulation, is established in this article. A compact
fourth order scheme is used to discretize the momentum
equation, and long-stencil fourth order operators are
applied to discretize the temperature transport equation. A
local vorticity boundary condition is used to enforce the
no-slip boundary condition for the velocity. One-sided
extrapolation is used near the boundary, dependent on the
type of boundary condition for the temperature, to prescribe
the temperature at "ghost" points lying outside of the
computational domain. Theoretical results of the stability
and accuracy of the method are also provided. In numerical
experiments the method has been shown to be capable of
producing highly resolved solutions at a reasonable
computational cost.},
Doi = {10.1007/s00211-003-0508-3},
Key = {fds246956}
}
@article{fds246954,
Author = {Lin, HE and Liu, JG and Xu, WQ},
Title = {Effects of small viscosity and far field boundary conditions
for hyperbolic systems},
Journal = {Communications on Pure and Applied Analysis},
Volume = {3},
Number = {2},
Pages = {267-290},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2004},
Month = {January},
ISSN = {1534-0392},
url = {http://dx.doi.org/10.3934/cpaa.2004.3.267},
Abstract = {In this paper we study the effects of small viscosity term
and the far-field boundary conditions for systems of
convection-diffusion equations in the zero viscosity limit.
The far-field boundary conditions are classified and the
corresponding solution structures are analyzed. It is
confirmed that the Neumann type of far-field boundary
condition is preferred. On the other hand, we also identify
a class of improperly coupled boundary conditions which lead
to catastrophic reflection waves dominating the inlet in the
zero viscosity limit. The analysis is performed on the
linearized convection-diffusion model which well describes
the behavior at the far field for many physical and
engineering systems such as fluid dynamical equations and
electro-magnetic equations. The results obtained here should
provide some theoretical guidance for designing effective
far field boundary conditions.},
Doi = {10.3934/cpaa.2004.3.267},
Key = {fds246954}
}
@article{fds246955,
Author = {Liu, JG and Xu, WQ},
Title = {Far field boundary condition for convection diffusion
equation at zero viscosity limit},
Journal = {Quarterly of Applied Mathematics},
Volume = {62},
Number = {1},
Pages = {27-52},
Publisher = {American Mathematical Society (AMS)},
Year = {2004},
Month = {January},
url = {http://dx.doi.org/10.1090/qam/2032571},
Abstract = {In this paper, we give a systematic study of the boundary
layer behavior for linear convection-diffusion equation in
the zero viscosity limit. We analyze the boundary layer
structures in the viscous solution and derive the boundary
condition satisfied by the viscosity limit as a solution of
the inviscid equation. The results confirm that the Neumann
type of far-field boundary condition is preferred in the
outlet and characteristic boundary dondition. Under some
appropriate regularity and compatibility conditions on the
initial and boundary data, we obtain optimal error estimates
between the full viscous solution and the inviscid solution
with suitable boundary layer corrections. These results hold
in arbitrary space dimensions and similar statements also
hold for the strip problem This model well describes the
behavior at the far-field for many physical and engineering
systems such as fluid dynamical equation and
electro-magnetic equation. The results obtained here should
provide some theoretical guidance for designing effective
far-field boundary conditions.},
Doi = {10.1090/qam/2032571},
Key = {fds246955}
}
@article{fds304583,
Author = {Liu, JG and Wang, C},
Title = {High order finite difference methods for unsteady
incompressible flows in multi-connected domains},
Journal = {Computers and Fluids},
Volume = {33},
Number = {2},
Pages = {223-255},
Publisher = {Elsevier BV},
Year = {2004},
Month = {January},
url = {http://dx.doi.org/10.1016/S0045-7930(03)00037-9},
Abstract = {Using the vorticity and stream function variables is an
effective way to compute 2-D incompressible flow due to the
facts that the incompressibility constraint for the velocity
is automatically satisfied, the pressure variable is
eliminated, and high order schemes can be efficiently
implemented. However, a difficulty arises in a
multi-connected computational domain in determining the
constants for the stream function on the boundary of the
"holes". This is an especially challenging task for the
calculation of unsteady flows, since these constants vary
with time to reflect the total fluxes of the flow in each
sub-channel. In this paper, we propose an efficient method
in a finite difference setting to solve this problem and
present some numerical experiments, including an accuracy
check of a Taylor vortex-type flow, flow past a
non-symmetric square, and flow in a heat exchanger. © 2003
Elsevier Ltd. All rights reserved.},
Doi = {10.1016/S0045-7930(03)00037-9},
Key = {fds304583}
}
@article{fds246959,
Author = {Li, B and Liu, JG},
Title = {Eptaxial growth without slope selection: energetics,
coarsening, and dynamic scaling},
Journal = {J. Nonlinear Sci.},
Volume = {14},
Number = {5},
Pages = {429-451},
Year = {2004},
ISSN = {0938-8974},
url = {http://dx.doi.org/10.1007/s00332-004-0634-9},
Abstract = {We study a continuum model for epitaxial growth of thin
films in which the slope of mound structure of film surface
increases. This model is a diffusion equation for the
surface height profile h which is assumed to satisfy the
periodic boundary condition. The equation happens to possess
a Liapunov or "free-energy" functional. This functional
consists of the term |Δ h| 2, which represents the
surface diffusion, and-log (1 + |∇ h| 2), which
describes the effect of kinetic asymmetry in the adatom
attachment-detachment. We first prove for large time t that
the interface width-the standard deviation of the height
profile-is bounded above by O(t 1/2), the averaged gradient
is bounded above by O(t 1/4), and the averaged energy is
bounded below by O(-log t). We then consider a small
coefficient ε 2 of |Δ h| 2 with ε = 1/L and L the
linear size of the underlying system, and study the energy
asymptotics in the large system limit ε → 0. We
show that global minimizers of the free-energy functional
exist for each ε > 0, the L 2-norm of the gradient of
any global minimizer scales as O(1/ε), and the global
minimum energy scales as O( log ε). The existence of
global energy minimizers and a scaling argument are used to
construct a sequence of equilibrium solutions with different
wavelengths. Finally, we apply our minimum energy estimates
to derive bounds in terms of the linear system size L for
the saturation interface width and the corresponding
saturation time. © 2005 Springer.},
Doi = {10.1007/s00332-004-0634-9},
Key = {fds246959}
}
@article{fds246965,
Author = {Liu, JG and Wang, C},
Title = {High order finite difference method for unsteady
incompressible flow on multi-connected domain in
vorticity-stream function formulation},
Journal = {Computer and Fluids},
Volume = {33},
Number = {2},
Pages = {223-255},
Year = {2004},
url = {http://dx.doi.org/10.1016/S0045-7930(03)00037-9},
Abstract = {Using the vorticity and stream function variables is an
effective way to compute 2-D incompressible flow due to the
facts that the incompressibility constraint for the velocity
is automatically satisfied, the pressure variable is
eliminated, and high order schemes can be efficiently
implemented. However, a difficulty arises in a
multi-connected computational domain in determining the
constants for the stream function on the boundary of the
"holes". This is an especially challenging task for the
calculation of unsteady flows, since these constants vary
with time to reflect the total fluxes of the flow in each
sub-channel. In this paper, we propose an efficient method
in a finite difference setting to solve this problem and
present some numerical experiments, including an accuracy
check of a Taylor vortex-type flow, flow past a
non-symmetric square, and flow in a heat exchanger. ©
2003 Elsevier Ltd. All rights reserved.},
Doi = {10.1016/S0045-7930(03)00037-9},
Key = {fds246965}
}
@article{fds246953,
Author = {Duraisamy, K and Baeder, JD and Liu, JG},
Title = {Concepts and Application of Time-Limiters to High Resolution
Schemes},
Journal = {Journal of Scientific Computing},
Volume = {19},
Number = {1-3},
Pages = {139-162},
Year = {2003},
Month = {December},
ISSN = {0885-7474},
url = {http://dx.doi.org/10.1023/A:1025395707090},
Abstract = {A new class of implicit high-order non-oscillatory time
integration schemes is introduced in a method-of-lines
framework. These schemes can be used in conjunction with an
appropriate spatial discretization scheme for the numerical
solution of time dependent conservation equations. The main
concept behind these schemes is that the order of accuracy
in time is dropped locally in regions where the time
evolution of the solution is not smooth. By doing this, an
attempt is made at locally satisfying monotonicity
conditions, while maintaining a high order of accuracy in
most of the solution domain. When a linear high order time
integration scheme is used along with a high order spatial
discretization, enforcement of monotonicity imposes severe
time-step restrictions. We propose to apply limiters to
these time-integration schemes, thus making them non-linear.
When these new schemes are used with high order spatial
discretizations, solutions remain non-oscillatory for much
larger time-steps as compared to linear time integration
schemes. Numerical results obtained on scalar conservation
equations and systems of conservation equations are highly
promising.},
Doi = {10.1023/A:1025395707090},
Key = {fds246953}
}
@article{fds246966,
Author = {Li, B and Liu, JG},
Title = {Thin film epitaxy with or without slope selection},
Journal = {European Journal of Applied Mathematics},
Volume = {14},
Number = {6},
Pages = {713-743},
Publisher = {Cambridge University Press (CUP)},
Year = {2003},
Month = {December},
url = {http://dx.doi.org/10.1017/S095679250300528X},
Abstract = {Two nonlinear diffusion equations for thin film epitaxy,
with or without slope selection, are studied in this work.
The nonlinearity models the Ehrlich-Schwoebel effect - the
kinetic asymmetry in attachment and detachment of adatoms to
and from terrace boundaries. Both perturbation analysis and
numerical simulation are presented to show that such an
atomistic effect is the origin of a nonlinear morphological
instability, in a rough-smooth-rough pattern, that has been
experimentally observed as transient in an early stage of
epitaxial growth on rough surfaces. Initial-boundary-value
problems for both equations are proven to be well-posed, and
the solution regularity is also obtained. Galerkin spectral
approximations are studied to provide both a priori bounds
for proving the well-posedness and numerical schemes for
simulation. Numerical results are presented to confirm part
of the analysis and to explore the difference between the
two models on coarsening dynamics.},
Doi = {10.1017/S095679250300528X},
Key = {fds246966}
}
@article{fds246968,
Author = {Liu, JG and Wang, C and Johnston, H},
Title = {A Fourth Order Scheme for Incompressible Boussinesq
Equations},
Journal = {Journal of Scientific Computing},
Volume = {18},
Number = {2},
Pages = {253-285},
Year = {2003},
Month = {April},
ISSN = {0885-7474},
url = {http://dx.doi.org/10.1023/A:1021168924020},
Abstract = {A fourth order finite difference method is presented for the
2D unsteady viscous incompressible Boussinesq equations in
vorticity-stream function formulation. The method is
especially suitable for moderate to large Reynolds number
flows. The momentum equation is discretized by a compact
fourth order scheme with the no-slip boundary condition
enforced using a local vorticity boundary condition. Fourth
order long-stencil discretizations are used for the
temperature transport equation with one-sided extrapolation
applied near the boundary. The time stepping scheme for both
equations is classical fourth order Runge-Kutta. The method
is highly efficient. The main computation consists of the
solution of two Poisson-like equations at each Runge-Kutta
time stage for which standard FFT based fast Poisson solvers
are used. An example of Lorenz flow is presented, in which
the full fourth order accuracy is checked. The numerical
simulation of a strong shear flow induced by a temperature
jump, is resolved by two perfectly matching resolutions.
Additionally, we present benchmark quality simulations of a
differentially-heated cavity problem. This flow was the
focus of a special session at the first MIT conference on
Computational Fluid and Solid Mechanics in June
2001.},
Doi = {10.1023/A:1021168924020},
Key = {fds246968}
}
@article{fds246951,
Author = {Wang, C and Liu, JG},
Title = {Positivity property of second-order flux-splitting schemes
for the compressible Euler equations},
Journal = {Discrete and Continuous Dynamical Systems - Series
B},
Volume = {3},
Number = {2},
Pages = {201-228},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2003},
Month = {January},
url = {http://dx.doi.org/10.3934/dcdsb.2003.3.201},
Abstract = {A class of upwind flux splitting methods in the Euler
equations of compressible flow is considered in this paper.
Using the property that Euler flux F(U) is a homogeneous
function of degree one in U, we reformulate the splitting
fluxes with F+ = A+U, F- = A -U, and the corresponding
matrices are either symmetric or symmetrizable and keep only
non-negative and non-positive eigenvalues. That leads to the
conclusion that the first order schemes are positive in the
sense of Lax-Liu [18], which implies that it is L2- stable
in some suitable sense. Moreover, the second order scheme is
a stable perturbation of the first order scheme, so that the
positivity of the second order schemes is also established,
under a CFL-like condition. In addition, these splitting
methods preserve the positivity of density and
energy.},
Doi = {10.3934/dcdsb.2003.3.201},
Key = {fds246951}
}
@article{fds246952,
Author = {Chainais-Hillairet, C and Liu, JG and Peng, YJ},
Title = {Finite volume scheme for multi-dimensional drift-diffusion
equations and convergence analysis},
Journal = {Mathematical Modelling and Numerical Analysis},
Volume = {37},
Number = {2},
Pages = {319-338},
Publisher = {E D P SCIENCES},
Year = {2003},
Month = {January},
url = {http://dx.doi.org/10.1051/m2an:2003028},
Abstract = {We introduce a finite volume scheme for multi-dimensional
drift-diffusion equations. Such equations arise from the
theory of semiconductors and are composed of two continuity
equations coupled with a Poisson equation. In the case that
the continuity equations are non degenerate, we prove the
convergence of the scheme and then the existence of
solutions to the problem. The key point of the proof relies
on the construction of an approximate gradient of the
electric potential which allows us to deal with coupled
terms in the continuity equations. Finally, a numerical
example is given to show the efficiency of the
scheme.},
Doi = {10.1051/m2an:2003028},
Key = {fds246952}
}
@article{fds366915,
Author = {Weinan, E and Liu, JG},
Title = {ADDENDUM TO “GAUGE METHOD FOR VISCOUS INCOMPRESSIBLE
FLOWS”*},
Journal = {Communications in Mathematical Sciences},
Volume = {1},
Number = {4},
Pages = {837-837},
Year = {2003},
Month = {January},
url = {http://dx.doi.org/10.4310/CMS.2003.v1.n4.a10},
Abstract = {Gauge transformation is a well-known concept in physics and
has been used as a computational tool also. In fluid
dynamics, Buttke was the first to use it as a computational
tool to design vortex methods [1], following earlier work of
Oseledets and others [3]. An alternative formulation was
found by Maddocks and Pego [2] using the impetus-striction
variables. This formulation does not seem to have the
problem of numerical instability at the linear level. These
authors are mainly concerned with writing down the
Hamiltonian formulation of Euler’s equation, whereas we
are mainly concerned with using the gauge freedom to
overcome the difficulties with boundary condition.},
Doi = {10.4310/CMS.2003.v1.n4.a10},
Key = {fds366915}
}
@article{fds246950,
Author = {Wang, C and Liu, JG},
Title = {Fourth order convergence of a compact difference solver for
incompressible flow},
Journal = {Commun. Appl. Anal.},
Volume = {7},
Pages = {171-191},
Year = {2003},
Key = {fds246950}
}
@article{fds246961,
Author = {Weinan, E and Liu, JG},
Title = {Gauge method for viscous incompressible flows},
Journal = {Comm. Math. Sci.},
Volume = {1},
Pages = {317-332},
Year = {2003},
Key = {fds246961}
}
@article{fds246967,
Author = {Chern, IL and Liu, JG and Wang, WC},
Title = {Accurate evaluation of electrostatics for macromolecules in
solution},
Journal = {Methods and Applications of Analysis},
Volume = {10},
Pages = {309-328},
Year = {2003},
Key = {fds246967}
}
@article{fds246939,
Author = {Johnston, H and Liu, JG},
Title = {Finite difference schemes for incompressible flow based on
local pressure boundary conditions},
Journal = {Journal of Computational Physics},
Volume = {180},
Number = {1},
Pages = {120-154},
Publisher = {Elsevier BV},
Year = {2002},
Month = {July},
ISSN = {0021-9991},
url = {http://dx.doi.org/10.1006/jcph.2002.7079},
Abstract = {In this paper we discuss the derivation and use of local
pressure boundary conditions for finite difference schemes
for the unsteady incompressible Navier-Stokes equations in
the velocity-pressure formulation. Their use is especially
well suited for the computation of moderate to large
Reynolds number flows. We explore the similarities between
the implementation and use of local pressure boundary
conditions and local vorticity boundary conditions in the
design of numerical schemes for incompressible flow in 2D.
In their respective formulations, when these local numerical
boundary conditions are coupled with a fully explicit
convectively stable time stepping procedure, the resulting
methods are, simple to implement and highly efficient.
Unlike the vorticity formulation, the use of the local
pressure boundary condition approach is readily applicable
to 3D flows. The simplicity of the local pressure boundary
condition approach and its easy adaptation to more general
flow settings make the resulting scheme an attractive
alternative to the more popular methods for solving the
Navier-Stokes equations in the velocity-pressure
formulation. We present numerical results of a second-order
finite difference scheme on a nonstaggered grid using local
pressure boundary conditions. Stability and accuracy of the
scheme applied to Stokes flow is demonstrated using normal
mode analysis. Also described is the extension of the method
to variable density flows. © 2002 Elsevier Science
(USA).},
Doi = {10.1006/jcph.2002.7079},
Key = {fds246939}
}
@article{fds246937,
Author = {Wang, C and Liu, JG},
Title = {Analysis of finite difference schemes for unsteady
Navier-Stokes equations in vorticity formulation},
Journal = {Numerische Mathematik},
Volume = {91},
Number = {3},
Pages = {543-576},
Year = {2002},
Month = {May},
url = {http://dx.doi.org/10.1007/s002110100311},
Abstract = {In this paper, we provide stability and convergence analysis
for a class of finite difference schemes for unsteady
incompressible Navier-Stokes equations in vorticity-stream
function formulation. The no-slip boundary condition for the
velocity is converted into local vorticity boundary
conditions. Thorn's formula, Wilkes' formula, or other local
formulas in the earlier literature can be used in the second
order method; while high order formulas, such as Briley's
formula, can be used in the fourth order compact difference
scheme proposed by E and Liu. The stability analysis of
these long-stencil formulas cannot be directly derived from
straightforward manipulations since more than one interior
point is involved in the formula. The main idea of the
stability analysis is to control local terms by global
quantities via discrete elliptic regularity for stream
function. We choose to analyze the second order scheme with
Wilkes' formula in detail. In this case, we can avoid the
complicated technique necessitated by the Strang-type high
order expansions. As a consequence, our analysis results in
almost optimal regularity assumption for the exact solution.
The above methodology is very general. We also give a
detailed analysis for the fourth order scheme using a 1-D
Stokes model.},
Doi = {10.1007/s002110100311},
Key = {fds246937}
}
@article{fds246938,
Author = {Weinan, E and Liu, JG},
Title = {Projection method III: Spatial discretization on the
staggered grid},
Journal = {Mathematics of Computation},
Volume = {71},
Number = {237},
Pages = {27-47},
Publisher = {American Mathematical Society (AMS)},
Year = {2002},
Month = {January},
url = {http://dx.doi.org/10.1090/S0025-5718-01-01313-8},
Abstract = {In E & Liu (SIAM J Numer. Anal., 1995), we studied
convergence and the structure of the error for several
projection methods when the spatial variable was kept
continuous (we call this the semi-discrete case). In this
paper, we address similar questions for the fully discrete
case when the spatial variables are discretized using a
staggered grid. We prove that the numerical solution in
velocity has full accuracy up to the boundary, despite the
fact that there are numerical boundary layers present in the
semi-discrete solutions.},
Doi = {10.1090/S0025-5718-01-01313-8},
Key = {fds246938}
}
@article{fds246934,
Author = {Liu, JG and Wang, WC},
Title = {An energy-preserving MAC-Yee scheme for the incompressible
MHD equation},
Journal = {Journal of Computational Physics},
Volume = {174},
Number = {1},
Pages = {12-37},
Publisher = {Elsevier BV},
Year = {2001},
Month = {November},
ISSN = {0021-9991},
url = {http://dx.doi.org/10.1006/jcph.2001.6772},
Abstract = {We propose a simple and efficient finite-difference method
for the incompressible MHD equation. The numerical method
combines the advantage of the MAC scheme for the
Navier-Stokes equation and Yee's scheme for the Maxwell
equation. In particular, the semi-discrete version of our
scheme introduces no numerical dissipation and preserves the
energy identity exactly. © 2001 Elsevier
Science.},
Doi = {10.1006/jcph.2001.6772},
Key = {fds246934}
}
@article{fds304582,
Author = {Liu, JG and Xin, Z},
Title = {Convergence of the point vortex method for 2-D vortex
sheet},
Journal = {Mathematics of Computation},
Volume = {70},
Number = {234},
Pages = {595-606},
Publisher = {American Mathematical Society (AMS)},
Year = {2001},
Month = {April},
url = {http://dx.doi.org/10.1090/S0025-5718-00-01271-0},
Abstract = {We give an elementary proof of the convergence of the point
vortex method (PVM) to a classical weak solution for the
two-dimensional incompressible Euler equations with initial
vorticity being a finite Radon measure of distinguished sign
and the initial velocity of locally bounded energy. This
includes the important example of vortex sheets, which
exhibits the classical Kelvin-Helmholtz instability. A
surprise fact is that although the velocity fields generated
by the point vortex method do not have bounded local kinetic
energy, the limiting velocity field is shown to have a
bounded local kinetic energy.},
Doi = {10.1090/S0025-5718-00-01271-0},
Key = {fds304582}
}
@article{fds246873,
Author = {Liu, JG and Weinan, E},
Title = {Simple finite element method in vorticity formulation for
incompressible flows},
Journal = {Mathematics of Computation},
Volume = {70},
Number = {234},
Pages = {579-593},
Publisher = {American Mathematical Society (AMS)},
Year = {2001},
Month = {April},
url = {http://dx.doi.org/10.1090/S0025-5718-00-01239-4},
Abstract = {A very simple and efficient finite element method is
introduced for two and three dimensional viscous
incompressible flows using the vorticity formulation. This
method relies on recasting the traditional finite element
method in the spirit of the high order accurate finite
difference methods introduced by the authors in another
work. Optimal accuracy of arbitrary order can be achieved
using standard finite element or spectral elements. The
method is convectively stable and is particularly suited for
moderate to high Reynolds number flows.},
Doi = {10.1090/S0025-5718-00-01239-4},
Key = {fds246873}
}
@article{fds246935,
Author = {Liu, JG and Weinan, E},
Title = {Simple finite element method in vorticity formulation for
incompressible flow},
Journal = {Math. Comp.},
Volume = {69},
Pages = {1385-1407},
Year = {2001},
Key = {fds246935}
}
@article{fds246936,
Author = {Liu, JG and Xin, Z},
Title = {Convergence of point vortex method for 2-D vortex
sheet},
Journal = {Math. Comp.},
Volume = {70},
Number = {234},
Pages = {565-606},
Year = {2001},
url = {http://dx.doi.org/10.1090/S0025-5718-00-01271-0},
Abstract = {We give an elementary proof of the convergence of the point
vortex method (PVM) to a classical weak solution for the
two-dimensional incompressible Euler equations with initial
vorticity being a finite Radon measure of distinguished sign
and the initial velocity of locally bounded energy. This
includes the important example of vortex sheets, which
exhibits the classical Kelvin-Helmholtz instability. A
surprise fact is that although the velocity fields generated
by the point vortex method do not have bounded local kinetic
energy, the limiting velocity field is shown to have a
bounded local kinetic energy.},
Doi = {10.1090/S0025-5718-00-01271-0},
Key = {fds246936}
}
@article{fds246933,
Author = {Weinan, E and Liu, JG},
Title = {Gauge finite element method for incompressible
flows},
Journal = {International Journal for Numerical Methods in
Fluids},
Volume = {34},
Number = {8},
Pages = {701-710},
Publisher = {WILEY},
Year = {2000},
Month = {December},
ISSN = {0271-2091},
url = {http://dx.doi.org/10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B},
Abstract = {A finite element method for computing viscous incompressible
flows based on the gauge formulation introduced in [Weinan
E. Liu J-G. Gauge method for viscous incompressible flows.
Journal of Computational Physics (submitted)] is presented.
This formulation replaces the pressure by a gauge variable.
This new gauge variable is a numerical tool and differs from
the standard gauge variable that arises from decomposing a
compressible velocity field. It has the advantage that an
additional boundary condition can be assigned to the gauge
variable, thus eliminating the issue of a pressure boundary
condition associated with the original primitive variable
formulation. The computational task is then reduced to
solving standard heat and Poisson equations, which are
approximated by straightforward, piecewise linear (or
higher-order) finite elements. This method can achieve
high-order accuracy at a cost comparable with that of
solving standard heat and Poisson equations. It is naturally
adapted to complex geometry and it is much simpler than
traditional finite elements methods for incompressible
flows. Several numerical examples on both structured and
unstructured grids are presented. Copyright © 2000 John
Wiley & Sons, Ltd.},
Doi = {10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B},
Key = {fds246933}
}
@article{fds246931,
Author = {Liu, JG and Shu, CW},
Title = {A High-Order Discontinuous Galerkin Method for 2D
Incompressible Flows},
Journal = {Journal of Computational Physics},
Volume = {160},
Number = {2},
Pages = {577-596},
Publisher = {Elsevier BV},
Year = {2000},
Month = {May},
url = {http://dx.doi.org/10.1006/jcph.2000.6475},
Abstract = {In this paper we introduce a high-order discontinuous
Galerkin method for two-dimensional incompressible flow in
the vorticity stream-function formulation. The momentum
equation is treated explicitly, utilizing the efficiency of
the discontinuous Galerkin method. The stream function is
obtained by a standard Poisson solver using continuous
finite elements. There is a natural matching between these
two finite element spaces, since the normal component of the
velocity field is continuous across element boundaries. This
allows for a correct upwinding gluing in the discontinuous
Galerkin framework, while still maintaining total energy
conservation with no numerical dissipation and total
enstrophy stability. The method is efficient for inviscid or
high Reynolds number flows. Optimal error estimates are
proved and verified by numerical experiments. © 2000
Academic Press.},
Doi = {10.1006/jcph.2000.6475},
Key = {fds246931}
}
@article{fds246930,
Author = {Liu, JG and Xin, Z},
Title = {Convergence of a Galerkin method for 2-D discontinuous Euler
flows},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {53},
Number = {6},
Pages = {786-798},
Publisher = {Wiley},
Year = {2000},
Month = {January},
url = {http://dx.doi.org/10.1002/(SICI)1097-0312(200006)53:6<786::AID-CPA3>3.0.CO;2-Y},
Abstract = {We prove the convergence of a discontinuous Galerkin method
approximating the 2-D incompressible Euler equations with
discontinuous initial vorticity: ω0 ∈ L2(Ω).
Furthermore, when ω0 ∈ L∞(Ω), the whole sequence is
shown to be strongly convergent. This is the first
convergence result in numerical approximations of this
general class of discontinuous flows. Some important flows
such as vortex patches belong to this class. © 2000 John
Wiley & Sons, Inc.},
Doi = {10.1002/(SICI)1097-0312(200006)53:6<786::AID-CPA3>3.0.CO;2-Y},
Key = {fds246930}
}
@article{fds246932,
Author = {Wang, C and Liu, JG},
Title = {Convergence of gauge method for incompressible
flow},
Journal = {Mathematics of Computation},
Volume = {69},
Number = {232},
Pages = {1385-1407},
Year = {2000},
Month = {January},
url = {http://dx.doi.org/10.1090/s0025-5718-00-01248-5},
Abstract = {A new formulation, a gauge formulation of the incompressible
Navier-Stokes equations in terms of an auxiliary field a and
a gauge variable φ, u = a + ∇φ, was proposed recently by
E and Liu. This paper provides a theoretical analysis of
their formulation and verifies the computational advantages.
We discuss the implicit gauge method, which uses backward
Euler or Crank-Nicolson in time discretization. However, the
boundary conditions for the auxiliary field a are
implemented explicitly (vertical extrapolation). The
resulting momentum equation is decoupled from the kinematic
equation, and the computational cost is reduced to solving a
standard heat and Poisson equation. Moreover, such explicit
boundary conditions for the auxiliary field a will be shown
to be unconditionally stable for Stokes equations. For the
full nonlinear Navier-Stokes equations the time stepping
constraint is reduced to the standard CFL constraint Δt/Δx
≤ C. We also prove first order convergence of the gauge
method when we use MAC grids as our spatial discretization.
The optimal error estimate for the velocity field is also
obtained.},
Doi = {10.1090/s0025-5718-00-01248-5},
Key = {fds246932}
}
@article{fds246927,
Author = {Lefloch, PG and Liu, JG},
Title = {Generalized monotone schemes, discrete paths of extrema, and
discrete entropy conditions},
Journal = {Mathematics of Computation},
Volume = {68},
Number = {227},
Pages = {1025-1055},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1090/s0025-5718-99-01062-5},
Abstract = {Solutions of conservation laws satisfy the monotonicity
property: the number of local extrema is a non-increasing
function of time, and local maximum/minimum values
decrease/increase monotonically in time. This paper
investigates this property from a numerical standpoint. We
introduce a class of fully discrete in space and time, high
order accurate, difference schemes, called generalized
monotone schemes. Convergence toward the entropy solution is
proven via a new technique of proof, assuming that the
initial data has a finite number of extremum values only,
and the flux-function is strictly convex. We define discrete
paths of extrema by tracking local extremum values in the
approximate solution. In the course of the analysis we
establish the pointwise convergence of the trace of the
solution along a path of extremum. As a corollary, we obtain
a proof of convergence for a MUSCL-type scheme that is
second order accurate away from sonic points and
extrema.},
Doi = {10.1090/s0025-5718-99-01062-5},
Key = {fds246927}
}
@article{fds246929,
Author = {Wang, ZJ and Liu, JG and Childress, S},
Title = {Connection between corner vortices and shear layer
instability in flow past an ellipse},
Journal = {Physics of Fluids},
Volume = {11},
Number = {9},
Pages = {2446-2448},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1063/1.870108},
Abstract = {We investigate, by numerical simulation, the shear layer
instability associated with the outer layer of a spiral
vortex formed behind an impulsively started thin ellipse.
The unstable free shear layer undergoes a secondary
instability. We connect this instability with the dynamics
of corner vortices adjacent to the tip of the ellipse by
observing that the typical turnover time of the corner
vortex matches the period of the unstable mode in the shear
layer. We suggest that the corner vortex acts as a signal
generator, and produces periodic perturbation which triggers
the instability. © 1999 American Institute of
Physics.},
Doi = {10.1063/1.870108},
Key = {fds246929}
}
@article{fds246926,
Author = {Choi, H and Liu, JG},
Title = {The Reconstruction of Upwind Fluxes for Conservation Laws:
Its Behavior in Dynamic and Steady State
Calculations},
Journal = {Journal of Computational Physics},
Volume = {144},
Number = {2},
Pages = {237-256},
Publisher = {Elsevier BV},
Year = {1998},
Month = {August},
url = {http://dx.doi.org/10.1006/jcph.1998.5970},
Abstract = {The Euler equation of compressible flows is solved by the
finite volume method, where high order accuracy is achieved
by the reconstruction of each component of upwind fluxes of
a flux splitting using the biased averaging procedure.
Compared to the solution reconstruction in Godunov-type
methods, its implementation is simple and easy, and the
computational complexity is relatively low. This approach is
parameter-free and requires neither a Riemann solver nor
field-by-field decomposition. The numerical results from
both dynamic and steady state calculations demonstrate the
accuracy and robustness of this approach. Some techniques
for the acceleration of the convergence to the steady state
are discussed, including multigrid and multistage
Runge-Kutta time methods. © 1998 Academic
Press.},
Doi = {10.1006/jcph.1998.5970},
Key = {fds246926}
}
@article{fds246925,
Author = {Xu, E and Liu, JG},
Title = {Pricing of mortgage-backed securities with option-adjusted
spread},
Journal = {Managerial Finance},
Volume = {24},
Pages = {94-109},
Year = {1998},
Key = {fds246925}
}
@article{fds246922,
Author = {E, W and Liu, JG},
Title = {Finite Difference Methods for 3D Viscous Incompressible
Flows in the Vorticity-Vector Potential Formulation on
Nonstaggered Grids},
Journal = {Journal of Computational Physics},
Volume = {138},
Number = {1},
Pages = {57-82},
Publisher = {Elsevier BV},
Year = {1997},
Month = {November},
url = {http://dx.doi.org/10.1006/jcph.1997.5815},
Abstract = {Simple, efficient, and accurate finite difference methods
are introduced for 3D unsteady viscous incompressible flows
in the vorticity-vector potential formulation on
nonstaggered grids. Two different types of methods are
discussed. They differ in the implementation of the normal
component of the vorticity boundary condition and
consequently the enforcement of the divergence free
condition for vorticity. Both second-order and fourth-order
accurate schemes are developed. A detailed accuracy test is
performed, revealing the structure of the error and the
effect of how the convective terms are discretized near the
boundary. The influence of the divergence free condition for
vorticity to the overall accuracy is studied. Results on the
cubic driven cavity flow at Reynolds number 500 and 3200 are
shown and compared with that of the MAC scheme. © 1997
Academic Press.},
Doi = {10.1006/jcph.1997.5815},
Key = {fds246922}
}
@article{fds246923,
Author = {Chen, GQ and Liu, JG},
Title = {Convergence of difference schemes with high resolution for
conservation laws},
Journal = {Mathematics of Computation},
Volume = {66},
Number = {219},
Pages = {1027-1053},
Year = {1997},
Month = {January},
url = {http://dx.doi.org/10.1090/s0025-5718-97-00859-4},
Abstract = {We are concerned with the convergence of Lax-Weridroff type
schemes with high resolution to the entropy solutions fo:
conservation laws. These schemes include the original
Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and
its two step versions-the Richtrayer scheme and the
MacCormack scheme. For the convex scalar conservation laws
with algebraic growth flux functions, we prove the
convergence of these schemes to the weak solutions
satisfying appropriate entropy inequalities. The proof is
based on detailed Lp estimates of the approximate solutions,
H-1 compactness estimates of the corresponding entropy
dissipation measures, and some compensated compactness
frameworks. Then these techniques are generalized to study
the convergence problem for the nonconvex scalar case and
the hyperbolic systems of conservation laws.},
Doi = {10.1090/s0025-5718-97-00859-4},
Key = {fds246923}
}
@article{fds246924,
Author = {Weinan, E and Liu, JG},
Title = {Finite difference schemes for incompressible flows in the
velocity - impulse density formulation},
Journal = {Journal of Computational Physics},
Volume = {130},
Number = {1},
Pages = {67-76},
Publisher = {Elsevier BV},
Year = {1997},
Month = {January},
url = {http://dx.doi.org/10.1006/jcph.1996.5537},
Abstract = {We consider finite difference schemes based on the impulse
density variable. We show that the original velocity -
impulse density formulation of Oseledets is marginally
ill-posed for the inviscid flow, and this has the
consequence that some ordinarily stable numerical methods in
other formulations become unstable in the velocity - impulse
density formulation. We present numerical evidence of this
instability. We then discuss the construction of stable
finite difference schemes by requiring that at the numerical
level the nonlinear terms be convertible to similar terms in
the primitive variable formulation. Finally we give a
simplified velocity - impulse density formulation which is
free of these complications and yet retains the nice
features of the original velocity - impulse density
formulation with regard to the treatment of boundary. We
present numerical results on this simplified formulation for
the driven cavity flow on both the staggered and
non-staggered grids. © 1997 Academic Press.},
Doi = {10.1006/jcph.1996.5537},
Key = {fds246924}
}
@article{fds246916,
Author = {Weinan, E and Liu, JG},
Title = {Vorticity boundary condition and related issues for finite
difference schemes},
Journal = {Journal of Computational Physics},
Volume = {124},
Number = {2},
Pages = {368-382},
Publisher = {Elsevier BV},
Year = {1996},
Month = {March},
url = {http://dx.doi.org/10.1006/jcph.1996.0066},
Abstract = {This paper discusses three basic issues related to the
design of finite difference schemes for unsteady viscous
incompressible flows using vorticity formulations: the
boundary condition for vorticity, an efficient time-stepping
procedure, and the relation between these schemes and the
ones based on velocity-pressure formulation. We show that
many of the newly developed global vorticity boundary
conditions can actually be written as some local formulas
derived earlier. We also show that if we couple a standard
centered difference scheme with third-or fourth-order
explicit Runge-Kutta methods, the resulting schemes have no
cell Reynolds number constraints. For high Reynolds number
flows, these schemes are stable under the CFL condition
given by the convective terms. Finally, we show that the
classical MAC scheme is the same as Thom's formula coupled
with second-order centered differences in the interior, in
the sense that one can define discrete vorticity in a
natural way for the MAC scheme and get the same values as
the ones computed from Thom's formula. We use this to derive
an efficient fourth-order Runge-Kutta time discretization
for the MAC scheme from the one for Thom's formula. We
present numerical results for driven cavity flow at high
Reynolds number (105). © 1996 Academic Press,
Inc.},
Doi = {10.1006/jcph.1996.0066},
Key = {fds246916}
}
@article{fds246915,
Author = {Jin, S and Liu, JG},
Title = {The effects of numerical viscosities: I. Slowly moving
shocks},
Journal = {Journal of Computational Physics},
Volume = {126},
Number = {2},
Pages = {373-389},
Publisher = {Elsevier BV},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1006/jcph.1996.0144},
Abstract = {We begin a systematical study on the effect of numerical
viscosities. In this paper we investigate the behavior of
shock-capturing methods for slowly moving shocks. It is
known that for slowly moving shocks even a first-order
scheme, such as the Godunov or Roe type methods, will
generate downstream oscillatory wave patterns that cannot be
effectively damped by the dissipation of these first-order
schemes. The purpose of this paper is to understand the
formation and behavior of these downstream patterns. Our
study shows that the downstream errors are generated by the
unsteady nature of the viscous shock profiles and behave
diffusively. The scenario is as follows. When solving the
compressible Euler equations by shock capturing methods, the
smeared density profile introduces a momentum spike at the
shock location if the shock moves slowly. Downstream waves
will necessarily emerge in order to balance the momentum
mass carried by the spike for the momentum conservation.
Although each family of waves decays in l∞ and l2 while
they preserve the same mass, the perturbing nature of the
viscous or spike profile is a constant source for the
generation of new downstream waves, causing spurious
solutions for all time. Higher order TVD or ENO type
interpolations accentuate this problem. © 1996 Academic
Press, Inc.},
Doi = {10.1006/jcph.1996.0144},
Key = {fds246915}
}
@article{fds246917,
Author = {Weinan, E and Liu, JG},
Title = {Essentially compact schemes for unsteady viscous
incompressible flows},
Journal = {Journal of Computational Physics},
Volume = {126},
Number = {1},
Pages = {122-138},
Publisher = {Elsevier BV},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1006/jcph.1996.0125},
Abstract = {A new fourth-order accurate finite difference scheme for the
computation of unsteady viscous incompressible flows is
introduced. The scheme is based on the vorticity-stream
function formulation. It is essentially compact and has the
nice features of a compact scheme with regard to the
treatment of boundary conditions. It is also very efficient,
at every time step or Runge-Kutta stage, only two
Poisson-like equations have to be solved. The Poisson-like
equations are amenable to standard fast Poisson solvers
usually designed for second order schemes. Detailed
comparison with the second-order scheme shows the clear
superiority of this new fourth-order scheme in resolving
both the boundary layers and the gross features of the flow.
This efficient fourth-order scheme also made it possible to
compute the driven cavity flow at Reynolds number 106 on a
10242 grid at a reasonable cost. Fourth-order convergence is
proved under mild regularity requirements. This is the first
such result to our knowledge. © 1996 Academic Press,
Inc.},
Doi = {10.1006/jcph.1996.0125},
Key = {fds246917}
}
@article{fds246918,
Author = {Weinan, E and Liu, JG},
Title = {Projection method II: Godunov-Ryabenki analysis},
Journal = {SIAM Journal on Numerical Analysis},
Volume = {33},
Number = {4},
Pages = {1597-1621},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1137/s003614299426450x},
Abstract = {This is the second of a series of papers on the subject of
projection methods for viscous incompressible flow
calculations. The purpose of the present paper is to explain
why the accuracy of the velocity approximation is not
affected by (1) the numerical boundary layers in the
approximation of pressure and the intermediate velocity
field and (2) the noncommutativity of the projection
operator and the laplacian. This is done by using a
Godunov-Ryabenki type of analysis in a rigorous fashion. By
doing so, we hope to be able to convey the message that
normal mode analysis is basically sufficient for
understanding the stability and accuracy of a
finite-difference method for the Navier-Stokes equation even
in the presence of boundaries. As an example, we analyze the
second-order projection method based on pressure increment
formulations used by van Kan and Bell, Colella, and Glaz.
The leading order error term in this case is of O(Δt) and
behaves as high frequency oscillations over the whole
domain, compared with the O(Δt1/2) numerical boundary
layers found in the second-order Kim-Moin
method.},
Doi = {10.1137/s003614299426450x},
Key = {fds246918}
}
@article{fds246919,
Author = {Levermore, CD and Liu, JG},
Title = {Large oscillations arising in a dispersive numerical
scheme},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {99},
Number = {2-3},
Pages = {191-216},
Publisher = {Elsevier BV},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1016/S0167-2789(96)00157-1},
Abstract = {We study the oscillatory behavior that arises in solutions
of a dispersive numerical scheme for the Hopf equation
whenever the classical solution of that equation develops a
singularity. Modulation equations are derived that describe
period-two oscillations so long as the solution of those
equations takes values for which the equations are
hyperbolic. However, those equations have an elliptic region
that may be entered by its solutions in a unite time, after
which the corresponding period-two oscillations are seen to
break down. This kind of phenomenon has not been observed
for integrable schemes. The generation and propagation of
period-two oscillations are asymptotically analyzed and a
matching formula is found for the transition between
oscillatory and nonoscillatory regions. Modulation equations
are also presented for period-three oscillations. Numerical
experiments are carried out that illustrate our analysis. ©
1996 Elsevier Science B.V. All rights reserved.},
Doi = {10.1016/S0167-2789(96)00157-1},
Key = {fds246919}
}
@article{fds246920,
Author = {Liu, JG and Xin, Z},
Title = {Kinetic and viscous boundary layers for broadwell
equations},
Journal = {Transport Theory and Statistical Physics},
Volume = {25},
Number = {3-5},
Pages = {447-461},
Publisher = {Informa UK Limited},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1080/00411459608220713},
Abstract = {In this paper, we investigate the boundary layer behavior of
solutions to the one dimensional Broadwell model of the
nonlinear Boltzmann equation for small mean free path. We
consider the analogue of Maxwell's diffusive and the
reflexive boundary conditions. It is found that even for
such a simple model, there are boundary layers due to purely
kinetic effects which cannot be detected by the
corresponding Navier-Stokes system. It is also found
numerically that a compressive boundary layer is not always
stable in the sense that it may detach from the boundary and
move into the interior of the gas as a shock
layer.},
Doi = {10.1080/00411459608220713},
Key = {fds246920}
}
@article{fds246914,
Author = {Jin, S and Liu, JG},
Title = {Oscillations induced by numerical viscosities},
Journal = {Mat. Contemp.},
Volume = {10},
Pages = {169-180},
Year = {1996},
Key = {fds246914}
}
@article{fds246921,
Author = {Liu, JG and Xin, Z},
Title = {Boundary layer behavior in the fluid-dynamic limit for a
nonlinear model Boltzmann equation},
Journal = {Arch. Rat. Mech. Anal.},
Volume = {135},
Number = {1},
Pages = {61-105},
Publisher = {Springer Nature},
Year = {1996},
url = {http://dx.doi.org/10.1007/BF02198435},
Abstract = {In this paper, we study the fluid-dynamic limit for the
one-dimensional Broadwell model of the nonlinear Boltzmann
equation in the presence of boundaries. We consider an
analogue of Maxwell's diffusive and reflective boundary
conditions. The boundary layers can be classified as either
compressive or expansive in terms of the associated
characteristic fields. We show that both expansive and
compressive boundary layers (before detachment) are
nonlinearly stable and that the layer effects are localized
so that the fluid dynamic approximation is valid away from
the boundary. We also show that the same conclusion holds
for short time without the structural conditions on the
boundary layers. A rigorous estimate for the distance
between the kinetic solution and the fluid-dynamic solution
in terms of the mean-free path in the L∞ -norm is obtained
provided that the interior fluid flow is smooth. The rate of
convergence is optimal.},
Doi = {10.1007/BF02198435},
Key = {fds246921}
}
@article{fds362427,
Author = {E, W and Liu, J-G},
Title = {Finite difference schemes for incompressible flows in
vorticity formulations},
Journal = {ESAIM: Proceedings},
Volume = {1},
Pages = {181-195},
Publisher = {EDP Sciences},
Editor = {Gagnon, Y and Cottet, G-H and G., D and F., A and Meiburg,
E},
Year = {1996},
url = {http://dx.doi.org/10.1051/proc:1996009},
Doi = {10.1051/proc:1996009},
Key = {fds362427}
}
@article{fds246912,
Author = {Weinan, E and Liu, JG},
Title = {Projection method I: convergence and numerical boundary
layers},
Journal = {SIAM J. Numer. Anal.},
Volume = {32},
Number = {4},
Pages = {1017-1057},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1995},
url = {http://dx.doi.org/10.1137/0732047},
Doi = {10.1137/0732047},
Key = {fds246912}
}
@article{fds246913,
Author = {Liu, JG and Xin, Z},
Title = {Convergence of vortex methods for weak solutions to the 2-D
Euler equations with vortex sheets data},
Journal = {Comm. Pure Appl. Math.},
Volume = {48},
Number = {6},
Pages = {611-628},
Year = {1995},
url = {http://dx.doi.org/10.1002/cpa.3160480603},
Abstract = {We prove the convergence of vortex blob methods to classical
weak solutions for the two‐dimensional incompressible
Euler equations with initial data satisfying the conditions
that the vorticity is a finite Radon measure of
distinguished sign and the kinetic energy is locally
bounded. This includes the important example of vortex
sheets. The result is valid as long as the computational
grid size h does not exceed the smoothing blob size ε,
i.e., h/ε ≦ C.. ©1995 John Wiley & Sons, Inc. Copyright
© 1995 Wiley Periodicals, Inc., A Wiley
Company},
Doi = {10.1002/cpa.3160480603},
Key = {fds246913}
}
@article{fds246911,
Author = {Jin, S and Liu, JG},
Title = {Relaxation and diffusion enhanced dispersive
waves},
Journal = {Proceedings of The Royal Society of London, Series A:
Mathematical and Physical Sciences},
Volume = {446},
Number = {1928},
Pages = {555-563},
Year = {1994},
Month = {January},
url = {http://dx.doi.org/10.1098/rspa.1994.0120},
Abstract = {The development of shocks in nonlinear hyperbolic
conservation laws may be regularized through either
diffusion or relaxation. However, we have observed
surprisingly that for some physical problems, when both of
the smoothing factors diffusion and relaxation coexist,
under appropriate asymptotic assumptions, the dispersive
waves are enhanced. This phenomenon is studied
asymptotically in the sense of the Chapman-Enskog expansion
and demonstrated numerically.},
Doi = {10.1098/rspa.1994.0120},
Key = {fds246911}
}
@article{fds246910,
Author = {Lefloch, P and Liu, JG},
Title = {Discrete entropy and monotonicity criteria for hyperbolic
conservation laws},
Journal = {C.R. Acad. Sci. Paris.},
Volume = {319},
Number = {8},
Pages = {881-886},
Publisher = {ELSEVIER FRANCE-EDITIONS SCIENTIFIQUES MEDICALES
ELSEVIER},
Year = {1994},
Key = {fds246910}
}
@article{fds359206,
Author = {Chen, G-Q and Liu, J-G},
Title = {Convergence of Second-Order Schemes for Isentropic Gas
Dynamics},
Journal = {Mathematics of Computation},
Volume = {61},
Number = {204},
Pages = {607-607},
Publisher = {JSTOR},
Year = {1993},
Month = {October},
url = {http://dx.doi.org/10.2307/2153243},
Doi = {10.2307/2153243},
Key = {fds359206}
}
@article{fds246909,
Author = {Liu, JG and Xin, Z},
Title = {Nonlinear stability of discrete shocks for systems of
conservation laws},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {125},
Number = {3},
Pages = {217-256},
Publisher = {Springer Nature},
Year = {1993},
Month = {September},
ISSN = {0003-9527},
url = {http://dx.doi.org/10.1007/BF00383220},
Abstract = {In this paper we study the asymptotic nonlinear stability of
discrete shocks for the Lax-Friedrichs scheme for
approximating general m×m systems of nonlinear hyperbolic
conservation laws. It is shown that weak single discrete
shocks for such a scheme are nonlinearly stable in the
Lp-norm for all p ≧ 1, provided that the sums of the
initial perturbations equal zero. These results should shed
light on the convergence of the numerical solution
constructed by the Lax-Friedrichs scheme for the
single-shock solution of system of hyperbolic conservation
laws. If the Riemann solution corresponding to the given
far-field states is a superposition of m single shocks from
each characteristic family, we show that the corresponding
multiple discrete shocks are nonlinearly stable in Lp (P ≧
2). These results are proved by using both a weighted
estimate and a characteristic energy method based on the
internal structures of the discrete shocks and the essential
monotonicity of the Lax-Friedrichs scheme. © 1993
Springer-Verlag.},
Doi = {10.1007/BF00383220},
Key = {fds246909}
}
@article{fds348002,
Author = {Liu, J-G and Xin, Z},
Title = {L 1 -Stability of Stationary Discrete Shocks},
Journal = {Mathematics of Computation},
Volume = {60},
Number = {201},
Pages = {233-233},
Publisher = {JSTOR},
Year = {1993},
Month = {January},
url = {http://dx.doi.org/10.2307/2153163},
Doi = {10.2307/2153163},
Key = {fds348002}
}
@article{fds246906,
Author = {Chen, GQ and Liu, JG},
Title = {Convergence of second-order schemes for isentropic gas
dynamics},
Journal = {Math. Comp.},
Volume = {61},
Number = {204},
Pages = {607-629},
Publisher = {AMER MATHEMATICAL SOC},
Year = {1993},
url = {http://dx.doi.org/10.2307/2153243},
Abstract = {Convergence of a second-order shock-capturing scheme for the
system of isentropic gas dynamics with L initial data is
established by analyzing the entropy dissipation measures.
This scheme is modified from the classical MUSCL scheme to
treat the vacuum problem in gas fluids and to capture local
entropy near shock waves. Convergence of this scheme for the
piston problem is also discussed. © 1993 American
Mathematical Society. ∞},
Doi = {10.2307/2153243},
Key = {fds246906}
}
@article{fds246907,
Author = {Engquist, B and Liu, JG},
Title = {Numerical methods for oscillatory solutions to hyperbolic
problems},
Journal = {Comm. Pure Appl. Math.},
Volume = {46},
Number = {10},
Pages = {1327-1361},
Publisher = {WILEY},
Year = {1993},
url = {http://dx.doi.org/10.1002/cpa.3160461003},
Abstract = {Difference approximations of hyperbolic partial differential
equations with highly oscillatory coefficients and initial
values are studied. Analysis of strong and weak convergence
is carried out in the practically interesting case when the
discretization step sizes are essentially independent of the
oscillatory wave lengths. © 1993 John Wiley & Sons, Inc.
Copyright © 1993 Wiley Periodicals, Inc., A Wiley
Company},
Doi = {10.1002/cpa.3160461003},
Key = {fds246907}
}
@article{fds246908,
Author = {Liu, JG and Xin, Z},
Title = {L1-stability of stationary discrete shocks},
Journal = {Math. Comp.},
Volume = {60},
Number = {201},
Pages = {233-244},
Publisher = {American Mathematical Society (AMS)},
Year = {1993},
url = {http://dx.doi.org/10.1090/S0025-5718-1993-1159170-7},
Abstract = {The nonlinear stability in the Lpnorm, p 1 , of stationary
weak discrete shocks for the Lax-Friedrichs scheme
approximating general m x m systems of nonlinear hyperbolic
conservation laws is proved, provided that the summations of
the initial perturbations equal zero. The result is proved
by using both a weighted estimate and characteristic energy
method based on the internal structures of the discrete
shocks and the essential monotonicity of the Lax-Friedrichs
scheme. © 1993 American Mathematical Society.},
Doi = {10.1090/S0025-5718-1993-1159170-7},
Key = {fds246908}
}
%% Papers Accepted
@article{fds320739,
Author = {P. Degond and J.-G. Liu and S. Merino-Aceituno and T.
Tardiveau},
Title = {Continuum dynamics of the intention field under weakly
cohesive social interactions},
Journal = {Math. Models Methods Appl. Sci.},
Year = {2016},
Key = {fds320739}
}
@article{fds320743,
Author = {Y. Gao and J.-G. Liu and J. Lu},
Title = {Continuum limit of a mesoscopic model of step motion on
vicinal surfaces},
Journal = {J. Nonlinear Science},
Year = {2016},
Key = {fds320743}
}
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