%% Books
@book{fds165493,
Title = {Multiscale 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 = {9789814273251},
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 = {9780821847282},
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{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 = {5467},
Year = {2019},
Month = {June},
url = {http://dx.doi.org/10.1016/j.physd.2019.01.004},
Abstract = {© 2019 Elsevier B.V. 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
brokenbond 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{fds340536,
Author = {Lafata, KJ and Hong, JC and Geng, R and Ackerson, BG and Liu, JG and Zhou,
Z and Torok, J and Kelsey, CR and Yin, FF},
Title = {Association of pretreatment 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/10.1088/13616560/aaf5a5},
Abstract = {The purpose of this work was to investigate the potential
relationship between radiomic features extracted from
pretreatment xray CT images and clinical outcomes
following stereotactic body radiation therapy (SBRT) for
nonsmallcell lung cancer (NSCLC). Seventy patients who
received SBRT for stage1 NSCLC were retrospectively
identified. The tumor was contoured on pretreatment
freebreathing CT images, from which 43 quantitative
radiomic features were extracted to collectively capture
tumor morphology, intensity, finetexture, and
coarsetexture. Treatment failure was defined based on
cancer recurrence, local cancer recurrence, and nonlocal
cancer recurrence following SBRT. The univariate association
between each radiomic feature and each clinical endpoint was
analyzed using Welch's ttest, and pvalues 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 LongRunHighGrayLevelEmphasis
(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 nonlocal recurrence endpoints, respectively. The
CTbased radiomic features used in this study may be more
associated with local failure than nonlocal failure
following SBRT for stage I NSCLC. This finding is supported
by both univariate and multivariate analyses.},
Doi = {10.1088/13616560/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 = {129},
Year = {2019},
Month = {January},
url = {http://dx.doi.org/10.1142/S0218202519500015},
Abstract = {© 2019 World Scientific Publishing Company. 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{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 = {44334453},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2018},
Month = {December},
url = {http://dx.doi.org/10.3934/dcdsb.2018170},
Abstract = {© 2018 American Institute of Mathematical Sciences. All
Rights Reserved. We study a continuum model for solid films
that arises from the modeling of onedimensional step flows
on a vicinal surface in the attachmentdetachmentlimited
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 = {31093135},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2018},
Month = {October},
url = {http://dx.doi.org/10.3934/dcdsb.2017210},
Abstract = {© 2018 American Institute of Mathematical Sciences. All
rights reserved. 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 blowup
criteria, we find relatively sharp bounds of the blowup
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 blowup time tends to zero while if the
initial value is small, the blowup 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 onedimensional time fractional
ODEs},
Journal = {Applied Mathematics Letters},
Volume = {83},
Pages = {8794},
Publisher = {Elsevier BV},
Year = {2018},
Month = {September},
url = {http://dx.doi.org/10.1016/j.aml.2018.03.015},
Abstract = {© 2018 Elsevier Ltd In this note, we prove or reprove
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 D cγ u=Au p . 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 = {10441096},
Publisher = {Elsevier BV},
Year = {2018},
Month = {August},
url = {http://dx.doi.org/10.1016/j.jde.2018.03.025},
Abstract = {© 2018 Elsevier Inc. 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)=−s n,γ ∫ R n [Formula presented]ρ(y)dy is
the Riesz potential with a singular kernel which takes into
account the long rang interaction. We first establish L r
−L q estimates and weighted estimates of the fundamental
solutions (P(x,t),Y(x,t)) (or equivalently, the solution
operators (S αβ (t),T αβ (t))). Then, we prove the
existence and uniqueness of the mild solutions when initial
data are in L p spaces, or the weighted spaces. Similar to
Keller–Segel equations, if the initial data are small in
critical space L p c (R n ) (p c =[Formula presented]), we
construct the global existence. Furthermore, we prove the L
1 integrability and integral preservation when the initial
data are in L 1 (R n )∩L p (R n ) or L 1 (R n )∩L p c (R
n ). 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{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 = {7394},
Publisher = {Elsevier BV},
Year = {2018},
Month = {July},
url = {http://dx.doi.org/10.1016/j.jcp.2018.03.013},
Abstract = {© 2018 Elsevier Inc. We consider a class of tumor growth
models under the combined effects of densitydependent
pressure and cell multiplication, with a free boundary model
as its singular limit when the pressuredensity 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 pressuredriven 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
predictioncorrection reformulation that can accurately
approximate the front propagation even when the nonlinearity
is extremely strong. We show that the semidiscrete 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 = {075010075010},
Publisher = {IOP Publishing},
Year = {2018},
Month = {May},
url = {http://dx.doi.org/10.1088/13616420/aac220},
Abstract = {© 2018 IOP Publishing Ltd. 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 incomingoutgoing pair of light intensity. We
formulate it as a PDEconstraint 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 = {10.1088/13616420/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 = {54895526},
Publisher = {ACADEMIC PRESS INC ELSEVIER SCIENCE},
Year = {2018},
Month = {April},
url = {http://dx.doi.org/10.1016/j.jde.2018.01.001},
Abstract = {© 2018 Elsevier Inc. 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 = {pEuler equations and pNavier–Stokes equations},
Journal = {Journal of Differential Equations},
Volume = {264},
Number = {7},
Pages = {47074748},
Publisher = {Elsevier BV},
Year = {2018},
Month = {April},
url = {http://dx.doi.org/10.1016/j.jde.2017.12.023},
Abstract = {© 2017 Elsevier Inc. We propose in this work new systems of
equations which we call pEuler equations and
pNavier–Stokes equations. pEuler equations are derived
as the Euler–Lagrange equations for the action represented
by the Benamou–Brenier characterization of Wassersteinp
distances, with incompressibility constraint. pEuler
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 pEuler equations
have streamfunctionvorticity formulation, where the
vorticity is given by the pLaplacian of the streamfunction.
By adding diffusion presented by γLaplacian of the
velocity, we obtain what we call pNavier–Stokes
equations. If γ=p, the a priori energy estimates for the
velocity and momentum have dual symmetries. Using these
energy estimates and a timeshift estimate, we show the
global existence of weak solutions for the pNavier–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/s005260181326x},
Abstract = {© 2018, SpringerVerlag GmbH Germany, part of Springer
Nature. 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 suitablydefined convex functional in a
nonreflexive space. Then by restricting it to a Hilbert
space and proving the uniqueness of its subdifferential, we
can apply the classical maximal monotone operator theory.
The mathematical difficulty is due to the fact that w hh 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 w hh is
replaced by its absolutely continuous part.},
Doi = {10.1007/s005260181326x},
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 = {777789},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.4310/CMS.2018.v16.n3.a7},
Abstract = {© 2018 International Press. 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 L 1
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.a7},
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 = {39633995},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1137/17M1145549},
Abstract = {© 2018 Society for Industrial and Applied Mathematics. The
AubinLions 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 AubinLions 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
Volterratype 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 NavierStokes equations
with constant density and time fractional KellerSegel
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 = {28072838},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1137/17M1132756},
Abstract = {© 2018 Society for Industrial and Applied Mathematics In
this paper, we present a dispersive regularization approach
to construct a global Npeakon 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 Npeakon 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
Npeakon 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 Npeakon
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 = {28672900},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1137/17M1160318},
Abstract = {© 2018 Society for Industrial and Applied Mathematics. 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
RiemannLiouville fractional calculus which agrees with the
traditional RiemannLiouville 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 = {189198},
Publisher = {American Mathematical Society (AMS)},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1090/qam/1479},
Abstract = {© 2017 Brown University. We study in this work convolution
groups generated by completely monotone sequences related to
the ubiquitous timedelay 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
timecontinuous RiemannLiouville 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 subcell shock capturing schemes via JinXin
relaxation and glimm front sampling for scalar conservation
laws},
Journal = {Mathematics of Computation},
Volume = {87},
Number = {311},
Pages = {10831126},
Publisher = {American Mathematical Society (AMS)},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1090/mcom/3253},
Abstract = {© 2017 American Mathematical Society. We introduce a
subcell shock capturing method for scalar conservation laws
built upon the JinXin relaxation framework. Here, subcell
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 selfsimilar solutions. These
solutions are then sampled using Glimm's random choice
method to advance in time. The resulting scheme combines the
simplicity of the JinXin 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 subcell 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 nonlinear flux
functions. Numerical results show that the proposed method
indeed captures shocksincluding interacting shockssharply
without any smearing.},
Doi = {10.1090/mcom/3253},
Key = {fds333568}
}
@article{fds333569,
Author = {Liu, JG and Wang, L and Zhou, Z},
Title = {Positivitypreserving and asymptotic preserving method for
2D KellerSegal equations},
Journal = {Mathematics of Computation},
Volume = {87},
Number = {311},
Pages = {11651189},
Publisher = {American Mathematical Society (AMS)},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1090/mcom/3250},
Abstract = {© 2017 American Mathematical Society. We propose a
semidiscrete scheme for 2D KellerSegel 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 quasistatic 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{fds340537,
Author = {Gao, Y and Liu, JG},
Title = {The modified CamassaHolm equation in Lagrangian
coordinates},
Journal = {Discrete & Continuous Dynamical Systems B},
Volume = {23},
Number = {6},
Pages = {25452592},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2018},
url = {http://dx.doi.org/10.3934/dcdsb.2018067},
Doi = {10.3934/dcdsb.2018067},
Key = {fds340537}
}
@article{fds329519,
Author = {Li, L and Liu, JG and Lu, J},
Title = {Fractional Stochastic Differential Equations Satisfying
FluctuationDissipation Theorem},
Journal = {Journal of Statistical Physics},
Volume = {169},
Number = {2},
Pages = {316339},
Publisher = {Springer Nature America, Inc},
Year = {2017},
Month = {October},
url = {http://dx.doi.org/10.1007/s109550171866z},
Abstract = {© 2017, Springer Science+Business Media, LLC. We propose in
this work a fractional stochastic differential equation
(FSDE) model consistent with the overdamped limit of the
generalized Langevin equation model. As a result of the
‘fluctuationdissipation 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 ‘fluctuationdissipation theorem’ is satisfied, and
this verifies that satisfying ‘fluctuationdissipation
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 powerlaw kernel and
subdiffusion behaviors.},
Doi = {10.1007/s109550171866z},
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 = {291313},
Publisher = {Springer Nature},
Year = {2017},
Month = {July},
url = {http://dx.doi.org/10.1007/s1091501703564},
Abstract = {© 2017, Springer Science+Business Media New York. 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 CFLtype 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/s1091501703564},
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 = {1325},
Publisher = {Elsevier BV},
Year = {2017},
Month = {July},
url = {http://dx.doi.org/10.1016/j.physd.2017.03.005},
Abstract = {© 2017 Elsevier B.V. Motivated by a model proposed by Peng
et al. (2014) for breakup of tear films on human eyes, we
study the dynamics of a generalized thin film model. The
governing equations form a fourthorder coupled system of
nonlinear parabolic PDEs for the film thickness and salt
concentration subject to nonconservative 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 finitetime rupture–shock phenomenon
due to the instabilities caused by locally elevated
evaporation rates, convergence to equilibrium and
infinitetime 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 = {873926},
Publisher = {Springer Nature},
Year = {2017},
Month = {June},
url = {http://dx.doi.org/10.1007/s0033201693541},
Abstract = {© 2016, Springer Science+Business Media New York. This work
considers the rigorous derivation of continuum models of
step motion starting from a mesoscopic Burton–Cabrera–Franktype
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 firstorder convergence
rate.},
Doi = {10.1007/s0033201693541},
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 = {14611492},
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 criticalcase unstable thin film equation in
onedimensional 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, 223256)
for n = 1. We obtain global existence of a nonnegative
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 blowup 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 longtime 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 GroupSize
Statistics},
Journal = {Journal of Nonlinear Science},
Volume = {27},
Number = {2},
Pages = {379424},
Publisher = {Springer Nature},
Year = {2017},
Month = {April},
url = {http://dx.doi.org/10.1007/s0033201693363},
Abstract = {© 2016, The Author(s). 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 Htheorem, we are able to develop a
rather complete description of equilibrium profiles and
largetime behavior, based on recent developments in complex
function theory for Bernstein and Pick functions. In the
largepopulation continuum limit, a scalinginvariant regime
is reached in which all equilibria are determined by a
single scaling profile. This universal profile exhibits
powerlaw behavior crossing over from exponent 23 for small
size to 32 for large size, with an exponential
cutoff.},
Doi = {10.1007/s0033201693363},
Key = {fds325700}
}
@article{fds329169,
Author = {Cong, W and Liu, JG},
Title = {Uniform L^{∞} boundedness for a degenerate
parabolicparabolic KellerSegel model},
Journal = {Discrete and Continuous Dynamical Systems Series
B},
Volume = {22},
Number = {2},
Pages = {307338},
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
parabolicparabolic KellerSegel model with the supercritical
diffusion exponent 0 < m < 2  2/d in the multidimensional
space ℝd under the condition that the L d(2m)/2 norm of
initial data is smaller than a universal constant. Moreover,
the weak entropy solution u(x,t) satisfies mass conservation
when m > 12/d. We also prove the local existence of weak
entropy solutions and a blowup 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 attachmentdetachmentlimited regime},
Journal = {Siam Journal on Mathematical Analysis},
Volume = {49},
Number = {3},
Pages = {17051731},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.1137/16M1094543},
Abstract = {© 2017 Society for Industrial and Applied Mathematics. We
study in this work a continuum model derived from a
onedimensional attachmentdetachmentlimited 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 spacetime 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 = {3560},
Publisher = {IOP Publishing},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.1088/09517715/30/1/35},
Abstract = {© 2016 IOP Publishing Ltd and London Mathematical Society
Printed in the UK. 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 hdx) a1 ∫ ℝd
∇h 2 dx/(∫ ℝd h m+1 dx) a+1/m+1 ≥ β 0 , where m
> 0 for d = 1, 2, 0 < m < d+2/d2 for d ≥ 3, and a =
d+2(m+1)/md. The EulerLagrange equation for critical points
of J(h) in the nonnegative radial decreasing function space
is given by a free boundary problem for a generalized
LaneEmden equation, which has a unique solution (denoted by
h c ) and the solution determines the best constant for the
above generalized Sz. Nagy inequality. The connection
between the critical mass M c = ∫ Rdbl; h c dx = 2√2π/3
for the thinfilm 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 22356). For the
following critical thin film equation in multidimension d
≥ 2 h t + ∇ · (h ∇ Delta; h) + ∇ · (h ∇ h m ) =
0, x ϵ ℝ d , where m = 1 + 2/d, the critical mass is also
given by M c := ∫ ℝd h c dx. A finite time blowup
occurs for solutions with the initial mass larger than M c .
On the other hand, if the initial mass is less than Mc and a
global nonnegative entropy weak solution exists, then the
second moment goes to infinity as t → ∞ or h(·, t k )
⇀ 0 in L 1 (ℝ d ) for some subsequence t k → ∞. This
shows that a part of the mass spreads to
infinity.},
Doi = {10.1088/09517715/30/1/35},
Key = {fds331396}
}
@article{fds329523,
Author = {Huang, H and Liu, JG},
Title = {Discreteintime random particle blob method for the
KellerSegel equation and convergence analysis},
Journal = {Communications in Mathematical Sciences},
Volume = {15},
Number = {7},
Pages = {18211842},
Publisher = {International Press of Boston},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.4310/CMS.2017.v15.n7.a2},
Abstract = {© 2017 International Press. We establish an error estimate
of a discreteintime random particle blob method for the
Keller{Segel (KS) equation in ℝd (d≥2). With a blob size
ε=N1/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 = {14031422},
Publisher = {International Press of Boston},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.4310/CMS.2017.v15.n5.a9},
Abstract = {© 2017 International Press. Meanfield games are games with
a continuum of players that incorporate the timedimension
through a controltheoretic approach. Recently, simpler
approaches relying on the BestReply 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 MeanField Games and the Best Reply Strategy
approach. This is done by introducing a Model Predictive
Control framework, which consists of setting the MeanField
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 MeanField Game
approach and the suboptimal BestReply Strategy.},
Doi = {10.4310/CMS.2017.v15.n5.a9},
Key = {fds330537}
}
@article{fds323838,
Author = {Degond, P and Liu, JG and MerinoAceituno, 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 = {159182},
Publisher = {World Scientific Pub Co Pte Lt},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.1142/S021820251740005X},
Abstract = {© 2017 The Author(s). We investigate the longTime dynamics
of an opinion formation model inspired by a work by
Borghesi, Bouchaud and Jensen. First, we derive a
FokkerPlanckType equation under the assumption that
interactions between individuals produce little consensus of
opinion (grazing collision approximation). Second, we study
conditions under which the FokkerPlanck equation has
nonTrivial equilibria and derive the macroscopic limit
(corresponding to the longTime 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 (nonsymmetric 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
nonconservative 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 kellersegel equation
and convergence analysis},
Journal = {Mathematics of Computation},
Volume = {86},
Number = {304},
Pages = {725745},
Publisher = {American Mathematical Society (AMS)},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.1090/mcom/3118},
Abstract = {© 2016 American Mathematical Society. In this paper, we
introduce a random particle blob method for the KellerSegel
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 CamassaHolm equation},
Journal = {Siam Journal on Mathematical Analysis},
Volume = {49},
Number = {2},
Pages = {12671294},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.1137/16M1102069},
Abstract = {© 2017 Society for Industrial and Applied Mathematics. In
this paper, we prove convergence of a sticky particle method
for the modified CamassaHolm 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 spacetime 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 = {22202245},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.1137/16M1098474},
Abstract = {© 2017 Society for Industrial and Applied Mathematics. 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. 17711784].},
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
KellerSegel equation},
Journal = {Mathematics of Computation},
Volume = {86},
Number = {308},
Pages = {27192744},
Publisher = {American Mathematical Society (AMS)},
Year = {2017},
Month = {January},
url = {http://dx.doi.org/10.1090/mcom/3174},
Abstract = {© 2017 American Mathematical Society. We establish an
optimal error estimate for a random particle blob method for
the KellerSegel 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/1κ) norm up to a probability
1hC 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 aggregationdiffusion splitting
algorithms for the KellerSegel equations},
Journal = {Discrete and Continuous Dynamical Systems Series
B},
Volume = {21},
Number = {10},
Pages = {34633478},
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 aggregationdiffusion splitting schemes for
the KellerSegel 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 timestep 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{fds318453,
Author = {Huang, H and Liu, JG},
Title = {A note on MongeAmpère KellerSegel equation},
Journal = {Applied Mathematics Letters},
Volume = {61},
Pages = {2634},
Publisher = {Elsevier BV},
Year = {2016},
Month = {November},
url = {http://dx.doi.org/10.1016/j.aml.2016.05.003},
Abstract = {© 2016 Elsevier Ltd. All rights reserved. This note studies
the MongeAmpère KellerSegel equation in a periodic domain
Td(d≥2), a fully nonlinear modification of the
KellerSegel equation where the MongeAmpère 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
KellerSegel Model},
Journal = {Acta Applicandae Mathematicae},
Volume = {142},
Number = {1},
Pages = {173188},
Publisher = {Springer Nature},
Year = {2016},
Month = {April},
ISSN = {01678019},
url = {http://dx.doi.org/10.1007/s1044001500225},
Abstract = {© 2015, Springer Science+Business Media Dordrecht. In this
note we establish the uniform (Formula presented.) bound
for the weak solutions to a degenerate KellerSegel 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/s1044001500225},
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 = {041902041902},
Publisher = {AIP Publishing},
Year = {2016},
Month = {April},
ISSN = {10706631},
url = {http://dx.doi.org/10.1063/1.4946005},
Abstract = {© 2016 Author(s). 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 channelreservoir 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, JG and Wang, J},
Title = {Refined hypercontractivity and uniqueness for the
Keller–Segel equations},
Journal = {Applied Mathematics Letters},
Volume = {52},
Pages = {212219},
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 SchrödingerLandauLifshitz system: Variational
structure and numerical methods},
Journal = {Multiscale Modeling & Simulation},
Volume = {14},
Number = {4},
Pages = {14631487},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2016},
Month = {January},
url = {http://dx.doi.org/10.1137/16M106947X},
Abstract = {© 2016 Society for Industrial and Applied Mathematics. From
a variational perspective, we derive a series of
magnetization and quantum spin current systems coupled via
an "sd" potential term, including the SchrödingerLandauLifshitz
Maxwell system, the PauliLandauLifshitz system, and the
SchrödingerLandauLifshitz system with successive
simplifications. For the latter two systems, we propose
using the time splitting spectral method for the quantum
spin current and the GaussSeidel 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
SchrödingerLandau Lifshitz system in different "sd"
coupling regimes are explored numerically.},
Doi = {10.1137/16M106947X},
Key = {fds329526}
}
@article{fds318454,
Author = {Huang, H and Liu, JG},
Title = {Wellposedness for the kellersegel equation with fractional
laplacian and the theory of propagation of
chaos},
Journal = {Kinetic and Related Models},
Volume = {9},
Number = {4},
Pages = {715748},
Year = {2016},
Month = {January},
url = {http://dx.doi.org/10.3934/krm.2016013},
Abstract = {© American Institute of Mathematical Sciences. This paper
investigates the generalized KellerSegel (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 hypercontractive 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
selfconsistent stochastic process driven by the
rotationally invariant αstable Lévy process Lα(t). Also,
we prove the weak solution is L1 bounded uniformly in time.
Lastly, we consider the Nparticle 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 cutoff
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 plaplacian kellersegel model},
Journal = {Kinetic and Related Models},
Volume = {9},
Number = {4},
Pages = {687714},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2016},
Month = {January},
url = {http://dx.doi.org/10.3934/krm.2016012},
Abstract = {© American Institute of Mathematical Sciences. This paper
investigates the existence of a uniform in time L∞ bounded
weak solution for the pLaplacian KellerSegel system with
the supercritical diffusion exponent 1 < p < 3d/d+1 in the
multidimensional space ℝd under the condition that the L
d(3p)/p norm of initial data is smaller than a universal
constant. We also prove the local existence of weak
solutions and a blowup 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 = {36673687},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2016},
Month = {January},
url = {http://dx.doi.org/10.1137/16M1059400},
Abstract = {© 2016 Society for Industrial and Applied Mathematics. In
this paper we study the existence assertion of the initial
boundary value problem for the equation @u/@t = ΔeΔu.
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 = {84998518},
Publisher = {American Mathematical Society (AMS)},
Year = {2016},
Month = {January},
url = {http://dx.doi.org/10.1090/tran/6618},
Abstract = {© 2016 American Mathematical Society. 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
FussCatalan 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 diffusiondrift 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 = {195223},
Publisher = {Springer International Publishing},
Year = {2016},
Month = {January},
ISBN = {9783319321424},
url = {http://dx.doi.org/10.1007/9783319321448_10},
Abstract = {© Springer International Publishing Switzerland 2016. 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 = {10.1007/9783319321448_10},
Key = {fds320553}
}
@article{fds320649,
Author = {J.G. Liu and R. Yang},
Title = {Propagation of chaos for large Brownian particle system with
Coulomb interaction},
Journal = {Research in the Mathematical Sciences},
Volume = {3},
Number = {40},
Year = {2016},
Key = {fds320649}
}
@article{fds320549,
Author = {Y. Duan and J.G. Liu},
Title = {Error estimate of the particle method for the
bequation},
Journal = {Methods and Applications of Analysis},
Volume = {23},
Pages = {119154},
Year = {2016},
Key = {fds320549}
}
@article{fds320556,
Author = {J.G. Liu and Y. Zhang},
Title = {Convergence of stochastic interacting particle systems in
probability under a Sobolev norm},
Journal = {Annals of Mathematical Sciences and Applications},
Volume = {1},
Pages = {251299},
Year = {2016},
Key = {fds320556}
}
@article{fds246842,
Author = {Xue, Y and Wang, C and Liu, JG},
Title = {Simple Finite Element Numerical Simulation of Incompressible
Flow Over Nonrectangular Domains and the SuperConvergence
Analysis},
Journal = {Journal of Scientific Computing},
Volume = {65},
Number = {3},
Pages = {11891216},
Publisher = {Springer Nature},
Year = {2015},
Month = {March},
ISSN = {08857474},
url = {http://dx.doi.org/10.1007/s1091501500058},
Abstract = {© 2015, Springer Science+Business Media New York. 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
Poissonlike 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 superconvergence analysis for the simple finite
element numerical scheme, using linear elements over a
uniform triangulation with right triangles.},
Doi = {10.1007/s1091501500058},
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 = {15393755},
url = {http://dx.doi.org/10.1103/PhysRevE.91.032403},
Abstract = {© 2015 American Physical Society. The BurtonCabreraFrank
(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
solidonsolid 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{fds341422,
Author = {Degond, P and Frouvelle, A and Liu, JG},
Title = {Phase Transitions, Hysteresis, and Hyperbolicity for
SelfOrganized Alignment Dynamics},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {216},
Number = {1},
Pages = {63115},
Year = {2015},
Month = {January},
url = {http://dx.doi.org/10.1007/s0020501408007},
Abstract = {© 2014, SpringerVerlag Berlin Heidelberg. We provide a
complete and rigorous description of phase transitions for
kinetic models of selfpropelled 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/s0020501408007},
Key = {fds341422}
}
@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 = {22462267},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2015},
Month = {January},
ISSN = {00361399},
url = {http://dx.doi.org/10.1137/140995854},
Abstract = {© 2015 Society for Industrial and Applied Mathematics. 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{fds246846,
Author = {Degond, P and Liu, JG and Ringhofer, C},
Title = {Evolution of wealth in a nonconservative economy driven by
local Nash equilibria.},
Journal = {Philosophical Transactions. Series A, Mathematical,
Physical, and Engineering Sciences},
Volume = {372},
Number = {2028},
Pages = {2013039420130394},
Publisher = {The Royal Society},
Year = {2014},
Month = {November},
ISSN = {1364503X},
url = {http://dx.doi.org/10.1098/rsta.2013.0394},
Abstract = {We develop a model for the evolution of wealth in a
nonconservative economic environment, extending a theory
developed in Degond et al. (2014 J. Stat. Phys. 154, 751780
(doi:10.1007/s1095501308884)). The model considers a
system of rational agents interacting in a gametheoretical
framework. This evolution drives the dynamics of the agents
in both wealth and economic configuration variables. The
cost function is chosen to represent a riskaverse 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 largescale 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 nonconservative property in the
hydrodynamic closure derivation of the largescale dynamics
for the evolution of wealth distribution. The result is a
system of gas dynamicstype 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{fds300223,
Author = {Degond, P and Frouvelle, A and Liu, JG},
Title = {Phase Transitions, Hysteresis, and Hyperbolicity for
SelfOrganized Alignment Dynamics},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {216},
Number = {1},
Pages = {63115},
Publisher = {Springer New York LLC},
Year = {2014},
Month = {October},
ISSN = {00039527},
url = {http://dx.doi.org/10.1007/s0020501408007},
Abstract = {We provide a complete and rigorous description of phase
transitions for kinetic models of selfpropelled 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/s0020501408007},
Key = {fds300223}
}
@article{fds246862,
Author = {Duan, Y and Liu, JG},
Title = {Convergence analysis of the vortex blob method for the
bequation},
Journal = {Discrete and Continuous Dynamical Systems Series
A},
Volume = {34},
Number = {5},
Pages = {19952011},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2014},
Month = {May},
ISSN = {10780947},
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 socalled bequation.
This kind of PDEs, including the CamassaHolm equation and
the DegasperisProcesi 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 bequation 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 = {Ultracontractivity for kellersegel model with diffusion
exponent m > 12/d},
Journal = {Kinetic and Related Models},
Volume = {7},
Number = {1},
Pages = {928},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2014},
Month = {March},
ISSN = {19375093},
url = {http://dx.doi.org/10.3934/krm.2014.7.9},
Abstract = {This paper establishes the hypercontractivity in L∞(ℝd)
(it's known as ultracontractivity) for the
multidimensional KellerSegel systems with the diffusion
exponent m > 12/d. The results show that for the super
critical and critical case 12/d < m ≤ 22/d, if
∥U0∥d(2m)/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 > 22/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{fds246848,
Author = {Coquel, F and Jin, S and Liu, JG and Wang, L},
Title = {WellPosedness and Singular Limit of a Semilinear Hyperbolic
Relaxation System with a TwoScale Discontinuous Relaxation
Rate},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {214},
Number = {3},
Pages = {10511084},
Year = {2014},
Month = {January},
ISSN = {00039527},
url = {http://dx.doi.org/10.1007/s0020501407736},
Abstract = {© 2014, SpringerVerlag Berlin Heidelberg. 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 twoscale 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
wellposedness and regularity (such as boundedness in sup
norm with bounded total variation and L 1 contraction) of
the coupling problem, thus providing a rigorous mathematical
foundation, in the general nonlinear setting, to the
multiscale domain decomposition method for this twoscale
problem originally proposed in Jin et al. in Math. Comp. 82,
749–779, 2013.},
Doi = {10.1007/s0020501407736},
Key = {fds246848}
}
@article{fds246849,
Author = {Degond, P and Herty, M and Liu, JG},
Title = {Flow on sweeping networks},
Journal = {Multiscale Modeling & Simulation},
Volume = {12},
Number = {2},
Pages = {538565},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2014},
Month = {January},
ISSN = {15403459},
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 rightgoing. Under
the assumptions of propagation of chaos and meanfield
limit, we derive a master equation and the corresponding
meanfield kinetic and macroscopic models. Steadystates 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 rodlike
particles},
Journal = {Communications in Mathematical Sciences},
Volume = {12},
Number = {8},
Pages = {15791601},
Publisher = {International Press of Boston},
Year = {2014},
Month = {January},
ISSN = {15396746},
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 rodlike particles under gravity,
deduced by Helzel, Otto, and Tzavaras (2011), which couples
the impressible (Navier)Stokes equation with the
FokkerPlanck equation. With a noflux 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, JG},
Title = {A NOTE ON PHASE TRANSITIONS FOR THE SMOLUCHOWSKI EQUATION
WITH DIPOLAR POTENTIAL},
Journal = {Hyperbolic Problems: Theory, Numerics, Applications},
Volume = {8},
Pages = {179192},
Booktitle = {Proceedings of the Fourteenth International Conference on
Hyperbolic Problems: Theory, Numerics and
Application},
Publisher = {AMER INST MATHEMATICAL SCIENCESAIMS},
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 = {Wellposedness for hallmagnetohydrodynamics},
Journal = {Annales De L'Institut Henri Poincare (C) Non Linear
Analysis},
Volume = {31},
Number = {3},
Pages = {555565},
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 HallMHD
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{fds300222,
Author = {Chertock, A and Liu, JG and Pendleton, T},
Title = {Elastic collisions among peakon solutions for the
CamassaHolm equation},
Journal = {Applied Numerical Mathematics},
Volume = {93},
Pages = {3046},
Publisher = {Elsevier BV},
Year = {2014},
ISSN = {01689274},
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
onedimensional CamassaHolm 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
CamassaHolm 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
CamassaHolm equation under a wide class of initial data. ©
2014 IMACS.},
Doi = {10.1016/j.apnum.2014.01.001},
Key = {fds300222}
}
@article{fds246856,
Author = {Goudon, T and Jin, S and Liu, JG and Yan, B},
Title = {Asymptoticpreserving schemes for kineticfluid modeling of
disperse twophase flows with variable fluid
density},
Journal = {International Journal for Numerical Methods in
Fluids},
Volume = {75},
Number = {2},
Pages = {81102},
Publisher = {WILEY},
Year = {2014},
ISSN = {02712091},
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
VlasovFokkerPlanck type, and the fluid is described by the
incompressible NavierStokes 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
asymptoticpreserving schemes suited to such regimes. We
extend the method introduced in [Goudon T, Jin S, Liu JG,
Yan B. Journal of Computational Physics 2013; 246:145164]
to the case of variable density in compressible flow. We
check the accuracy and the asymptoticpreserving 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{fds246857,
Author = {Johnston, H and Wang, C and Liu, JG},
Title = {A Local Pressure Boundary Condition Spectral Collocation
Scheme for the ThreeDimensional Navier–Stokes
Equations},
Journal = {Journal of Scientific Computing},
Volume = {60},
Number = {3},
Pages = {612626},
Publisher = {Springer Nature},
Year = {2014},
ISSN = {08857474},
url = {http://dx.doi.org/10.1007/s1091501398087},
Abstract = {© 2014, Springer Science+Business Media New York.A spectral
collocation scheme for the threedimensional incompressible
(u,p) formulation of the Navier–Stokes equations, in
domains Ω with a nonperiodic 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 liddriven cavity flow and a differentially heated
cavity flow, to demonstrate the scheme produces accurate
threedimensional results at a reasonable computational
cost.},
Doi = {10.1007/s1091501398087},
Key = {fds246857}
}
@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 = {B953B986},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2013},
Month = {November},
ISSN = {10648275},
url = {http://dx.doi.org/10.1137/120875508},
Abstract = {We propose a simple finite difference scheme for
NavierStokes 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 infsup
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{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 = {11791215},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2013},
Month = {October},
ISSN = {00361410},
url = {http://dx.doi.org/10.1137/120887850},
Abstract = {We studied coupled systems of the FokkerPlanck equation and
the NavierStokes 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 centerofmass 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 = {Asymptoticpreserving schemes for kineticfluid modeling of
disperse twophase flows},
Journal = {Journal of Computational Physics},
Volume = {246},
Pages = {145164},
Publisher = {Elsevier BV},
Year = {2013},
Month = {August},
ISSN = {00219991},
url = {http://dx.doi.org/10.1016/j.jcp.2013.03.038},
Abstract = {We consider a system coupling the incompressible
NavierStokes equations to the VlasovFokkerPlanck
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
asymptoticpreserving, thus efficient in both the kinetic
and hydrodynamic regimes. It has a numerical stability
condition controlled by the nonstiff convection operator,
with an implicit treatment of the stiff drag term and the
FokkerPlanck operator. Yet, consistent to a standard
asymptoticpreserving FokkerPlanck solver or an
incompressible NavierStokes solver, only the
conjugategradient 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{fds246895,
Author = {Degond, P and Frouvelle, A and Liu, JG},
Title = {Macroscopic limits and phase transition in a system of
selfpropelled particles},
Journal = {Journal of Nonlinear Science},
Volume = {23},
Number = {3},
Pages = {427456},
Publisher = {Springer Nature},
Year = {2013},
Month = {June},
ISSN = {09388974},
url = {http://dx.doi.org/10.1007/s003320129157y},
Abstract = {We investigate systems of selfpropelled particles with
alignment interaction. Compared to previous work (Degond and
Motsch, Math. Models Methods Appl. Sci. 18:11931215, 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 spaceinhomogeneous 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 selfalignment interactions as
derived in (Degond and Motsch, Math. Models Methods Appl.
Sci. 18:11931215, 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/s003320129157y},
Key = {fds246895}
}
@article{fds246870,
Author = {Chen, X and Liu, JG},
Title = {Global weak entropy solution to DoiSaintillanShelley model
for active and passive rodlike and ellipsoidal particle
suspensions},
Journal = {Journal of Differential Equations},
Volume = {254},
Number = {7},
Pages = {27642802},
Publisher = {Elsevier BV},
Year = {2013},
Month = {April},
ISSN = {00220396},
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 DoiSaintillanShelley model for active and passive
rodlike particle suspensions, which couples a FokkerPlanck
equation with the incompressible NavierStokes or Stokes
equation, under the noflux boundary conditions,
L2(Ω;L1(Sd1)) 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×Sd1) 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{fds246896,
Author = {Jin, S and Liu, JG and Wang, L},
Title = {A domain decomposition method for semilinear hyperbolic
systems with twoscale relaxations},
Journal = {Mathematics of Computation},
Volume = {82},
Number = {282},
Pages = {749779},
Publisher = {American Mathematical Society (AMS)},
Year = {2013},
Month = {February},
url = {http://dx.doi.org/10.1090/S002557182012026433},
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/S002557182012026433},
Key = {fds246896}
}
@article{fds220117,
Author = {P. Degond and A. Frouvelle and J.G. Liu and S Motsch and L
Navoret},
Title = {Macroscopic models of collective motion and
selforganization},
Journal = {Seminaire Laurent Schwartz  EDP et applicatios},
Volume = {2012  2013},
Pages = {127},
Year = {2013},
Key = {fds220117}
}
@article{fds220119,
Author = {P. Degond and J.G, Liu and S. Motsch and V. Panferov},
Title = {Hydrodynamic models of selforganized dynamics: derivation
and existence theory},
Journal = {Methods Anal. Appl.},
Volume = {20},
Pages = {89114},
Year = {2013},
Key = {fds220119}
}
@article{fds246859,
Author = {Degond, P and Liu, JG 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 = {130},
Publisher = {Springer Nature},
Year = {2013},
ISSN = {00224715},
url = {http://dx.doi.org/10.1007/s1095501308884},
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 dynamicstype 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/s1095501308884},
Key = {fds246859}
}
@article{fds246860,
Author = {Chen, X and Jüngel, A and Liu, JG},
Title = {A Note on AubinLionsDubinskiǐ Lemmas},
Journal = {Acta Applicandae Mathematicae},
Volume = {133},
Number = {1},
Pages = {111},
Year = {2013},
ISSN = {01678019},
url = {http://dx.doi.org/10.1007/s1044001398588},
Abstract = {Strong compactness results for families of functions in
seminormed nonnegative cones in the spirit of the
AubinLionsDubinskiǐ lemma are proven, refining some
recent results in the literature. The first theorem sharpens
slightly a result of Dubinskiǐ (in Mat. Sb.
67(109):609642, 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. 75:30723077, 2012) and Chen and Liu (in
Appl. Math. Lett. 25:22522257, 2012). An application is
given, which is useful in the study of porousmedium or
fastdiffusion type equations. © 2013 Springer
Science+Business Media.},
Doi = {10.1007/s1044001398588},
Key = {fds246860}
}
@article{fds246863,
Author = {Degond, P and Liu, JG and Ringhofer, C},
Title = {LargeScale Dynamics of MeanField Games Driven by Local
Nash Equilibria},
Journal = {Journal of Nonlinear Science},
Volume = {24},
Number = {1},
Pages = {123},
Year = {2013},
ISSN = {09388974},
url = {http://dx.doi.org/10.1007/s0033201391852},
Abstract = {We introduce a new mean field kinetic model for systems of
rational agents interacting in a gametheoretical framework.
This model is inspired from noncooperative anonymous games
with a continuum of players and MeanField 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/s0033201391852},
Key = {fds246863}
}
@article{fds246866,
Author = {Bian, S and Liu, JG},
Title = {Dynamic and Steady States for MultiDimensional KellerSegel
Model with Diffusion Exponent m > 0},
Journal = {Communications in Mathematical Physics},
Volume = {323},
Number = {3},
Pages = {154},
Publisher = {Springer Nature},
Year = {2013},
ISSN = {00103616},
url = {http://dx.doi.org/10.1007/s002200131777z},
Abstract = {This paper investigates infinitetime spreading and
finitetime blowup for the KellerSegel system. For 0 <
m ≤ 2  2 / d, the Lp space for both dynamic and steady
solutions are detected with {Mathematical expression} .
Firstly, the global existence of the weak solution is proved
for small initial data in Lp. Moreover, when m > 1  2
/ d, the weak solution preserves mass and satisfies the
hypercontractive estimates in Lq 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
hypercontractive estimates are also obtained. Finally, we
focus on the Lp norm of the steady solutions, it is shown
that the energy critical exponent m = 2 d/(d + 2) is the
critical exponent separating finite Lp norm and infinite Lp
norm for the steady state solutions. © 2013 SpringerVerlag
Berlin Heidelberg.},
Doi = {10.1007/s002200131777z},
Key = {fds246866}
}
@article{fds246867,
Author = {Chae, D and Degond, P and Liu, JG},
Title = {Wellposedness for Hallmagnetohydrodynamics},
Journal = {Annales De L'Institut Henri Poincare (C) Non Linear
Analysis},
Volume = {31},
Number = {3},
Pages = {555565},
Publisher = {Elsevier BV},
Year = {2013},
ISSN = {02941449},
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 HallMHD
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{fds340251,
Author = {Degond, P and Frouvelle, A and Liu, JG},
Title = {Macroscopic limits and phase transition in a system of
selfpropelled particles},
Journal = {Journal of Nonlinear Science},
Volume = {23},
Number = {3},
Pages = {427456},
Year = {2013},
url = {http://dx.doi.org/10.1007/s003320129157y},
Abstract = {We investigate systems of selfpropelled particles with
alignment interaction. Compared to previous work (Degond and
Motsch, Math. Models Methods Appl. Sci. 18:11931215, 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 spaceinhomogeneous 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 selfalignment interactions as
derived in (Degond and Motsch, Math. Models Methods Appl.
Sci. 18:11931215, 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/s003320129157y},
Key = {fds340251}
}
@article{fds246894,
Author = {Haack, J and Jin, S and Liu, JG},
Title = {An allspeed asymptoticpreserving method for the isentropic
Euler and NavierStokes equations},
Journal = {Communications in Computational Physics},
Volume = {12},
Number = {4},
Pages = {955980},
Publisher = {Global Science Press},
Year = {2012},
Month = {October},
ISSN = {18152406},
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 allspeed asymptotic preserving (AP)
numerical scheme for the compressible isentropic Euler and
NavierStokes 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
methodlike incompressible solver. We present numerical
results in one and two dimensions in both compressible and
incompressible regimes. © 2012 GlobalScience
Press.},
Doi = {10.4208/cicp.250910.131011a},
Key = {fds246894}
}
@article{fds246887,
Author = {Chae, D and Liu, JG},
Title = {Blowup, Zero α Limit and the Liouville Type Theorem for
the EulerPoincaré Equations},
Journal = {Communications in Mathematical Physics},
Volume = {314},
Number = {3},
Pages = {671687},
Publisher = {Springer Nature},
Year = {2012},
Month = {September},
ISSN = {00103616},
url = {http://dx.doi.org/10.1007/s0022001215348},
Abstract = {In this paper we study the EulerPoincaré 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 blowup criterion. For the
zero dispersion equation (α = 0) we prove a finite time
blowup 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 EulerPoincaré 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
SpringerVerlag.},
Doi = {10.1007/s0022001215348},
Key = {fds246887}
}
@article{fds246891,
Author = {Carrillo, JA and Chen, L and Liu, JG and Wang, J},
Title = {A note on the subcritical two dimensional KellerSegel
system},
Journal = {Acta Applicandae Mathematicae},
Volume = {119},
Number = {1},
Pages = {4355},
Publisher = {Springer Nature},
Year = {2012},
Month = {June},
ISSN = {01678019},
url = {http://dx.doi.org/10.1007/s1044001196604},
Abstract = {The existence of solution for the 2DKellerSegel 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.
46:691721, 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. 4:553386,
1991). © 2011 Springer Science+Business Media
B.V.},
Doi = {10.1007/s1044001196604},
Key = {fds246891}
}
@article{fds246889,
Author = {Chen, L and Liu, JG and Wang, J},
Title = {Multidimensional degenerate KellerSegel system with
critical diffusion exponent 2n/(n + 2)},
Journal = {Siam Journal on Mathematical Analysis},
Volume = {44},
Number = {2},
Pages = {10771102},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2012},
Month = {May},
ISSN = {00361410},
url = {http://dx.doi.org/10.1137/110839102},
Abstract = {This paper deals with a degenerate diffusion
PatlakKellerSegel 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* = 22/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+xx0 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 blowup 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 Journal on Mathematical Analysis},
Volume = {44},
Number = {2},
Pages = {791826},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2012},
Month = {May},
ISSN = {00361410},
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
meanfield equation which is nothing more than the
Smoluchowski equation on the sphere with dipolar potential.
In this selfcontained 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 Fishervon 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{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 Journal on Numerical Analysis},
Volume = {50},
Number = {1},
Pages = {121},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2012},
Month = {May},
ISSN = {00361429},
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
selfcontained method for proving its convergence. The
latter is accomplished by using the concept of spacetime
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{fds246892,
Author = {Degond, P and Liu, JG},
Title = {Hydrodynamics of selfalignment interactions with precession
and derivation of the LandauLifschitzGilbert
equation},
Journal = {Mathematical Models and Methods in Applied
Sciences},
Volume = {22},
Number = {SUPPL.1},
Pages = {11400011140001},
Publisher = {World Scientific Pub Co Pte Lt},
Year = {2012},
Month = {April},
ISSN = {02182025},
url = {http://dx.doi.org/10.1142/S021820251140001X},
Abstract = {We consider a kinetic model of selfpropelled 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 nonconservative equation for the orientation.
First, we assume that the alignment interaction is purely
local and derive a firstorder system. However, we show that
this system may lose its hyperbolicity. Under the assumption
of weakly nonlocal interaction, we derive diffusive
corrections to the firstorder system which lead to the
combination of a heat flow of the harmonic map and
LandauLifschitzGilbert dynamics. In the particular case of
zero selfpropelling 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{fds246888,
Author = {Chen, X and Liu, JG},
Title = {Two nonlinear compactness theorems in L^{p}(0,T;B)},
Journal = {Applied Mathematics Letters},
Volume = {25},
Number = {12},
Pages = {22522257},
Publisher = {Elsevier BV},
Year = {2012},
Month = {January},
ISSN = {08939659},
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 FENEtype beadspring
chains model). © 2012 Elsevier Ltd. All rights
reserved.},
Doi = {10.1016/j.aml.2012.06.012},
Key = {fds246888}
}
@article{fds220112,
Author = {A. Chertock and J.G. Liu and T. Pendleton},
Title = {Convergence analysis of the particle method for the
CamassaHolm equation},
Pages = {365373},
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{fds246898,
Author = {Acheritogaray, M and Degond, P and Frouvelle, A and Liu,
JG},
Title = {Kinetic formulation and global existence for the
hallmagnetohydrodynamics system},
Journal = {Kinetic and Related Models},
Volume = {4},
Number = {4},
Pages = {901918},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2011},
Month = {December},
ISSN = {19375093},
url = {http://dx.doi.org/10.3934/krm.2011.4.901},
Abstract = {This paper deals with the derivation and analysis of the the
Hall MagnetoHydrodynamic equations. We first provide a
derivation of this system from a twofluids EulerMaxwell
system for electrons and ions, through a set of scaling
limits. We also propose a kinetic formulation for the
HallMHD equa tions which contains as fluid closure
different variants of the HallMHD model. Then, we prove the
existence of global weak solutions for the incompressible
viscous resistive HallMHD 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{fds246899,
Author = {Zheng, W and Gao, H and Liu, JG and Zhang, Y and Ye, Q and Swank,
C},
Title = {General solution to gradientinduced transverse and
longitudinal relaxation of spins undergoing restricted
diffusion},
Journal = {Physical Review A},
Volume = {84},
Number = {5},
Pages = {0534118},
Publisher = {American Physical Society (APS)},
Year = {2011},
Month = {November},
ISSN = {10502947},
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
gradientinduced transverse (T 2 ) 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 wellknown slow diffusion result of Torrey and
the fast diffusion result. We also perform free induction
decay measurements on spinexchange 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{fds246904,
Author = {Huang, YL and Liu, JG and Wang, WC},
Title = {An FFT based fast poisson solver on spherical
shells},
Journal = {Communications in Computational Physics},
Volume = {9},
Number = {3},
Pages = {649667},
Publisher = {Global Science Press},
Year = {2011},
Month = {March},
ISSN = {18152406},
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 differential operator.
As a result, the Fast fourier Transform can be applied to
solve the Poisson equation with O(N3log N) operations.
Numerical examples have confirmed the accuracy and
robustness of the new scheme. © 2011 GlobalScience
Press.},
Doi = {10.4208/cicp.060509.080609s},
Key = {fds246904}
}
@article{fds246897,
Author = {Liu, JG and Lorz, A},
Title = {A coupled chemotaxisfluid model: Global
existence},
Journal = {Annales De L'Institut Henri Poincare (C) Non Linear
Analysis},
Volume = {28},
Number = {5},
Pages = {643652},
Publisher = {Elsevier BV},
Year = {2011},
Month = {January},
ISSN = {02941449},
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
chemotaxisNavier 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 chemotaxisStokes 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{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 = {34283453},
Publisher = {Elsevier BV},
Year = {2010},
Month = {May},
ISSN = {00219991},
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 NavierStokes
equations with noslip 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 timediscrete projection
methods implicit in viscosity only, and (b) devise new
higherorder accurate timediscrete projection methods that
extend a slipcorrection idea behind the wellknown
finitedifference 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 thirdorder accuracy for
both pressure and velocity without a timestep stability
restriction of diffusive type. Furthermore, two kinds of
projection methods are found stable using piecewiselinear
elements for both velocity and pressure. © 2010 Elsevier
Inc.},
Doi = {10.1016/j.jcp.2010.01.010},
Key = {fds246900}
}
@article{fds246928,
Author = {Liu, JG and Pego, RL},
Title = {STABLE DISCRETIZATION OF MAGNETOHYDRODYNAMICS IN BOUNDED
DOMAINS},
Journal = {Communications in Mathematical Sciences},
Volume = {8},
Number = {1},
Pages = {234251},
Publisher = {INT PRESS BOSTON, INC},
Year = {2010},
Month = {March},
ISSN = {15396746},
Abstract = {We study a semiimplicit timedifference 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
developed formula for the NavierStokes pressure involving
the commutator of Laplacian and Leray projection operators.
We prove stability of the timedifference scheme, and deduce
a localtime wellposedness theorem for MHD dynamics
extended to ignore the divergencefree constraint on
velocity and magnetic fields. These fields are
divergencefree for all later time if they are initially so.
Â© 2010 International Press.},
Key = {fds246928}
}
@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 Journal on Numerical Analysis},
Volume = {48},
Number = {4},
Pages = {14741491},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2010},
Month = {January},
ISSN = {00361429},
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. 334368] 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. © 2010 Society for
Industrial and Applied Mathematics.},
Doi = {10.1137/090772770},
Key = {fds246905}
}
@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 = {235251},
Publisher = {International Press of Boston},
Year = {2010},
Month = {January},
ISSN = {15396746},
url = {http://dx.doi.org/10.4310/CMS.2010.v8.n1.a12},
Abstract = {We study a semiimplicit timedifference 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
developed formula for the NavierStokes pressure involving
the commutator of Laplacian and Leray projection operators.
We prove stability of the timedifference scheme, and deduce
a localtime wellposedness theorem for MHD dynamics
extended to ignore the divergencefree constraint on
velocity and magnetic fields. These fields are
divergencefree for all later time if they are initially so.
© 2010 International Press.},
Doi = {10.4310/CMS.2010.v8.n1.a12},
Key = {fds304584}
}
@article{fds246943,
Author = {Liu, JG and Liu, J and Pego, RL},
Title = {Error estimates for finiteelement NavierStokes solvers
without standard InfSup conditions},
Journal = {Chinese Annals of Mathematics, Series B},
Volume = {30},
Number = {6},
Pages = {743768},
Publisher = {Springer Nature},
Year = {2009},
Month = {December},
ISSN = {02529599},
url = {http://dx.doi.org/10.1007/s1140100901163},
Abstract = {The authors establish error estimates for recently developed
finiteelement methods for incompressible viscous flow in
domains with noslip boundary conditions. The methods arise
by discretization of a wellposed extended NavierStokes
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 finiteelement projection method
close to classical timesplitting methods of Orszag,
Israeli, DeVille and Karniadakis. © Editorial Office of CAM
and SpringerVerlag Berlin Heidelberg 2009.},
Doi = {10.1007/s1140100901163},
Key = {fds246943}
}
@article{fds246944,
Author = {Liu, JG and Wang, WC},
Title = {Characterization and regularity for axisymmetric solenoidal
vector fields with application to navierstokes
equation},
Journal = {Siam Journal on Mathematical Analysis},
Volume = {41},
Number = {5},
Pages = {18251850},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2009},
Month = {December},
ISSN = {00361410},
url = {http://dx.doi.org/10.1137/080739744},
Abstract = {We consider the vorticitystream 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 NavierStokes 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 CuckerSmale flocking dynamics and
meanfield limit},
Journal = {Communications in Mathematical Sciences},
Volume = {7},
Number = {2},
Pages = {297325},
Publisher = {International Press of Boston},
Year = {2009},
Month = {January},
ISSN = {15396746},
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 CuckerSmale 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 meanfield
limit from the many particle CuckerSmale 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
EulerPoisson system in the quasineutral
limit},
Journal = {Siam Journal on Numerical Analysis},
Volume = {46},
Number = {3},
Pages = {12981322},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2008},
Month = {November},
ISSN = {00361429},
url = {http://dx.doi.org/10.1137/070690584},
Keywords = {stiffness • Debye length • electron plasma period
• BurgersPoisson • sheath problem •
KleinGordon},
Abstract = {In a previous work [P. Crispel, P. Degond, and M.H. Vignal,
J. Comput. Phys., 223 (2007), pp. 208234], a new numerical
discretization of the EulerPoisson 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
spacedecentered scheme is used (which brings in some
numerical dissipation) rather than a spacecentered 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
BurgersPoisson problem, are also discussed. © 2008 Society
for Industrial and Applied Mathematics.},
Doi = {10.1137/070690584},
Key = {fds246946}
}
@article{fds246941,
Author = {Lin, P and Liu, JG and Lu, X},
Title = {Long time numerical solution of the NavierStokes equations
based on a sequential regularization formulation},
Journal = {Siam Journal on Scientific Computing},
Volume = {31},
Number = {1},
Pages = {398419},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2008},
Month = {November},
ISSN = {10648275},
url = {http://dx.doi.org/10.1137/060673722},
Abstract = {The sequential regularization method is a reformulation of
the unsteady NavierStokes equations from the viewpoint of
constrained dynamical systems or the approximate
HelmholtzHodge projection. In this paper we study the long
time behavior of the sequential regularization formulation.
We give a uniformintime estimate between the solution of
the reformulated system and that of the NavierStokes
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 = {2655},
Year = {2008},
Month = {July},
ISSN = {18152406},
Abstract = {A fourthorder finite difference method is proposed and
studied for the primitive equations (PEs) of largescale
atmospheric and oceanic flow based on mean vorticity
formulation. Since the vertical average of the horizontal
velocity field is divergencefree, we can introduce mean
vorticity and mean stream function which are connected by a
2D 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
fourthorder longstencil 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
fourthorder accuracy is checked by an example of the
reformulated PEs with force terms. Additionally, numerical
results of a largescale oceanic circulation are presented.
© 2008 GlobalScience Press.},
Key = {fds246942}
}
@article{fds246948,
Author = {Lu, X and Lin, P and Liu, JG},
Title = {Analysis of a sequential regularization method for the
unsteady NavierStokes equations},
Journal = {Mathematics of Computation},
Volume = {77},
Number = {263},
Pages = {14671494},
Publisher = {American Mathematical Society (AMS)},
Year = {2008},
Month = {July},
ISSN = {00255718},
url = {http://dx.doi.org/10.1090/S0025571808020875},
Keywords = {NavierStokes equations • iterative penalty method
• implicit parabolic PDE • error estimates •
constrained dynamical system • stabilization
method},
Abstract = {The incompressibility constraint makes NavierStokes
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 NavierStokes equations. We
also give error estimates for the time discretized SRM
formulation. ©2008 American Mathematical
Society.},
Doi = {10.1090/S0025571808020875},
Key = {fds246948}
}
@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 = {495512},
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 = {135176},
Publisher = {Springer Nature},
Year = {2007},
Month = {December},
ISSN = {14249286},
url = {http://dx.doi.org/10.1007/s0003200700686},
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/s0003200700686},
Key = {fds246949}
}
@article{fds246958,
Author = {Moore, J and Cheng, Z and Hao, J and Guo, G and Liu, JG and Lin, C and Yu,
LL},
Title = {Effects of solidstate 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 = {1017310182},
Year = {2007},
Month = {December},
ISSN = {00218561},
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 solidstate 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. Solidstate 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 1112% but did not significantly alter the
fiber composition of wheat bran. Effects of solidstate
yeast treatment on both ORAC and TPC of wheat bran were
altered by yeast dose, treatment time, and their
interaction. Results suggest that solidstate yeast
treatment may be a commercially viable postharvest procedure
for improving the health beneficial properties of wheat bran
and other wheatbased 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 NavierStokes solvers
via a commutator estimate},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {60},
Number = {10},
Pages = {14431487},
Publisher = {WILEY},
Year = {2007},
Month = {October},
ISSN = {00103640},
url = {http://dx.doi.org/10.1002/cpa.20178},
Abstract = {For strong solutions of the incompressible NavierStokes
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 infsup 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 NavierStokes 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 = {251270},
Booktitle = {Kyoto Conference on the NavierStokes 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 NavierStokes equation with noslip 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 nonhomogeneous Stokes
system.},
Key = {fds139011}
}
@article{fds246903,
Author = {Liu, JG and Liu, J and Pego, R},
Title = {Stability and convergence of efficient NavierStokes solvers
via a commutator estimate via a commutator
estimate},
Journal = {Comm. Pure Appl. Math.},
Volume = {60},
Pages = {14431487},
Year = {2007},
Key = {fds246903}
}
@article{fds246947,
Author = {Degond, P and Jin, S and Liu, JG},
Title = {Machnumber uniform asymptotic preserving Gauge schemes for
compressible flows},
Journal = {Bulletin of the Institute of Mathematics Academia Sinica
(New Series)},
Volume = {2},
Pages = {851892},
Year = {2007},
Keywords = {Mach number uniform method • Euler equations •
NavierStokes equations • Asymptotic Preserving schemes
• gauge schemes • compressible fluids •
LowMach number limit • macromicro decomposition
• semiimplicit scheme • EulerPoisson system
• NavierStokesPoisson system},
Abstract = {We present novel algorithms for compressible flows that are
efficient for all Mach numbers. The approach is based on
several ingredients: semiimplicit 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 NavierStokes case).
Additionally, we show that our approach corresponds to a
micromacro 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 NavierStokes case, and the
isentropic NavierStokesPoisson 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 = {24562480},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2006},
Month = {December},
ISSN = {00361429},
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 & Simulation},
Volume = {5},
Number = {3},
Pages = {940979},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2006},
Month = {September},
ISSN = {15403459},
url = {http://dx.doi.org/10.1137/060651574},
Keywords = {KineticFluid 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, JG 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 = {53135322},
Year = {2006},
Month = {July},
ISSN = {00218561},
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{fds246964,
Author = {Liu, JG and Samelson, R and Wang, C},
Title = {Global weak solution of planetary geostrophic equations with
inviscid geostrophic balance},
Journal = {International Journal of Phytoremediation},
Volume = {85},
Number = {67},
Pages = {593605},
Year = {2006},
Month = {January},
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 firstorder 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{fds139013,
Author = {J.G. Liu and Jie Liu and R. Pego},
Title = {On incompressible NavierStokes dynamics: a new approach for
analysis and computation},
Pages = {2944},
Booktitle = {Proceedings of the Tenth International Conference on
Hyperbolic Problems},
Publisher = {Yokohama Publishers, Inc.},
Editor = {F. Asakura and etc},
Year = {2006},
Key = {fds139013}
}
@article{fds246902,
Author = {Liu, JG and Wang, WC},
Title = {Energy and helicity preserving schemes for hydro and
magnetohydrodynamics flows with symmetry},
Journal = {Journal of Computational Physics},
Volume = {200},
Number = {1},
Pages = {833},
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 vorticitystream formulation,
the divergence free constraints are automatically satisfied.
In addition, with explicit treatment of the nonlinear terms
and local vorticity boundary condition, the NavierStokes
(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
electromotive 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 = {Boundarylayer separation and adverse pressure gradient for
2D viscous incompressible flow},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {197},
Number = {12},
Pages = {149173},
Publisher = {Elsevier BV},
Year = {2004},
Month = {October},
ISSN = {01672789},
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 twodimensional (2D)
incompressible flow subject to noslip boundary conditions
and its connection with boundarylayer separation. The
boundarylayer separation theory of M. Ghil, T. Ma and S.
Wang, based on the structuralbifurcation 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 secondorder 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 2D drivencavity flow, governed by the
Navier Stokes equations, is presented; boundarylayer
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 = {429451},
Publisher = {Springer Nature},
Year = {2004},
Month = {October},
ISSN = {09388974},
url = {http://dx.doi.org/10.1007/s0033200406349},
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 "freeenergy" functional. This functional
consists of the term Δ h 2, which represents the surface
diffusion, andlog (1 + ∇ h 2), which describes the
effect of kinetic asymmetry in the adatom
attachmentdetachment. We first prove for large time t that
the interface widththe standard deviation of the height
profileis 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 freeenergy functional exist for
each ε > 0, the L 2norm 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/s0033200406349},
Key = {fds304585}
}
@article{fds246962,
Author = {Johnston, H and Liu, JG},
Title = {Accurate, stable and efficient NavierStokes solvers based
on explicit treatment of the pressure term},
Journal = {Journal of Computational Physics},
Volume = {199},
Number = {1},
Pages = {221259},
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
NavierStokes 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 NavierStokes
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 NavierStokes 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{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 = {267290},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2004},
Month = {June},
ISSN = {15340392},
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 farfield boundary conditions for systems of
convectiondiffusion equations in the zero viscosity limit.
The farfield boundary conditions are classified and the
corresponding solution structures are analyzed. It is
confirmed that the Neumann type of farfield 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 convectiondiffusion model which well describes
the behavior at the far field for many physical and
engineering systems such as fluid dynamical equations and
electromagnetic 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{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 = {555594},
Publisher = {Springer Nature},
Year = {2004},
Month = {May},
url = {http://dx.doi.org/10.1007/s0021100305083},
Abstract = {The convergence of a fourth order finite difference method
for the 2D unsteady, viscous incompressible Boussinesq
equations, based on the vorticitystream function
formulation, is established in this article. A compact
fourth order scheme is used to discretize the momentum
equation, and longstencil fourth order operators are
applied to discretize the temperature transport equation. A
local vorticity boundary condition is used to enforce the
noslip boundary condition for the velocity. Onesided
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/s0021100305083},
Key = {fds246956}
}
@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 = {2752},
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 convectiondiffusion 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 farfield 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 farfield for many physical and engineering
systems such as fluid dynamical equation and
electromagnetic equation. The results obtained here should
provide some theoretical guidance for designing effective
farfield 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 multiconnected domains},
Journal = {Computers & Fluids},
Volume = {33},
Number = {2},
Pages = {223255},
Publisher = {Elsevier BV},
Year = {2004},
Month = {January},
url = {http://dx.doi.org/10.1016/S00457930(03)000379},
Abstract = {Using the vorticity and stream function variables is an
effective way to compute 2D 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
multiconnected 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
subchannel. 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 vortextype flow, flow past a
nonsymmetric square, and flow in a heat exchanger. © 2003
Elsevier Ltd. All rights reserved.},
Doi = {10.1016/S00457930(03)000379},
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 = {429451},
Year = {2004},
ISSN = {09388974},
url = {http://dx.doi.org/10.1007/s0033200406349},
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 "freeenergy" functional. This functional
consists of the term Î” h 2, which represents the
surface diffusion, andlog (1 + âˆ‡ h 2), which
describes the effect of kinetic asymmetry in the adatom
attachmentdetachment. We first prove for large time t that
the interface widththe standard deviation of the height
profileis 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 Îµ 2 of Î” h 2 with Îµ = 1/L and L the
linear size of the underlying system, and study the energy
asymptotics in the large system limit Îµ â†’ 0. We
show that global minimizers of the freeenergy functional
exist for each Îµ > 0, the L 2norm of the gradient of
any global minimizer scales as O(1/Îµ), and the global
minimum energy scales as O( log Îµ). 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. Â© 2005 Springer.},
Doi = {10.1007/s0033200406349},
Key = {fds246959}
}
@article{fds246965,
Author = {Liu, JG and Wang, C},
Title = {High order finite difference method for unsteady
incompressible flow on multiconnected domain in
vorticitystream function formulation},
Journal = {Computer and Fluids},
Volume = {33},
Number = {2},
Pages = {223255},
Year = {2004},
url = {http://dx.doi.org/10.1016/S00457930(03)000379},
Abstract = {Using the vorticity and stream function variables is an
effective way to compute 2D 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
multiconnected 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
subchannel. 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 vortextype flow, flow past a
nonsymmetric square, and flow in a heat exchanger. Â©
2003 Elsevier Ltd. All rights reserved.},
Doi = {10.1016/S00457930(03)000379},
Key = {fds246965}
}
@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 = {713743},
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 EhrlichSchwoebel 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 roughsmoothrough pattern, that has been
experimentally observed as transient in an early stage of
epitaxial growth on rough surfaces. Initialboundaryvalue
problems for both equations are proven to be wellposed, and
the solution regularity is also obtained. Galerkin spectral
approximations are studied to provide both a priori bounds
for proving the wellposedness 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{fds246951,
Author = {Wang, C and Liu, JG},
Title = {Positivity property of secondorder fluxsplitting schemes
for the compressible Euler equations},
Journal = {Discrete and Continuous Dynamical Systems Series
B},
Volume = {3},
Number = {2},
Pages = {201228},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2003},
Month = {May},
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
nonnegative and nonpositive eigenvalues. That leads to the
conclusion that the first order schemes are positive in the
sense of LaxLiu [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 CFLlike condition. In addition, these splitting
methods preserve the positivity of density and
energy.},
Doi = {10.3934/dcdsb.2003.3.201},
Key = {fds246951}
}
@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 = {253285},
Year = {2003},
Month = {April},
ISSN = {08857474},
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
vorticitystream 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 noslip boundary condition
enforced using a local vorticity boundary condition. Fourth
order longstencil discretizations are used for the
temperature transport equation with onesided extrapolation
applied near the boundary. The time stepping scheme for both
equations is classical fourth order RungeKutta. The method
is highly efficient. The main computation consists of the
solution of two Poissonlike equations at each RungeKutta
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
differentiallyheated 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{fds246952,
Author = {ChainaisHillairet, C and Liu, JG and Peng, YJ},
Title = {Finite volume scheme for multidimensional driftdiffusion
equations and convergence analysis},
Journal = {Esaim: Mathematical Modelling and Numerical
Analysis},
Volume = {37},
Number = {2},
Pages = {319338},
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 multidimensional
driftdiffusion 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{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 = {171191},
Year = {2003},
Key = {fds246950}
}
@article{fds246953,
Author = {Duraisamy, K and Baeder, JD and Liu, JG},
Title = {Concepts and Application of TimeLimiters to High Resolution
Schemes},
Journal = {Journal of Scientific Computing},
Volume = {19},
Number = {13},
Pages = {139162},
Year = {2003},
ISSN = {08857474},
url = {http://dx.doi.org/10.1023/A:1025395707090},
Abstract = {A new class of implicit highorder nonoscillatory time
integration schemes is introduced in a methodoflines
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
timestep restrictions. We propose to apply limiters to
these timeintegration schemes, thus making them nonlinear.
When these new schemes are used with high order spatial
discretizations, solutions remain nonoscillatory for much
larger timesteps 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{fds246961,
Author = {Weinan, E and Liu, JG},
Title = {Gauge method for viscous incompressible flows},
Journal = {Comm. Math. Sci.},
Volume = {1},
Pages = {317332},
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 = {309328},
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 = {120154},
Publisher = {Elsevier BV},
Year = {2002},
Month = {July},
ISSN = {00219991},
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 NavierStokes equations in
the velocitypressure 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
NavierStokes equations in the velocitypressure
formulation. We present numerical results of a secondorder
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
NavierStokes equations in vorticity formulation},
Journal = {Numerische Mathematik},
Volume = {91},
Number = {3},
Pages = {543576},
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 NavierStokes equations in vorticitystream
function formulation. The noslip 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 longstencil 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 Strangtype 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 1D
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 = {2747},
Publisher = {American Mathematical Society (AMS)},
Year = {2002},
Month = {January},
url = {http://dx.doi.org/10.1090/S0025571801013138},
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 semidiscrete 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
semidiscrete solutions.},
Doi = {10.1090/S0025571801013138},
Key = {fds246938}
}
@article{fds246934,
Author = {Liu, JG and Wang, WC},
Title = {An energypreserving MACYee scheme for the incompressible
MHD equation},
Journal = {Journal of Computational Physics},
Volume = {174},
Number = {1},
Pages = {1237},
Publisher = {Elsevier BV},
Year = {2001},
Month = {November},
ISSN = {00219991},
url = {http://dx.doi.org/10.1006/jcph.2001.6772},
Abstract = {We propose a simple and efficient finitedifference method
for the incompressible MHD equation. The numerical method
combines the advantage of the MAC scheme for the
NavierStokes equation and Yee's scheme for the Maxwell
equation. In particular, the semidiscrete 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 2D vortex
sheet},
Journal = {Mathematics of Computation},
Volume = {70},
Number = {234},
Pages = {595606},
Publisher = {American Mathematical Society (AMS)},
Year = {2001},
Month = {April},
url = {http://dx.doi.org/10.1090/S0025571800012710},
Abstract = {We give an elementary proof of the convergence of the point
vortex method (PVM) to a classical weak solution for the
twodimensional 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 KelvinHelmholtz 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/S0025571800012710},
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 = {579593},
Publisher = {American Mathematical Society (AMS)},
Year = {2001},
Month = {April},
url = {http://dx.doi.org/10.1090/S0025571800012394},
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/S0025571800012394},
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 = {13851407},
Year = {2001},
Key = {fds246935}
}
@article{fds246936,
Author = {Liu, JG and Xin, Z},
Title = {Convergence of point vortex method for 2D vortex
sheet},
Journal = {Math. Comp.},
Volume = {70},
Number = {234},
Pages = {565606},
Year = {2001},
url = {http://dx.doi.org/10.1090/S0025571800012710},
Abstract = {We give an elementary proof of the convergence of the point
vortex method (PVM) to a classical weak solution for the
twodimensional 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 KelvinHelmholtz 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/S0025571800012710},
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 = {701710},
Publisher = {WILEY},
Year = {2000},
Month = {December},
ISSN = {02712091},
url = {http://dx.doi.org/10.1002/10970363(20001230)34:8<701::AIDFLD76>3.0.CO;2B},
Abstract = {A finite element method for computing viscous incompressible
flows based on the gauge formulation introduced in [Weinan
E. Liu JG. 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
higherorder) finite elements. This method can achieve
highorder 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 = {10.1002/10970363(20001230)34:8<701::AIDFLD76>3.0.CO;2B},
Key = {fds246933}
}
@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 = {13851407},
Year = {2000},
Month = {October},
Abstract = {A new formulation, a gauge formulation of the incompressible
NavierStokes 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 CrankNicolson 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 NavierStokes 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.},
Key = {fds246932}
}
@article{fds246931,
Author = {Liu, JG and Shu, CW},
Title = {A HighOrder Discontinuous Galerkin Method for 2D
Incompressible Flows},
Journal = {Journal of Computational Physics},
Volume = {160},
Number = {2},
Pages = {577596},
Publisher = {Elsevier BV},
Year = {2000},
Month = {May},
url = {http://dx.doi.org/10.1006/jcph.2000.6475},
Abstract = {In this paper we introduce a highorder discontinuous
Galerkin method for twodimensional incompressible flow in
the vorticity streamfunction 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 2D discontinuous Euler
flows},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {53},
Number = {6},
Pages = {786798},
Publisher = {WILEY},
Year = {2000},
Month = {January},
url = {http://dx.doi.org/10.1002/(SICI)10970312(200006)53:6<786::AIDCPA3>3.0.CO;2Y},
Abstract = {We prove the convergence of a discontinuous Galerkin method
approximating the 2D 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)10970312(200006)53:6<786::AIDCPA3>3.0.CO;2Y},
Key = {fds246930}
}
@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 = {10251055},
Year = {1999},
Month = {July},
Abstract = {Solutions of conservation laws satisfy the monotonicity
property: the number of local extrema is a nonincreasing
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 fluxfunction 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 MUSCLtype scheme that is
second order accurate away from sonic points and
extrema.},
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 = {24462448},
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 = {237256},
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 Godunovtype
methods, its implementation is simple and easy, and the
computational complexity is relatively low. This approach is
parameterfree and requires neither a Riemann solver nor
fieldbyfield 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
RungeKutta time methods. © 1998 Academic
Press.},
Doi = {10.1006/jcph.1998.5970},
Key = {fds246926}
}
@article{fds246925,
Author = {Xu, E and Liu, JG},
Title = {Pricing of mortgagebacked securities with optionadjusted
spread},
Journal = {Managerial Finance},
Volume = {24},
Pages = {94109},
Year = {1998},
Key = {fds246925}
}
@article{fds246922,
Author = {E, W and Liu, JG},
Title = {Finite Difference Methods for 3D Viscous Incompressible
Flows in the VorticityVector Potential Formulation on
Nonstaggered Grids},
Journal = {Journal of Computational Physics},
Volume = {138},
Number = {1},
Pages = {5782},
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 vorticityvector 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 secondorder and fourthorder
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 = {10271053},
Year = {1997},
Month = {July},
Abstract = {We are concerned with the convergence of LaxWeridroff type
schemes with high resolution to the entropy solutions fo:
conservation laws. These schemes include the original
LaxWendroff scheme proposed by Lax and Wendroff in 1960 and
its two step versionsthe 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 Lpestimates of the approximate solutions,
H1compactness 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.},
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 = {6776},
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
illposed 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
nonstaggered grids. © 1997 Academic Press.},
Doi = {10.1006/jcph.1996.5537},
Key = {fds246924}
}
@article{fds246921,
Author = {Liu, JG and Xin, Z},
Title = {Boundarylayer behavior in the fluiddynamic limit for a
nonlinear model Boltzmann equation},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {135},
Number = {1},
Pages = {61105},
Publisher = {Springer Nature},
Year = {1996},
Month = {October},
url = {http://dx.doi.org/10.1007/BF02198435},
Abstract = {In this paper, we study the fluiddynamic limit for the
onedimensional 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 fluiddynamic solution
in terms of the meanfree 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{fds246918,
Author = {Weinan, E and Liu, JG},
Title = {Projection method II: GodunovRyabenki analysis},
Journal = {Siam Journal on Numerical Analysis},
Volume = {33},
Number = {4},
Pages = {15971621},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1996},
Month = {August},
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
GodunovRyabenki 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
finitedifference method for the NavierStokes equation even
in the presence of boundaries. As an example, we analyze the
secondorder 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 secondorder KimMoin
method.},
Doi = {10.1137/s003614299426450x},
Key = {fds246918}
}
@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 = {368382},
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 timestepping
procedure, and the relation between these schemes and the
ones based on velocitypressure 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 thirdor fourthorder
explicit RungeKutta 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 secondorder 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 fourthorder RungeKutta 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 = {373389},
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
shockcapturing methods for slowly moving shocks. It is
known that for slowly moving shocks even a firstorder
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 firstorder
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 = {122138},
Publisher = {Elsevier BV},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1006/jcph.1996.0125},
Abstract = {A new fourthorder accurate finite difference scheme for the
computation of unsteady viscous incompressible flows is
introduced. The scheme is based on the vorticitystream
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 RungeKutta stage, only two
Poissonlike equations have to be solved. The Poissonlike
equations are amenable to standard fast Poisson solvers
usually designed for second order schemes. Detailed
comparison with the secondorder scheme shows the clear
superiority of this new fourthorder scheme in resolving
both the boundary layers and the gross features of the flow.
This efficient fourthorder scheme also made it possible to
compute the driven cavity flow at Reynolds number 106 on a
10242 grid at a reasonable cost. Fourthorder 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{fds246919,
Author = {Levermore, CD and Liu, JG},
Title = {Large oscillations arising in a dispersive numerical
scheme},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {99},
Number = {23},
Pages = {191216},
Publisher = {Elsevier BV},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1016/S01672789(96)001571},
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
periodtwo 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 periodtwo oscillations are seen to
break down. This kind of phenomenon has not been observed
for integrable schemes. The generation and propagation of
periodtwo oscillations are asymptotically analyzed and a
matching formula is found for the transition between
oscillatory and nonoscillatory regions. Modulation equations
are also presented for periodthree oscillations. Numerical
experiments are carried out that illustrate our analysis. ©
1996 Elsevier Science B.V. All rights reserved.},
Doi = {10.1016/S01672789(96)001571},
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 = {35},
Pages = {447461},
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 NavierStokes 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 = {169180},
Year = {1996},
Key = {fds246914}
}
@article{fds246913,
Author = {Liu, J and Xin, Z},
Title = {Convergence of vortex methods for weak solutions to the
2‐D euler equations with vortex sheet data},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {48},
Number = {6},
Pages = {611628},
Year = {1995},
Month = {January},
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{fds246912,
Author = {Weinan, E and Liu, JG},
Title = {Projection method I: convergence and numerical boundary
layers},
Journal = {Siam J. Numer. Anal.},
Volume = {32},
Pages = {10171057},
Year = {1995},
Key = {fds246912}
}
@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},
Pages = {881886},
Year = {1994},
Key = {fds246910}
}
@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 = {555563},
Year = {1994},
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 ChapmanEnskog expansion
and demonstrated numerically.},
Key = {fds246911}
}
@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 = {217256},
Publisher = {Springer Nature},
Year = {1993},
Month = {September},
ISSN = {00039527},
url = {http://dx.doi.org/10.1007/BF00383220},
Abstract = {In this paper we study the asymptotic nonlinear stability of
discrete shocks for the LaxFriedrichs 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
Lpnorm 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 LaxFriedrichs scheme for the
singleshock solution of system of hyperbolic conservation
laws. If the Riemann solution corresponding to the given
farfield 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 LaxFriedrichs scheme. © 1993
SpringerVerlag.},
Doi = {10.1007/BF00383220},
Key = {fds246909}
}
@article{fds246906,
Author = {Chen, GQ and Liu, JG},
Title = {Convergence of secondorder schemes for isentropic gas
dynamics},
Journal = {Mathematics of Computation},
Volume = {61},
Number = {204},
Pages = {607627},
Publisher = {American Mathematical Society (AMS)},
Year = {1993},
Month = {January},
url = {http://dx.doi.org/10.1090/S00255718199311852397},
Abstract = {Convergence of a secondorder shockcapturing 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 = {10.1090/S00255718199311852397},
Key = {fds246906}
}
@article{fds246907,
Author = {Engquist, B and Liu, J},
Title = {Numerical methods for oscillatory solutions to hyperbolic
problems},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {46},
Number = {10},
Pages = {13271361},
Publisher = {WILEY},
Year = {1993},
Month = {January},
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 = {L^{1}stability of stationary discrete
shocks},
Journal = {Mathematics of Computation},
Volume = {60},
Number = {201},
Pages = {233244},
Publisher = {American Mathematical Society (AMS)},
Year = {1993},
Month = {January},
url = {http://dx.doi.org/10.1090/S00255718199311591707},
Abstract = {The nonlinear stability in the Lpnorm, p 1 , of stationary
weak discrete shocks for the LaxFriedrichs 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 LaxFriedrichs
scheme. © 1993 American Mathematical Society.},
Doi = {10.1090/S00255718199311591707},
Key = {fds246908}
}
%% Papers Accepted
@article{fds320739,
Author = {P. Degond and J.G. Liu and S. MerinoAceituno 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}
}
