CMS Forthcoming Papers
Forthcoming Papers
Jianfeng Lu and Jeremy Marzuola
Strang splitting methods for a quasilinear Schrodinger equation -- convergence, instability and dynamics
We study the Strang splitting scheme for quasilinear Schrodinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear
instability of the numerical scheme, which explains the numerical blow-up of large data solutions
and connects to analytical breakdown of regularity of solutions to quasilinear Schrodinger equations. Numerical tests are performed for a modified version of the superfluid thin film equation.
Anne C. Bronzi, Milton C. Lopes Filho and Helena J. Nussenzveig Lopes
Wild solutions for 2D incompressible ideal flow with passive tracer
In [C. De Lellis and L. Szekelyhidi, Ann. of Math. 170, m 2009] C. De Lellis and L. Szekelyhidi
Jr. constructed wild solutions of the incompressible Euler
equations using a reformulation of the Euler equations as a differential inclusion
together with convex integration. In this article we adapt their construction to the system consisting
of adding the transport of a passive scalar to the two-dimensional incompressible Euler equations.
Jacques Schneider
On a well-posed simulation model for
multicomponent reacting gases
We aim to present a relaxation model that can be used in real simulations of dilute
multicomponent reacting gases. The kinetic framework is the semi-classical approach with only
one variable for the internal energy modes. The relaxation times for the internal energy modes
are assumed to be smaller than the chemistry characteristic times. The strategy is the same as
in [S. Brull, J. Schneider, Comm. Math. Sci. to appear]. That is a sum of operators for respectively the mechanical and chemical processes. The
mechanical operator(s) is the "natural" extension to polyatomic gases of the method of moment
relaxations presented in [S. Brull, J. Schneider, Cont. Mech. Thermodyn. 20, 63-74, 2008] and
[S. Brull, V. Pavan and J. Schnieder, Eur. J. Mech. (B-Fluids) 33, 74-86, 2012]. The derivation of the chemical model lies on the chemical processes
at thermal equilibria. It is shown that this BGK approach features the same properties as the
Boltzmann equation: conservations and entropy production. Moreover null entropy production states
are characterized by vanishing chemical production rates. We also study the hydrodynamic limit in
the slow chemistry regime. Finally we show that the whole set of parameters that are used in the
derivation of the model can be calculated by softwares such as EGlib or STANJAN.
Giuliano Lazzaroni, Mariapia Palombaro and Anja Schloemerkemper
A discrete to continuum analysis of dislocations in nanowire heterostructures
Epitaxially grown heterogeneous nanowires present dislocations at
the interface between the phases if their radius is big. We consider a corresponding
variational discrete model with quadratic pairwise atomic interaction energy.
By employing the notion of Gamma-convergence and a geometric rigidity estimate,
we perform a discrete to continuum limit and a dimension reduction to
a one-dimensional system. Moreover, we compare a defect-free model and models
with dislocations at the interface and show that the latter are energetically
convenient if the thickness of the wire is sufficiently large.
Stefan Possanner, Luigi Barletti, Florian Hehats and Claudia
Numerical study of a quantum-diffusive spin model for two-dimensional
electron gases
We investigate the time evolution of spin densities in a two-dimensional electron gas subjected to Rashba spin-orbit coupling on the basis of the quantum drift-diffusive model derived in
[L. barletti and F. Mehats, J. Math. Phys. 51, 10]. This model assumes the electrons to be in a quantum equilibrium state in the form of a Maxwellian operator. The resulting quantum drift-diffusion equations for spin-up and spin-down densities are coupled in a non-local manner via two spin chemical potentials (Lagrange multipliers) and via off-diagonal elements of the equilibrium spin density and spin current matrices, respectively. We present two space-time discretizations of the model, one semi-implicit and one explicit, which comprise also the Poisson equation in order to account for electron-electron interactions. In a first step pure time discretization is applied in order to prove the well-posedness of the two schemes, both of which are based on a functional formalism to treat the non-local relations between spin densities. We then use the
fully space-time discrete schemes to
simulate the time evolution of a Rashba electron gas confined in a bounded domain and subjected to spin-dependent external potentials. Finite difference approximations are first order in time and second order in space. The discrete functionals introduced are minimized with the help of a conjugate gradient-based algorithm, where the Newton method is applied in order to find the respective line minima. The numerical convergence in the long-time limit of a Gaussian initial condition towards the solution of the corresponding stationary Schr\"odinger-Poisson problem is demonstrated for different values of the parameters $\eps$ (semiclassical parameter), $\alpha$ (Rashba coupling parameter), $\Delta x$ (grid spacing) and $\Delta t$ (time step). Moreover, the performances of the semi-implicit and the explicit scheme are compared.
Solene Ozere and Carole Le Guyader
Topology preservation for image-registration-related deformation fields
In this paper, we address the issue of designing a theoretically well-
motivated and computationally efficient method ensuring topology preservation
on image-registration-related deformation elds. The model is motivated by a mathematical characterization of topology preservation for
a deformation eld mapping two subsets of Z^2, namely, positivity of the
four approximations to the Jacobian determinant of the deformation on
a square patch. The first step of the proposed algorithm thus consists in
correcting the gradient vector field of the deformation (that does not comply with the topology preservation criteria) at the discrete level in order to
fulfill this positivity condition. Once this step is achieved, it thus remains
to reconstruct the deformation field, given its full set of discrete gradient
vectors. We propose to decompose the reconstruction problem into independent problems of smaller dimensions, yielding a natural parallelization
of the computations and enabling us to reduce drastically the computational time (up to 80 in some applications). For each subdomain, a functional minimization problem under Lagrange interpolation constraints is
introduced and its well-posedness is studied: existence/uniqueness of the
solution, characterization of the solution, convergence of the method when
the number of data increases to infinity, discretization with the Finite Element Method and discussion on the properties of the matrix involved in
the linear system. Numerical simulations based on OpenMP parallelization and MKL multi-threading demonstrating the ability of the model to
handle large deformations (contrary to classical methods) and the interest
of having decomposed the problem into smaller ones are provided.
Jose Raul Quintero Henao
Stability of 2D solitons
for a sixth order
Boussinesq type model
We study orbital stability of solitary wave of the least energy for a nonlinear 2D
Benney-Luke model of higher order related with long water waves with small amplitude in the presence
of strong surface tension. We follow a variational approach which includes the characterization
of the ground state solution set associated with solitary waves. We use the Hamiltonian structure
of this model to establish the existence of an energy functional conserved in time for the modulated
equation associated with this Benney-Luke type model. For wave speed near to zero or one, and in
the regime of strong surface tension, we prove the orbital stability result by following a variational
Pierre Degond, Giacomo Dimarco, Thi Bich Ngoc Mac, Nan Wang
Macroscopic models of collective motion with repulsion
We study a system of self-propelled particles which interact with their neighbors
via alignment and repulsion. The particle velocities result from self-propulsion and repulsion by
close neighbors. The direction of self-propulsion is continuously aligned to that of the neighbors,
up to some noise. A continuum model is derived starting from a mean-field kinetic description of
the particle system. It leads to a set of non conservative hydrodynamic equations. We provide a
numerical validation of the continuum model by comparison with the particle model. We also provide
comparisons with other self-propelled particle models with alignment and repulsion.
Giacomo Albi, Michael Herty, and Lorenzo Pareschi
Kinetic description of optimal control problems and applications
to opinion consensus
In this paper an optimal control problem for a large system of interacting agents
is considered using a kinetic perspective. As a prototype model we analyze a microscopic model
of opinion formation under constraints. For this problem a Boltzmann type equation based on a
model predictive control formulation is introduced and discussed. In particular, the receding horizon
strategy permits to embed the minimization of suitable cost functional into binary particle interactions. The corresponding Fokker-Planck asymptotic limit is also derived and explicit expressions
of stationary solutions are given. Several numerical results showing the robustness of the present
approach are finally reported.
G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani
On Rosenau-type approximations to fractional diffusion equations
Owing to the Rosenau argument [Phys. Rev. A 46, 12-15, 1992], originally proposed to obtain a regularized
version of the Chapman-Enskog expansion of hydrodynamics, we introduce a non-local linear kinetic
equation which approximates a fractional diffusion equation. We then show that the solution to
this approximation, apart of a rapidly vanishing in time perturbation, approaches the fundamental
solution of the fractional diffusion (a Levy stable law) at large times.
Emeric Bouin and Sepideh Mirrahimi
A Hamilton-Jacobi approach for a model of population
structured by space and trait
We study a non-local parabolic Lotka-Volterra type equation describing a population struc-
tured by a space variable x\in R^d and a phenotypical trait \theta \in {\Cal \Theta}.
Considering diffusion, mu-
tations and space-local competition between the individuals, we analyze the asymptotic (long
time/longreal phase WKB ansatz, we prove that the propagation of the population in space can be de-
scribed by a Hamilton-Jacobi equation with obstacle which is independent of \theta. The effective
Hamiltonian is derived from an eigenvalue problem.
The main difficulties are the lack of regularity estimates in the space variable, and the lack of
comparison principle due to the non-local term.
Tadele Mengesha and Qiang Du
Multiscale analysis of linearized peridynamics
In this paper, we study the asymptotic behavior of a state-based multiscale heterogeneous
peridynamic model. The model involves nonlocal interaction forces with highly oscillatory
perturbations representing the presence of heterogeneities on a finer spatial length scale. The
two-scale convergence theory is established for a steady state variational problem associated
with the multiscale linear model. We also examine the regularity of the limit nonlocal equation
and present the strong approximation to the solution of the peridyanmic model via a suitably
scaled two-scale limit.
Nicola Bellomo and Abdelghani Bellouquid
On Multiscale Models of Pedestrian Crowds
From Mesoscopic to Macroscopic
This paper deals with the derivation of macroscopic equations from the underlying
mesoscopic description that is suitable to capture the main features of pedestrian crowd dynamics.
The interactions are modeled by means of theoretical tools of game theory, while the macroscopic
equations are derived from asymptotic limits.
Jian Zu, Shihong Shao, Huazhong Tang and Dongyi Wei
Multi-hump solitary waves of a nonlinear Dirac equation
This paper concentrates on a (1+1)-dimensional nonlinear Dirac (NLD) equation
with a general self-interaction, being a linear combination of the scalar, pseudoscalar, vector and
axial vector self-interactions to the power of the integer k+1. The solitary wave solutions to the
NLD equation are analytically derived, and the upper bounds of the hump number in the charge,
energy and momentum densities for the solitary waves are proved analytically in theory. The results
show that: (1) for a given integer k, the hump number in the charge density is not bigger than 4,
while that in the energy density is not bigger than 3; (2) those upper bounds can only be achieved in
the situation of higher nonlinearity, namely k\in {5,6,7, ...} for the charge density and k \in
{3,5,7 ...}
fo (3) the momentum density has the same multi-hump structure as the energy
(4) more than two humps (resp. one hump) in the charge (resp. energy) density can only
happen under the linear combination of the pseudoscalar self-interaction and at least one of the
scalar and vector (or axial vector) self-interactions. Our results on the multi-hump structure will be
interesting in the interaction dynamics for the NLD solitary waves.
Stefano Bosia, Monica Conti and Maurizio Grasselli
On the Cahn-Hilliard-Brinkman system
We consider a diffuse interface model for phase separation of an isothermal
incompressible binary fluid in a Brinkman porous medium. The coupled system consists
of a convective Cahn-Hilliard equation for the phase field \phi, i.e., the difference of
the (relative) concentrations of the two phases, coupled with a modified Darcy equation
proposed by H.C. Brinkman in 1947 for the fluid velocity u. This equation incorporates
a diffuse interface surface force proportional to \phi \nabla \mu, where \mu where 5 is the so-called chemical
potential. We analyze the well-posedness of the resulting Cahn-Hilliard-Brinkman
(CHB) system for (\phi, u). Then we establish the existence of a global attractor and the
convergence of a given (weak) solution to a single equilibrium via
Lojasiewicz-Simon
inequality. Furthermore, we study the behavior of the solutions as the viscosity goes to
zero, that is, when the CHB system approaches the Cahn-Hilliard-Hele-Shaw (CHHS)
system. We first prove the existence of a weak solution to the CHHS system as limit of
CHB solutions. Then, in dimension two, we estimate the difference of the solutions to
CHB and CHHS systems in terms of the viscosity constant appearing in CHB.
Isabelle Tristani
Fractional Fokker-Planck equation
This paper deals with the long time behavior of solutions to a "fractional
Fokker-Planck" equation of the form \partial_t f= I[f] + \div(xf) where the operator I stands for
a fractional Laplacian. We prove an exponential in time convergence towards equilibrium
in new spaces. Indeed, such a result was already obtained in a L^2 space with a weight
prescribed by the equilibrium in [Gentil, I. and ]Imbert, C., Asymp. Anal. 59, 3-4 (2008), 125-138] .
We improve this result obtaining the convergence
in a L^1 space with a polynomial weight. To do that, we take advantage of the recent
paper [Gualdani, M.P., Mischler, S. and Mouhot, C., http://hal.archives-ouvertes.fr/ccsd-10).]
in which an abstract theory of enlargement of the functional space of the
semigroup decay is developed.
Benjamin Gess and Panagiotis E. Souganidis
Scalar conservation laws with multiple rough fluxes
We study pathwise entropy solutions for scalar conservation laws with inhomogeneous
fluxes and quasilinear multiplicative rough path dependence. This extends the previous work of Lions,
Perthame and Souganidis who considered spatially independent and inhomogeneous fluxes with multiple paths and a single driving singular path respectively. The approach is motivated by the theory
of stochastic viscosity solutions which relies on special test functions constructed by inverting locally
the flow of the stochastic characteristics. For conservation laws this is best implemented at the level
of the kinetic formulation which we follow here.
Jack Spencer and Ke Chen
A Convex and Selective Variational Model for Image Segmentation
Selective image segmentation is the task of extracting one object of interest from an image,
based on minimal user input. Recent level-set based variational models have shown to
be effective and reliable, although they can be sensitive to initialization due to the minimization problems being nonconvex. This sometimes means that successful segmentation
relies too heavily on user input or a solution found is only a local minimizer, i.e. not the
correct solution. The same principle applies to variational models that extract all objects
in an image (global segmentation); however, in recent years, some have been successfully
reformulated as convex optimization problems, allowing global minimizers to be found.
There are, however, problems associated with extending the convex formulation to the
current selective models, which provides the motivation for the proposal of
a new selective
model. In this paper we propose a new selective segmentation model combining ideas from
global segmentation that can be reformulated as convex such that a global minimizer can
be found independent of initialization. Numerical results are given that demonstrate its
reliability in terms of removing the sensitivity to initialization present in previous models,
and its robustness to user input.
Xinglong Wu
On the global well-posedness of the stochastic plasma equations with
magnetic-curvature-driven in R^3
The present paper is devoted to the study of the Cauchy problem
for the magnetic-curvature-driven electromagnetic fluid equation with
random effects in a bounded domain of R^3. We first obtain a crucial
property of the solution to O.U. process, thanks to the lemma, the local
well-posedness of the equation with the initial and boundary value is
established by the contraction mapping argument. Finally, by virtue
of a priori estimates, the existence and uniqueness of global solution
to the stochastic plasma equation is proved.
Yanqiu Guo, Konrad Simon and Edriss Titi
Global well-posedness of a system of nonlinearly
coupled KdV equations of Majda and Biello
This paper addresses the problem of global well-posedness of a cou-
pled system of Korteweg-de Vries equations, derived by Majda and Biello in the
context of nonlinear resonant interaction of Rossby waves, in a periodic setting in
homogeneous Sobolev spaces H^s, for s\ge 0. Our approach is based on a
sussessive time-averaging method developed by Babin, Ilyin and Titi
[A.V. Babin, A.A. Ilyin and E.S. Titi, Comm. Pure Appl. Math. 64, 591-648,
Zhong Tan and Yong Wang
Asymptotic behavior of solutions to the compressible bipolar
Euler-Maxwell system in R^3
We study the large time behavior of solutions near a constant equilibrium
state to the compressible Euler-Maxwell system in R^3. We first refine
the global existence of solutions by assuming that the
initial data
is small in the H^3 norm but its higher order derivatives could be
large. If further the initial data belongs to H^{-s}
(0\le s \le 3/2) or
B^{-s}_{2, \infty} (0\le s \le 3/2), then we obtain the various time decay
rates of the solution and its higher order derivatives. As an immediate
byproduct the L^p-L^2 (1\le p \le 2) type of the decay rates
follows without requiring the smallness for L^p norm of the initial data.
So far, our decay results are
most comprehensive ones for the bipolar Euler-Maxwell system in R^3.
Honghu Liu, Taylan Sengul, Shouhong Wang and Pingwen Zhang
Dynamic Transitions and Pattern Formations for Cahn-Hilliard
Model with Long-Range Repulsive Interactions
The main objective of this article is to study the order-disorder
phase transition and pattern formation for systems with long-range repulsive
interactions. The main focus is on a Cahn-Hilliard model with a nonlocal
term in the corresponding energy functional, representing certain long-range
repulsive interaction. We show that as soon as the trivial steady state loses
its linear stability, the system always undergoes a dynamic transition to one
of the three types-- continuous, catastrophic, or random-- forming different
patterns/structures, such as lamellae, hexagonally packed cylinders,
rectangles, and spheres. The types of transitions are dictated by a non-dimensional
parameter, measuring the interactions between the long-range repulsive term
and the quadratic and cubic nonlinearities in the model. In particular, the
hexagonal pattern is unique to this long-range interaction, and it is captured
by the corresponding two-dimensional reduced equations on the center
manifold, which involve (degenerate) quadratic terms and non-degenerate cubic
terms. Explicit information on the metastability and basins of attraction of
different ordered states, corresponding to different patterns, are derived as well.
Existence of regular solutions to an Ericksen-Leslie model of liquid crytsal system
We study a general Ericksen-Leslie system with non-constant density, which
describes the flow of nematic liquid crystal. In particular the model investigated here is associated with Parodi's relation. We prove that: in two dimension, the solutions are globally
regul in three dimension, the solutions are globally regular with small
initial data, or for short time with large data. Moreover, a weak-strong type of uniqueness
result is obtained.
Karl Yngve Lervag and John Lowengrub
Towards a second-order
diffuse-domain method for solving PDEs in complex geometries
In recent work, Li et al. (Comm.\ Math.\ Sci., 7:81-107, 2009) developed
a diffuse-domain method (DDM) for solving partial differential equations in
complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary
conditions.
The diffuse-domain method uses an implicit representation of the
geometry where the sharp boundary is replaced by a diffuse layer with
thickness $\epsilon$ that is typically proportional to the minimum grid size.
The original equations are reformulated on a larger regular domain and the
boundary conditions are incorporated via singular source terms.
resulting equations can be solved with standard finite difference and finite
element software packages.
Here, we present a matched asymptotic analysis of
general diffuse-domain methods for Neumann and Robin boundary conditions.
Our analysis shows that for certain choices of the boundary condition
approximations, the DDM is second-order accurate in $\epsilon$.
However, for
other choices the DDM is only first-order accurate.
This helps to explain
why the choice of boundary-condition approximation is important for rapid
global convergence and high accuracy.
Our analysis also suggests correction
terms that may be added to yield more accurate diffuse-domain methods.
Simple modifications of first-order boundary condition approximations are
proposed to achieve asymptotically second-order accurate schemes.
analytic results are confirmed numerically in the $L^2$ and $L^\infty$ norms
for selected test problems.
Chunxiong Zheng
Global geometrical optics approximation to the high frequency Helmholtz equation
with discontinuous media
Global geometrical optics method is a new semi-classical approach for the high frequency linear waves proposed
by the author in [Commun. Math. Sci., 11(1), 105-140, 2013]. In this paper, we rederive it in a more concise way. It is shown
that the right candidate of solution ansatz for the high frequency wave equations is the extended WKB function, other than the
WKB function used in the classical geometrical optics approximation. A new and main contribution of this paper is an interface
analysis for the Helmholtz equation when the incident wave is of extended WKB-type. We derive asymptotic expressions for
the reflected and/or transmitted propagating waves in the general case. These expressions are valid even when the incident
rays include caustic points.
Laurent Gosse
MUSCL reconstruction and Harr wavelets
MUSCL extensions (Monotone Upstream-centered Schemes for Conservation Laws)
of the Godunov numerical scheme for scalar conservation laws are shown
to admit a rather simple reformulation when recast in the formalism of the
Harr multi-resolution analysis of L^2(R). By pursuing this wavelet
reformulation, a seeimingly new MUSCI-WB scheme is derived for
advection-reaction equations which is stable for a Courant number up to 1
(instead of roughly 1/2). However these high-order reconstructions aren't
likely to improve the handling of delication nonlinear wave interactions
in the involved case of systems of Conservation/Balance laws.
Fenghang Yin and Jack Xin
Phaseliftoff: an accurate and stable phase retrieval method
based on difference of trace and Frobenius norms
Phase retrieval aims to recover a signal x\in C^m from its amplitude
measurements |(x, a_i)|^2, i=1,2,...m, where a_i's are over-complete basis
vectors, with m at least 3n-2 to ensure a unique solution up to a constant phase factor.
The quadratic measurement becomes linear in terms of the rank-one matrix X=xx^*.
Phase retrieval is then a rank-
one minimization problem subject to linear constraint for which a convex relaxation based on trace-norm minimization
(PhaseLift) has been extensively studied recently. At m=O(n),
PhaseLift recovers with high probability the rank-one
solution. In this paper, we present a precise proxy of rank-one condition via the dierence of trace and Frobenius norms
which we call PhaseLiftO. The associated least squares minimization with this penalty as regularization is equivalent
to the rank-one least squares problem under a mild condition on the measurement noise. Stable recovery error estimates
are valid at m=O(n) with high probability. Computation of PhaseLiftO minimization is carried out by a convergent
dierence of convex functions algorithm. In our numerical example, a_i's
are Gaussian distributed. Numerical results show
that PhaseLiftO outperforms PhaseLift and its nonconvex variant (log-determinant regularization), and successfully
recovers signals near the theoretical lower limit on the number of measurements without the noise.
Denise Aregba
Godunov scheme for Maxwell equations with kerr nonlinearity
We study the Godunov scheme for a nonlinear Maxwell model arising
in nonlinear optics, the Kerr model. This is a hyperbolic system of
conservation laws with some eigenvalues of variable multiplicity,
neither genuinely nonlinear nor linearly degenerate. The solution of
the Riemann problem for the full-vector 6\times 6 system is constrcuted
and proved to exist for all data. This solution is comprated to the one of the reduced Transverse Magnetic model. The scheme is implemented in one
and two space dimensions. The results are very close to the ones
obtained with a Kerr-Debye relaxation approximation.
Nengsheng Fang, Xinfeng Ruan and Caiciu Liao
Real option model upon dynamic growth process with construction
A real option model is built upon a stochastic process for some real investment
decision making in incomplete markets. Typically, optimal consumption level is obtained under logarithm
utility constraint, and a partial integro-dierential equation (PIDE) of the real option is deduced by
martingale methods. Analytical formulation of the PIDE is solved by Fourier transformation. Two
types of decision making strategies, i.e.: option price and IRP (inner risk primium) comparisons,
are provided. Monte Carlo simulation and numerical computation are provided at last to verify the
conclusion.
Jason Kaye, Lin Lin and Chao Yang
A posteriori error estimator for adaptive local basis functions
to solve Kohn-Sham density functional theory
Kohn-Sham density functional theory is one of the most widely used electronic
structure theories. The recently developed adaptive local basis functions
form an accurate and
systematically improvable basis set for solving Kohn-Sham density
functional theory using discontinuous Galerkin methods, requiring a small number of basis functions per atom. In this paper we
develop residual-based a posteriori error estimates for the adaptive
local basis approach, which can
be used to guide non-uniform basis refinement for highly inhomogeneous
systems such as surfaces
and large molecules. The adaptive local basis functions are non-polynomial basis functions, and
standard a posteriori error estimates for
hp-refinement using polynomial basis functions do not directly apply. We generalize the error estimates for
hp-refinement to non-polynomial basis functions.
We demonstrate the practical use of the a posteriori error estimator in
performing three-dimensional
Kohn-Sham density functional theory calculations for quasi-2D aluminum surfaces and a single-layer
graphene oxide system in water.
Ewelina Zatorska, Piotr Bogus law Mucha
Multicomponent Mixture Model. The Issue of Existence via Time Discretization
We prove the existence of global-in-time weak solutions to a model of chemically reacting
mixture. We consider a coupling between the compressible Navier-Stokes system and the reaction
diusion equations for chemical species when the thermal eects are neglected. We rst prove
the existence of weak solutions to the semi-discretization in time. Based of this, the existence of
solutions to the evolutionary system is proved.
Hong Cai and Zhong Tan
Weak time-periodic solutions to the compressible Navier-Stokes-Poisson equations
The compressible Navier{Stokes{Poisson equations driven by a time{periodic external force are considered in this paper. The system takes into account the eect of self{gravitation.
We establish the existence of weak time{periodic solutions on condition that the adiabatic constant
satisfices \gamma>5/3.
D. Benedetto, E. Caglioti, U. Montemagno
On the complete phase synchronization for the Kuramoto model
in the mean-field limit
We study the Kuramoto model for coupled oscillators.
For the case of identical natural frequencies, we
give a new
proof of the complete frequency synchronization
extending this result to the continuous version of the model,
we manage to prove the complete phase
synchronization for any non-atomic measure-valued initial datum.
We also discuss the relation between the boundedness of the entropy
and the convergence to an incoherent state, for the case of non
identical natural frequencies.
Christophe Gomez
Wave Propagation in Underwater Acoustic Waveguides with Rough Boundaries
In underwater acoustic waveguides a pressure field can be decomposed over three kinds of modes: the propagating modes, the radiating modes and the evanescent
modes. In this paper, we analyze the effects produced by a randomly perturbed
free surface and an uneven bottom topography on the coupling mechanism
between these three kinds of modes. Using an asymptotic analysis based on
a separation of scales technique we derive the asymptotic form of the
distribution of the forward mode amplitudes. We show that the surface
and bottom fluctuations affect the propagating-mode amplitudes mainly in the
same way. We observe an effective amplitude attenuation which is mainly
due to the coupling between the propagating modes themselves. However, for the
highest propagating modes this mechanism is stronger and due to an efficient
coupling with the radiating modes.
Igor Kukavica and Fei Wang
Weighted Decay for the Surface
Quasi-Geostrophic Equation
We address the weighted decay for the solution of the surface quasi-geostrophic
(SQG) equation. The first moment decay was obtained by M. and T. Schonbek.
Here we obtain new decay rates of the first moment and the rate of increase
of this quantity under natural assumption on the initial data.
Subsonic potential flows in general smooth bounded domains
In this paper, we study the existence and uniqueness of subsonic potential flows in general smooth bounded domains when the normal component of the momentum on the boundary is prescribed. It is showed that if the Bernoulli constant is given larger than a critical number, there exists a unique subsonic potential flow. Moreover, as the Bernoulli constants decrease to the critical number, the subsonic flows converge to a subsonic-sonic flow.
Qiang Tao, Jincheng Gao and Zheng-an Yao
Global strong
solutions of the compressible nematic liquid crystal flow with the cylinder symmetry
In this paper, we consider the well-posedness of the compressible nematic liquid
crystal flow with the cylinder symmetry in R^n.
By establishing a uniform point-wise positive lower
and upper bounds of the density, we derive the global existence and uniqueness of strong solution
and show the long time behavior of the global solution. Our results do not need the smallness of the
initial data. Furthermore, a regularity result of global strong solution is given as well.
Rongjie Lai, Jianfeng Lu and Stanley Osher
Density matrix minimization with l_1 regularization
We propose a convex variational principle to find sparse representation of
low-lying eigenspace of symmetric metrices in the context of electronic
structure calculation, this corresponds to a sparse density matrix
minimization algorithm with l_1 regularization. The minimization
problem can be efficiently solved by a split Bregman iteration type
algorithm. We further prove that from any initial condition, the algorithm
converges to a minimizer of the variational principle.
Qian Tao and Hui Zhang
Noninear continuum theory of smectic-C liquid crystals
Here we develop a model of smectic-C liquid crystals by forming their hydrostatic and hydrodynamic theories, which are motivated by the work of W. E
[Arch. Rational Mech. Anal., 137(1997)].
A simplied model is also presented. In order to prove the rationality of the model, we establish
the energy dissipative relation of the new model. Meanwhile, we verify that the system can also
be obtained using asymptotic analysis when both the fluid and layers are incompressible.
Russel E. Caflisch, Stanley J. Osher, Hayden Schaeffer and Giang Tran
PDEs with compressed solutions
Sparsity plays a central role in recent developments in signal processing, linear algebra, statistics,
optimization, and other fields. In these developments, sparsity is promoted through the addition of an L^1 norm (or related quantity) as a constraint or penalty in a variational principle. We apply this approach
to partial differential equations that come from a variational quantity, either by minimization (to obtain
an elliptic PDE) or by gradientfl ow (to obtain a parabolic PDE). Also, we show that some PDEs can be
rewritten in an L^1 form, such as the divisible sandlile problem and signum-Gordon. Addition of an L^1 term in the variational principle leads to a modied PDE where a subgradient term appears. It is known
that modified PDEs of this form will often have solutions with compact support, which corresponds to the
discrete solution being sparse. We show that this is advantageous numerically through the use of efficient
algorithms for solving L^1 based problems.
Armin Lechleiter and Stefan Peters
Determining transmission eigenvalues of anisotropic inhomogeneous media from
far field data
We characterize interior transmission eigenvalues of penetrable anisotropic acoustic
scattering objects by a technique known as inside-outside duality. This method has recently been
identified to be able to link interior eigenvalues of the penetrable scatterer with the behavior of
the eigenvalues of the far field operator for the corresponding exterior time-harmonic scattering
problem. A basic ingredient for the resulting connection is a suitable self-adjoint factorization of the
far field operator based on wave number-dependent function spaces. Under certain conditions on
the anisotropic material coeffi cients of the scatterer, the inside-outside duality allows to rigorously
characterize interior transmission eigenvalues from multi-frequency far field data. This theoretical
characterization moreover allows to derive a simple numerical algorithm for the approximation of
interior transmission eigenvalues. Since it is merely based on far field data, the resulting eigenvalue
solver does not require knowledge on the scatterer or its
several numerical
examples show its feasibility and accuracy for noisy data.
Bojan Popov and Vladimir Tomov
Central schemes for Mean Field Games
Mean field type models have been recently introduced and analyzed by
Lasry and Lions. They describe a limiting behavior of stochastic dierential games as
the number of players tends to infinity. Numerical methods for the approximation of
such models have been developed by Achdou, Camilli, Capuzzo-Dolcetta, Gueant, and
others. Efficient algorithms for such problems require special efforts and so far all methods
introduced have been first order accurate. In this manuscript we design a second order
accurate numerical method for time dependent Mean Field Games. The discretization is
based on central schemes which are widely used in hyperbolic conservation laws.
John Barrett, Harald Garcke and Robert Nurnberg
Stable Numerical Approximation of Two-Phase Flow
with a Boussinesq?Scriven Surface Fluid
We consider two-phase Navier-Stokes flow with a Boussinesq-Scriven
fluid. In such a fluid the rheological behaviour at the interface includes surface
viscosity effects, in addition to the classical surface tension effects. We introduce
and analyze parametric finite element approximations, and show, in particular,
stability results for semidiscrete versions of the methods, by demonstrating that a
free energy inequality also holds on the discrete level. We perform several numerical
simulations for various scenarios in two and three dimensions, which illustrate the
effects of the surface viscosity.
Celment Cances, Frederic Coquel, Edwige Godlewski, Helene Mathis, and Noclas Seguin
Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations
In order to validate theoretically a dynamic model adaptation method, we propose
to consider a simple case where the
model error
can be thoroughly analyzed. The dynamic model
adaptation consists in detecting at each time step the region where a given
fine model
replaced by a corresponding
coarse model
in an automatic way, without deteriorating the accuracy
of the result, and to couple the two models, each being computed on its respective region. Our fine
model is 2 \times 2 system which involves a small time scale and setting this
time scale to 0 leads to a
classical conservation law, the
coarse model, with a flux which depends on the unknown and on space
and time. The adaptation method provides an intermediate
solution which results from the
coupling of both models at each time step. In order to obtain sharp and rigorous error estimates for
the model adaptation procedure, a simple fine model is investigated and smooth transitions between
fine and coarse models are considered. We refine existing stability results for conservation laws with
respect to the flux function which enables us to know how to balance the time step, the threshold
for the domain decomposition and the size of the transition zone. Numerical results are presented
at the end and show that our estimate is optimal.
Christophe Berthon, Bruno Debroca and Afeintou Sangam
An entropy preserving relaxation scheme for the ten-moments equations with source terms
The present paper concerns the derivation of finite volume methods to approximate weak solutions of Ten-Moments equations
with source
terms. These equations model compressible anisotropic flows. A relaxation
type scheme is proposed to approximate such flows. Both robustness and
stability conditions of the suggested finite volume methods
are established. To
prove discrete entropy inequalities, we derive a new strategy based on local
minimum entropy principle and never use some approximate PDE's auxiliary
model as usually recommended. Moreover, numerical simulations in 1D and
in 2D illustrate our approach.
Alberto Bressan and Tao Huang
Representation of Dissipative Solutions to a Nonlinear Variational Wave
The paper introduces a new way to construct dissipative solutions to a second order
variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case,
here the source terms are discontinuous and the discontinuities are not always crossed
transversally. Solutions to the semilinear system are obtained by an approximation argument, relying on Kolmogorov's compactness theorem. Reverting to the original variables,
one recovers a solution to the nonlinear wave equation where the total energy is a monotone
decreasing function of time.
N. Ben Abdallah, E. Fouassier, C. Jourdana and D. Sanchez
On a model of magnetization switching driven by a spin current: a
multiscale approach
We study a model of magnetization switching driven by a spin current: the magnetization reversal
can be induced without applying an external magnetic field. We first write our one dimensional
model in an adimensionalized form, using a small parameter $\epsilon$.
We then explain the various time
and space scales involved in the studied phenomena. Taking into account these scales, we first construct an appropriate numerical scheme, that allows us to recover numerically various results of
physical experiments. We then perform a formal asymptotic study as $\epsilon$
tends to 0, using a multiscale
approach and asymptotic expansions. We thus obtain approximate limit models that we compare
with the original model via numerical simulation.}