WELL-POSEDNESS IN CRITICAL SPACES FOR THE FULL COMPRESSIBLE MHD EQUATIONS

In this paper we prove local well-posedness in critical Besov spaces for the full compressible MHD equations in R^N, N≥ 2, under the assumptions that the initialdensity is bounded away from zero. The proof relies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term.     

Existence of strong solutions in critical spaces for barotropic viscous fluids in larger spaces

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N 2.We address the question of well-posedness for large data having critical Besov regularity.Our result improves the analysis of Danchin and of the author inasmuch as we may take initial density in B N p p,1 with 1 p < +∞.Our result relies on a new a priori estimate for the velocity,where we introduce a new unknown called effective velocity to weaken one of the couplings between the density and the velocity.In particular,our result is the first in which we obtain uniqueness without imposing hypothesis on the gradient of the density.     

Well-posedness of a non-local abstract Cauchy problem with a singular integral

A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of the evolution system is proved under some boundedness and smoothness conditions on the coefficient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with a singular convolution kernel, which is a generalized form of the stationary Wigner equation. Our investigation considerably improves the understanding of the open problem concerning the well-posedness of the stationary Wigner equation with in ow boundary conditions.     

Global Well-posedness for the 2D Micropolar Rayleigh–Bénard Convection Problem without Velocity Dissipation

In this article, we study the Cauchy problem to the micropolar Rayleigh–Bénard convection problem without velocity dissipation in two dimension. We first prove the local well-posedness of a smooth solution, and then establish a blow up criterion in terms of the gradient of scalar temperature field. At last, we obtain the global well-posedness to the system.     

Well- and Ill-Posedness Issues for a Class of 2D Wave Equation with Exponential Growth

Extending the previous work [1], we establish well-posedness results for a more general class of semilinear wave equations with exponential growth. First, we investigate the well-posedness in the energy space. Then, we prove the propagation of the regularity in the Sobolev spaces HS（IR^2） with s 〉 1. Finally, an ill-posedness result is obtained in HS（IR^2） for s 〈 1.     

Time-Periodic Solution to the Compressible Navier–Stokes/Allen–Cahn System

In this paper,we investigate the time-periodic solution to a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids with a time periodic external force in a periodic domain in R^N.The existence of the time-periodic solution to the system is established by using an approach of parabolic regularization and combining with the topology degree theory,and then the uniqueness of the period solution is obtained under some smallness and symmetry assumptions on the external force.     

TIME PERIODIC SOLUTION TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS IN A PERIODIC DOMAIN

This article is concerned with the time periodic solution to the isentropic compressible Navier-Stokes equations in a periodic domain. Using an approach of parabolic regularization, we first obtain the existence of the time periodic solution to a regularized problem under some smallness and symmetry assumptions on the external force. The result for the original compressible Navier-Stokes equations is then obtained by a limiting process.The uniqueness of the periodic solution is also given.     

WELL-POSEDNESS FOR THE CAUCHY PROBLEM TO THE HIROTA EQUATION IN SOBOLEV SPACES OF NEGATIVE INDICES

The local well-posedness of the Cauchy problem for the Hirota equation is established for low regularity data in Sobolev spaces Hs(s≥-1/4). Moreover, the global well-posedness for L2 data follows from the local well-posedness and the conserved quantity. For data in Hs(s > 0), the global well-posedness is also proved. The main idea is to use the generalized trilinear estimates, associated with the Fourier restriction norm method.     

Small Alfvén number limit for incompressible magneto-hydrodynamics in a domain with boundaries

For any fixed Alfvén number, the local well-posedness is proved for the equations of threedimensional ideal incompressible magneto-hydrodynamics in a domain with boundaries. Under appropriate conditions, a smooth solution is shown to exist in a time interval independent of the Alfvén number, and the solutions of the original system tend to the solutions of a two-dimensional Euler flow coupled with a linear transport equation as the Alfvén number goes to zero.     

COUPLING OF FINITE ELEMENT AND BOUNDARY ELEMENT METHODS FOR THE SCATTERING BY PERIODIC CHIRAL STRUCTURES

Consider a time-harmonic electromagnetic plane wave incident on a biperiodic structure in R^3. The periodic structure separates two homogeneous regions. The medium inside the structure is chiral and nonhomogeneous. In this paper, variational formulations coupling finite element methods in the chiral medium with a method of integral equations on the periodic interfaces are studied. The well-posedness of the continuous and discretized problems is established. Uniform convergence for the coupling variational approximations of the model problem is obtained.     

Global well-posedness for the 3-D MHD equations with partial diffusion in the periodic domain

In this paper, we prove the global well-posedness of the 3-D magnetohydrodynamics(MHD) equations with partial diffusion in the periodic domain when the initial velocity is small and the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition.     

Existence of weak solutions to the three-dimensional steady compressible magnetohydrodynamic equations

The purpose of this paper is to prove the existence of a spatially periodic weak solution to the steady compressible isentropic MHD equations in R3 for any specific heat ratio γ 1.The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density,and the method of weak convergence.According to the author's knowledge,it is the first result that treats in three dimensions the existence of weak solutions to the steady compressible MHD equations with γ 1.     

LOCAL WELL-POSEDNESS IN SOBOLEV SPACES WITH NEGATIVE INDICES FOR A SEVENTH ORDER DISPERSIVE EQUATION

This paper is concerned with the Cauchy problem of a seventh order dispersive equation. We prove local well-posedness with initial data in Sobolev spaces Hs（R） for negative indices of s〉-114 .     

INVERSE CONDUCTIVE MEDIUM SCATTERING WITH UNKNOWN BURIED OBJECTS

This paper is concerned with inverse acoustic scattering in an inhomogeneous medium with a conductive boundary condition and the unknown buried impenetrable objects inside.Using a variational approach,we establish the well-posedness of the direct problem.For the inverse problem,we shall numerically reconstruct the inhomogeneous medium from the far-field data for different kinds of cases.For the case when a Dirichlet boundary condition is imposed on the buried object,the classical factorization m...     

Well-posedness of equations with fractional derivative

We study the well-posedness of the equations with fractional derivative D^αu（t） = Au（t） ＋ f（t）,0≤ t ≤ 2π, where A is a closed operator in a Banach space X, α 〉 0 and D^α is the fractional derivative in the sense of Weyl. Using known results on LP-multipliers, we give necessary and/or sufficient conditions for the LP-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.     

A mathematical model for unsteady mixed flows in closed water pipes

We present the formal derivation of a new unidirectional model for unsteady mixed flows in nonuniform closed water pipes.In the case of free surface incompressible flows,the FS-model is formally obtained,using formal asymptotic analysis,which is an extension to more classical shallow water models.In the same way,when the pipe is full,we propose the P-model,which describes the evolution of a compressible inviscid flow,close to gas dynamics equations in a nozzle.In order to cope with the transition between a free surface state and a pressured(i.e.,compressible) state,we propose a mixed model,the PFS-model,taking into account changes of section and slope variation.     

CONVERGENCE OF NUMERICAL SCHEMES FOR A CONSERVATION EQUATION WITH CONVECTION AND DEGENERATE DIFFUSION

The approximation of problems with linear convection and degenerate nonlinear difFusion,which arise in the framework of the transport of energy in porous media with thermodynamic transitions,is done usingθ-scheme based on the centred gradient discretisation method.The convergence of the numerical scheme is proved,although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field,owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting.Some numerical examples,using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme,show the role ofθfor stabilising the scheme.     

GLOBAL WELL-POSEDNESS FOR THE KLEIN-GORDON EQUATION BELOW THE ENERGY NORM

We study global well-posedness below the energy norm of the Cauchyproblem for the Klein-Gordon equation in R^n with n≥3. By means of Bourgain‘s method along with the endpoint Strichartz estimates of Keel and Tao, we prove the H^s-global well-posedness with s&lt;1 of the Cauchy problem for the Klein-Gordon equation. This we do by establishing a series of nonlinear a priori estimates in the setting of Besov spaces.     

A THEORETICAL AND NUMERICAL STUDY FOR THE ROD-LIKE MODEL OF A POLYMERIC FLUID

The first part of this paper is concerned with the well-posedness for the rigid rod-like model in shear flow of a polymeric fluid. The constitutive relations considered in this work are motivated by the kinetic theory. The stress tensor is given by an integral which involves the solution of the Fokker-Planck equation. A novel numerical scheme for the Fokker-Planck equation is proposed, which preserves the positivity of the distribution function.Another part of this work establishes the convergence theory of the fully discretized schemes for a simple micro-macro simulation of a polymeric flow.     

Incompressible Limit of the Compressible Q-tensor System of Liquid Crystals

We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more specific, the convergence of the weak solutions of the compressible nematic liquid crystal model to the incompressible one is proved as the Mach number approaches zero, and we also obtain the similar results in the stochastic setting when the equations are driven by a stochastic fo...