On the stochastic quasi-linear symmetric hyperbolic system

We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for a quasi-linear symmetric hyperbolic system with random noise in Rd. When the noise is multiplicative satisfying some nondegenerate conditions and the initial data are sufficiently small, we show that the solution exists globally in time in probability, i.e., the probability of global existence can be made arbitrarily close to one if the initial date are small accordingly.     

Robin Boundary Value Problem for One-Dimensional Landau–Lifshitz Equations

In this paper, we are concerned with the existence and uniqueness of global smooth solution for the Robin boundary value problem of Landau-Lifshitz equations in one dimension when the boundary value depends on time t. Furthermore, by viscosity vanishing approach, we get the existence and uniqueness of the problem without Gilbert damping term when the boundary value is independent of t.     

A nonlinear singular diffusion equation with source

In this paper, the existence, uniqueness and dependence on initial value of solution for a singular diffusion equation with nonlinear boundary condition are discussed. It is proved that there exists a unique global smooth solution which depends on initial data in L¹ continuously.     

Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given.     

Global solution to the incompressible flow of liquid crystals

The initial–boundary value problem for the three-dimensional incompressible flow of liquid crystals is considered in a bounded smooth domain. The existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small data. It is also proved that when the strong solution exists, a weak solution must be equal to the unique strong solution with the same data.     

Global smooth solution for the compressible Landau-Lifshitz-Bloch equation

The Landau-Lifshitz-Bloch equation is often used to describe micromagnetic phenomenon under high temperature. In this paper, we establish the existence and uniqueness of global smooth solution for the initial problem of the compressible Landau-Lifshitz-Bloch equation in dimension one.     

Asymptotic Behavior of the Solution to a 3-D Simplified Energy-Transport Model for Semiconductors

The well-posedness of smooth solution to a 3-Dsimplified Energy-Transport model is discussed in this paper. We prove the local existence, uniqueness, and asymptotic behavior of solution to the equations with hybrid cross-diffusion. The smooth solution convergences to a stationary solution with an exponential rate as time tends to infinity when the initial date is a small perturbation of the stationary solution.     

Etude d'une equation non lineaire d'evolution de type divergentiel

We give a result of existence and uniqueness for a class of nonlinear divergential evolution equations, firstly in the case of a smooth initial data, then in the general case. Under suitable conditions, the dercrease and the regularity of the solution are specified.     

Global existence of weak discontinuous solutions to the Cauchy problem with small BV initial data for quasilinear hyperbolic systems

In this paper, we study the Cauchy problem for quasilinear hyperbolic system with a kind of non‐smooth initial data. Under the assumption that the initial data possess a suitably small bounded variation norm, a necessary and sufficient condition is obtained to guarantee the existence and uniqueness of global weak discontinuous solution. Copyright © 2014 John Wiley & Sons, Ltd.     

Hyperbolic stefan problem in a curvilinear sector

The problem with unknown boundaries for a first-order semilinear hyperbolic system is studied in the case where the curve of definition of the initial conditions degenerates to a point. An existence and uniqueness theorem for a classical solution of the problem is proved for small t.     

Well-posedness,global existence,and blowup phenomena for a periodic quasi-linear hyperbolic equation

We establish the local well-posedness of a recently derived model for small-amplitude, shallow water waves. For a large class of initial data we prove global existence of the corresponding solution. Criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained, and an existence and uniqueness result for a class of global weak solutions is also given. © 1998 John Wiley & Sons, Inc.     

高维非线性四阶抛物型方程初值问题全局解

对一类高维非线性四阶抛物型方程初值问题,在初始数据适当小及非线性项适当光滑的情况下,获得了其古典解全局存在性。     

Mathematical analysis of absorbing boundary conditions for the wave equation: the corner problem

Our goal in this work is to establish the existence and the uniqueness of a smooth solution to what we call in this paper the corner problem, that is to say, the wave equation together with absorbing conditions at two orthogonal boundaries. First we set the existence of a very smooth solution to this initial boundary value problem. Then we show the decay in time of energies of high order--higher than the order of the boundary conditions. This result shows that the corner problem is strongly well-posed in spaces smaller than in the half-plane case. Finally, specific corner conditions are derived to select the smooth solution among less regular solutions. These conditions are required to derive complete numerical schemes.

     

Poche de tourbillon pour Euler 2D dans un ouvert à bord

We consider Euler equations for an homogeneous incompressible non viscous fluid inside a smooth bounded domain of the plane. For an initial data of smooth vortex patch type, we obtain existence and uniqueness of a solution of the same type, locally in time if the initial patch is tangent to the boundary of the domain, and globally in time if the patch is far away from the boundary. We use pseudo-differential calculus to take care of the boundary condition. For the tangent limit case, we show that the velocity gradient of a vortex patch is Hölder continuous up to the boundary of the patch, using singular integrals. Our method provide also a result for several mutually tangent vortex patches in the plane.     

Boundary value problems for the Vlasov-Maxwell equation in one dimension

Global existence in time and uniqueness of solutions are proved for the Cauchy problem for the Vlasov-Maxwell system of equations in one dimension. The limiting values of the field ±(x, t) as the space variable x → E ∞ are shown to be uniquely determined by the initial data. This result then yields existence of solutions of various boundary value problems. Solutions periodic in x are also discussed in this same framework.     

Local Solutions to the Navier–Stokes Equations with Mixed Boundary Conditions

We study the existence and uniqueness of solutions to the system of three-dimensionalNavier-Stokes equations and the continuity equation for incompressible fluid with mixedboundary conditions. It is known that if the Dirichlet boundary conditions are prescribed on thewhole boundary and the total influx equals zero, then weak solutions exist globally in time andthey are even unique and smooth in the case of two-dimensional domains. The methods that havebeen used to prove these results fail if non-Dirichlet conditions are applied on a part of theboundary, since there is then no control over the energy flux on this part of the boundary. In thispaper, we prove the existence and the uniqueness of solutions on a (short) time interval. Theproof is performed for Lipschitz domains and a wide class of initial data. The length of the timeinterval on which the solution exists depends only on certain norms of the data.     

On second grade fluids with vanishing viscosity

We consider in ℝⁿ (n = 2, 3) the equation of a second grade fluid with vanishing viscosity, also known as Camassa-Holm equation. We prove local existence and uniqueness of solutions for smooth initial data. We also give a blow-up condition which implies that the solution is global for n = 2. Finally, we prove the convergence of the solutions of second grade fluid equation to the solution of the Camassa-Holm equation as the viscosity tends to zero.     

Existence and regularity for a chemotaxis model involved in the modeling of multiple sclerosis

We study in this work the global existence of solutions to a system of reaction cross diffusion equations appearing in the modeling of multiple sclerosis, in the one-dimensional case. Weak solutions are obtained for general initial data, and existence, uniqueness, stability and smoothness are proven when initial data are smooth.

     

Existence and Uniqueness of Smooth Solution for a Four-waves Coupled System

In this paper, we consider a four-waves coupled system which describes the interaction between particles. Based on the uniform bound and strong convergence property in lower order norm, local existence and uniqueness of smooth solution is established by a limiting argument. Moreover, we show the solution exists globally in two dimensional case under certain condition on the size for $L^2$ norm of the initial data.     

Hereditary differential systems with constant delays. I. General case

This paper presents a discussion of the structure of hereditary differential systems defined on a Banach space with initial data in the space of p-integrable maps. Both finite and infinite time histories are allowed. A unified approach to Global and Local Cauchy problems on finite or infinite time intervals is presented. An existence theorem for Carathéodory systems and an existence and uniqueness theorem for Lipschitz systems are derived. In both cases continuity of a solution with respect to the initial data is established.