带地形作用的正压方程组的Cauchy问题适定性(Ⅰ):无粘

带地形作用的无粘正压方程组是Cauchy-Kowalewska型,可用Cauchy-Kowalewska定理分析,但是分层理论给出更多的信息,例如该方程组是稳定的,该方程组的解空间结构以及Cauchy问题适定性的判别标准．证明了在超平面{t=t^1}包含R^2上粘性正压方程组是适定的。     

带地形作用的正压方程组的Cauchy问题适定性(Ⅱ):粘性

带地形作用的粘性正压方程组不是Cauchy-Kowalewska型,但利用分层理论可证明它是稳定的,并且给出解空间的分层结构以及Cauchy问题适定性的判别标准．证明了在超平面{t=t^0}包含R^2上粘性正压方程组是不适定的。     

半导体磁流体动力学模型经典解的整体适定性

研究一类半导体磁流体动力学模型,它是由关于电子的质量和速度的守恒律方程耦合Maxwell方程构成的流体动力学方程组.在小初值条件下,运用经典的双曲能量方法,得到了磁流体动力学模型Cauchy问题经典解的整体适定性.     

高维变系数自共轭线性偏微分方程组Cauchy问题的一致适定性

本文基于文献[1]-[7],研究自共扼高维线性偏微分方程组的Cauchy问题一致适定性的充分条件,导出了一类抛物型方程组,并推广了文[7]的结果。     

耦合Schrdinger-KdV方程组Cauchy问题的适定性

本文研究了耦合Schrodinger－KdV方程组的Cauchy问题，此耦合方程组刻化了一维Langmuir和离子声波相互作用的非线性动力学行为．本文建立了此问题在Hk×Hk中的整体适定性理论（k∈Z+）．     

N阶线性卷积方程组Cauchy问题的适定性和求解公式

在本文中提出N阶线性卷积方程组,并在一定条件下证明了其Cauchy问题的适定性,还找到了求解公式。     

三维Navier-Stokes方程组解的正则性相关问题的一些探索

本文回顾了近年来作者团队对三维不可压缩Navier-Stokes方程组Cauchy问题所作的一些探索.众所周知,三维不可压缩Navier-Stokes系统存在整体Leray-Hopf弱解.当弱解满足Prodi-Serrin条件时,解是正则的.本文在解正则性条件的判别方面取得了一些新结果.特别对于轴对称系统,当旋转速度为零时,系统的整体适定性结论是众所周知的.本文在研究中发现了一个新的守恒量,进而得到了旋转速度非零时其轴对称解正则性条件的一些新进展,还得到了一个系统只要求初始旋转速度小的整体适定性结果,进一步还将结果推广到变密度的系统.最后,考虑了一类超耗散广义Navier-Stokes系统的整体适定性,其中水平黏性项具有更高阶导数D_h~(2α),α≥4/3.     

具定重特征的一类双曲型全特征算子的Cauchy问题的Gevrey类适定性

本文研究了全特征Cauchy问题(1.1)的Gevrey类适定性。得到如下的两个主要结果: 1.在条件(Ⅰ)—(Ⅵ)下,对任意的s≥1,全特征Cauchy问题(1.1)均在B([O,T],G_(L~2)~s(R~n))内适定。 2.在条件(Ⅰ)—(Ⅴ)及(Ⅶ)下,若1≤s<θ~(-1)(θ由(1.10)式定义),则全特征Cauchy问题(1.1)在B([O,T],G_(L~3)~8(R~n))内适定;若s=θ~(-1),则存在s>0充分小,使得(1.1)在B([O,s],G_(L~3)~(θ-1)(R~n)内有唯一解。     

大气运动基本方程组的稳定性分析

以分层理论提供的基本方法分析大气运动基本方程组的拓扑学特征;证明局地直角坐标系中的大气运动基本方程组在无穷可微函数类中是稳定方程;给出局部解意义下使方程组典型定解问题适定的充要条件;讨论大气动力学中有关“以过去推测未来”以及当涉及应用问题时如何修改定解条件和下垫面的选择等问题;指出在通常假设下,基本方程组中的3个运动方程和连续方程完全决定了这个方程组的性质．     

抽象Cauchy问题的适定性与算子半群

在算子A非稠定、问题解非指数有界的情况下，研究抽象Cauchy问题的适定性及其与A生成的算子族之间的关系．首先，引进（ACP1）的C适定性概念和C半群生成元的全新定义，证明：（ACP1）是C适定的充要条件是A生成C半群．并给出A生成非指数有界C半群的充分条件．另外，引进（ACP2）的（n，k）适定性定义，并讨论（n，k）适定性与积分余弦函数的关系．     

Well-posedness of the analytic Cauchy problem in classes of entire functions of finite order

We study the Cauchy problem for a system of complex linear differential equations in scales of spaces of functions of exponential type with an integral metric. Conditions under which this problem is well posed are obtained. These sufficient conditions are shown to be also necessary for the well-posedness of the Cauchy problem in the case of systems of ordinary differential equations with a parameter.     

Somk remarks on the navier-stokes equations with a pressure-dependent viscosity

We discuss the Navier-Stokes equations for an incompressible fluid wit a viscosity that is allowed to depend on the pressure. Ellipticity and the complementing condition of Agmon, Douglis and Nirenberg [l] are discussec It is found that the pressure dependence of viscosity leads to the possibilit of a change of type. It is shown that the Dirichlet initial-boundary valu problem is well posed as long as the equations do not change type     

Regularization for a class of ill-posed evolution problems in Banach space

We study evolution equations in Banach space, and provide a general framework for regularizing a wide class of ill-posed Cauchy problems by proving continuous dependence on modeling for nonautonomous equations. We approximate the ill-posed problem by a well-posed one, and obtain H?lder-continuous dependence results that provide estimates of the error for a class of solutions under certain stabilizing conditions. For examples that include the linearized Korteweg-de Vries equation and the Schr?dinger equation in L ^p ,p??2, we obtain a family of regularizing operators for the ill-posed problem. This work extends to the nonautonomous case several recent results for ill-posed problems with constant coefficients.     

Regularization for Ill-posed Cauchy problems associated with generators of analytic semigroups

This paper is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. Our main result is that if −A is the generator of an analytic semigroup, then there exists a family of regularizing operators for such an ill-posed Cauchy problem by using the quasi-reversibility method, fractional powers and semigroups of linear operators. The applications to ill-posed partial differential equations are also given.     

Elliptic-Hyperbolic Systems and the Einstein Equations

The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasi-linear elliptic-hyperbolic system of evolution equations. We prove that the Cauchy problem is locally strongly well posed and that a continuation principle holds.¶For initial data satisfying the Einstein constraint and gauge conditions, the solutions to the elliptic-hyperbolic system defined by the gauge fixed Einstein evolution equations are shown to give vacuum space-times.     

A note on the derivation of filter regularization operators for nonlinear evolution equations

Despite the strong focus of regularization on ill-posed problems, the general construction of such methods has not been fully explored. Moreover, many previous studies cannot be clearly adapted to handle more complex scenarios, albeit the greatly increasing concerns on the improvement of wider classes. In this note, we rigorously study a general theory for filter regularized operators in a Hilbert space for nonlinear evolution equations which have occurred naturally in different areas of science. The starting point lies in problems that are in principle ill-posed with respect to the initial/final data – these basically include the Cauchy problem for nonlinear elliptic equations and the backward-in-time nonlinear parabolic equations. We derive general filters that can be used to stabilize those problems. Essentially, we establish the corresponding well-posed problem whose solution converges to the solution of the ill-posed problem. The approximation can be confirmed by the error estimates in the Hilbert space. This work improves very much many papers in the same field of research.     

On the Cauchy Problem for D t 2-D xa(t,x)D x in the Gevrey Class of Order s > 2

We study the Cauchy problem for second order hyperbolic equations with non negative characteristic form of two independent variables. We show that for such equations in divergence-free form, the Cauchy problem is well posed in the Gevrey class of order less than 5/2.     

Missing boundary data reconstruction by the factorization method

We consider the data completion problem for the Laplace equation in a cylindrical domain. The Neumann and Dirichlet boundary conditions are given on one face of the cylinder while there is no condition on the other face. This Cauchy problem has been known since Hadamard (1953) to be ill-posed. Here it is set as an optimal control problem with a regularized cost function. We use the factorization method for elliptic boundary value problems. For each set of Cauchy data, to obtain the estimate of the missing data one has to solve a parabolic Cauchy problem in the cylinder and a linear equation. The operator appearing in these problems satisfy a Riccati equation that does not depend on the data. To cite this article: A. Ben Abda et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Asymptotic Solutions of Hamilton-Jacobi Equations with State Constraints

We study Hamilton-Jacobi equations in a bounded domain with the state constraint boundary condition. We establish a general convergence result for viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with the state constraint boundary condition to asymptotic solutions as time goes to infinity.     

Estimates for solutions of Burgers type equations and some applications

We obtain precise large time asymptotics for the Cauchy problem for Burgers type equations satisfying shock profile condition. The proofs are based on the exact a priori estimates for (local) solutions of these equations and the result of [G.M. Henkin, A.A. Shananin, Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations, J. Math. Pures Appl. 83 (2004) 1457–1500].