与可压Euler方程耦合的几类偏微分方程的数学理论研究

本项目主要围绕与可压Euler方程耦合的几类偏微分方程，研究其解的局部与整体适定性、解的性质等数学理论问题。这几类方程包括可压Euler-Piosson方程、可压Euler-Maxwell方程、可压Euler-Korteweg方程以及与可压Euler方程耦合的磁流体力学方程组。这几类方程都具有强烈的物理背景，对它们的研究具有重要的实际意义。近年来，人们主要研究与Nacier-Stokes方程耦合的偏微分方程理论。但是与可压Euler方程耦合的方程的数学问题，比之难度更大，目前还没有固定的数学框架可循。尽管最近在可压Euler-Piosson方程、可压Euler-Maxwell方程的适定性和解的衰减估计方面有突破，但仍有许多问题需要解决。涉及Korteweg项的方程，其适定性和解的性质仅在几个特殊情形下才有部分的结果，理论远未完善。因此，对它们的研究具有完善偏微分方程理论的重要意义。

The project mainly around some PDEs coupled to the compressible Euler equations,and study its local and global well-posedness and the properties of its solutions. These equations are the compressible Euler-Piosson equations、the compressible Euler-Maxwell equations、the compressible Euler-Korteweg equations and the magnetohydrodynamic equation coupled to the compressible Euler equations.These equations have strong Physics background.In recently,We mainly studied the theory of PDEs coupled to the compressible Navier-Stokes equations.Dut the PDEs coupled to the compressible Euler equations are much more complicated than the PDEs coupled to the compressible Navier-Stokes equations,for which no general mathematical framework.Despite recent the wellposedness and time decay estimates of the compressible Euler-Poisson equations、the compressible Euler-Maxwell Equations have breakthrough,but there are still many issues needed to be addressed. The PDEs involving Korteweg,their well-posedness and the properties of the solutions have only part results in a few exceptional cases, the theory is far from perfect.Thus,the study of the PDEs coupled to the compressible Euler equations has perfect significance of the theory of partial differential equations.