高维双曲守恒律组几类特征间断的性态研究

The mathematical study of hyperbolic conservation laws has been an important branch of the theory of partial differential equations. Compressible Euler equations and compressible magetohydrodynamic (MHD) equations are the prototypes of nonlinear hyperbolic conservation laws. This project mainly concerns entropy waves for two-dimensional compressible Euler equations and contact discontinuities for compressible MHD equations. These problems are characteristic free boundary problems with the symbol associated to the free boundary being non-elliptic. How to solve these difficulties for obtaining stability results of these characteristic discontinuities is the main goal of this project.

双曲守恒律组的数学研究一直是偏微分方程理论的一个重要分支。可压缩Euler方程组和可压缩MHD方程组是非线性双曲守恒律组的典型特例。本项目拟主要研究二维可压缩Euler方程组的熵波以及可压缩MHD方程组的接触间断。这两个问题是特征自由边界问题，与这两种特征间断阵面相关联的象征是非椭圆的。如何克服这些难点来得到这两种特征间断的稳定性结果是本项目的主要研究目标。

双曲守恒律组是偏微分方程理论中的一个主要分支。作为高维双曲守恒律组熵解的一类基本波，关于特征间断的研究近年来受到了众多专家学者的关注。在本项目的资助下，申请人证明了两个空间变量下非等熵可压缩欧拉方程组以及相对论欧拉方程组的涡片解的非线性稳定性，分别发表在偏微分方程领域国际知名期刊Journal of Differential Equations和Archive for Rational Mechanics and Analysis上。