一类高维拟线性双曲型守恒律组初边值问题的研究

The Chaplygin gas is a good approximation to a real gas. The weak solution theory of it in multi-dimensional case is now an international hotspot in the research field of partial differential equations. This project mainly focuses on: 1) studying the two-dimensional Riemann initial-boundary value problems of Euler system for the Chaplygin gas in an angle domain bounded by two walls, the construction and classification of the global solutions; 2) The study of shock diffraction on a backward-facing step, analyzing the properties of solution of this unsteady problem in finite time. The character of this project is that: 1）focus on studying the theory of weak solutions and the structure of nonliear waves; 2) concerning the study of nonlinear second order elliptic equation with mixed boundary condition on a domain with an angle point. This project will utilize the methods of characteristic analysis, the classical estimate skills of second order elliptic equations, etc., and closely combining the theory and method developed latest to research these problems and also pay attention to the connection between physical background and mathematical results. The study of this project will help us to research the weak solution theory of more general initial-boundary value problems for multi-dimensional conservation laws.

Chaplygin 气体源于物理力学中对实际气体近似的一个常用模型，其高维弱解理论是国际偏微分方程研究领域的前沿和热点之一。本项目主要研究：1）Chaplygin气体Euler方程组在一个以两壁为边形成的角状区域上的二维黎曼初边值问题，分析边界条件对解的影响，构造整体解并分类；2）研究激波在后台阶上的绕射问题，分析该非定常问题的解在有限时间内的性态。本项目的特点：1）侧重于Euler方程初边值问题的弱解理论和波的结构的研究；2）注重带角点的非线性二阶退化椭圆型方程混合型边值问题的研究。本项目将综合利用特征分析法以及二阶椭圆方程的一些经典估计技巧等，并且紧密结合最新发展的有关理论和方法对问题进行研究，尤其注重物理背景与数学理论之间的联系。本项目的研究有助于探索更一般的高维守恒律方程组初边值问题的弱解理论。

描述理想流体的可压缩欧拉方程组是最重要的双曲方程组，也是航天航空领域的基本方程组。该方程高维情形不仅具有重要的应用背景，而且呈现出非常丰富的结构，然而相关数学理论极具挑战性。目前相关数学弱解理论研究主要集中在包含一些比较典型的波的结构的理论分析或者是含真空情形。本项目执行以来，围绕欧拉方程组初边值问题的数学理论展开：研究了Chaplygin 气体的黎曼初值问题，黎曼初边值问题，得到了解的整体结构，并对不同初值对应的波结构进行了分类；并对可能含真空的扩张管中的超音速流问题进行了分析。在SCI期刊上发表标注了该项目资助的论文3篇，其中一篇在国际权威杂志“Archive for Rational Mechanics and Analysis”上发表。