超音速绕流中若干问题的研究

超音速气体绕楔形流是流体力学数学理论中一个长期悬而未决的基本问题，诱导出了偏微分方程和纯数学研究中大量极富挑战性的问题。本项目主要研究：1、高维定常超音速绕流中强波的相互作用解的存在性及稳定性，探索波的连续反射现象及耗散机制；2、定常高维高超音速极限流绕三维楔面问题的Radon测度解理论。本项目研究要克服的典型难点包括强非线性波的相互作用及双边反射的耗散机制、高维拟线性双曲方程组特征边值问题、支撑在曲面上的Dirac测度的刻画等。这些研究涉及特征分解、Glimm格式、波前跟踪等方法以及微分几何、几何测度论等知识和理论的应用和发展。本项目的研究不仅有助于帮助人们更深入地理解一些物理现象，同时也将有助于促进高维守恒律方程组数学理论的发展。

The problem of supersonic flow onto solid wedges is a longstanding fundamental problem in mathematical fluid mechanics, which leads to a large number of challenging problems in the study of partial differential equations and pure mathematics. The main problems of this project are as follows: 1. The existence and stability of solutions of strong wave interactions in multi-dimensional steady supersonic flow around solid bodies, and exploring the continuous reflection phenomenon and dissipation mechanism of the flow field; 2. Radon measure solution theory of steady multi-dimensional hypersonic limit flow passing a three-dimensional wedge. The main difficulties lie in the interaction of strong nonlinear waves and the dissipation mechanism of bilateral reflections of the waves, the study of the characteristic boundary value problem of multi-dimensional quasilinear hyperbolic equations, and the study of the Dirac measure supported on a surface. The methods we use involve characteristic decomposition, Glimm scheme, wave-front tracking method, differential geometry, geometric measure theory, etc. The research of this project not only contributes to the in-depth understanding of some physical phenomena, but also promotes the development of mathematical theory of multi-dimensional conservation laws.