超音速绕流及线性退化双曲组中若干问题的研究

双曲守恒律方程组研究中的高维活塞问题及线性退化方程组大初值问题具有重要的物理背景与应用价值，近年来受到国内外同行的高度关注。本项目一方面研究Chaplygin气体和位势流的高维活塞问题的可解性及解的结构，另一方面，基于Chaplygin气体研究线性退化双曲组大初值问题解的适定性等。相关理论分析涉及混合型方程的可解性以及测度论在偏微分方程弱解理论中的应用等，具有一定难度。本项目将综合利用特征分析法及混合型方程的一些处理方法，充分发掘所考察模型的物理背景与数学理论之间的联系，并引入新的研究思路和方法。本项目的研究有助于对一般高维守恒律方程组初边值问题弱解理论的进一步探索。

Multi-dimensional piston problems and initial value problems for linearly degenerate partial differential equations in quasi-linear hyperbolic conservation laws have physical backgrounds and are useful in applications. They have attracted a lot of attention in recent years. This project investigates the solvability and the structure of the solutions of piston problems both for Chaplygin gas and potential flow. Based on Chaplygin gas, we study the existence of solutions of linearly degenerate huperbolic equations with large initial data. Since the theoretical study of them involves the analysis of equations of mixed type as well as the application of measure theory in the weak theory of hyperbolic conservation laws, the study on these problems is challenging. We will overcome these difficulties by using the method of character analysis, the way of analyzing equations of mixed type, and the relationship between physical backgrounds and mathematics, as well as developing some new ideas and methods. Our research will make contributions to the development of the weak theory of initial boundary value problems for multi-dimensional conservation laws.