双曲型偏微分方程初边值问题理论及应用

This project is mainly devoted to studying the long time well-posed theory as well as its application to aerodynamics on a large group of solutions to quasi-linear hyperbolic partial differential equations with initial-boundary value conditions. This topic arises from the study of long time stability of various wave patterns when compressible gases pass through solid bodies. Meanwhile, it also comes from investigation on the structures of various of piecewise smooth solutions for quasi-linear hyperbolic partial differential equations, such as structures of free characteristic surfaces of sound wave solutions, structures of shock waves formed when smooth solutions blow up and so on. Due to its profound physical background and mathematical theory, this topic becomes one of the fundamental subjects in theories as well as their applications of quasi-linear hyperbolic partial differential equations. Based on the pioneering work and our academic accumulation, this project wants to expand the above theme in the following several aspects of research: (1) Long time well-posedness of smooth solutions to quasi-linear hyperbolic partial differential equations with initial-boundary value conditions in exterior domains or half spaces; (2) Formation and construction of shock fronts formed by supersonic steady potential flow equations; (3) Formation of shock fronts formed when smooth solutions of two dimensional Euler system blow up; (4) Global stability of rarefaction waves formed when supersonic gases pass through bending downward ramps; (5) Long time stability of supersonic oblique shocks formed when supersonic gases pass through sharp wedges.

本项目主要研究拟线性双曲型方程初边值问题解的长时间适定性理论及其在空气动力学理论中的应用。该类问题来源于可压缩气体流经固定物体时所产生的各类波型长时间稳定性研究；同时，也来源于拟线性双曲型方程初值问题分片光滑解自由特征面稳定性或光滑解爆破后所对应的激波生成理论研究等。其深刻的物理背景及数学理论，使得此类问题成为双曲型偏微分方程理论及应用领域中的基本问题。基于前人及我们研究工作的积累，本项目围绕上述主题展开以下几个方面的研究: (1) 外区域及半空间区域中，拟线性双曲型方程初边值问题光滑解的长时间适定性; (2) 超音速定常位势流方程的激波生成及构造; (3) 二维Euler方程组光滑解爆破对应的激波生成及构造; (4) 超音速流体流经下弯物体时疏散波的整体稳定性; (5) 超音速流体流经尖锐楔形物体时产生的超音速斜激波的长时间稳定性。

拟线性双曲方程组经典解的时间整体适定性或有限时间爆破一直是偏微分方程领域基本问题，其中初值问题或初边值问题经典整体解的存在性、经典解的有限时间爆破以及相关奇性结构的刻画都是目前此领域重点关注的问题。在此背景下， 我们认真开展了项目所拟定的五个问题的研究工作，建立了如下三个研究结果：(1)满足真正非线性条件下的一般一维严格双曲方程组经典解的有限时间爆破机制；(2)三维乘积空间上Klein-Gordon方程经典解的几乎整体唯一存在性；(3)三维乘积空间上Klein-Gordon方程经典解的整体唯一存在性。