高维接触间断的稳定机理分析

This project will study the mathematical theory of stability of multi-dimensional contact discontinuities in the quasi-linear hyperbolic conservation law systems which have valuable application background. We will investigate the stability condition and structural stability of contact discontinuities of 3D compressible nonisentropic steady Euler equations. By developing the theories of free boundary value problem of nonlinear hyperbolic equations and micro-local analysis, we will analyze the coupling of fluid velocity, temperature and the discontinuous front to discover main factors in the stability mechanism. We will also study the stability of contact discontinuities of multi-dimensional magnetohydrodynamic (MHD) equations. Detail analysis of the role of the magnetic field in the stability mechanism will help us establish the stability theory of contact discontinuities of MHD equations. Moreover, in the view of wave interactions, stability of contact discontinuities with high frequency oscillating waves in steady Euler equations and MHD equations will be discussed by employing the method of nonlinear geometric optics. This project will improve the mathematical theory of well-posedness of weak solutions to conservation law systems, especially the stability analysis of multi-dimensional contact discontinuities with many unstable factors. It will provide theoretical basis for the research of various singular wave interactions and Mach reflection. Meanwhile, it will develop the theory of boundary value problems of nonlinear partial differential equations and facilitate the mathematical research on aerodynamics.

本项目将研究具有重要应用背景的几类拟线性双曲守恒律方程组高维接触间断稳定性的数学理论。重点考察三维可压缩非等熵稳态流体中接触间断的稳定性条件及其结构稳定性，分析速度场、温度场和间断面的相互耦合作用，通过发展非线性双曲方程组自由边界问题、微局部分析等数学理论，揭示制约接触间断稳定性的重要因素；考察磁流体力学方程组中接触间断的稳定性，重点分析磁场在稳定机制中的角色，建立磁场作用下接触间断的稳定性理论；进一步从波的干扰角度出发，通过发展非线性几何光学方法等理论，研究理想流体和磁流体中高维接触间断在高频振荡波干扰下的稳定性。本项目的研究将丰富完善双曲守恒律系统的弱解理论,尤其是具有诸多不稳定因素的高维接触间断的稳定性分析，为进一步研究各种奇性波的干扰、激波的马赫反射等热点问题提供相关的研究基础。同时，将发展非线性偏微分方程边值问题理论，促进空气动力学数学基础的研究。

本项目主要研究了流体力学方程组和磁流体力学方程组定解问题中解的适定性。此类方程组具有较强的物理背景，是偏微分方程研究中的重要课题之一。项目资助期间，负责人与合作者研究了理想磁流体力学方程组电流涡量片在高频振荡波扰动下的稳定性，推导了相应的渐近展开式。考察了一类含参数的广义水波方程组，通过守恒律理论建立了初值问题解的局部适定性，并且在一定条件下得到了爆破准则。另外，采用包含等离子体剪切流的磁流体力学模型，研究了多电流片系统中亚阿尔芬类型的涡流对三种典型的磁流体不稳定模态的性质的影响。通过本征值方法进行分析，发现对称型和反对称型的剪切流对不同模态的磁流体不稳定性影响有本质的区别。在该项目的资助下，在国际期刊发表了2篇SCI科研论文，相关成果完善了部分双曲守恒律系统的数学理论并且为解释观测到的物理现象提供了一定的理论依据。