非线性双曲守恒律的Riemann问题和激波反射问题

Riemann solutions of nonlinear hyperbolic conservation laws and problems on shock reflection are studied in this project. For one dimension problem, the asymptotic properties of Riemann solutions will be considered; stability of solutions and the existence of weak solutions of Cauchy problem for systems of conservation laws will be considered respectively by perturbed Riemann problems and viscosity method. For multi-dimensional problems, the project will focus on two aspects: the one is two dimensional Riemann problem and the other is problems on shock reflection. As for the former, we concentrate on self-similar solutions of Riemann problem, make sure the supersonic and subsonic regions in transonic flow, obtain the configurations and well posedness of solutions in supersonic region by generalized characteristic analysis, and discuss the existence of solutions in subsonic region by studying free boundary problems for elliptic equations. For the latter, that is, problems on shock reflection, we mainly focus our attention on critical conditions of appearing some certain reflection configurations and the conditions of transitions between different reflection configurations. Dividing the types of shock reflection by considering the angle of wedge and the strength of incident shock, then discuss the position of the triple point and the solutions in supersonic region around, by using the Rankine-Hugoniot conditions to obtain a multivariate polynomial equations.

本项目研究非线性双曲型守恒律方程组的Riemann解及激波反射问题。研究一维双曲守恒律组相关模型的Riemann解的渐近性质，通过对扰动Riemann问题的研究讨论解的稳定性，用粘性消失的方法研究守恒律方程组初值问题的弱解的存在性。对于高维问题，主要集中于以下两方面：其一为二维Riemann问题，其二为激波反射问题。前者主要研究二维Riemann问题的自相似解，确定跨音流的超音区和亚音区，用广义特征分析的方法确定解在超音区的结构及适定性，通过对椭圆型方程的自由边界问题的讨论分析解在亚音区的存在性。后者则着重于研究激波反射过程中，某些特定反射结构发生的临界条件以及不同反射结构之间过渡的转换条件。以斜坡角度的大小以及入射激波的强弱作为依据划分激波反射的类型，利用Rankine-Hugoniot条件，建立多元高次方程，进而确定三波点的位置及其附近超音区解的状态。

本项目研究了双曲型守恒律方程组黎曼解的相关问题。对于一维问题，集中考虑了带源项的非线性双曲守恒律方程组。由于右端项的影响，方程组不再是自相似的，因此在构造黎曼解的过程中需要一些不同的处理。研究了黎曼解的小扰动问题，解的极限性态，解的整体结构以及初值含delta波时黎曼解的稳定性等。对于高维问题，主要研究了超音速流通过有限长的凸管道向真空扩张的问题以及磁流体力学方程组通过尖角的绕流问题。 前者的流体由两维等熵定常欧拉方程组给出。后者由二维磁流体力学方程组的一个简化模型描述。在这个过程中，会出现平面疏散波和中心波。通过求解古尔莎问题，混合边值问题，中心波问题以及波的相互作用问题，利用特征分析和特征分解进行解的先验估计，最终得到此问题解的整体存在性。