Mathematical Sciences: Riemann Problems for Nonlinear Conservation Laws in One and Two Space Dimensions

9403598 CANIC该奖项支持专注于求解非线性守恒律系统的数学研究。调查有两个目标。第一个是理解二维守恒律系的黎曼问题的解的结构，该守恒律系模拟弱棚屋反射。二是阐明了一维系统激波解的物理可容许性。弱收缩波的反射问题已经在实验和数值上得到了广泛的研究，但相应的可压缩欧拉方程的二维Riemann问题在理论上是难以解决的。这项工作将考虑非定常小扰动方程的解，该方程产生于欧拉系统的渐近极限。在一维守恒律系统的研究中，激波的物理可容许性是一个中心问题。最近研究了一般双参数扰动下二次守恒律的粘性可容许激波的稳定性。这些结果使用了基波流形框架内的矢量场展开理论。在这个项目中，我们将努力将这些结果推广到在这种情况下自然产生的三参数族中的摄动。对波动现象和激波的研究在数学上用偏微分方程组或这种方程组来表示。这些方程反映了支配解的某些物理(守恒)定律。对这些解决方案的数学和数值分析是这笔赠款支持的研究的基础。***

9403598 Canic This award supports mathematical research focusing on solutions of nonlinear systems of conservation laws. The investigation has two goals. The first is to understand the structure of solutions of Riemann problems for a two-dimensional system of conservation laws that model weak shack reflection. The second is to elucidate the physical admissibility of shockwave solutions of one-dimensional systems. The reflection of weak shack waves has been extensively studied both experimentally and numerically, but the corresponding two-dimensional Riemann problems for the compressible Euler equations are intractable on a theoretical level. This work will consider solutions of the unsteady small disturbance equation that arises in an asymptotic limit of the Euler system. In the study of one-dimensional systems of conservation laws, the physical admissibility of shock waves is a central question. A recent study of stability of viscous-admissible shock waves for quadratic conservation laws under generic two-parameter perturbations. These results use the theory of unfoldings of vector fields within the framework of the fundamental wave manifold. In this project, work will be done to extend these results to perturbations within the three-parameter family that arises naturally in this context. The study of wave phenomena and shocks is represented mathematically by partial differential equations or systems of such equations. The equations reflect certain physical (conservation) laws which govern the solutions. The mathematical and numerical analysis of these solutions is the basis for studies supported by this grant. ***