Stability of Shock Waves and Related Structures in Combustion Models, Thin Film Flows, and General Conservative Systems

在燃烧模型、薄膜流动和一般守恒系统中激波及其相关结构的稳定性。特别是，我们将研究在燃烧模型中出现的简并粘性激波的稳定性，在薄膜流动模型中出现的压缩和欠压缩粘性冲击波的稳定性，以及更一般地，在可压缩N-S方程组中出现的压缩、过压和欠压缩粘性激波的稳定性。我们还将研究相关系统中稀疏波的稳定性，以及在燃烧理论中出现的稀疏-压缩图案的稳定性，以及在研究薄膜流动时出现的欠压缩-压缩图案。将使用的方法涉及格林函数和埃文斯函数的应用。这项研究的更广泛的影响在于对我们的模型及其所基于的物理过程的洞察。通过对稳定性的研究，我们描述了我们的系统的基本方面，如可能出现的物理可观察现象的类型以及解的长期行为。我们还获得了对我们的模型捕捉它们所代表的潜在物理的准确性的关键洞察力。此外，所开发的技术应该在一系列相关系统的分析中得到应用。

Stability of Shock Waves and Related Structures in Combustion Models,Thin Film Flows and General Conservative Systems.Abstract of proposed researchPeter B HowardUnder this award the stability of nonlinear waves arising in the context of regularized and partially regularized conservation laws will be studied. In particular we shall investigate the stability of degenerate viscous shock waves such as arise in combustion models, the stability of compressive and undercompressive viscous shock waves that arise in models of thin film flows, and more generally, compressive, over- and undercompressive viscous shock waves that arise in systems such as the compressible Navier--Stokes equations with only partial regularization. We shall also investigate the stability of rarefaction waves in related systems, and the stability of rarefaction-compressive patterns that arise in combustion theory as well as the undercompressive-compressive patterns that arise in the study of thin film flows. The methods to be used involve the application of Green's functions and the Evans function.The broader impact of this research lies in the insight gained into our models and the physical processes on which they are based. Through a study of stability, we characterize such fundamental aspects of our systems as the types of physically observable phenomena that can arise and the long time behavior of solutions. We also gain critical insight into the accuracy with which our models capture the underlying physics they represent. In addition, the techniques developed should find application in the analysis of a range of related systems.