Nonlinear Stability of Multidimensional Structures in Fluid Dynamics

提案DMS-0401252PI：Mark Williams，北卡罗来纳大学(教堂山)标题：流体动力学中多维结构的非线性稳定性摘要该项目的第一个目标是研究Navier-Stokes方程的小粘性极限。激波数学中的一个长期存在的问题是，欧拉方程的曲线多维激波解可以作为Navier-Stokes方程光滑解的极限得到。另一个目的是研究燃烧的Chapman-Jouget模型和ZND模型解(如强爆轰波)的存在性、唯一性和稳定性，并阐明这些简化模型的解何时以及在多大程度上逼近完整的Navier-Stokes燃烧系统的精确解。所提出的研究还将考察激波和涡片的中性稳定区，特别是将试图为Artola和Majda的理论提供一个严格的基础，该理论描述了超音速涡片中导致托卷的扭结模式的发展。作者打算对可压缩流体的数学研究中出现的某些重要的多维结构的存在性和非线性稳定性进行严格的研究。这些结构包括激波、爆轰波阵面和涡流片。在这些主题上有大量的应用文献，但直到最近，他们还抵制严格的数学分析，特别是在多维情况下。在作者、他的合作者和其他人最近的工作中，已经有了新的工具，这些工具允许对在研究这种结构的稳定性时出现的高度奇异摄动问题进行严格的研究。

Proposal DMS-0401252PI: Mark Williams, University of North Carolina (Chapel Hill)Title: Nonlinear stability of multidimensional structures in Fluid DynamicsABSTRACTThe first goal of the project is to investigate the small viscosity limit for the Navier-Stokes equations. A long-standing question in shock wave mathematics isto show that curved multidimensional shock solutions to the Eulerequations can be obtained as limits of smooth solutions to theNavier-Stokes equations. Another goal is to study the existence,uniqueness, and stability of solutions (like strong detonations)to the Chapman-Jouget and ZND models of combustion, and to clarifywhen and how well solutions to these simplified models approximatetrue exact solutions of the full Navier-Stokes-combustion system.The proposed research will also examine the neutral stabilityregime for both shocks and vortex sheets, and in particular willattempt to provide a rigorous foundation for the theory of Artolaand Majda which describes the development of kink modes leading toroll-up in supersonic vortex sheets.The proposer intends to carry out a rigorous investigation of theexistence and nonlinear stability of certain importantmultidimensional structures arising in the mathematical study ofcompressible fluids. These structures include shock waves,detonation fronts, and vortex sheets. There is a vast appliedliterature on these topics, but until relatively recently theyhave resisted rigorous mathematical analysis, particularly in themultidimensional case. In recent work by the proposer, hiscollaborators, and others new tools have become available thatpermit a rigorous study of the highly singular perturbationproblems that arise in investigating the stability of suchstructures.