Nonlinear Stability of Multidimensional Structures in Fluid Dynamics

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

Proposal DMS-0401252 PI: Mark Williams, University of North Carolina (Chapel Hill) Title: Nonlinear stability of multidimensional structures in Fluid Dynamics ABSTRACT The 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 is to show that curved multidimensional shock solutions to the Euler equations can be obtained as limits of smooth solutions to the Navier-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 clarify when and how well solutions to these simplified models approximate true exact solutions of the full Navier-Stokes-combustion system. The proposed research will also examine the neutral stability regime for both shocks and vortex sheets, and in particular will attempt to provide a rigorous foundation for the theory of Artola and Majda which describes the development of kink modes leading to roll-up in supersonic vortex sheets. The proposer intends to carry out a rigorous investigation of the existence and nonlinear stability of certain important multidimensional structures arising in the mathematical study of compressible fluids. These structures include shock waves, detonation fronts, and vortex sheets. There is a vast applied literature on these topics, but until relatively recently they have resisted rigorous mathematical analysis, particularly in the multidimensional case. In recent work by the proposer, his collaborators, and others new tools have become available that permit a rigorous study of the highly singular perturbation problems that arise in investigating the stability of such structures.