Nonlinear Stability of Multidimensional Structures in Fluid Dynamics

Nonlinear Stability of Multidimensional Structures in Fluid Dynamics

Abstract of Proposed Research

Mark Williams

This project is to conduct 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. One example of a problem that is now within reach is the rigorous investigation of the process, proposed in 1987 by Artola and Majda, through which kink modes lead to roll-up in supersonic vortex sheets. The proposed strategy in this case draws on new degenerate Kreiss symmetrizers for the vortex sheet problem and a calculus of singular paradifferential operators that has already proved useful in studying highly oscillatory multidimensional shocks. We would also study long-time stability of multidimensional curved shocks, and investigate the effects of curvature on stability. 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.

There is a vast applied literature on these topics in which the arguments are often merely formal and not rigorous. Until relatively recently these topics have resisted careful mathematical analysis, particularly in the more complex multidimensional case. It is important to complement the formal work with rigorous analysis not only to insure the correctness of the formal work, but also because the rigorous analysis provides new insight, new analytical tools, and sometimes uncovers unexpected phenomena. 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 shocks, detonations, and vortex sheets.