Microlocal Analysis and Nonlinear Waves

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

ABSTRACT A major goal of this proposal is to understand the role of fixed and free boundaries in rigorous nonlinear geometric optics. The central importance of boundaries is indicated, for example, by the fact that multidimensional shock and vortex sheet problems can be formulated as nonlinear hyperbolic free boundary problems. The author proposes to continue his work on geometric optics for quasilinear boundary problems and multidimensional strong shocks, and to extend it to weak shocks; to study the evolution of boundary layers in quasilinear fixed and free boundary problems, especially the role of glancing boundary layers in a new mechanism for shock instability; to construct Rayleigh waves in nonlinear elasticity as propagating elliptic boundary layers and to investigate their nonlinear (resonant) interactions with other waves. The construction of formal, asymptotic solutions to nonlinear PDEs (nonlinear geometric optics) has long been an important tool of applied mathematics. Formal solutions yield qualititive information about many complex wave phenomena such as resonance, shocks, Mach stems, and vortex sheets, and can be used to predict and explain the results of physical or numerical experiments. An important task for pure mathematicians is to rigorously justify such expansions, that is, to show that they are close to genuine exact solutions. Rigorous analysis also leads to the discovery of unexpected phenomena, such as new blow-up mechanisms, for example.