First Order Hyperbolic Boundary Value Problems With a Large Oscillatory Zero Order Term

TL;DR

This paper analyzes hyperbolic boundary value problems with large oscillatory zero order terms, establishing uniform energy estimates and constructing high order approximate solutions for applications like Mach stems and vortex sheets.

Contribution

It extends existing methods to handle more general oscillatory hyperbolic problems and constructs high order geometric optics solutions without frequency restrictions.

Findings

Positive energy estimates in small/medium frequency regions

Construction of high order approximate solutions via geometric optics

Overcoming obstacles posed by large oscillatory zero order terms

Abstract

We study the weakly stable hyperbolic boundary value problem with a large zero order oscillatory coefficient. This problem is related to linearized problems in the study of Mach stem and vortex sheets. We wish to establish a uniform energy estimate with respect to {\epsilon}, which is needed in mentioned applications and justification of geometric optics solutions, but the zero order oscillatory term gives rise to great obstacles. In this paper we obtain positive results in the small/medium frequency region by adapting the approach of Williams to a more general situation than the one treated there. We also show that it is possible to construct high order approximate solutions by the method of geometric optics for those systems without any restriction on frequencies.