Resonant leading term geometric optics expansions with boundary layers for quasilinear hyperbolic boundary problems

TL;DR

This paper develops and justifies leading order geometric optics expansions with boundary layers for nonlinear hyperbolic boundary problems, including Euler equations, accommodating both real and nonreal phases to handle generic boundary frequencies.

Contribution

It introduces a novel approach incorporating nonreal phases into geometric optics expansions for hyperbolic problems with boundary layers, extending previous work limited to real phases.

Findings

Constructed weakly nonlinear geometric optics expansions for hyperbolic PDEs.

Incorporated nonreal phases to handle generic boundary frequencies.

Demonstrated convergence of approximate solutions to exact solutions in small wavelength limit.

Abstract

We construct and justify leading order weakly nonlinear geometric optics expansions for nonlinear hyperbolic initial value problems, including the compressible Euler equations. The technique of simultaneous Picard iteration is employed to show approximate solutions tend to the exact solutions in the small wavelength limit. Recent work [2] by Coulombel, Gues, and Williams studied the case of reflecting wave trains whose expansions involve only real phases. We treat generic boundary frequencies by incorporating into our expansions both real and nonreal phases. Nonreal phases introduce difficulties such as approximately solving complex transport equations and result in the addition of boundary layers with exponential decay. This also prevents us from doing an error analysis based on almost-periodic profiles as in [2].