Nonlinear geometric optics for reflecting uniformly stable pulses

TL;DR

This paper rigorously justifies geometric optics approximations for reflecting pulses in hyperbolic systems, providing error estimates and handling multiple group velocities, with applications to Euler equations and shock stability.

Contribution

It offers the first rigorous construction and error analysis of weakly nonlinear geometric optics for reflecting pulses with multiple velocities, extending formal methods.

Findings

Error estimates show convergence of approximate to exact solutions as wavelength decreases.

Construction of pulse profiles accounts for multiple group velocities without resonance issues.

Results apply to quasilinear systems like the Euler equations, including shock stability under oscillatory perturbations.

Abstract

We provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting pulses is close to the uniquely determined exact solution for small wavelengths. Pulses reflecting off fixed noncharacteristic boundaries are considered under the assumption that the underlying boundary problem is uniformly spectrally stable in the sense of Kreiss. There are two respects in which these results make rigorous earlier formal treatments of pulses. First, we give a rigorous construction of leading pulse profiles in problems where pulses traveling with many distinct group velocities are, unavoidably, present; and second, we provide a rigorous error analysis which yields a rate of convergence of approximate to exact solutions as the wavelength approaches zero. Unlike wavetrains, interacting pulses do not produce resonances that affect leading order profiles. However, our error analysis shows the importance of estimating pulse interactions in the construction and estimation of correctors. Our results apply to a general class of systems that includes quasilinear problems like the compressible Euler equations; moreover, the same methods yield a stability result for uniformly stable Euler shocks perturbed by highly oscillatory pulses.