Transverse instability of high frequency weakly stable quasilinear boundary value problems

TL;DR

This paper demonstrates that high frequency perturbations in weakly stable quasilinear hyperbolic boundary value problems can lead to strong instabilities due to resonances, especially when boundary frequencies are at the zero locus of the Lopatinskii determinant.

Contribution

It provides an analytical proof of boundary-induced instabilities in high order geometric optics expansions for weakly stable problems, including a model solution and application to Euler equations.

Findings

High frequency boundary perturbations can cause instabilities.

Resonances amplify boundary effects leading to instability.

Instability configurations can occur in 3D Euler equations.

Abstract

This work intends to prove that strong instabilities may appear for high order geometric optics expansions of weakly stable quasilinear hyperbolic boundary value problems, when the forcing boundary term is perturbed by a small amplitude oscillating function, with a transverse frequency. Since the boundary frequencies lie in the locus where the so-called Lopatinskii determinant is zero, the amplifications on the boundary give rise to a highly coupled system of equations for the profiles. A simplified model for this system is solved in an analytical framework using the Cauchy-Kovalevskaya theorem as well as a version of it ensuring analyticity in space and time for the solution. Then it is proven that, through resonances and amplification, a particular configuration for the phases may create an instability, in the sense that the small perturbation of the forcing term on the boundary interferes at the leading order in the asymptotic expansion of the solution. Finally we study the possibility for such a configuration of frequencies to happen for the isentropic Euler equations in space dimension three.