Weakly nonlinear multiphase geometric optics for hyperbolic quasilinear boundary value problems: Construction of a leading profile

TL;DR

This paper develops a weakly nonlinear geometric optics framework for hyperbolic quasilinear boundary value problems with multiple phases, constructing a leading profile using a novel approach that handles infinite resonances and lacks symmetry.

Contribution

It introduces a new method for constructing the leading profile in weakly nonlinear geometric optics with multiple phases and infinite resonances, using a priori estimates without loss of derivatives.

Findings

Successfully constructs a leading profile for complex boundary problems.

Handles infinitely many resonances in a weakly nonlinear setting.

Provides an example with isentropic Euler equations.

Abstract

We investigate in this paper the existence of the leading profile of a WKB expansion for quasilinear initial boundary value problems with a highly oscillating forcing boundary term. The framework is weakly nonlinear, as the boundary term is of order O($\epsilon$) where the frequencies are of order O(1/$\epsilon$). We consider here multiple phases on the boundary, generating a countable infinite number of phases inside the domain, and we therefore use an almost periodic functional framework. The major difficulties of this work are the lack of symmetry in the leading profile equation and the occurrence of infinitely many resonances (opposite to the simple phase case studied earlier) The leading profile is constructed as the solution of a quasilinear problem, which is solved using a priori estimates without loss of derivatives. The assumptions of this work are illustrated with the example of isentropic Euler equations in space dimension two.