The Mach stem equation and amplification in strongly nonlinear geometric optics

TL;DR

This paper investigates the solvability and stability of highly oscillating solutions in strongly nonlinear hyperbolic boundary problems, extending previous weakly nonlinear results to a more complex regime with new theoretical insights.

Contribution

It proves the well-posedness of the Mach stem equation in a strongly nonlinear setting and generalizes the derivation to various hyperbolic boundary value problems with periodic forcing.

Findings

Proved solvability of the WKB cascade equations in strongly nonlinear regime.

Established well-posedness of the Mach stem equation.

Extended formal expansions to a broader class of problems with periodic forcing.

Abstract

We study highly oscillating solutions to a class of weakly well-posed hyperbolic initial boundary value problems. Weak well-posedness is associated with an amplification phenomenon of oscillating waves on the boundary. In the previous works [CGW14, CW14], we have rigorously justified a weakly nonlinear regime for semilinear problems. In that case, the forcing term on the boundary has amplitude O($\epsilon$^2) and oscillates at a frequency O(1/$\epsilon$). The corresponding exact solution, which has been shown to exist on a time interval that is independent of $\epsilon$ $\in$ (0,1], has amplitude O($\epsilon$). In this paper, we deal with the exact same scaling, namely O($\epsilon$^2) forcing term on the boundary and O($\epsilon$) solution, for quasilinear problems. In analogy with [CGM03], this corresponds to a strongly nonlinear regime, and our main result proves solvability for the corresponding WKB cascade of equations, which yields existence of approximate solutions on a time interval that is independent of $\epsilon$ $\in$ (0,1]. Existence of exact solutions close to approximate ones is a stability issue which, as shown in [CGM03], highly depends on the hyperbolic system and on the boundary conditions; we do not address that question here. This work encompasses previous formal expansions in the case of weakly stable shock waves [MR83] and two-dimensional compressible vortex sheets [AM87]. In particular, we prove well-posedness for the leading amplitude equation (the "Mach stem equation") of [MR83] and generalize its derivation to a large class of hyperbolic boundary value problems and to periodic forcing terms. The latter case is solved under a crucial nonresonant assumption and a small divisor condition.