The Mach stem equation and amplification in strongly nonlinear geometric optics - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue American Journal of Mathematics Année : 2017

The Mach stem equation and amplification in strongly nonlinear geometric optics

Résumé

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(ε^2) and oscillates at a frequency O(1/ε). The corresponding exact solution, which has been shown to exist on a time interval that is independent of ε ∈ (0,1], has amplitude O(ε). In this paper, we deal with the exact same scaling, namely O(ε^2) forcing term on the boundary and O(ε) 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 ε ∈ (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.
Fichier principal
Vignette du fichier
MachStems.pdf (715.48 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-01102079 , version 1 (12-01-2015)

Identifiants

Citer

Jean-François Coulombel, Mark Williams. The Mach stem equation and amplification in strongly nonlinear geometric optics. American Journal of Mathematics, 2017, 139 (4), pp.967-1046. ⟨hal-01102079⟩
294 Consultations
110 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More