Title :

Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems

Author(s) :

Coulombel, Jean-François [Auteur]
SImulations and Modeling for PArticles and Fluids [SIMPAF]
Guès, Olivier [Auteur]
Laboratoire d'Analyse, Topologie, Probabilités [LATP]
Williams, Mark [Auteur]
Department of Mathematics [Chapel Hill]

Journal title :

Communications in Partial Differential Equations

Pages :

1797-1859

Publisher :

Taylor & Francis

Publication date :

2011-12-31

ISSN :

0360-5302

English keyword(s) :

Hyperbolic systems
Profile equations
Geometric optics
Reflection of oscillations

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

We provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting wavetrains is close to the uniquely determined exact ...
Show more >We provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting wavetrains is close to the uniquely determined exact solution for small wavelengths. Waves reflecting off of fixed noncharacteristic boundaries and off of multidimensional shocks are considered under the assumption that the underlying fixed (respectively, free) boundary problem is uniformly spectrally stable in the sense of Kreiss (respectively, Majda). Our results apply to a general class of problems that includes the compressible Euler equations; as a corollary we rigorously justify the leading term in the geometric optics expansion of highly oscillatory multidimensional shock solutions of the Euler equations. An earlier stability result of this type was obtained by a method that required the construction of high-order approximate solutions. That construction in turn was possible only under a generically valid (absence of) small divisors assumption. Here we are able to remove that assumption and avoid the need for high-order expansions by studying associated singular fixed and free boundary problems. The analysis applies equally to systems that cannot be written in conservative form.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Interaction d'ondes compressibles

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL