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Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites

Olivier Guès — 1995

Annales de l'institut Fourier

Ce travail concerne le problème de Cauchy-Dirichlet pour des systèmes hyperboliques semilinéaires multidimensionnels perturbés par une “petite viscosité". Les solutions considérées sont C et locales en temps, le but étant de décrire le comportement de la solution lorsque le paramètre de viscosité ( ϵ > 0 ) tend vers zéro. Il s’agit d’un problème de perturbation singulière pour lequel une “couche limite" se forme au voisinage du bord. Par des méthodes inspirées de l’optique géométrique non linéaire, nous...

A Transmission Strategy for Hyperbolic Internal Waves of Small Width

Olivier GuesJeffrey Rauch

Séminaire Équations aux dérivées partielles

Semilinear hyperbolic problems with source terms piecewise smooth and discontinuous across characteristic surfaces yield similarly piecewise smooth solutions. If the discontinuous source is replaced with a smooth transition layer, the discontinuity of the solution is replaced by a smooth internal layer. In this paper we describe how the layer structure of the solution can be computed from the layer structure of the source in the limit of thin layers. The key idea is to use a transmission problem...

Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems

Jean-François CoulombelOlivier Guès — 2010

Annales de l’institut Fourier

We compute and justify rigorous geometric optics expansions for linear hyperbolic boundary value problems that do not satisfy the uniform Lopatinskii condition. We exhibit an amplification phenomenon for the reflection of small high frequency oscillations at the boundary. Our analysis has two important consequences for such hyperbolic boundary value problems. Firstly, we make precise the optimal energy estimate in Sobolev spaces showing that losses of derivatives must occur from the source terms...

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