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

Jean-François Coulombel[1]; Olivier Guès[2]

  • [1] CNRS & Université Lille 1 Laboratoire Paul Painlevé (UMR CNRS 8524) and Project Team SIMPAF of INRIA Lille Nord Europe Cité scientifique, Bâtiment M2 59655 VILLENEUVE D’ASCQ Cedex (France)
  • [2] Université de Provence Laboratoire d’Analyse, Topologie et Probabilités (UMR CNRS 6632) Technopôle Château-Gombert 39 rue F. Joliot Curie 13453 MARSEILLE Cedex 13 (France)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 6, page 2183-2233
  • ISSN: 0373-0956

Abstract

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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 to the solution. Secondly, we are able to derive a lower bound for the finite speed of propagation, showing that waves may propagate faster than for the propagation in free space. We illustrate our analysis with some examples.

How to cite

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Coulombel, Jean-François, and Guès, Olivier. "Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems." Annales de l’institut Fourier 60.6 (2010): 2183-2233. <http://eudml.org/doc/116330>.

@article{Coulombel2010,
abstract = {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 to the solution. Secondly, we are able to derive a lower bound for the finite speed of propagation, showing that waves may propagate faster than for the propagation in free space. We illustrate our analysis with some examples.},
affiliation = {CNRS & Université Lille 1 Laboratoire Paul Painlevé (UMR CNRS 8524) and Project Team SIMPAF of INRIA Lille Nord Europe Cité scientifique, Bâtiment M2 59655 VILLENEUVE D’ASCQ Cedex (France); Université de Provence Laboratoire d’Analyse, Topologie et Probabilités (UMR CNRS 6632) Technopôle Château-Gombert 39 rue F. Joliot Curie 13453 MARSEILLE Cedex 13 (France)},
author = {Coulombel, Jean-François, Guès, Olivier},
journal = {Annales de l’institut Fourier},
keywords = {Hyperbolic systems; boundary value problems; geometric optics; hyperbolic systems; high frequency expansions; optical loss of regularity; strong stability condition; speed of propagation},
language = {eng},
number = {6},
pages = {2183-2233},
publisher = {Association des Annales de l’institut Fourier},
title = {Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems},
url = {http://eudml.org/doc/116330},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Coulombel, Jean-François
AU - Guès, Olivier
TI - Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 6
SP - 2183
EP - 2233
AB - 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 to the solution. Secondly, we are able to derive a lower bound for the finite speed of propagation, showing that waves may propagate faster than for the propagation in free space. We illustrate our analysis with some examples.
LA - eng
KW - Hyperbolic systems; boundary value problems; geometric optics; hyperbolic systems; high frequency expansions; optical loss of regularity; strong stability condition; speed of propagation
UR - http://eudml.org/doc/116330
ER -

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