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

top
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

top

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 -

References

top
  1. M. Artola, Nonlinear development of instabilities in supersonic vortex sheets. I. The basic kink modes, Phys. D 28 (1987), 253-281 Zbl0632.76074MR914450
  2. S. Benzoni-Gavage, F. Rousset, D. Serre, K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), 1073-1104 Zbl1029.35165MR1938714
  3. S. Benzoni-Gavage, Multidimensional hyperbolic partial differential equations, (2007), Oxford University Press Zbl1113.35001MR2284507
  4. J. Chazarain, Caractérisation des problèmes mixtes hyperboliques bien posés, Ann. Inst. Fourier (Grenoble) 22 (1972), 193-237 Zbl0234.35052MR333469
  5. J. Chikhi, Sur la réflexion des oscillations pour un système à deux vitesses, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 675-678 Zbl0746.35020MR1135437
  6. J.-F. Coulombel, Well-posedness of hyperbolic initial boundary value problems, J. Math. Pures Appl. (9) 84 (2005), 786-818 Zbl1078.35066MR2138641
  7. J.-F. Coulombel, The hyperbolic region for hyperbolic boundary value problems, (2008) Zbl1237.35106
  8. W. Domański, Surface and boundary waves for linear hyperbolic systems: applications to basic equations of electrodynamics and mechanics of continuum, J. Tech. Phys. 30 (1989), 283-300 MR1081053
  9. O. Guès, Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal. 6 (1993), 241-269 Zbl0780.35017MR1201195
  10. M. Ikawa, Mixed problem for the wave equation with an oblique derivative boundary condition, Osaka J. Math. 7 (1970), 495-525 Zbl0218.35060MR289947
  11. J.-L. Joly, G. Métivier, J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics, Ann. Sci. École Norm. Sup. (4) 28 (1995), 51-113 Zbl0836.35087MR1305424
  12. H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277-298 Zbl0193.06902MR437941
  13. P. D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24 (1957), 627-646 Zbl0083.31801MR97628
  14. V. Lescarret, Wave transmission in dispersive media, Math. Models Methods Appl. Sci. 17 (2007), 485-535 Zbl1220.35170MR2316297
  15. A. Majda, Nonlinear geometric optics for hyperbolic mixed problems, Analyse mathématique et applications (1988), 319-356, Gauthier-Villars Zbl0674.35057MR956966
  16. A. Majda, A theory for spontaneous Mach stem formation in reacting shock fronts. I. The basic perturbation analysis, SIAM J. Appl. Math. 43 (1983), 1310-1334 Zbl0544.76135MR722944
  17. A. Marcou, Rigorous weakly nonlinear geometric optics for surface waves, (2009) Zbl1222.35118
  18. G. Métivier, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc. 32 (2000), 689-702 Zbl1073.35525MR1781581
  19. T. Ohkubo, On structures of certain L 2 -well-posed mixed problems for hyperbolic systems of first order, Hokkaido Math. J. 4 (1975), 82-158 Zbl0304.35067MR380131
  20. Jeffrey Rauch, Markus Keel, Lectures on geometric optics, Hyperbolic equations and frequency interactions (Park City, UT, 1995) 5 (1999), 383-466, Amer. Math. Soc., Providence, RI Zbl0926.35003MR1662833
  21. M. Sablé-Tougeron, Existence pour un problème de l’élastodynamique Neumann non linéaire en dimension 2 , Arch. Rational Mech. Anal. 101 (1988), 261-292 Zbl0652.73019MR930125
  22. M. Williams, Nonlinear geometric optics for hyperbolic boundary problems, Comm. Partial Differential Equations 21 (1996), 1829-1895 Zbl0881.35068MR1421213
  23. M. Williams, Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. École Norm. Sup. (4) 33 (2000), 383-432 Zbl0962.35118MR1775187
  24. M. Williams, Singular pseudodifferential operators, symmetrizers, and oscillatory multidimensional shocks, J. Funct. Anal. 191 (2002), 132-209 Zbl1028.35174MR1909266

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.