Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems
[Développements d’optique géométrique avec amplification pour des problèmes aux limites hyperboliques linéaires]
Coulombel, Jean-François ; Guès, Olivier
Annales de l'Institut Fourier, Tome 60 (2010), p. 2183-2233 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Nous calculons et justifions rigoureusement des développements d’optique géométrique pour des problèmes aux limites hyperboliques ne satisfaisant pas la condition de Lopatinskii uniforme. Nous mettons en évidence un phénomène d’amplification pour la réflexion au bord d’oscillations haute fréquence et de petite amplitude. Notre analyse induit deux conséquences importantes pour de tels problèmes aux limites. Tout d’abord, nous précisons la perte de régularité optimale dans l’échelle des espaces de Sobolev entre les termes source et la solution du problème. Ensuite, nous donnons une borne inférieure pour la vitesse finie de propagation, celle-ci pouvant être supérieure à la vitesse de propagation libre dans tout l’espace. Nous illustrons notre analyse par quelques exemples.

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.

Publié le : 2010-01-01
DOI : https://doi.org/10.5802/aif.2581
Classification:  35L50,  78A05
Mots clés: systèmes hyperboliques, problèmes aux limites, optique géométrique 
@article{AIF_2010__60_6_2183_0,
     author = {Coulombel, Jean-Fran\c cois and Gu\`es, Olivier},
     title = {Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems},
     journal = {Annales de l'Institut Fourier},
     volume = {60},
     year = {2010},
     pages = {2183-2233},
     doi = {10.5802/aif.2581},
     zbl = {1218.35137},
     mrnumber = {2791655},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2010__60_6_2183_0}
}

Coulombel, Jean-François; Guès, Olivier. Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems. Annales de l'Institut Fourier, Tome 60 (2010) pp. 2183-2233. doi : 10.5802/aif.2581. http://gdmltest.u-ga.fr/item/AIF_2010__60_6_2183_0/

[1] Artola, M. A. Majda Nonlinear development of instabilities in supersonic vortex sheets. I. The basic kink modes, Phys. D, Tome 28 (1987) no. 3, pp. 253-281 | Article | MR 914450 | Zbl 0632.76074

[2] Benzoni-Gavage, S.; Rousset, F.; Serre, D.; Zumbrun, K. Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A, Tome 132 (2002) no. 5, pp. 1073-1104 | Article | MR 1938714 | Zbl 1029.35165

[3] Benzoni-Gavage, S. D. Serre Multidimensional hyperbolic partial differential equations, Oxford University Press, Oxford Mathematical Monographs (2007) | MR 2284507 | Zbl 1113.35001

[4] Chazarain, J. A. Piriou Caractérisation des problèmes mixtes hyperboliques bien posés, Ann. Inst. Fourier (Grenoble), Tome 22 (1972) no. 4, pp. 193-237 | Article | Numdam | MR 333469 | Zbl 0234.35052

[5] Chikhi, J. Sur la réflexion des oscillations pour un système à deux vitesses, C. R. Acad. Sci. Paris Sér. I Math., Tome 313 (1991) no. 10, pp. 675-678 | MR 1135437 | Zbl 0746.35020

[6] Coulombel, J.-F. Well-posedness of hyperbolic initial boundary value problems, J. Math. Pures Appl. (9), Tome 84 (2005) no. 6, pp. 786-818 | MR 2138641 | Zbl 1078.35066

[7] Coulombel, J.-F. The hyperbolic region for hyperbolic boundary value problems (2008) (Preprint)

[8] Domański, W. Surface and boundary waves for linear hyperbolic systems: applications to basic equations of electrodynamics and mechanics of continuum, J. Tech. Phys., Tome 30 (1989) no. 3-4, pp. 283-300 | MR 1081053

[9] Guès, O. Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal., Tome 6 (1993) no. 3, pp. 241-269 | MR 1201195 | Zbl 0780.35017

[10] Ikawa, M. Mixed problem for the wave equation with an oblique derivative boundary condition, Osaka J. Math., Tome 7 (1970), pp. 495-525 | MR 289947 | Zbl 0218.35060

[11] Joly, J.-L.; Métivier, G.; Rauch, J. Coherent and focusing multidimensional nonlinear geometric optics, Ann. Sci. École Norm. Sup. (4), Tome 28 (1995) no. 1, pp. 51-113 | Numdam | MR 1305424 | Zbl 0836.35087

[12] Kreiss, H.-O. Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., Tome 23 (1970), pp. 277-298 | Article | MR 437941 | Zbl 0188.41102 | Zbl 0193.06902

[13] Lax, P. D. Asymptotic solutions of oscillatory initial value problems, Duke Math. J., Tome 24 (1957), pp. 627-646 | Article | MR 97628 | Zbl 0083.31801

[14] Lescarret, V. Wave transmission in dispersive media, Math. Models Methods Appl. Sci., Tome 17 (2007) no. 4, pp. 485-535 | Article | MR 2316297 | Zbl 1220.35170

[15] Majda, A. M. Artola Nonlinear geometric optics for hyperbolic mixed problems, Analyse mathématique et applications, Gauthier-Villars (1988), pp. 319-356 | MR 956966 | Zbl 0674.35057

[16] Majda, A. R. Rosales A theory for spontaneous Mach stem formation in reacting shock fronts. I. The basic perturbation analysis, SIAM J. Appl. Math., Tome 43 (1983) no. 6, pp. 1310-1334 | Article | MR 722944 | Zbl 0544.76135

[17] Marcou, A. Rigorous weakly nonlinear geometric optics for surface waves (2009) (Preprint) | MR 2760337 | Zbl 1222.35118

[18] Métivier, G. The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., Tome 32 (2000) no. 6, pp. 689-702 | Article | MR 1781581 | Zbl 1073.35525

[19] Ohkubo, T. T. Shirota On structures of certain L 2 -well-posed mixed problems for hyperbolic systems of first order, Hokkaido Math. J., Tome 4 (1975), pp. 82-158 | MR 380131 | Zbl 0304.35067

[20] Rauch, Jeffrey; Keel, Markus Lectures on geometric optics, Hyperbolic equations and frequency interactions (Park City, UT, 1995), Amer. Math. Soc., Providence, RI (IAS/Park City Math. Ser.) Tome 5 (1999), pp. 383-466 | MR 1662833 | Zbl 0926.35003

[21] Sablé-Tougeron, M. Existence pour un problème de l’élastodynamique Neumann non linéaire en dimension 2, Arch. Rational Mech. Anal., Tome 101 (1988) no. 3, pp. 261-292 | Article | MR 930125 | Zbl 0652.73019

[22] Williams, M. Nonlinear geometric optics for hyperbolic boundary problems, Comm. Partial Differential Equations, Tome 21 (1996) no. 11-12, pp. 1829-1895 | MR 1421213 | Zbl 0881.35068

[23] Williams, M. Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. École Norm. Sup. (4), Tome 33 (2000) no. 3, pp. 383-432 | Numdam | MR 1775187 | Zbl 0962.35118

[24] Williams, M. Singular pseudodifferential operators, symmetrizers, and oscillatory multidimensional shocks, J. Funct. Anal., Tome 191 (2002) no. 1, pp. 132-209 | Article | MR 1909266 | Zbl 1028.35174