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

Coulombel, Jean-François; Guès, Olivier

Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2183-2233.

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.

Classification : 35L50, 78A05
Keywords: Hyperbolic systems, boundary value problems, geometric optics
Mot clés : systèmes hyperboliques, problèmes aux limites, optique géométrique

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

[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, Volume 132 (2002) no. 5, pp. 1073-1104 | DOI | MR | Zbl

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

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

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

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

[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., Volume 30 (1989) no. 3-4, pp. 283-300 | MR

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

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

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

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

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

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

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

[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., Volume 43 (1983) no. 6, pp. 1310-1334 | DOI | MR | Zbl

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

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

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

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

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

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

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

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