# A Transmission Strategy for Hyperbolic Internal Waves of Small Width

Olivier Gues[1]; Jeffrey Rauch[2]

• [1] Université de Provence, Marseille, France
• [2] University of Michigan, Ann Arbor MI, USA

Séminaire Équations aux dérivées partielles (2005-2006)

• Volume: 2005-2006, page 1-9

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## Abstract

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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 strategy for the problem with the smooth internal layer. That leads to an ansatz different from the obvious candidates. The obvious candidates lead to overdetermined equations for correctors. With the transmission problem strategy we compute infinitely accurate expansions.

## How to cite

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Gues, Olivier, and Rauch, Jeffrey. "A Transmission Strategy for Hyperbolic Internal Waves of Small Width." Séminaire Équations aux dérivées partielles 2005-2006 (2005-2006): 1-9. <http://eudml.org/doc/11125>.

@article{Gues2005-2006,
abstract = {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 strategy for the problem with the smooth internal layer. That leads to an ansatz different from the obvious candidates. The obvious candidates lead to overdetermined equations for correctors. With the transmission problem strategy we compute infinitely accurate expansions.},
affiliation = {Université de Provence, Marseille, France; University of Michigan, Ann Arbor MI, USA},
author = {Gues, Olivier, Rauch, Jeffrey},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {discontinuous source terms; piecewise smooth solutions; thin layers; transmission problem strategy; overdetermined equations for correctors},
language = {eng},
pages = {1-9},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {A Transmission Strategy for Hyperbolic Internal Waves of Small Width},
url = {http://eudml.org/doc/11125},
volume = {2005-2006},
year = {2005-2006},
}

TY - JOUR
AU - Gues, Olivier
AU - Rauch, Jeffrey
TI - A Transmission Strategy for Hyperbolic Internal Waves of Small Width
JO - Séminaire Équations aux dérivées partielles
PY - 2005-2006
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2005-2006
SP - 1
EP - 9
AB - 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 strategy for the problem with the smooth internal layer. That leads to an ansatz different from the obvious candidates. The obvious candidates lead to overdetermined equations for correctors. With the transmission problem strategy we compute infinitely accurate expansions.
LA - eng
KW - discontinuous source terms; piecewise smooth solutions; thin layers; transmission problem strategy; overdetermined equations for correctors
UR - http://eudml.org/doc/11125
ER -

## References

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1. S. Alinhac, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations 14, no. 2, 173–230, 1989. Zbl0692.35063
2. D. Alterman, J. Rauch, Nonlinear geometric optics for short pulses, J. Differential Equations 178 (2002), no. 2, 437–465. Zbl1006.35015MR1879833
3. K. Barrailh, D. Lannes, A general framework for diffractive optics and its applications to lasers with large spectrum and short pulses, SIAM, Journal on Mathematical Analysis 34 , no. 3, 636-674, 2003. Zbl1032.78015MR1970887
4. K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 7, 517-550, 1954. Zbl0059.08902MR62932
5. O. Guès, Problèmes mixtes hyperboliques caractéristiques semi-linéaires, in Thèse, Univ. of Rennes 1, 1989.
6. O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations 15, no. 5, 595-645, 1990. Zbl0712.35061MR1070840
7. O. Guès, Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites, Ann. Inst. Fourier, 45, no. 4, 973-1006, 1995. Zbl0831.34023MR1359836
8. O. Guès, J. Rauch Nonlinear asymptotics for hyperbolic internal waves of small width, Journal of Hyperbolic PDE, to appear, (see http://www.math.lsa.umich.edu/~rauch). Zbl1096.35083MR2229857
9. O. Guès, G. Métivier, M. Williams, K. Zumbrun, Multidimensional viscous shocks. II: The small viscosity limit, Comm. Pure Appl. Math. 57 (2004), no. 2, 141–218. Zbl1073.35162MR2012648
10. P. Lax, R. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math., 13, 427-455, 1960. Zbl0094.07502MR118949
11. A. Majda, S. Osher, Initial boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28, 607-676, 1975. Zbl0314.35061MR410107
12. G. Métivier, Ondes discontinues pour les systèmes hyperboliques semi-linéaires, Recent developments in hyperbolic equations, 159–169, Pitman Res. Notes Math. Ser., 183, Longman Sci. Tech., Harlow, 1988. Zbl0724.35065MR984367
13. G. Métivier, The Cauchy problem for semilinear hyperbolic systems with discontinuous data, Duke Math. J., 53, no. 4, 983-1011, 1986. Zbl0631.35056MR874678
14. J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc. 291, no. 1, 167–187, 1985. Zbl0549.35099MR797053
15. J. Rauch, M. Keel, Lectures on geometric optics. Hyperbolic equations and frequency interactions, 383–466, IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, 1999. Zbl0926.35003MR1662833
16. J. Rauch, M. Reed, Bounded, stratified and striated solutions of hyperbolic systems, Nonlinear partial differential equations and their applications. College de France Seminar, Vol. IX, 334–351, Pitman Res. Notes Math. Ser., 181, Longman Sci. Tech., Harlow, 1988. Zbl0695.35124MR992654
17. J. Rauch, M. Reed, Discontinuous progressing waves for semilinear systems, Comm. Partial Differential Equations 10, no. 9, 1033–1075, 1985. Zbl0598.35069MR806255
18. F. Sueur, Approche visqueuse de solutions discontinues de systèmes hyperboliques semi linéaires, Annales Institut Fourier, Grenoble, to appear. Zbl1094.35024
19. B. Texier, The short wave limit for symmetric hyperbolic systems, Advances in Differential Equations, 9 no. 1-2, 1–52. 2004. Zbl1108.35372MR2099605

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