A Transmission Strategy for Hyperbolic Internal Waves of Small Width
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 13, 9 p.

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.

Gues, Olivier 1 ; Rauch, Jeffrey 2

1 Université de Provence, Marseille, France
2 University of Michigan, Ann Arbor MI, USA
@article{SEDP_2005-2006____A13_0,
     author = {Gues, Olivier and Rauch, Jeffrey},
     title = {A {Transmission} {Strategy} for {Hyperbolic} {Internal} {Waves} of {Small} {Width}},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:13},
     pages = {1--9},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2005-2006},
     mrnumber = {2276078},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_2005-2006____A13_0/}
}
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 (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:13
PY  - 2005-2006
SP  - 1
EP  - 9
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/item/SEDP_2005-2006____A13_0/
LA  - en
ID  - SEDP_2005-2006____A13_0
ER  - 
%0 Journal Article
%A Gues, Olivier
%A Rauch, Jeffrey
%T A Transmission Strategy for Hyperbolic Internal Waves of Small Width
%J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
%Z talk:13
%D 2005-2006
%P 1-9
%I Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://www.numdam.org/item/SEDP_2005-2006____A13_0/
%G en
%F SEDP_2005-2006____A13_0
Gues, Olivier; Rauch, Jeffrey. A Transmission Strategy for Hyperbolic Internal Waves of Small Width. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 13, 9 p. http://www.numdam.org/item/SEDP_2005-2006____A13_0/

[A] S. Alinhac, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations 14, no. 2, 173–230, 1989. | Zbl

[AR] D. Alterman, J. Rauch, Nonlinear geometric optics for short pulses, J. Differential Equations 178 (2002), no. 2, 437–465. | MR | Zbl

[BR] 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. | MR | Zbl

[F] K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 7, 517-550, 1954. | MR | Zbl

[G1] O. Guès, Problèmes mixtes hyperboliques caractéristiques semi-linéaires, in Thèse, Univ. of Rennes 1, 1989.

[G2] O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations 15, no. 5, 595-645, 1990. | MR | Zbl

[G3] O. Guès, Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites, Ann. Inst. Fourier, 45, no. 4, 973-1006, 1995. | Numdam | MR | Zbl

[GR] 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). | MR | Zbl

[GMWZ] 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. | MR | Zbl

[LP] P. Lax, R. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math., 13, 427-455, 1960. | MR | Zbl

[MO] A. Majda, S. Osher, Initial boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28, 607-676, 1975. | MR | Zbl

[M1] 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. | MR | Zbl

[M2] G. Métivier, The Cauchy problem for semilinear hyperbolic systems with discontinuous data, Duke Math. J., 53, no. 4, 983-1011, 1986. | MR | Zbl

[R] J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc. 291, no. 1, 167–187, 1985. | MR | Zbl

[RK] 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. | MR | Zbl

[RR1] 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. | MR | Zbl

[RR2] J. Rauch, M. Reed, Discontinuous progressing waves for semilinear systems, Comm. Partial Differential Equations 10, no. 9, 1033–1075, 1985. | MR | Zbl

[S] F. Sueur, Approche visqueuse de solutions discontinues de systèmes hyperboliques semi linéaires, Annales Institut Fourier, Grenoble, to appear. | Numdam | Zbl

[T] B. Texier, The short wave limit for symmetric hyperbolic systems, Advances in Differential Equations, 9 no. 1-2, 1–52. 2004. | MR | Zbl