On s’intéresse à des problèmes mixtes pour des systèmes symétriques hyperboliques multidimensionnels semilinéaires perturbés par une petite viscosité. La description à la limite non visqueuse recquiert des développements du type BKW mettant en évidence une couche limite caractéristique (CLC) et une couche limite non caractéristique (CLNC). Ce thème traité dans [12] est ici enrichi de trois améliorations :
- l’étude inclut des développements ayant peu de termes (comme un seul terme),
- on étudie aussi bien la propagation que le problème de Cauchy et les conditions de compatibilité des données,
- l’étude de l’interaction CLC-CLNC est approfondie.
In this article, we consider boundary problem for semilinear symmetric hyperbolic systems in several space dimensions perturbated by a small viscosity. This theme is tackled in [12] and the inviscid limit is described by WKB-like asymptotic expansions. The latter involve characteristic and non characteristic boundary layers. Here, we give three improvements :
- we consider expansions with a few terms (for example with one term),
- we also look at the initial boundary value problem and at compatibilities between initial and boundaries data,
- the interaction between the non characteristic boundary layer and the characteristic one is pushed further.
@article{AFST_2006_6_15_2_323_0, author = {Sueur, Franck}, title = {Couches limites semilin\'eaires}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {323--380}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {6e s{\'e}rie, 15}, number = {2}, year = {2006}, doi = {10.5802/afst.1124}, mrnumber = {2244220}, language = {fr}, url = {http://archive.numdam.org/articles/10.5802/afst.1124/} }
TY - JOUR AU - Sueur, Franck TI - Couches limites semilinéaires JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2006 SP - 323 EP - 380 VL - 15 IS - 2 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1124/ DO - 10.5802/afst.1124 LA - fr ID - AFST_2006_6_15_2_323_0 ER -
%0 Journal Article %A Sueur, Franck %T Couches limites semilinéaires %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2006 %P 323-380 %V 15 %N 2 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://archive.numdam.org/articles/10.5802/afst.1124/ %R 10.5802/afst.1124 %G fr %F AFST_2006_6_15_2_323_0
Sueur, Franck. Couches limites semilinéaires. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 2, pp. 323-380. doi : 10.5802/afst.1124. http://archive.numdam.org/articles/10.5802/afst.1124/
[1] Perturbations singulières et prolongements maximaux d’opérateurs positifs, Arch. Rational Mech. Anal., Volume 53 (1973/74), pp. 69-100 | MR | Zbl
[2] Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels ; théorèmes d’approximation ; application à l’équation de transport, Ann. Sci. École Norm. Sup. (4), Volume 3 (1970), pp. 185-233 | Numdam | MR | Zbl
[3] Maximal positive boundary value problems as limits of singular perturbation problems, Trans. Amer. Math. Soc. (1982) | MR | Zbl
[4] Couche limite dans un modèle de ferromagnétisme, Comm. Partial Differential Equations, Volume 27 (2002) no. (7-8, pp. 1467-1495 | MR | Zbl
[5] Quelques contributions mathématiques en optique non linéaire, École Polytechnique (1994) (Ph. D. Thesis)
[6] On the stability of boundary layers of incompressible Euler equations, J. Differential Equations, Volume 164 (2000) no. 1, pp. 180-222 | MR | Zbl
[7] On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., Volume 53 (2000) no. 9, pp. 1067-1091 | MR | Zbl
[8] Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations, Volume 143 (1998) no. 1, pp. 110-146 | MR | Zbl
[9] Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations, Volume 15 (1990) no. 5, pp. 595-645 | MR | Zbl
[10] Ondes multidimensionnelles -stratifiées et oscillations, Duke Math. J., Volume 68 (1992) no. 3, pp. 401-446 | MR | Zbl
[11] Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal., Volume 6 (1993) no. 3, pp. 241-269 | MR | Zbl
[12] Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites, Ann. Inst. Fourier (Grenoble), Volume 45 (1995) no. 4, pp. 973-1006 | EuDML | Numdam | MR | Zbl
[13] Linear partial differential operators, Springer Verlag, Berlin, 1976 | MR | Zbl
[14] Justification of multidimensional single phase semilinear geometric optics, Trans. Amer. Math. Soc., Volume 330 (1992) no. 2, pp. 599-623 | MR | Zbl
[15] Nonstationary flows of viscous and ideal fluids in , J. Functional Analysis, Volume 9 (1972), pp. 296-305 | MR | Zbl
[16] Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., Volume 34 (1981) no. 4, pp. 481-524 | MR | Zbl
[17] Stability of multidimensional shocks (Cours de DEA)
[18] Stability of small viscosity noncharacteristic boundary layers (Cours de DEA)
[19] Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems (Preprint) | MR | Zbl
[20] Characteristic initial-boundary value problems for symmetric hyperbolic systems, Osaka J. Math., Volume 35 (1998) no. 3, pp. 629-657 | MR | Zbl
[21] Singular symmetric positive first order differential operators, J. Math. Mech., Volume 15 (1966), pp. 235-271 | MR | Zbl
[22] Boundary value problems as limits of problems in all space, Séminaire Goulaouic-Schwartz (1978/1979), École Polytechnique, Palaiseau, 1979, pp. 1-17 (Exp. No. 3) | EuDML | Numdam | MR | Zbl
[23] Boundary value problems with nonuniformly characteristic boundary, J. Math. Pures Appl. (9), Volume 73 (1994) no. 4, pp. 347-353 | MR | Zbl
[24] Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., Volume 291 (1985) no. 1, pp. 167-187 | MR | Zbl
[25] Inviscid boundary conditions and stability of viscous boundary layers, Asymptot. Anal., Volume 26 (2001) no. 3-4, pp. 285-306 | MR | Zbl
[26] A symmetric positive system with nonuniformly characteristic boundary, Differential Integral Equations, Volume 11 (1998) no. 4, pp. 605-621 | MR | Zbl
[27] Full regularity of solutions to a nonuniformly characteristic boundary value problem for symmetric positive systems, Adv. Math. Sci. Appl., Volume 10 (2000) no. 1, pp. 39-55 | MR | Zbl
[28] Highly oscillatory multidimensional shocks, Comm. Pure Appl. Math., Volume 52 (1999) no. 2, pp. 129-192 | MR | Zbl
[29] 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
Cité par Sources :