Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites
Guès, Olivier
Annales de l'Institut Fourier, Tome 45 (1995), p. 973-1006 / Harvested from Numdam
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Ce travail concerne le problème de Cauchy-Dirichlet pour des systèmes hyperboliques semilinéaires multidimensionnels perturbés par une “petite viscosité". Les solutions considérées sont C ∞ et locales en temps, le but étant de décrire le comportement de la solution lorsque le paramètre de viscosité (ε>0) tend vers zéro. Il s’agit d’un problème de perturbation singulière pour lequel une “couche limite" se forme au voisinage du bord. Par des méthodes inspirées de l’optique géométrique non linéaire, nous construisons et justifions un développement asymptotique (à tout ordre) de la solution. On obtient ainsi une description très précise de la singularité qui se forme au voisinage du bord (la “couche limite"). C’est cette analyse qui permet de montrer que la solution existe sur un intervalle de temps indépendant de ε>0. On en déduit la convergence de la solution (ε>0,ε→0) vers la solution d’un problème mixte hyperbolique limite. Dans le cas linéaire, ces résultats améliorent et complètent les résultats de convergence connus, obtenus par C. Bardos, D. Brézis, H. Brézis en 1973, J.-L. Lions en 1973 et C. Bardos et J. Rauch en 1982.

This work concerns the initial-boundary value problem for viscous perturbations of semilinear hyperbolic problems in several dimensions. We are interested in smooth and local in time solutions, the goal being to describe their behaviour as the viscosity coefficient (ε>0) goes to zero. This is a singular perturbation problem for which a boundary layer forms in the vicinity of the boundary. We prove that the solution exists on a domain independant of ε>0, and we give a complete three scale asymptotic expansion of BKW type of the solution, constructing correctors at any order. These results are reached simultaneously: the construction of the correctors is used to prove the existence of the solution. This asymptotic expansion provides a very accurate description of the viscosity boundary layer. It also implies the convergence of the solution (ε>0,ε→0) towards the solution of some semilinear hyperbolic initial-boundary value problem. Even in the linear case these results improve the former results obtained by C. Bardos, D. Brézis, H. Brézis in 1973, J.-L. Lions in 1973, and C. Bardos and J. Rauch in 1982.

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     author = {Gu\`es, Olivier},
     title = {Perturbations visqueuses de probl\`emes mixtes hyperboliques et couches limites},
     journal = {Annales de l'Institut Fourier},
     volume = {45},
     year = {1995},
     pages = {973-1006},
     doi = {10.5802/aif.1481},
     mrnumber = {97i:35103},
     zbl = {0831.34023},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1995__45_4_973_0}
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Guès, Olivier. Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites. Annales de l'Institut Fourier, Tome 45 (1995) pp. 973-1006. doi : 10.5802/aif.1481. http://gdmltest.u-ga.fr/item/AIF_1995__45_4_973_0/

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