Approche visqueuse de solutions discontinues de systèmes hyperboliques semilinéaires
Annales de l'Institut Fourier, Tome 56 (2006) no. 1, p. 183-245
On s’intéresse à des systèmes symétriques hyperboliques multidimensionnels en présence d’une semilinéarité. Il est bien connu que ces systèmes admettent des solutions discontinues, régulières de part et d’autre d’une hypersurface lisse caractéristique de multiplicité constante. Une telle solution u 0 étant donnée, on montre que u 0 est limite quand ε0 de solutions (u ε ) ε]0,1] du système perturbé par une viscosité de taille ε. La preuve utilise un problème mixte parabolique et des développements de couches limites. On s’intéresse aussi à des singularités plus faibles comme des sauts de dérivées.
We are interested in some multidimensional semilinear symmetric hyperbolic systems. It is well known that these systems have some discontinuous solutions which are regular outside of a smooth hypersurface characteristic of constant multiplicity. We suppose that such a solution u 0 is given and we show that u 0 is the limit, when ε0, of solutions (u ε ) ε]0,1] of the system perturbated by a viscosity of size ε. The key tools of the proof are a parabolic boundary problem and boundary layers expansions. We also consider weaker singularities as derivatives jumps.
DOI : https://doi.org/10.5802/aif.2177
Classification:  35F30,  35K50,  35R05
Mots clés: approche visqueuse, couches limites, solutions discontinues
@article{AIF_2006__56_1_183_0,
     author = {Sueur, Franck},
     title = {Approche visqueuse de solutions discontinues de syst\`emes hyperboliques semilin\'eaires},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {1},
     year = {2006},
     pages = {183-245},
     doi = {10.5802/aif.2177},
     mrnumber = {2228685},
     zbl = {1094.35024},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2006__56_1_183_0}
}
Sueur, Franck. Approche visqueuse de solutions discontinues de systèmes hyperboliques semilinéaires. Annales de l'Institut Fourier, Tome 56 (2006) no. 1, pp. 183-245. doi : 10.5802/aif.2177. http://www.numdam.org/item/AIF_2006__56_1_183_0/

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