Existence of quasilinear relaxation shock profiles in systems with characteristic velocities

Métivier, Guy; Texier, Benjamin; Zumbrun, Kevin

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 1, pp. 1-23.

Pour des systèmes de relaxation quasi-linéaires, dans le cas dégénéré où la vitesse est un mode caractéristique, nous donnons un résultat d’existence de profils de relaxation de petite amplitude, avec des taux de décroissance. Comme dans le cas semi-linéaire traité dans un travail antérieur de Métivier et Zumbrun, nous construisons un profil approché par un développement de Chapman-Enskog et nous utilisons la décomposition “micro-macro" de Liu et Yu. L’ingrédient nouveau dans le cas quasi-linéaire est le recours à un théorème de Nash-Moser à paramètre, du à Texier et Zumbrun, par opposition au cas semi-linéaire dans lequel un simple argument de point fixe permet de conclure la preuve.

We revisit the existence problem for shock profiles in quasilinear relaxation systems in the case that the velocity is a characteristic mode, implying that the profile ODE is degenerate. Our result states existence, with sharp rates of decay and distance from the Chapman–Enskog approximation, of small-amplitude quasilinear relaxation shocks. Our method of analysis follows the general approach used by Métivier and Zumbrun in the semilinear case, based on Chapman–Enskog expansion and the macro–micro decomposition of Liu and Yu. In the quasilinear case, however, in order to close the analysis, we find it necessary to apply a parameter-dependent Nash-Moser iteration due to Texier and Zumbrun, whereas, in the semilinear case, a simple contraction-mapping argument sufficed.

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