Couches limites: un problème inverse.

Franck Sueur (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

We consider quasilinear symmetric hyperbolic boundary problems in several space dimen- sions, with maximal dissipative conditions on a boundary noncharacteristic or characteristic of constant multiplicity. We suppose that a regular solution of such a problem is given on the time interval (0, T0), where T0 > 0. We consider parabolic perturbations, by introducing in the equation a family ("E)"2]0,1] where E is a given nonlinear uniformly elliptic viscosity. We prescribe some very particular nonlinear boundary conditions of Dirichlet-Neumann type. We show, for small ", the existence of regular solutions u" of these problems on the time interval (0, T0). Moreover, we show that u0 is the limit in C(0, T0; L1 \H1), when " ! 0+, of the u". The existence and the convergence to u0 of the u" untill T0 are the result of an original property of transparency. Indeed, we give a very accurate asymptotic description, for " ! 0+, of the u" by using WKB expansions which reveal small amplitude boundary lay- ers. The smallness of the boundary layers is linked to the choice of the boundary conditions for the viscous perturbations.

Domaines: Equations aux dérivées partielles [math.AP]

Franck Sueur. Couches limites: un problème inverse.. 2005. ⟨hal-00002137v2⟩