Limited layers: An inverse problem

摘要

We consider quasilinear symmetric hyperbolic boundary problems in several space dimensions, 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, T-0), where T-0 > 0. We consider parabolic perturbations, by introducing in the equation a family (epsilon E)(epsilon is an element of) (]0, 1]) where is a given nonlinear uniformly elliptic viscosity. We prescribe some very particular nonlinear boundary conditions of Dirichlet-Neumann type. We show, for small epsilon, the existence of regular solutions u(epsilon) of these problems on the time interval (0, T-0). Moreover, we show that u(0) is the limit in C (0, T-0; L-infinity boolean AND H-1 ), when epsilon -> 0(+) , of the u(epsilon) . The existence and the convergene to u(0) of the u(epsilon) untill T-0 are the result of an original property of transparency. Indeed, we give a very accurate asymptotic description, for epsilon -> 0(+) , of the u(epsilon) by using WKB (Wentzel, 1926, Kramers, 1926, Brillouin, 1926) expansions which reveal small amplitude boundary layers. The smallness of the boundary layers is linked to the choice of the boundary conditions for the viscous perturbations.