July  2002, 8(3): 605-625. doi: 10.3934/dcds.2002.8.606

The residual boundary conditions coming from the real vanishing viscosity method

1. 

Unité de Mathématiques Pures et Appliquées, Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, 69364 Lyon cedex 07, France

Received  June 2001 Revised  November 2001 Published  April 2002

We consider an initial boundary value problem set in $\{x > 0\}$ for a mixed hyperbolic parabolic system of conservation laws with a small parameter $\varepsilon$, $u_t +F(u)_x =\varepsilon(B(u)u_x)_x$. In the non-characteristic case a boundary layers analysis gives a set of boundary conditions, the set of residual boundary conditions $\mathcal C$, for the inviscid system $u_t+F(u)_x = 0$. We generalize the results of [16] obtained for strictly parabolic perturbations to a realistic setting. We show that the set $\mathcal C$ has the suitable geometric property to construct a solution of the inviscid sytem in a vicinity of a point where the Evans function of the corresponding profile of boundary layer does not vanish at zero. Next we consider multidimensional systems. We show that the Kreiss-Lopatinski determinant for the hyperbolic system linearized about a constant state in $\mathcal C$ is equal to the reduced Evans function for the viscous system linearized about the corresponding profile of boundary layer.
Citation: Frederic Rousset. The residual boundary conditions coming from the real vanishing viscosity method. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 605-625. doi: 10.3934/dcds.2002.8.606
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