Inviscid boundary conditions and stability of viscous boundary layers

Article type: Research Article

Authors: Rousset, Frédéric

Affiliations: UMPA, CNRS, UMR 5669, École Normale Superieure de Lyon, 46, allée d'Italie, 69364 Lyon cedex 7, France

Abstract: We study the set of residual boundary conditions for a one‐dimensional hyperbolic system of conservation laws set in x>0 which appears when the diffusion coefficient of Dirichlet's problem for a parabolic perturbation tends to zero. We show that this set is a submanifold in a vicinity of a point where the Evans function of the associated profile of boundary layer is such that D(0)≠0. Next we linearize a multidimensional hyperbolic problem about a constant state in the set of residual conditions and a viscous approximation about the associated profile of boundary layer. We show that the Evans function for the viscous problem reduces in the long‐wave limit to the Lopatinsky determinant. We deduce that inviscid well‐posedness is necessary for stability of the boundary layer.

Journal: Asymptotic Analysis, vol. 26, no. 3-4, pp. 285-306, 2001