The stability of viscous boundary layers

We consider viscous systems of conservation laws, posed on a half-space x > 0. The initial-boundary value problem, with a Dirichlet boundary data, may admit steady solutions which tend to some constant as x --> +infinity. Such a solution U may be view;ed as the profile of a boundary layer in the viscous approximation of the underlying first order system of conservation laws. The stability of this layer is closely tied with the linear asymptotic stability of U. To study this stability, we define a kind of Evans function. This is a holomorphic function defined on Rz > 0, which takes real values on the real axis and can be analytically extended to a neighbourhood of the origin. in some cases, we are able to compute its value for z = 0, and to relate it to its signum near +infinity. Therefore we obtain an index in Z/2Z, which equals the parity of the set of unstable eigenvalues. This allows us to built explicit examples of unstable boundary layers. This explains the smallness assumption encountered by Gisclon-Serre (1994) and Grenier-Gues (1998), when proving stability theorems.