The Evans function for the stability analysis of boundary layers

One considers the vanishing viscosity method for hyperbolic systems of conservation laws. The main application is the Euler equations of an inviscid, non-conducting gas, approached by the Navier-Stokes equations. We restrict ourselves to a non-characteristic boundary, in particular, the net flow accross it is not zero. In this context, a boundary layer (B.L.) forms, which role is to match the Dirichlet boundary condition with the hyperbolic solution in the interior (I.S). The evolution of the B.L. and of the I.S. are coupled.

The convergence as the dissipation parameters tend to zero was proved in 1-d by Gisclon and Serre (1994) and in any dimension by Grenier and Gues (1998), under the asumption that the amplitude of the B.L. is moderate. We understood that instability might occur for larger B.L.s. The problem has been identified : the linearized IBVP around the B.L. may have eigenvalues with positive real part, which behave like the inverse of dissipation.

The search of such eigenvalues is a difficult task, which can be partly solved using an Evans function. In some cases of physical interest, one exhibits unstable B.L.s. Typically, such B.L.s are close to strong steady viscous shock waves.