Plane Wave Stability

Abstract

In this chapter we analyze the plane wave stability of profiles w(x). We start with general remarks about plane wave stability, deriving necessary conditions for energy estimates. These conditions are expressed in terms of a Lopatinski determinant in the constant coefficient case and an Evans function when the coefficients depend on the normal variable. We refer to the Introduction for references concerning these notions. A key point in this chapter is the theorem of F. Rousset [Ro1] asserting that the uniform Evans condition implies that the limiting hyperbolic boundary value problem satisfies the uniform Lopatinski condition (see also [Zu-Se] for viscous shocks).