Evaluation of the Lopatinski Determinant for Multi-dimensional Euler Equations

There is a large body of literature on the problem of stability for inviscid shock waves and in particular for gas dynamics, see [2, 17, 3, 5, 6, 9, 10, 16, 4, 7, 21, 12, 13, 1, 14, 15, 24]. The purpose of this appendix is to calculate the Lopatinski determinant, or “stability function,” for the Euler equations of compressible gas dynamics. We describe two approaches to this problem. In the first we use a change of variables to simplify the computation. In the second method we take advantage of the Galilean invariance of the Euler equations and exploit a relationship between the eigenvectors of a particular pair of matrices. As a preliminary step for the first technique, we discuss how the Lopatinski determinant behaves under a change of coordinates. Consider an inviscid system of n hyperbolic conservation laws in d space dimensions,