The hyperbolic/elliptic transition in the multi-dimensional Riemann Problem

For a continuous self-similar solution to a system of conservation laws, genuine nonlinearity yields Lipschitz continuity at points where the type of the governing system changes. This is a well-known fact in one space dimension, where a constant state ū bifurcates towards a rarefaction wave at a point x/t that equals an eigenvalue λj(ū). We extend this observation to several space dimensions. The result generalizes a calculation that Bae, Chen and Feldman carried out in two space dimensions for an irrotational gas in their paper [3] (see their Theorem 4.2). As a corollary, we find the astonishing fact that a genuinely 3-D rarefaction wave matches a constant state in a C1-way, instead of a Lipschitz way! 2010 MSC: 35L67, 35L65