article

Title: The hyperbolic/elliptic transition in the multi-dimensional Riemann problem

Authors: Denis Serre and Heinrich Freistuehler

Issue: Volume 62 (2013), Issue 2, 465-485

Abstract:

$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 $\bar{u}$ bifurcates towards a rarefaction wave at a point $x/t$ that equals an eigenvalue $\lambda_j(\bar{u})$. We extend this observation to several space dimensions. The result generalizes a calculation that Bae, Chen, and Feldman [\textit{Regularity of solutions to regular shock reflection for potential flow}, Invent. Math. \textbf{175} (2009), no. 3, 505--543] carried out in their paper (see their Theorem 4.2) in two space dimensions for an irrotational gas. As a corollary, we find the astonishing fact that a genuinely $3$D rarefaction wave matches a constant state in a $C^1$-way, rather than a Lipschitz-way.$