Weak Stability of Nonuniformly Stable Multidimensional Shocks

Abstract

The aim of this paper is to investigate the linear stability of multidimensional shock waves that violate the uniform stability condition derived by Majda [Mem. Amer. Math. Soc., 41 (1983)]. Two examples of such shock waves are studied: (1) planar Lax shocks in isentropic gas dynamics and (2) phase transitions in an isothermal van der Waals fluid. In both cases we prove an energy estimate on the resulting linearized system. Special attention is paid to the losses of derivatives arising from the failure of the uniform stability condition.