Traveling waves in a relaxation system

Certain phase transitions are modeled as conservation laws with a non-monotone stress-strain relation. The governing equations are, then, of mixed type. Additional constitutive assumptions are required to formulate a mathematically well posed system of equations. Examples of such regularization include viscosity, visco-capillarity, viscoelasticity, and kinetic phase boundary relations. Here we regularize by adding a viscoelastic relaxation term. We examine Riemann problems and traveling wave solutions of the mixed system. In particular, we show that for certain data, the relaxation system admits a solution not seen by viscosity - visco-capillarity admissibility criteria. For these data, a generalization of the Oleinik-Liu entropy condition admits the same solution as does viscoelastic relaxation. Published by Elsevier Science Ltd.