Nonclassical Riemann solvers and kinetic relations I. A nonconvex hyperbolic model of phase transitions

Abstract

We consider a nonlinear hyperbolic model describing phase transitions in nonlinear elastodynamics. The Riemann problem is solved uniquely, provided we supplement the fundamental conservation laws (mass, momentum) with a kinetic relation. The latter takes into account small-scale mechanisms, such as the viscosity and capillary effects in the material under consideration. Our construction generalizes, to solutions of arbitrary large amplitude of the model under study, an approach proposed by Hayes and LeFloch for general systems of conservation laws. The Riemann solutions may contain rarefaction waves, (compressive) classical shock waves, as well as (undercompressive) nonclassical shocks.