Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues

Abstract

Kinetic relations are required in order to characterize nonclassical undercompressive shock waves and formulate a well-posed initial value problem for nonlinear hyperbolic systems of conservation laws. Such nonclassical waves arise in weak solutions of a large variety of physical models: phase transitions, thin liquid films, magnetohydrodynamics, Camassa-Holm model, martensite-austenite materials, semi-conductors, combustion theory, etc. This review presents the research done in the last fifteen years which led the development of the theory of kinetic relations for undercompressive shocks and has now covered many physical, mathematical, and numerical issues. The main difficulty overcome here in our analysis of nonclassical entropy solutions comes from their lack of monotonicity with respect to initial data. Undercompressive shocks of hyperbolic conservation laws turn out to exhibit features that are very similar to shocks of nonconservative hyperbolic systems, who were investigated earlier by the author.