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

TL;DR

This paper reviews the development of kinetic relations for undercompressive shock waves in nonlinear hyperbolic systems, addressing physical, mathematical, and numerical challenges over the past fifteen years.

Contribution

It synthesizes fifteen years of research on kinetic relations for nonclassical shocks, highlighting new theoretical and numerical insights into their properties and solutions.

Findings

Kinetic relations are essential for well-posedness of nonclassical shock solutions.

Nonclassical shocks exhibit non-monotonic behavior with respect to initial data.

The analysis links nonclassical shocks to nonconservative hyperbolic systems.

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.