Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues. (arXiv:1002.2950v1 [math.AP])

Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues. (arXiv:1002.2950v1 [math.AP]): "

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.

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