An introduction to nonclassical shocks of systems of conservation laws

We review a recent activity on nonclassical shock waves of strictly hyperbolic systems of conservation laws, generated by balanced diffusion and dispersion effects. These shocks do not satisfy the standard Lax and Liu entropy criteria, and in fact are undercompressive and satisfy a single entropy inequality. The selection of admissible nonclassical shocks requires a strengthened version of the entropy inequality, called a kinetic relation, which constrains the entropy dissipation. The kinetic function is determined from traveling wave solutions to a system of equations augmented with diffusion and dispersion. For nonconvex scalar conservation laws and non-genuinely nonlinear, strictly hyperbolic systems, the existence and uniqueness of nonclassical shocks is investigated using successively the traveling wave analysis, the front tracking algorithm and the compensated compactness method. Nonclassical shocks may also be generated by finite difference schemes. The kinetic relation provides a useful tool to study the properties of nonclassical shocks and, in particular, their sensitivity to regularization parameters.