Nonclassical shocks and kinetic relations: Strictly hyperbolic systems

We consider strictly hyperbolic systems of conservation laws whose characte ristic fields are not genuinely nonlinear, and we introduce a framework for the nonclassical shocks generated by diffusive or diffusive-dispersive app roximations. A nonclassical shock does not fulfill the Liu entropy criterio n and turns out to be undercompressive. We study the Riemann problem in the class of solutions satisfying a single entropy inequality, the only such constraint available for general diffusiv e-dispersive approximations. Each non-genuinely nonlinear characteristic fi eld admits a two-dimensional wave set, instead of the classical one-dimensi onal wave curve. In specific applications, these wave sets are narrow and r esemble the classical curves. We nd that even in strictly hyperbolic system s, nonclassical shocks with arbitrarily small amplitudes occur. The Riemann problem can be solved uniquely using nonclassical shocks, provided an addi tional constraint is imposed: we stipulate that the entropy dissipation acr oss any nonclassical shock be a given constitutive function. We call this a dmissibility criterion a kinetic relation, by analogy with similar laws int roduced in material science for propagating phase boundaries. In particular , the kinetic relation may be expressed as a function of the propagation sp eed. It is derived from traveling waves and, typically, depends on the rati o of the diffusion and dispersion parameters.