Nonclassical Shocks and Kinetic Relations: Strictly Hyperbolic Systems

Abstract

We consider strictly hyperbolic systems of conservation laws whose characteristic fields are not genuinely nonlinear, and we introduce a framework for the nonclassical shocks generated by diffusive or diffusive-dispersive approximations. A nonclassical shock does not fulfill the Liu entropy criterion 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 diffusive-dispersive approximations. Each non-genuinely nonlinear characteristic field admits a two-dimensional wave set, instead of the classical one-dimensional wave curve. In specific applications, these wave sets are narrow and resemble the classical curves. We find that even in strictly hyperbolic systems, nonclassical shocks with arbitrarily small amplitudes occur. The Riemann problem can be solved uniquely using nonclassical shocks, provided an additional constraint is imposed: we stipulate that the entropy dissipation across any nonclassical shock be a given constitutive function. We call this admissibility criterion a kinetic relation, by analogy with similar laws introduced in material science for propagating phase boundaries. In particular, the kinetic relation may be expressed as a function of the propagation speed. It is derived from traveling waves and, typically, depends on the ratioof the diffusion and dispersion parameters.