Parabolic Approximations of Diffusive-Dispersive Equations

We consider a lower-order approximation for a third-order diffusive-dispersive conservation law with nonlinear flux. It consists of a system of two second-order parabolic equations; a coupling parameter is also added. If the flux has an inflection point it is well-known, on the one hand, that the diffusive-dispersive law admits traveling-wave solutions whose end states are also connected by undercompressive shock waves of the underlying hyperbolic conservation law. On the other hand, if the diffusive-dispersive regularization vanishes, the solutions of the corresponding initial-value problem converge to a weak solution of the hyperbolic conservation law. We show that both of these properties also hold for the lower-order approximation. Furthermore, when the coupling parameter tends to infinity, we prove that solutions of initial value problems for the approximation converge to a weak solution of the diffusive-dispersive law. The proofs rely on new a-priori energy estimates for higher-order derivatives and the technique of compensated compactness.