Scalar conservation laws with mixed local and nonlocal diffusion-dispersion terms

We consider a nonlinear scalar conservation law that is regularized by a local viscous term and a nonlocal dispersive term. This nonstandard regularization is motivated by phase transition problems that take into account long range interactions close to the interface. We identify a parameter regime such that this mixed-type regularization provides a new example that is able to drive nonclassical undercompressive shock waves in the limit of vanishing regularization parameter. In view of the applications this shows that nonlocal regularizations can be used to model dynamical phase transition processes. In the next step we establish the existence and uniqueness of classical solutions for the Cauchy problem in multiple space dimensions. In the main part of the paper we then deduce appropriate a priori estimates to analyze the sharp-interface limit for vanishing regularization parameter with the method of compensated compactness in one space dimension and, using measure-valued solutions, in multiple space dimensions. It is shown that the limits exist and are weak solutions of the corresponding Cauchy problem for the hyperbolic conservation law.