On nonlinear conservation laws with a nonlocal diffusion term

Scalar one-dimensional conservation laws with a nonlocal diffusion term corresponding to a Riesz–Feller differential operator are considered. Solvability results for the Cauchy problem in L∞ are adapted from the case of a fractional derivative with homogeneous symbol. The main interest of this work is the investigation of smooth shock profiles. In the case of a genuinely nonlinear smooth flux function we prove the existence of such travelling waves, which are monotone and satisfy the standard entropy condition. Moreover, the dynamic nonlinear stability of the travelling waves under small perturbations is proven, similarly to the case of the standard diffusive regularisation, by constructing a Lyapunov functional.