Vanishing viscosity versus Rosenau approximation for scalar conservation laws: The fractional case

We consider approximations of scalar conservation laws by adding nonlocal diffusive operators. In particular, we consider solutions associated to fractional Laplacian and fractional Rosenau perturbations and show that, for any t > 0, the mutual L1 distance of their profiles is negligible as compared to their common distance to the underlying inviscid entropy solution. We provide explicit examples showing that our rates are optimal in the supercritical and critical cases, in one space dimension and for the Burgers’ flux. For subcritical equations, our rates are not optimal but they remain explicit.