Regularity results for the solutions of a non-local model of traffic

We consider a non-local traffic model involving a convolution product. Unlike other studies, the considered kernel is discontinuous on R. We prove Sobolev estimates and prove the convergence of approximate solutions solving a viscous and regularized non-local equation. It leads to weak, $C([0,T],L^2(\R))$, and smooth, $W^{2,2N}([0,T]\times\R)$, solutions for the non-local traffic model.