Local limit of nonlocal traffic models: Convergence results and total variation blow-up

Consider a nonlocal conservation where the flux function depends on convolution of solution with given kernel. In singular local limit obtained by letting kernel converge to Dirac delta one formally recovers law. However, recent counter-examples show that in general solutions equations do not this work we focus laws modeling vehicular traffic: case, is anisotropic. We that, under fairly assumptions (anisotropic) kernel, nonlocal-to-local can be rigorously justified provided initial datum satisfies one-sided Lipschitz condition and bounded away from $0$. also exhibit counter-example showing if attains value $0$, then there are severe obstructions convergence proof.