Ole\uınik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

Abstract

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:=1_{(-\infty ,0]}(\cdot )\exp (\cdot )\ast \ddot{I}$ satisfy an Ole\uınik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a one-sided Lipschitz condition and that $V^{\prime }(W)W{\partial }_{x}W$ satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.