*Submitted Paper*

**Inserted:** 28 mar 2023

**Last Updated:** 28 mar 2023

**Year:** 2023

**Abstract:**

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:={1}_{(-\infty,0]}(\cdot)\exp(\cdot) \ast \rho$ satisfy an Oleĭnik-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 $V'(W) W \partial_x W$ 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.

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