Ole\u{\i}nik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

TL;DR

This paper establishes Olednik-type estimates for nonlocal conservation laws with exponential kernels and demonstrates their convergence to local conservation law solutions as the kernel becomes singular.

Contribution

It introduces Olednik-type estimates for nonlocal laws with exponential kernels and proves convergence to local laws without requiring bounded variation of initial data.

Findings

Nonlocal quantities satisfy Olednik-type entropy conditions.

Under assumptions on the velocity function, certain bounds hold for the nonlocal terms.

Weak solutions converge to local entropy solutions as the kernel approaches a delta distribution.

Abstract

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