Calculus of Variations and Geometric Measure Theory

G. M. Coclite - M. Colombo - G. Crippa - N. De Nitti - A. Keimer - E. Marconi - L. Pflug - L. V. Spinolo

Oleĭnik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

created by denitti on 28 Mar 2023


Submitted Paper

Inserted: 28 mar 2023
Last Updated: 28 mar 2023

Year: 2023


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.