A note on nonlocal approximations of sign-unrestricted solutions of conservation laws

TL;DR

This paper investigates how nonlocal conservation laws with unrestricted initial signs approximate local entropy solutions as the nonlocal kernel becomes singular, extending previous results to more general cases.

Contribution

It generalizes the sign-restricted singular limit problem to include sign-unrestricted initial data and covers models like generalized Burgers' and Keyfitz--Kranzer systems.

Findings

Nonlocal solutions converge to local entropy solutions as the kernel tends to a Dirac delta.

The results extend existing theories to sign-unrestricted initial data.

Includes special cases like generalized Burgers' equation.

Abstract

We study the singular limit problem for nonlocal conservation laws in which the sign of the initial datum is unrestricted and the velocity of the conservation law depends on a nonlocal approximation of the absolute value of the density. We demonstrate that the nonlocal solutions converge to the local entropy solution when the nonlocal kernel tends to a Dirac distribution, and thus obtain an approximation result for local unsigned conservation laws, generalizing the current results on the so-called sign-restricted singular limit problem. The considered model class covers special cases like a generalized Burgers' equation and scalar versions of the Keyfitz--Kranzer system.