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

Alexander Keimer, Lukas Pflug

Published 2023-12-20Version 1

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.