Solutions for a Nonlocal Conservation Law with Fading Memory

Abstract

Global entropy solutions in $BV$ for a scalar nonlocal conservation law with fading memory are constructed as limits of vanishing viscosity approximate solutions. The uniqueness and stability of entropy solutions in $BV$ are established, which also yield the existence of entropy solutions in $L^{\infty }$ while the initial data is only in $L^{\infty }$. Moreover, if the memory kernel depends on a relaxation parameter $\de>0$ and tends to a delta measure weakly as measures when $\de\to 0+$, then the global entropy solution sequence in $BV$ converges to an admissible solution in $BV$ for the corresponding local conservation law.

11 pages. Proceedings of American Mathematical Society, 2006 (to appear)