An integro-differential conservation law arising in a model of granular flow

We study a scalar integro-differential conservation law which was recently derived by the authors as the slow erosion limit of a granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for which one cannot adapt the standard theory of conservation laws. We construct approximate solutions with a fractional step method, by recomputing the integral term at each time step. A priori L ^∞ bound and total variation estimates yield the convergence and global existence of solutions with bounded variation. Furthermore, we present a well-posedness analysis which establishes that these solutions are stable in the L ¹ norm with respect to the initial data.