An Integro-Differential Conservation Law arising in a Model of Granular Flow

TL;DR

This paper investigates a scalar integro-differential conservation law modeling granular flow erosion, establishing existence, stability, and convergence of solutions with a novel fractional step approximation method.

Contribution

It introduces a new approach to analyze a complex integro-differential conservation law with general erosion functions, extending standard theory.

Findings

Constructed approximate solutions via fractional step method

Proved global existence of BV solutions

Established stability in L^1 with respect to initial data

Abstract

We study a scalar integro-differential conservation law. The equation was first derived in [2] as the slow erosion limit of granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for which one can not 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^\infty bounds and BV estimates yield convergence and global existence of BV solutions. Furthermore, we present a well-posedness analysis, showing that the solutions are stable in L^1 with respect to the initial data.