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

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 $L^∞$ 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.

Reference
[2] Amadori, D. and Shen, W.; The Slow Erosion Limit in a Model of Granular Flow. [Preprint 2009-003] Arch. Ration. Mech. Anal. 199 (2011), 1–31.
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