# American Institute of Mathematical Sciences

May  2012, 32(5): 1481-1502. doi: 10.3934/dcds.2012.32.1481

## Front tracking approximations for slow erosion

 1 Dipartimento di Matematica Pura & Applicata, University of L’Aquila, Italy 2 Mathematics Department, Penn State University, University Park, PA 16802, United States

Received  December 2010 Revised  May 2011 Published  January 2012

In this paper we study an integro-differential equation describing slow erosion, in a model of granular flow. In this equation the flux is non local and depends on $x$, $t$. We define approximate solutions by using a front tracking technique, adapted to this special equation. Convergence of the approximate solutions is established by means of suitable a priori estimates. In turn, these yield the global existence of entropy solutions in BV. Such entropy solutions are shown to be unique.
We also prove the continuous dependence on initial data and on the erosion function, for the approximate as well as for the exact solutions. This establishes the well-posedness of the Cauchy problem.
Citation: Debora Amadori, Wen Shen. Front tracking approximations for slow erosion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1481-1502. doi: 10.3934/dcds.2012.32.1481
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