Front tracking approximations for slow erosion

Debora Amadori; Wen Shen

doi:10.3934/dcds.2012.32.1481

Article Contents

2012, Volume 32, Issue 5: 1481-1502. Doi: 10.3934/dcds.2012.32.1481

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Front tracking approximations for slow erosion

Debora Amadori^1, and
Wen Shen^2,

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

Received: November 30, 2010

Revised: April 30, 2011

Published: April 2012

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Abstract

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.

Keywords:

Granular flow,
slow erosion,
conservation laws,
front tracking,
existence and uniqueness of BV solutions.

Mathematics Subject Classification: Primary: 35L65; Secondary: 35L67, 35Q70, 35L60, 35L03.

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