Archive for Rational Mechanics and Analysis

, Volume 199, Issue 1, pp 1–31

The Slow Erosion Limit in a Model of Granular Flow

Article

Abstract

We study a 2 × 2 system of balance laws that describes the evolution of a granular material (avalanche) flowing downhill. The original model was proposed by Hadeler and Kuttler (Granul Matter 2:9–18, 1999). The Cauchy problem for this system has been studied by the authors in recent papers (Amadori and Shen in Commun Partial Differ Equ 34:1003–1040, 2009; Shen in J Math Anal Appl 339:828–838, 2008). In this paper, we first consider an initial-boundary value problem. The boundary condition is given by the flow of the incoming material. For this problem we prove the global existence of BV solutions for a suitable class of data, with bounded but possibly large total variations. We then study the “slow erosion (or deposition) limit”. We show that, if the thickness of the moving layer remains small, then the profile of the standing layer depends only on the total mass of the avalanche flowing downhill, not on the time-law describing the rate at which the material slides down. More precisely, in the limit as the thickness of the moving layer tends to zero, the slope of the mountain is provided by an entropy solution to a scalar integro-differential conservation law.

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversity of L’AquilaL’AquilaItaly
  2. 2.Department of MathematicsPenn State UniversityUniversity ParkUSA

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