An existence result for the sandpile problem on flat tables with walls

TL;DR

This paper proves the existence of equilibrium solutions for a granular matter model involving walls, extending previous results for open tables, and demonstrates how to numerically compute these solutions with simulations.

Contribution

It generalizes existing theorems to include boundary walls and provides a numerical method for computing equilibrium solutions in the sandpile model.

Findings

Existence of solutions with walls at the boundary.

Explicit characterization of surface flow density.

Numerical simulations of stationary solutions.

Abstract

We derive an existence result for solutions of a differential system which characterizes the equilibria of a particular model in granular matter theory, the so-called partially open table problem for growing sandpiles. Such result generalizes a recent theorem of Cannarsa and Cardaliaguet established for the totally open table problem. Here, due to the presence of walls at the boundary, the surface flow density at the equilibrium may result no more continuous nor bounded, and its explicit mathematical characterization is obtained by domain decomposition techniques. At the same time we show how these solutions can be numerically computed as stationary solutions of a dynamical two-layer model for growing sandpiles and we present the results of some simulations.