Exponential Stability of Large BV Solutions in a Model of Granular flow

TL;DR

This paper proves exponential stability of large bounded variation solutions for a hyperbolic system modeling granular flow, extending previous existence results by establishing Lipschitz continuous dependence on initial data.

Contribution

It introduces a Lyapunov-like functional to demonstrate the exponential Lipschitz stability of solutions in the L1 norm for a complex granular flow model.

Findings

Solutions depend Lipschitz continuously on initial data with exponential growth in time.

Established the stability of large BV solutions beyond small initial height assumptions.

Constructed a Lyapunov functional similar to those used in conservation law systems.

Abstract

We consider a $2\times 2$ system of hyperbolic balance laws, in one-space dimension, that describes the evolution of a granular material with slow erosion and deposition. The dynamics is expressed in terms of the thickness of a moving layer on top and of a standing layer at the bottom. The system is linearly degenerate along two straight lines in the phase plane and genuinely nonlinear in the subdomains confined by such lines. In particular, the characteristic speed of the first characteristic family is strictly increasing in the region above the line of linear degeneracy and strictly decreasing in the region below such a line. The non dissipative source term is the product of two quantities that are transported with the two different characteristic speeds. The global existence of entropy weak solutions of the Cauchy problem for such a system was established by Amadori and Shen for initial data with bounded but possibly large total variation, under the assumption that the initial height of the moving layer be sufficiently small. In this paper we establish the Lipschitz ${\bf L^1}$-continuous dependence of the solutions on the initial data with a Lipschitz constant that grows exponentially in time. The proof of the ${\bf L^1}$-stability of solutions is based on the construction of a Lyapunov like functional equivalent to the ${\bf L^1}$-distance, in the same spirit of the functional introduced by Liu and Yang and then developed by Bressan, Liu, Yang for systems of conservation laws with genuinely nonlinear or linearly degenerate characteristic fields.