On L1-stability of BV solutions for a model of granular flow

We are concerned with the well-posedness of a model of granular flow that consists of a hyperbolic system of two balance laws in one-space dimension, which is linearly degenerate along two straight lines in the phase plane and genuinely nonlinear in the subdomains confined by such lines. This note provides a survey of recent results on the Lipschitz L1-continuous dependence of the entropy weak solutions on the initial data, with a Lipschitz constant that grows exponentially in time. Our analysis relies on the extension of a Lyapunov like functional and provide the first construction of a Lipschitz semigroup of entropy weak solutions to the regime of hyperbolic systems of balance laws (i) with characteristic families that are neither genuinely nonlinear nor linearly degenerate and (ii) initial data of arbitrarily large total variation.