Weak Solutions for a Dense Granular Model with Drucker-Prager Plasticity and Bagnold Viscosity Scaling

Abstract

This article addresses the existence of solutions for partial differential equation (PDE) models describing granular flows. We emphasize the essential role of flow dilatation, coupled with complex rheology, in ensuring both stability and the existence of dissipative energy. A central focus of the paper is to understand how this energy, arising from strongly nonlinear and singular terms, contributes to the existence of weak solutions. We first establish an existence result for a model that reflects some mathematical difficulties of the complete system. In particular, the dilatancy law describes local volume changes in terms of the velocity divergence, which depends on the shear rate and the square root of the pressure, reflecting a balance between these two quantities. While the model rigorously studied in this article does not address all the difficulties of the full physical model - specifically, the variable volume fraction case is handled here solely through regularisation - this work represents a significant step forward in the mathematical analysis of models for such complex flows.