Weak solution for granular model

TL;DR

This paper investigates the existence of weak solutions for complex PDE models of granular flows, incorporating threshold rheology and dilatance effects, and demonstrates a significant mathematical breakthrough in analyzing such models.

Contribution

It provides the first comprehensive proof of weak solution existence for coupled granular flow models with complex rheology and dilatance effects.

Findings

Established existence of solutions for simplified models.

Extended results to more general granular flow models.

Demonstrated stability and energy dissipation in solutions.

Abstract

This article is devoted to questions concerning the existence of solutions for partial differential equation problems modeling granular flows. The models studied take into account the complex threshold rheology of these flows, as well as the dilatance effects. It is the coupling of these two physical phenomena that ensures stability and the existence of dissipated energy. The key point of the article is to understand how this energy can ensure the existence of a weak solution. We first establish a complete result on a simplified model, then demonstrate how it can be extended to more general cases. This work represents a real breakthrough in the mathematical analysis of this type of models for complex flows.