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Weak solution for granular model

Laurent Chupin, Thierry Dubois

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.

Weak solution for granular model

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.

Paper Structure

This paper contains 29 sections, 11 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. From grain to continuous model
  3. Issues and open questions
  4. Results and article outline
  5. Modeling dense granular flow
  6. Mathematical results
  7. Mathematical framework and notations
  8. Notations -
  9. Weak formulation and statement of result
  10. Approximations system
  11. Uniform estimates
  12. Limit process
  13. Step 1: Limit in \ref{['weak1k']}
  14. Step 2: Limit in \ref{['weak2k']}
  15. Step 3: Limit in \ref{['weak3k']}
  16. ...and 14 more sections

Key Result

Proposition 1

If $({\bf u},p,\boldsymbol{\sigma})$ is a weak solution of pb1--pb4 then the relations pb2 are satisfied almost everywhere.

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 1: weak solution
  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • ...and 8 more