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Minimum Flow Steepest Descent Approach for Nonlinear PDE

Noureddine Igbida

Abstract

This paper presents a minimum flow approach applicable to a wide range of doubly nonlinear diffusion problems. We introduce a minimum flow steepest descent algorithm that seeks an optimal traffic flow by minimizing an internal energy function, while incorporating a specific minimum flow constraint for the transition work. This flexible framework surpasses traditional methods by generalizing (among other things) the established $H^{-1}$-theory for linear diffusion to the nonlinear setting. It offers distinct advantages in handling diverse applications, leveraging both the intrinsic internal energy and the inherent traffic flow concepts. The approach demonstrably tackles various applications, including fluid flow in porous media and many other scenarios like the Stefan/Hele-Shaw problem. Notably, it can handle diverse differential operators, encompassing even nonlinear ones like the $p-$Laplacian and Leray-Lions operators. Furthermore, it allows for the simultaneous handling convection/transport phenomena and complex boundary conditions within both the energy and transition work functions, contrasting favorably with other steepest descent algorithm like the JKO scheme in Wasserstein spaces. The paper delves into details of this comparison and offers additional theoretical insights involving $p'-$curve and metric gradient flow in Sobolev dual spaces.

Minimum Flow Steepest Descent Approach for Nonlinear PDE

Abstract

This paper presents a minimum flow approach applicable to a wide range of doubly nonlinear diffusion problems. We introduce a minimum flow steepest descent algorithm that seeks an optimal traffic flow by minimizing an internal energy function, while incorporating a specific minimum flow constraint for the transition work. This flexible framework surpasses traditional methods by generalizing (among other things) the established -theory for linear diffusion to the nonlinear setting. It offers distinct advantages in handling diverse applications, leveraging both the intrinsic internal energy and the inherent traffic flow concepts. The approach demonstrably tackles various applications, including fluid flow in porous media and many other scenarios like the Stefan/Hele-Shaw problem. Notably, it can handle diverse differential operators, encompassing even nonlinear ones like the Laplacian and Leray-Lions operators. Furthermore, it allows for the simultaneous handling convection/transport phenomena and complex boundary conditions within both the energy and transition work functions, contrasting favorably with other steepest descent algorithm like the JKO scheme in Wasserstein spaces. The paper delves into details of this comparison and offers additional theoretical insights involving curve and metric gradient flow in Sobolev dual spaces.
Paper Structure (47 sections, 17 theorems, 197 equations)

This paper contains 47 sections, 17 theorems, 197 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Introduction
  3. Preliminaries
  4. Plan of the paper
  5. Overview : Wasserstein distance vs $H^{-1}-$norm
  6. Proximity through $W_2-$Wasserstein distance
  7. Proximity through $H^{-1}-$norm
  8. Minimum flow concept and related steepest descent
  9. Notations and assumptions
  10. Reminder
  11. Proof :
  12. Assumptions
  13. Proximal energy minimization
  14. Main results
  15. Proofs of Theorem \ref{['tduality1']} and Theorem \ref{['tduality2']}
  16. ...and 32 more sections

Key Result

Lemma 1

For any $\mu\in L^{p'}(\Omega),$$\chi\in L^{p'}(\Omega)^N$and $\eta\in W^{-1/{p'},{p'}}(\Gamma_N),$ there exists $\phi \in H^{p'}(div,\Omega)$ solution of the following PDE : in the sense that

Theorems & Definitions (27)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 1
  • Theorem 2
  • Remark 7
  • ...and 17 more