Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A gradient flow for the Porous Medium Equations with Dirichlet boundary conditions

Dongkwang Kim, Dowan Koo, Geuntaek Seo

TL;DR

This work establishes a gradient-flow formulation for the porous medium equation with Dirichlet boundary data by employing the Figalli–Gigli distance $Wb_2$ in a minimizing movement scheme. It introduces a tailored entropy functional $\mathcal{E}$ and proves the existence of weak solutions as limits of the JKO scheme; these solutions form curves of maximal slope for $\mathcal{E}$ in the metric space $(\mathcal{M}_2(\Omega), Wb_2)$. The analysis handles the nonconservative boundary behavior, proves boundary trace results ensuring the Dirichlet data, and shows convergence to a weak PME formulation with a Dirichlet condition, while situating the results within the broader gradient-flow framework and relating to recent work on drift–diffusion extensions. Overall, the paper provides a variational PDE approach to non-mass-conserving evolution in bounded domains and clarifies boundary interactions via the $Wb_2$ metric, with potential extensions to more general drivers.

Abstract

We consider the gradient flow structure of the porous medium equations with non-negative constant Dirichlet boundary conditions. We construct weak solutions to the equations via the minimizing movement scheme by considering an entropy functional with respect to $Wb_2$ distance, which is a modified Wasserstein distance introduced by Figalli and Gigli [J. Math. Pures Appl. 94, (2010), pp. 107-130]. Furthermore, the constructed solutions are characterized as curves of maximal slope in a suitable sense.

A gradient flow for the Porous Medium Equations with Dirichlet boundary conditions

TL;DR

This work establishes a gradient-flow formulation for the porous medium equation with Dirichlet boundary data by employing the Figalli–Gigli distance in a minimizing movement scheme. It introduces a tailored entropy functional and proves the existence of weak solutions as limits of the JKO scheme; these solutions form curves of maximal slope for in the metric space . The analysis handles the nonconservative boundary behavior, proves boundary trace results ensuring the Dirichlet data, and shows convergence to a weak PME formulation with a Dirichlet condition, while situating the results within the broader gradient-flow framework and relating to recent work on drift–diffusion extensions. Overall, the paper provides a variational PDE approach to non-mass-conserving evolution in bounded domains and clarifies boundary interactions via the metric, with potential extensions to more general drivers.

Abstract

We consider the gradient flow structure of the porous medium equations with non-negative constant Dirichlet boundary conditions. We construct weak solutions to the equations via the minimizing movement scheme by considering an entropy functional with respect to distance, which is a modified Wasserstein distance introduced by Figalli and Gigli [J. Math. Pures Appl. 94, (2010), pp. 107-130]. Furthermore, the constructed solutions are characterized as curves of maximal slope in a suitable sense.
Paper Structure (19 sections, 13 theorems, 171 equations)

This paper contains 19 sections, 13 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Historical background
  3. Main results
  4. Preliminaries
  5. The space $({\mathcal{M}}_2(\Omega), Wb_2)$
  6. Metric Derivatives and Slopes
  7. Analytical tools
  8. Auxiliary Lemmas
  9. Well-definedness of JKO scheme
  10. Euler--Lagrange equation
  11. Fundamental estimates
  12. Analysis near the boundary
  13. Proof of Theorem \ref{['main']}
  14. Convergence
  15. PME equation
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose that $\mu_0:=\rho_0 \mathcal{L}^d |_{ \Omega } \in \mathcal{M}_2({ \Omega })$ and ${{\mathcal{E}}}(\mu_0)<\infty$. Then we have

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 22 more