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The Verigin problem with phase transition as a Wasserstein flow

Anna Kubin, Tim Laux, Alice Marveggio

TL;DR

The paper reframes the Verigin problem with phase transition as a Wasserstein gradient flow and constructs weak solutions via a time-discrete minimizing movements scheme. The energy functional combines a surface term, a density-dependent bulk energy, and a phase-coupling term, producing an energy–dissipation structure that drives the analysis. The main result is the existence of distributional solutions (ρ,χ,u) that satisfy a transport equation, a weak Young–Laplace interface condition, and an energy-dissipation balance; with d ≥ 3 and uniform Muckenhoupt weights, χ becomes the indicator of a finite-perimeter set in the positively dense region, enabling perimeter convergence. This variational approach provides a rigorous link between mass diffusion, phase transition, and free-boundary dynamics, yielding strong compactness and BV control that are essential for the mathematical well-posedness of this coupled PDE-free boundary problem.

Abstract

We study the modeling of a compressible two-phase flow in a porous medium. The governing free boundary problem is known as the Verigin problem with phase transition. We introduce a novel variational framework to construct weak solutions. Our approach reveals the gradient-flow structure of the system by adopting a minimizing movement scheme using the Wasserstein distance. We prove the convergence of the scheme, obtaining ``relaxed" distributional solutions in the limit that satisfy an optimal energy-dissipation rate. Under the additional assumptions that $d \geq 3$ and that the discrete mass densities are uniformly Muckenhoupt weights, we show that the limit is the characteristic function of a set of finite perimeter in the region where there is no vacuum.

The Verigin problem with phase transition as a Wasserstein flow

TL;DR

The paper reframes the Verigin problem with phase transition as a Wasserstein gradient flow and constructs weak solutions via a time-discrete minimizing movements scheme. The energy functional combines a surface term, a density-dependent bulk energy, and a phase-coupling term, producing an energy–dissipation structure that drives the analysis. The main result is the existence of distributional solutions (ρ,χ,u) that satisfy a transport equation, a weak Young–Laplace interface condition, and an energy-dissipation balance; with d ≥ 3 and uniform Muckenhoupt weights, χ becomes the indicator of a finite-perimeter set in the positively dense region, enabling perimeter convergence. This variational approach provides a rigorous link between mass diffusion, phase transition, and free-boundary dynamics, yielding strong compactness and BV control that are essential for the mathematical well-posedness of this coupled PDE-free boundary problem.

Abstract

We study the modeling of a compressible two-phase flow in a porous medium. The governing free boundary problem is known as the Verigin problem with phase transition. We introduce a novel variational framework to construct weak solutions. Our approach reveals the gradient-flow structure of the system by adopting a minimizing movement scheme using the Wasserstein distance. We prove the convergence of the scheme, obtaining ``relaxed" distributional solutions in the limit that satisfy an optimal energy-dissipation rate. Under the additional assumptions that and that the discrete mass densities are uniformly Muckenhoupt weights, we show that the limit is the characteristic function of a set of finite perimeter in the region where there is no vacuum.
Paper Structure (16 sections, 12 theorems, 149 equations, 1 figure)

This paper contains 16 sections, 12 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation.
  3. Derivation of the model
  4. Our setting
  5. Minimization problem and main results
  6. Time-discrete problem
  7. Compactness
  8. Proof of Theorem \ref{['thm:main']}: Existence of distributional solutions
  9. Construction of the velocity field and transport equation \ref{['eq:transport']}
  10. Perimeter convergence
  11. Energy dissipation rate \ref{['eq:energydiss']}
  12. Passage to the limit in the Euler-Lagrange equation
  13. Optimal energy dissipation relation \ref{['eq:optimalenrel']}
  14. Proof of Theorem \ref{['cor']}: Strong compactness for $\chi_{E^h}$
  15. Compactness in measure
  16. ...and 1 more sections

Key Result

Theorem 1

Let $T>0$ and $\Omega \subset \mathbb{R}^d$ be an open set. Let $E_0 \subset \Omega$ be a measurable set, and $\rho_0 \in L^1(\Omega)$ be initial conditions such that $\int_\Omega \rho_0=1$ and $\mathcal{E}(\rho_0,\chi_{E_0})< \infty$, and let $(\rho^h, E^h)$ be constructed as in minpb-eq:defdisc fo and for all $\eta \in C^\infty_c(\overline\Omega \times [0, T))$. Moreover, $(\rho, \chi)$ satisfi

Figures (1)

  • Figure 1: Example of $\Phi$

Theorems & Definitions (23)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Proposition 3
  • proof
  • Lemma 4: Euler--Lagrange equations
  • proof : Proof of Lemma \ref{['lemma:EL']}
  • Lemma 5: Discrete almost minimizing property
  • proof
  • ...and 13 more