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The Cahn--Hilliard--Darcy--Forchheimer system with surfactant: Existence and long-time behavior of global weak solutions

Maurizio Grasselli, Bohan Ouyang, Andrea Poiatti, Hao Wu

Abstract

We consider a diffuse-interface model for two-phase incompressible viscous flows with a soluble surfactant in a bounded porous medium. This hydrodynamic system consists of a Darcy--Forchheimer equation for the seepage velocity $\boldsymbol{u}$ coupled with two Cahn--Hilliard equations involving Flory--Huggins type singular potentials, one for the phase-field variable $φ$, the difference in volume fractions of the two fluids, and the other for the surfactant concentration $ψ$. We study the initial boundary value problem in two or three dimensions, with impermeability boundary conditions for $\boldsymbol{u}$ and homogeneous Neumann boundary conditions for $(φ, ψ)$ and their associated chemical potentials. First, we establish the existence of global weak solutions via an implicit-explicit time-discretization scheme based on the energy dissipation law. Furthermore, applying the seminal results of the first and third authors (arXiv:2510.17296), we prove that every weak solution satisfying an energy inequality converges to a single equilibrium as time tends to infinity. In sharp contrast with the available literature on similar models, in this case weak solutions are enough to guarantee the uniqueness of asymptotic limits, without the necessity of any further eventual regularization.

The Cahn--Hilliard--Darcy--Forchheimer system with surfactant: Existence and long-time behavior of global weak solutions

Abstract

We consider a diffuse-interface model for two-phase incompressible viscous flows with a soluble surfactant in a bounded porous medium. This hydrodynamic system consists of a Darcy--Forchheimer equation for the seepage velocity coupled with two Cahn--Hilliard equations involving Flory--Huggins type singular potentials, one for the phase-field variable , the difference in volume fractions of the two fluids, and the other for the surfactant concentration . We study the initial boundary value problem in two or three dimensions, with impermeability boundary conditions for and homogeneous Neumann boundary conditions for and their associated chemical potentials. First, we establish the existence of global weak solutions via an implicit-explicit time-discretization scheme based on the energy dissipation law. Furthermore, applying the seminal results of the first and third authors (arXiv:2510.17296), we prove that every weak solution satisfying an energy inequality converges to a single equilibrium as time tends to infinity. In sharp contrast with the available literature on similar models, in this case weak solutions are enough to guarantee the uniqueness of asymptotic limits, without the necessity of any further eventual regularization.
Paper Structure (8 sections, 7 theorems, 162 equations)

This paper contains 8 sections, 7 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Statement of results
  5. Existence of Global Weak Solutions
  6. Implicit-explicit time discretization
  7. Proof of Theorem \ref{['ws_chdf_thm']}
  8. Convergence to a Single Equilibrium

Key Result

Theorem 2.1

Let Assumption assumptions be satisfied. (1) For any initial data $\psi_0, \phi_0 \in V$ with $(\phi_0,\psi_0) \in [-1,1]\times[0,1]$ almost everywhere in $\Omega$ and $\overline{\phi_0} \in (-1,1)$ and $\overline{\psi_0} \in (0,1)$ (and $\bm{u}_0 \in \bm{L}^2_{\sigma} (\Omega)$ if $\alpha >0$), pro (2) For every global weak solution $( \bm{u},\phi,\psi,\mu_\phi,\mu_\psi )$ in $I=[0,\infty)$ in t

Theorems & Definitions (14)

  • Remark 2.1
  • Definition 2.1
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.2
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • Proposition 3.1
  • ...and 4 more