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

TL;DR

This paper proves the existence and long-term convergence of weak solutions for a complex two-phase flow model with surfactant in porous media, highlighting the sufficiency of weak solutions for predicting equilibrium states.

Contribution

It establishes the existence of global weak solutions and their convergence to equilibrium without requiring additional regularity assumptions, advancing understanding of such diffuse-interface models.

Findings

Global weak solutions exist for the model.

Weak solutions converge to a unique equilibrium.

No additional regularization needed for asymptotic analysis.

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 $\phi$, the difference in volume fractions of the two fluids, and the other for the surfactant concentration $\psi$. 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 $(\phi, \psi)$ 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.