Well-posedness for a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase Flows with Surfactant

TL;DR

This paper proves the existence and uniqueness of solutions for a complex Navier-Stokes-Cahn-Hilliard model describing incompressible two-phase flows with surfactant, including regularity and stability properties.

Contribution

It introduces a well-posedness analysis for a coupled sixth- and fourth-order PDE system modeling two-phase flows with surfactant, including existence, uniqueness, and regularization results.

Findings

Existence of global weak solutions in 2D

Uniqueness of solutions in 2D

Surfactant concentration remains bounded away from pure states after positive time

Abstract

We investigate a diffuse-interface model that describes the dynamics of incompressible two-phase viscous flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn-Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn-Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier-Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (resp. local) strong solution in two (resp. three) dimensions. In the two dimensional case, we can derive a continuous dependence estimate with respect to the norms controlled by the total energy. Then we establish instantaneous regularization properties of global weak solutions for $t>0$. In particular, we show that the surfactant concentration stays uniformly away from the pure states $0$ and $1$ after some positive time.