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Global Weak Solutions of a Thermodynamically Consistent Diffuse Interface Model for Nonhomogeneous Incompressible Two-phase Flows with a Soluble Surfactant

We study a thermodynamically consistent diffuse interface model that describes the motion of a two-phase flow of two viscous incompressible Newtonian fluids with unmatched densities and a soluble surfactant in a bounded domain of two or three dimensions. The resulting hydrodynamic system consists of a nonhomogeneous Navier-Stokes system for the (volume averaged) velocity $\mathbf{u}$ and a coupled Cahn-Hilliard system for the phase-field variables $ϕ$ and $ψ$ that represent the difference in volume fractions of the binary fluids and the surfactant concentration, respectively. For the initial boundary value problem with physically relevant singular potentials subject to a no-slip boundary condition for the fluid velocity and homogeneous Neumann boundary conditions for the phase-field variables and the chemical potentials, we first establish the existence of global weak solutions in the case of non-degenerate mobilities based on a suitable semi-implicit time discretization. Next, we prove the existence of global weak solutions for a class of general degenerate mobilities, with the aid of a new type of approximations for both the mobilities and the singular parts of the potential densities.