Diffuse Interface Model for Two-Phase Flows on Evolving Surfaces with Different Densities: Global Well-Posedness

TL;DR

This paper proves the existence, uniqueness, and global well-posedness of solutions for a diffuse interface model describing two-phase flows with different densities on evolving surfaces, including strict phase separation.

Contribution

It establishes global well-posedness and strict separation for a Navier-Stokes/Cahn-Hilliard system on evolving surfaces with different densities, using novel regularity and energy estimates.

Findings

Existence and uniqueness of strong solutions.

Global well-posedness of the system.

Validation of the strict separation property.

Abstract

We show existence and uniqueness of strong solutions to a Navier-Stokes/Cahn-Hilliard type system on a given two-dimensional evolving surface in the case of different densities and a singular (logarithmic) potential. The system describes a diffuse interface model for a two-phase flow of viscous incompressible fluids on an evolving surface. We also establish the validity of the instantaneous strict separation property from the pure phases. To show these results we use our previous achievements on local well-posedness together with suitable novel regularity results for the convective Cahn-Hilliard equation. The latter allows to obtain higher-order energy estimates to extend the local solution globally in time. To this aim the time evolution of energy type quantities has to be calculated and estimated carefully.