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

TL;DR

This paper develops a mathematical model for two-phase flows with different densities on evolving surfaces, proving local well-posedness and regularity results using advanced PDE techniques.

Contribution

It introduces a new Cahn-Hilliard-Navier-Stokes model for evolving surfaces with non-matched densities and establishes well-posedness and regularity results.

Findings

Proved short-time existence and uniqueness of strong solutions.

Established maximal regularity for surface Stokes and Cahn-Hilliard operators.

Reformulated the problem via pullback to initial surface.

Abstract

A Cahn-Hilliard-Navier-Stokes system for two-phase flow on an evolving surface with non-matched densities is derived using methods from rational thermodynamics. For a Cahn-Hilliard energy with a singular (logarithmic) potential short time well-posedness of strong solutions together with a separation property is shown, under the assumption of a priori prescribed surface evolution. The problem is reformulated with the help of a pullback to the initial surface. Then a suitable linearization and a contraction mapping argument for the pullback system are used. In order to deal with the linearized system, it is necessary to show maximal $L^2$-regularity for the surface Stokes operator in the case of variable viscosity and to obtain maximal $L^p$-regularity for the linearized Cahn-Hilliard system.