On a linearized Mullins-Sekerka/Stokes system for two-phase flows

TL;DR

This paper analyzes a linearized Mullins-Sekerka/Stokes system crucial for understanding the convergence of phase field models to sharp interface limits in two-phase flows, establishing solvability and regularity results.

Contribution

It provides the first solvability and maximal regularity results for a linearized Mullins-Sekerka/Stokes system with boundary conditions, aiding the analysis of two-phase flow models.

Findings

Proves solvability of the linearized system in Sobolev spaces.

Establishes maximal regularity for non-autonomous evolution equations.

Supports convergence analysis of Stokes/Cahn-Hilliard to sharp interface models.

Abstract

We study a linearized Mullins-Sekerka/Stokes system in a bounded domain with various boundary conditions. This system plays an important role to prove the convergence of a Stokes/Cahn-Hilliard systemto its sharp interface limit, which is a Stokes/Mullins-Sekerka system, and to prove solvability of the latter system locally in time. We prove solvability of the linearized system in suitable $L^2$-Sobolev spaces with the aid of a maximal regularity result for non-autonomous abstract linear evolution equations.