Well-Posedness and Qualitative Behaviour of Solutions for a Two-Phase Navier-Stokes-Mullins-Sekerka System

TL;DR

This paper studies a two-phase fluid system with a sharp interface, establishing local existence of solutions and demonstrating exponential convergence of the interface to a sphere over time.

Contribution

It provides new results on the well-posedness and long-term behavior of solutions for a two-phase Navier-Stokes-Mullins-Sekerka system, including convergence to equilibrium.

Findings

Velocity field converges to zero over time

Interface becomes spherical exponentially fast

Solutions exist locally and remain well-behaved

Abstract

We consider a two-phase problem for two incompressible, viscous and immiscible fluids which are separated by a sharp interface. The problem arises as a sharp interface limit of a diffuse interface model. We present results on local existence of strong solutions and on the long-time behavior of solutions which start close to an equilibrium. To be precise, we show that as time tends to infinity, the velocity field converges to zero and the interface converges to a sphere at an exponential rate.