Sharp Interface Limit for a Mass-Conserving Navier-Stokes/Allen-Cahn System with Different Viscosities

TL;DR

This paper rigorously analyzes the limit where a coupled Navier-Stokes and Allen-Cahn system with mass conservation converges to a sharp interface model, revealing the interface's evolution as mean curvature flow with convection.

Contribution

It provides a rigorous proof of convergence from a diffuse interface model to a sharp interface limit in a coupled Navier-Stokes/Allen-Cahn system with mass conservation.

Findings

Solutions converge to a sharp interface model as interface thickness tends to zero.

The interface evolves according to mass-conserving mean curvature flow with convection.

A new spectral estimate aids in error analysis between approximate and exact solutions.

Abstract

We perform a rigorous examination of the sharp interface limit of a coupled Navier-Stokes and mass-conserving Allen-Cahn system in a two-dimensional, bounded, and smooth domain as the parameter $\varepsilon > 0$, representing the thickness of the diffuse interface, tends to zero. We prove the convergence of solutions from the mass-conserving Navier-Stokes/Allen-Cahn system to those of its sharp interface limit. In this limit, the interface evolves according to mass-conserving mean curvature flow with a convection term and is coupled to a two-phase Navier-Stokes system with surface tension. Our approach entails the construction of an approximate solution for the limiting system through the use of matched asymptotic expansions, complemented by a special ansatz for the leading-order term. In order to estimate the error between this approximate solution and the exact solution, we employ a refined spectral estimate for the linearized Allen-Cahn operator near the approximate solution.