The sharp interface limit of a Navier--Stokes/Allen--Cahn system with constant mobility: Convergence rates by a relative energy approach

TL;DR

This paper rigorously analyzes the sharp interface limit of a coupled Navier--Stokes/Allen--Cahn system, establishing convergence to a two-phase flow with surface tension and providing optimal rates under certain conditions.

Contribution

It introduces a novel relative energy approach to prove convergence rates for the sharp interface limit in all relevant dimensions, a first in the field.

Findings

Solutions converge to a two-phase fluid system with surface tension.

Optimal convergence rates are established for well-prepared initial data.

The results are valid in both 2D and 3D settings.

Abstract

We investigate the sharp interface limit of a diffuse interface system that couples the Allen--Cahn equation with the instationary Navier--Stokes system in a bounded domain in $\mathbb{R}^d$ with $d \in \{2,3\}$. This model is used to describe a propagating front in a viscous incompressible flow with the width of the transition layer being characterized by a small parameter $\varepsilon>0$. We show that the solutions converge to a limit two-phase fluid system with surface tension that couples the mean curvature flow and the Navier--Stokes system. The main assumptions are that the evolution of the limit system is sufficiently regular and that the associated evolving interface does not intersect the boundary of the container. For quantitatively well-prepared initial data, we even establish an optimal convergence rate. This is the first rigorous result of this kind which is valid in all physically relevant ambient dimensions.