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

Abstract

We investigate the sharp interface limit of a diffusive interface system that couples the Allen-Cahn equation with the instationary Stokes system in a bounded domain in R 2 . 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 ε > 0. For well-prepared initial data, we show that the solution converges to a limit system that couples the curve-shortening flow and the Stokes system.