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We investigate the sharp interface limit of a diffuse interface system thatcouples the Allen--Cahn equation with the instationary Navier--Stokes system ina bounded domain in $\mathbb{R}^d$ with $d \in \{2,3\}$. This model is used todescribe a propagating front in a viscous incompressible flow with the width ofthe transition layer being characterized by a small parameter $\varepsilon>0$.We show that the solutions converge to a limit two-phase fluid system withsurface tension that couples the mean curvature flow and the Navier--Stokessystem. The main assumptions are that the evolution of the limit system issufficiently regular and that the associated evolving interface does notintersect the boundary of the container. For quantitatively well-preparedinitial data, we even establish an optimal convergence rate. This is the firstrigorous result of this kind which is valid in all physically relevant ambientdimensions.

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