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We discuss the sharp interface limit of a coupled Navier-Stokes/Allen-Cahnsystem in a two dimensional, bounded and smooth domain, when a parameter$\varepsilon>0$ that is proportional to the thickness of the diffuse interfacetends to zero rigorously. We prove convergence of the solutions of theNavier-Stokes/Allen-Cahn system to solutions of a sharp interface model, wherethe interface evolution is given by the mean curvature flow with an additionalconvection term coupled to a two-phase Navier-Stokes system with surfacetension. This is done by constructing an approximate solution from the limitingsystem via matched asymptotic expansions together with a novel Ansatz for thehighest order term, and then estimating its difference with the real solutionwith the aid of a refined spectral estimate of the linearized Allen-Cahnoperator near the approximate solution.

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