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

.We discuss the sharp interface limit of a coupled Navier–Stokes/Allen–Cahn system in a two dimensional, bounded and smooth domain, when a parameter \(ε > 0\) that is proportional to the thickness of the diffuse interface tends to zero, rigorously. We prove convergence of the solutions of the Navier–Stokes/Allen–Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature flow with an additional convection term coupled to a two-phase Navier–Stokes system with surface tension. This is done by constructing an approximate solution from the limiting system via matched asymptotic expansions together with a novel ansatz for the highest order term, and then estimating its difference with the real solution with the aid of a refined spectral estimate of the linearized Allen–Cahn operator near the approximate solution.