Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier-Stokes/Allen-Cahn System

We show convergence of the Navier-Stokes/Allen-Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility m(epsilon)>0 in the Allen-Cahn equation tends to zero in a subcritical way, i.e., m(epsilon)=m(0)epsilon(beta) for some beta is an element of (0,2) and m(0)>0. The proof proceeds by showing via a relative entropy argument that the solution to the Navier-Stokes/Allen-Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term m(epsilon)H(Gamma t )in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow.