  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 $m_\varepsilon>0$ in the Allen-Cahn
equation tends to zero in a subcritical way, i.e., $m_\varepsilon= m_0
\varepsilon^\beta$ for some $\beta\in (0,2)$ 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 $m_\varepsilon H_{\Gamma_t}$ 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.
