  We consider the sharp interface limit of a Navier-Stokes/Allen Cahn equation
in a bounded smooth domain in two space dimensions, in the case of vanishing
mobility $m_\varepsilon=\sqrt{\varepsilon}$, where the small parameter
$\varepsilon>0$ related to the thickness of the diffuse interface is sent to
zero. For well-prepared initial data and sufficiently small times, we
rigorously prove convergence to the classical two-phase Navier-Stokes system
with surface tension. The idea of the proof is to use asymptotic expansions to
construct an approximate solution and to estimate the difference of the exact
and approximate solutions with a spectral estimate for the (at the approximate
solution) linearized Allen-Cahn operator. In the calculations we use a
fractional order ansatz and new ansatz terms in higher orders leading to a
suitable $\varepsilon$-scaled and coupled model problem. Moreover, we apply the
novel idea of introducing $\varepsilon$-dependent coordinates.
