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

Sharp Interface Limit for a Navier-Stokes/Allen-Cahn System in the Case of a Vanishing Mobility