Sharp Interface Limit of a Navier-Stokes/Allen-Cahn System with Vanishing Mobility

We consider the sharp interface limit of a Navier-Stokes/Allen-Cahn system, when a parameter $\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. In dependence on the mobility coefficient in the Allen-Cahn equation in dependence on $\varepsilon>0$ different limit systems or non-convergence can occur. In the case that the mobility vanishes as $\varepsilon$ tends to zero slower than quadratic we prove convergence of solutions to a smooth solution of a classical sharp interface model for well-prepared and sufficiently smooth initial data. The proof is based on a relative entropy method and the construction of sufficiently smooth solutions of a suitable perturbed sharp interface limit system. This is a joint work with Julian Fischer and Maximilian Moser (ISTA Klosterneuburg, Austria)