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

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 or does not vanish we prove convergence of solutions to a smooth solution of a corresponding sharp interface model for well-prepared and sufficiently smooth initial data. In the first case the proof is based on a relative entropy method and the construction of sufficiently smooth solutions of a suitable perturbed sharp interface limit system. In the second case it is based on the construction of a suitable approximate solution and estimates for the linearized operator.

This is a joint work with Julian Fischer and Maximilian Moser (ISTA Klosterneuburg, Austria) and Maximilian Moser and Mingwen Fei (Anhui Normal University, Wuhu, China), respectively.