M643

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 $\varepsilon$ different limit systems or non-convergence can occur. In the case that the mobility is proportional to the square root of $\varepsilon$ solutions convergence to a solutions of a classical sharp interface model on short times for well-prepared and sufficiently smooth initial data. To this end we construct a suitable approximation of the solution of the Navier-Stokes/Allen-Cahn system using an expansion with half-integer powers of $\varepsilon$ and suitable solutions of a linearized limit system. Then the difference of approximate and exact solution is estimated with the aid of a suitable spectral estimate of the linearized Allen-Cahn operator. This is a joint work with Mingwen Fei (Anhui Normal University, China), and Maximilian Moser (ISTA Klosterneuburg, Austria).

Helmut Abels

Universität Regensburg