| Abstract: |
| We consider the sharp interface limit of the Allen-Cahn equation, when a parameter $\varepsilon>0" that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain with Neumann boundary conditions. We prove convergence of the solutions of the Allen-Cahn equation to solutions of the sharp interface limit, which is the mean curvature flow with a 90 degree contact angle, provided the limit problem possesses a smooth solution on a certain time interval. To this end we construct a suitable approximation of the Allen-Cahn equation, using three levels of the terms in the formally matched asymptotic calculations, and estimate the difference with the aid of a suitable spectral estimates of the linearized Allen-Cahn operator. This is a joint-work with Maximilian MoserAllen-Cahn operator. Moreover, we will discuss recent extensions of this results e.g.\ to the Navier-Stokes/Allen-Cahn system or the Stokes/Cahn-Hilliard system. |
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