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Sharp Interface Limit for a Stokes/Allen–Cahn System

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Abstract

We consider the sharp interface limit of a coupled Stokes/Allen–Cahn system, when a parameter \({\epsilon > 0}\) that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. For sufficiently small times we prove convergence of the solutions of the Stokes/Allen–Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two-phase Stokes system with an additional contribution to the stress tensor, which describes the capillary stress. To this end we construct a suitable approximation of the solution of the Stokes/Allen–Cahn system, using three levels of the terms in the formally matched asymptotic calculations, and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized Allen–Cahn operator. Moreover, a careful treatment of the coupling terms is needed.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Helmut Abels

  2. NYU Shanghai, 1555 Century Avenue, Shanghai, 200122, China

    Yuning Liu

  3. NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China

    Yuning Liu

Authors
  1. Helmut Abels
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  2. Yuning Liu
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Correspondence to Helmut Abels.

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Communicated by F. Lin

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Abels, H., Liu, Y. Sharp Interface Limit for a Stokes/Allen–Cahn System. Arch Rational Mech Anal 229, 417–502 (2018). https://doi.org/10.1007/s00205-018-1220-x

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