Abels, Helmut and Moser, Maximilian (2022) CONVERGENCE OF THE ALLEN-CAHN EQUATION WITH A NONLINEAR ROBIN BOUNDARY CONDITION TO MEAN CURVATURE FLOW WITH CONTACT ANGLE CLOSE TO 90 degrees. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 54 (1). pp. 114-172. ISSN 0036-1410, 1095-7154
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This paper is concerned with the sharp interface limit for the Allen-Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain Omega subset of R-2. We assume that a diffuse interface already has developed and that it is in contact with the boundary partial derivative Omega. The boundary condition is designed in such a way that the limit problem is given by the mean curvature flow with constant alpha-contact angle. For alpha close to 90 degrees we prove a local in time convergence result for well-prepared initial data for times when a smooth solution to the limit problem exists. Based on the latter we construct a suitable curvilinear coordinate system and carry out a rigorous asymptotic expansion for the Allen-Cahn equation with the nonlinear Robin boundary condition. Moreover, we show a spectral estimate for the corresponding linearized Allen-Cahn operator and with its aid we derive strong norm estimates for the difference of the exact and approximate solutions using a Gronwall-type argument.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | PHASE-TRANSITIONS; GENERALIZED MOTION; GRADIENT THEORY; PROPAGATION; HILLIARD; ENERGY; LIMIT; sharp interface limit; Allen-Cahn equation; Robin boundary condition; mean curvature flow; contact angle |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics Mathematics > Prof. Dr. Helmut Abels |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 09 Jan 2024 07:58 |
| Last Modified: | 29 Jan 2024 14:12 |
| URI: | https://pred.uni-regensburg.de/id/eprint/57078 |
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