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arXiv · 1606.07318

Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow

Tim Laux·Thilo Simon
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Abstract

We present a convergence result for solutions of the vector-valued Allen-Cahn Equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multi-phase mean-curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time-integrated energies of the approximations converge to those of the limit. Furthermore, we apply our proof to two variants of the equation, incorporating external forces and a volume constraint.

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Tim Laux, Thilo Simon. 2016-06-23. Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow. https://arxiv.org/abs/1606.07318

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