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A modified phase field approximation for mean curvature flow with conservation of the volume
Author(s) -
Brassel M.,
Bretin E.
Publication year - 2011
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1426
Subject(s) - mathematics , allen–cahn equation , hypersurface , curvature , mathematical analysis , context (archaeology) , mean curvature flow , vector field , mean curvature , flow (mathematics) , field (mathematics) , geometry , pure mathematics , paleontology , biology
This paper is concerned with the motion of a time‐dependent hypersurface ∂Ω( t ) in ℝ d that evolves with a normal velocitywhere κ is the mean curvature of ∂Ω( t ), and g is an external forcing term. Phase field approximation of this motion leads to the Allen–Cahn equationwhere ε is an approximation parameter, W a double well potential and c W a constant that depends only on W . We study here a modified version of this equationand we prove its convergence to the same geometric motion. We then make use of this modified equation in the context of mean curvature flow with conservation of the volume, and we show that it has better volume‐preserving properties than the traditional nonlocal Allen–Cahn equation. Copyright © 2011 John Wiley & Sons, Ltd.