Convergence of the Allen-Cahn Equation to Multiphase Mean Curvature Flow

Tim Laux, Thilo M. Simon

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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 multiphase 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 volume constraints.

Original languageEnglish (US)
Pages (from-to)1597-1647
Number of pages51
JournalCommunications on Pure and Applied Mathematics
Volume71
Issue number8
DOIs
StatePublished - Aug 2018

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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