Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature

Matthieu Alfaro, Jerome Droniou, Hiroshi Matano

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)

Abstract

We investigate the singular limit, as , of the Allen-Cahn equation , with f a balanced bistable nonlinearity. We consider rather general initial data u (0) that is independent of . It is known that this equation converges to the generalized motion by mean curvature - in the sense of viscosity solutions-defined by Evans, Spruck and Chen, Giga, Goto. However, the convergence rate has not been known. We prove that the transition layers of the solutions are sandwiched between two sharp interfaces moving by mean curvature, provided that these interfaces sandwich at t = 0 an neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an estimate of the location and the thickness measured in space-time of the transition layers. A result on the regularity of the generalized motion by mean curvature is also provided in the Appendix.
Original languageEnglish
Pages (from-to)267 - 294
Number of pages28
JournalJournal of Evolution Equations
Volume12
Issue number2
DOIs
Publication statusPublished - 2012

Cite this

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abstract = "We investigate the singular limit, as , of the Allen-Cahn equation , with f a balanced bistable nonlinearity. We consider rather general initial data u (0) that is independent of . It is known that this equation converges to the generalized motion by mean curvature - in the sense of viscosity solutions-defined by Evans, Spruck and Chen, Giga, Goto. However, the convergence rate has not been known. We prove that the transition layers of the solutions are sandwiched between two sharp interfaces moving by mean curvature, provided that these interfaces sandwich at t = 0 an neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an estimate of the location and the thickness measured in space-time of the transition layers. A result on the regularity of the generalized motion by mean curvature is also provided in the Appendix.",
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Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature. / Alfaro, Matthieu; Droniou, Jerome; Matano, Hiroshi.

In: Journal of Evolution Equations, Vol. 12, No. 2, 2012, p. 267 - 294.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Alfaro, Matthieu

AU - Droniou, Jerome

AU - Matano, Hiroshi

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AB - We investigate the singular limit, as , of the Allen-Cahn equation , with f a balanced bistable nonlinearity. We consider rather general initial data u (0) that is independent of . It is known that this equation converges to the generalized motion by mean curvature - in the sense of viscosity solutions-defined by Evans, Spruck and Chen, Giga, Goto. However, the convergence rate has not been known. We prove that the transition layers of the solutions are sandwiched between two sharp interfaces moving by mean curvature, provided that these interfaces sandwich at t = 0 an neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an estimate of the location and the thickness measured in space-time of the transition layers. A result on the regularity of the generalized motion by mean curvature is also provided in the Appendix.

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