TY - JOUR

T1 - Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature

AU - Alfaro, Matthieu

AU - Droniou, Jerome

AU - Matano, Hiroshi

PY - 2012

Y1 - 2012

N2 - 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.

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.

UR - http://www.springerlink.com/content/hh222x0m83221055/?MUD=MP

U2 - 10.1007/s00028-011-0132-0

DO - 10.1007/s00028-011-0132-0

M3 - Article

VL - 12

SP - 267

EP - 294

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

SN - 1424-3199

IS - 2

ER -