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

We investigate the singular limit, as \({\varepsilon \to 0}\), of the Allen-Cahn equation \({u^\varepsilon_t=\Delta u^\varepsilon+\varepsilon^{-2}f(u^\varepsilon)}\), with f a balanced bistable nonlinearity. We consider rather general initial data u0 that is independent of \({{\varepsilon}}\). 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 \({u^{\varepsilon}}\) are sandwiched between two sharp “interfaces” moving by mean curvature, provided that these “interfaces” sandwich at t = 0 an \({\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}\) neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an \({\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}\) 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.