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Motion of sharp interface of Allen-Cahn equation with anisotropic nonlinear diffusion

Tadahisa Funaki, Hyunjoon Park

Abstract

We consider the Allen-Cahn equation with nonlinear anisotropic diffusion and derive anisotropic direction-dependent curvature flow under the sharp interface limit. The anisotropic curvature flow was already studied, but its derivation is new. We prove both generation and propagation of the interface. For the proof we construct sub- and super-solutions applying the comparison theorem. The problem discussed in this article naturally appeared in the study of the interacting particle systems, especially of non-gradient type. The Allen-Cahn equation obtained from systems of gradient type has a simpler nonlinearity in diffusion and leads to isotropic mean-curvature flow. We extend those results to anisotropic situations.

Motion of sharp interface of Allen-Cahn equation with anisotropic nonlinear diffusion

Abstract

We consider the Allen-Cahn equation with nonlinear anisotropic diffusion and derive anisotropic direction-dependent curvature flow under the sharp interface limit. The anisotropic curvature flow was already studied, but its derivation is new. We prove both generation and propagation of the interface. For the proof we construct sub- and super-solutions applying the comparison theorem. The problem discussed in this article naturally appeared in the study of the interacting particle systems, especially of non-gradient type. The Allen-Cahn equation obtained from systems of gradient type has a simpler nonlinearity in diffusion and leads to isotropic mean-curvature flow. We extend those results to anisotropic situations.
Paper Structure (8 sections, 10 theorems, 83 equations)

This paper contains 8 sections, 10 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Formal asymptotic expansion
  3. Generation of the interface
  4. Motion of the interface
  5. Modified signed distance function
  6. Estimates of ${U_0}$ and linearized solution $\ul$
  7. Construction of sub- and super-solutions
  8. Proof of Theorem \ref{['thm:prop']}

Key Result

Theorem 1.1

Let $u^\varepsilon$ be the solution of the problem $(P^\varepsilon)$, $\eta_g$ be an arbitrary constant satisfying $0 < \eta_g < \eta_0$, where $\eta_0 := \min \{ {\alpha_+} - \alpha, \alpha - {\alpha_-} \}$. Then, there exist positive constants $\varepsilon_0$ and $M_0$ such that, for all $\varepsi Here $t^\varepsilon = \nu^{-1} \varepsilon^2 \ln \varepsilon$, and recall cond:f-bistable for $\nu$

Theorems & Definitions (17)

  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof : Proof of Theorem \ref{['thm:gen']}
  • Lemma 4.1
  • ...and 7 more