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Motion of interfaces for a damped hyperbolic Allen-Cahn equation

Raffaele Folino, Corrado Lattanzio, Corrado Mascia

Abstract

Consider the Allen-Cahn equation $u_t=\varepsilon^2Δu-F'(u)$, where $F$ is a double well potential with wells of equal depth, located at $\pm1$. There are a lot of papers devoted to the study of the limiting behavior of the solutions as the diffusion coefficient $\varepsilon\to0^+$, and it is well known that, if the initial datum $u(\cdot,0)$ takes the values $+1$ and $-1$ in the regions $Ω_+$ and $Ω_-$, then the "interface" connecting $Ω_+$ and $Ω_-$ moves with normal velocity equal to the sum of its principal curvatures, i.e. the interface moves by mean curvature flow. This paper concerns with the motion of the inteface for a damped hyperbolic Allen-Cahn equation, in a bounded domain of $\mathbb{R}^n$, for $n=2$ or $n=3$. In particular, we focus the attention on radially simmetric solutions, studying in detail the differences with the classic parabolic case, and we prove that, under appropriate assumptions on the initial data $u(\cdot,0)$ and $u_t(\cdot,0)$, the interface moves by mean curvature as $\varepsilon\to0^+$ also in the hyperbolic framework.

Motion of interfaces for a damped hyperbolic Allen-Cahn equation

Abstract

Consider the Allen-Cahn equation , where is a double well potential with wells of equal depth, located at . There are a lot of papers devoted to the study of the limiting behavior of the solutions as the diffusion coefficient , and it is well known that, if the initial datum takes the values and in the regions and , then the "interface" connecting and moves with normal velocity equal to the sum of its principal curvatures, i.e. the interface moves by mean curvature flow. This paper concerns with the motion of the inteface for a damped hyperbolic Allen-Cahn equation, in a bounded domain of , for or . In particular, we focus the attention on radially simmetric solutions, studying in detail the differences with the classic parabolic case, and we prove that, under appropriate assumptions on the initial data and , the interface moves by mean curvature as also in the hyperbolic framework.

Paper Structure

This paper contains 13 sections, 16 theorems, 280 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Limiting behavior as $\varepsilon\to0$ in the general case
  3. Existence framework
  4. The compactness theorem
  5. Higher order estimates
  6. Formal derivation of the interface motion equation
  7. Motion of the interface in the radial case
  8. Study of the ODE
  9. Change of variables
  10. Properties of $\phi$
  11. The energy functional
  12. Dynamics of $v^\varepsilon$
  13. Proof of the main results

Key Result

Lemma 2.1

Let $(u,u_t)\in C\left([0,T],H^2(\Omega)\times H^1(\Omega)\right)$ be a solution to eq:hypalca-multiD with $f,g:\mathbb{R}\to\mathbb{R}$, $f=-F'$ for some $F:\mathbb{R}\to\mathbb{R}$ and either Neumann eq:neumann or Dirichlet eq:dirichlet boundary conditions. Then, for any $0\leq t_1<t_2\leq T$

Figures (6)

  • Figure 1: Initial datum $u_0$ with transition at $\rho_0=0.6$.
  • Figure 2: Solution for $\tau=1$, $\varepsilon=0.02$ and different values of $t$. Top left: $t=100$, top right: $t=250$, bottom left: $t=400$, bottom right: $t=450$. The initial datum is as in Figure \ref{['fig:t=0']}.
  • Figure 3: Solution for $\tau=1$, $\varepsilon=0.01$ and different values of $t$. Left: $t=50$, right: $t=250$. The initial datum is as in Figure \ref{['fig:t=0']}.
  • Figure 4: Solution to \ref{['eq:rho']} with $\tau=1$, $\varepsilon=0.02$ and initial data: $\rho_0=0.6$, $\nu_0=0$.
  • Figure 5: Solution to \ref{['eq:rho']} with initial data $\rho_0=0.6$, $\nu_0=0$ and $\tau=1$ for different values of $\varepsilon$: left $\varepsilon=0.03$, right $\varepsilon=0.01$.
  • ...and 1 more figures

Theorems & Definitions (39)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 29 more