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Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity

Raffaele Folino, César Hernández Melo, Luis López Ríos, Ramón Plaza

Abstract

This paper considers a one-dimensional generalized Allen-Cahn equation of the form \[ u_t = \varepsilon^2 (D(u)u_x)_x - f(u), \] where $\varepsilon>0$ is constant, $D=D(u)$ is a positive, uniformly bounded below diffusivity coefficient that depends on the phase field $u$ and $f(u)$ is a reaction function that can be derived from a double-well potential with minima at two pure phases $u = α$ and $u = β$. It is shown that interface layers (namely, solutions that are equal to $α$ or $β$ except at a finite number of thin transitions of width $\varepsilon$) persist for an exponentially long time proportional to $\exp(C/\varepsilon)$, where $C > 0$ is a constant. In other words, the emergence and persistence of \emph{metastable patterns} for this class of equations is established. For that purpose, we prove energy bounds for a renormalized effective energy potential of Ginzburg-Landau type. Numerical simulations, which confirm the analytical results, are also provided.

Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity

Abstract

This paper considers a one-dimensional generalized Allen-Cahn equation of the form where is constant, is a positive, uniformly bounded below diffusivity coefficient that depends on the phase field and is a reaction function that can be derived from a double-well potential with minima at two pure phases and . It is shown that interface layers (namely, solutions that are equal to or except at a finite number of thin transitions of width ) persist for an exponentially long time proportional to , where is a constant. In other words, the emergence and persistence of \emph{metastable patterns} for this class of equations is established. For that purpose, we prove energy bounds for a renormalized effective energy potential of Ginzburg-Landau type. Numerical simulations, which confirm the analytical results, are also provided.

Paper Structure

This paper contains 15 sections, 7 theorems, 115 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Allen-Cahn model
  3. Phase-dependent diffusivity
  4. Assumptions and main results
  5. Persistence of metastable patterns
  6. Metastable patterns and speed of the layers
  7. Existence of metastable patterns
  8. Layers speed
  9. Numerical experiments
  10. Classical Allen-Cahn equation
  11. Mullins diffusion
  12. Exponential diffusion
  13. The degenerate case $f'(\alpha)=f'(\beta)=0$.
  14. The degenerate case $D(\alpha)=0$.
  15. Discussion

Key Result

Lemma 2.2

Let $u\in C([0,T],H^2(a,b))$ be solution to equation eq:D-model with homogeneous Neumann boundary conditions eq:Neu. If $E_\varepsilon$ is the functional defined in eq:energy, then

Figures (5)

  • Figure 1: Numerical solutions to \ref{['eq:D-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $D(u)=1$, $f(u)=u(u^2-1)$, $\varepsilon=0.1$ and initial datum $u_0$ with $N = 6$ transitions, located at $-3.4, -2, -0.5, 0.8, 2.2,3.2$.
  • Figure 2: Numerical solutions to \ref{['eq:D-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $D(u)=(1+u^2)^{-1}$, $f(u)=u(u^2-1)$ and $\varepsilon=0.1$. The initial datum $u_0$ with 6 transitions is the same of Figure \ref{['fig:classic']}.
  • Figure 3: Numerical solutions to \ref{['eq:D-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $D(u)=e^u$, $f(u)=(u-\frac{e^2-7}{2})(u^2-1)$ and two different values for $\varepsilon$. The initial datum $u_0$ is the same of Figures \ref{['fig:classic']}-\ref{['fig:Mullins']}.
  • Figure 4: Numerical solutions to \ref{['eq:D-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $D(u)=(1+u^2)^{-1}$, $\varepsilon=0.1$ and two different reaction terms satisfying $f'(\alpha)=f'(\beta)=0$. The initial datum $u_0$ has $6$ transitions as in Figure \ref{['fig:Mullins']}.
  • Figure 5: Numerical solutions to \ref{['eq:D-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $D(u)=u^2$, $f(u)=u(u-\frac{2}{3})(u-1)$ and two different values for $\varepsilon$. The initial datum has 6 transitions located at the same positions of Figures \ref{['fig:classic']}-\ref{['fig:Mullins']}-\ref{['fig:exp']}-\ref{['fig:f-deg']}.

Theorems & Definitions (17)

  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 7 more