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Metastable dynamics for a hyperbolic variant of the mass conserving Allen-Cahn equation in one space dimension

Raffaele Folino

Abstract

In this paper, we consider some hyperbolic variants of the mass conserving Allen-Cahn equation, which is a nonlocal reaction-diffusion equation, introduced (as a simpler alternative to the Cahn-Hilliard equation) to describe phase separation in binary mixtures. In particular, we focus our attention on the metastable dynamics of some solutions to the equation in a bounded interval of the real line with homogeneous Neumann boundary conditions. It is shown that the evolution of profiles with $N+1$ transition layers is very slow and we derive a system of ODEs, which describes the exponentially slow motion of the layers. A comparison with the classical Allen-Cahn and Cahn-Hilliard equations and theirs hyperbolic variations is also performed.

Metastable dynamics for a hyperbolic variant of the mass conserving Allen-Cahn equation in one space dimension

Abstract

In this paper, we consider some hyperbolic variants of the mass conserving Allen-Cahn equation, which is a nonlocal reaction-diffusion equation, introduced (as a simpler alternative to the Cahn-Hilliard equation) to describe phase separation in binary mixtures. In particular, we focus our attention on the metastable dynamics of some solutions to the equation in a bounded interval of the real line with homogeneous Neumann boundary conditions. It is shown that the evolution of profiles with transition layers is very slow and we derive a system of ODEs, which describes the exponentially slow motion of the layers. A comparison with the classical Allen-Cahn and Cahn-Hilliard equations and theirs hyperbolic variations is also performed.

Paper Structure

This paper contains 12 sections, 9 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Mass conserving Allen--Cahn equation
  3. Hyperbolic mass conserving Allen--Cahn equation
  4. Main results
  5. Mass conservation and energy estimates
  6. Approximately invariant manifold
  7. ODE for the layers
  8. The coordinate system close to the submanifold $\mathcal{M}$
  9. Preliminaries
  10. The constant mass submanifold
  11. Existence of the slow channel
  12. Layer dynamics

Key Result

Lemma 2.1

Assume that $g$ satisfies eq:ass-g. If $(u,u_t)\in C\left([0,T],H^2(0,1)\times H^1(0,1)\right)$ is solution to eq:hyp-nonlocal-eq:Neumann-eq:initial for some $T>0$, with $u_1$ satisfying eq:ass-u1, then for any $t\in[0,T]$.

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 3.1: Carr--Pego Carr-Pego
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 10 more