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Metastability and layer dynamics for the hyperbolic relaxation of the Cahn-Hilliard equation

Raffaele Folino, Corrado Lattanzio, Corrado Mascia

Abstract

The goal of this paper is to accurately describe the metastable dynamics of the solutions to the hyperbolic relaxation of the Cahn-Hilliard equation in a bounded interval of the real line, subject to homogeneous Neumann boundary conditions. We prove the existence of an "approximately invariant manifold" $\mathcal{M}_0$ for such boundary value problem, that is we construct a narrow channel containing $\mathcal{M}_0$ and satisfying the following property: a solution starting from the channel evolves very slowly and leaves the channel only after an exponentially long time. Moreover, in the channel the solution has a "transition layer structure" and we derive a system of ODEs, which accurately describes the slow dynamics of the layers. A comparison with the layer dynamics of the classic Cahn-Hilliard equation is also performed.

Metastability and layer dynamics for the hyperbolic relaxation of the Cahn-Hilliard equation

Abstract

The goal of this paper is to accurately describe the metastable dynamics of the solutions to the hyperbolic relaxation of the Cahn-Hilliard equation in a bounded interval of the real line, subject to homogeneous Neumann boundary conditions. We prove the existence of an "approximately invariant manifold" for such boundary value problem, that is we construct a narrow channel containing and satisfying the following property: a solution starting from the channel evolves very slowly and leaves the channel only after an exponentially long time. Moreover, in the channel the solution has a "transition layer structure" and we derive a system of ODEs, which accurately describes the slow dynamics of the layers. A comparison with the layer dynamics of the classic Cahn-Hilliard equation is also performed.

Paper Structure

This paper contains 10 sections, 9 theorems, 217 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximate metastable states
  4. Base Manifold
  5. Slow dynamics in a neighborhood of the base manifold
  6. Equations of motion and slow channel
  7. Main result
  8. Layer dynamics
  9. Equations of transition layers
  10. Comparison with the classic Cahn--Hilliard equation

Key Result

Proposition 2.1

Let $F$ be such that eq:ass-F holds and set There exists $r_0>0$ such that if $0<r<r_0$, then with corresponding asymptotic formulae for the derivatives of $\alpha_\pm$ and $\beta_\pm$.

Figures (1)

  • Figure 1: The graph of $h_1(t)$ for $\varepsilon=0.07$ in the case of systems \ref{['eq:h_i-CahnHilliard']} (left) and \ref{['eq:h2layers']} (right) with $\tau=50$.

Theorems & Definitions (16)

  • Proposition 2.1: Carr--Pego Carr-Pego
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • ...and 6 more