Layer dynamics for the one dimensional $\varepsilon$-dependent Cahn-Hilliard / Allen-Cahn equation

TL;DR

This paper analyzes the layer dynamics of the one-dimensional $ ext{Cahn-Hilliard/Allen-Cahn}$ equation near equilibrium states with multiple transition layers, considering mass conservation constraints and spectral stability in the small $ ext{epsilon}$ limit.

Contribution

It develops an $N$-dimensional manifold approximation for layered metastable states and derives the layer dynamics and spectral stability conditions, highlighting the impact of mass conservation.

Findings

Derived the spectrum of the linearized operator.

Identified conditions for exponential stability of layer dynamics.

Compared Allen-Cahn and Cahn-Hilliard profiles under mass constraints.

Abstract

We study the dynamics of the one-dimensional $\varepsilon$-dependent Cahn-Hilliard / Allen-Cahn equation within a neighborhood of an equilibrium of $N$ transition layers, that in general does not conserve mass. Two different settings are considered which differ in that, for the second, we impose a mass-conservation constraint in place of one of the zero-mass flux boundary conditions at $x=1$. Motivated by the study of Carr and Pego on the layered metastable patterns of Allen-Cahn in [10], and by this of Bates and Xun in [5] for the Cahn-Hilliard equation, we implement an $N$-dimensional, and a mass-conservative $ N-1$-dimensional manifold respectively; therein, a metastable state with $N$ transition layers is approximated. We then determine, for both cases, the essential dynamics of the layers (ode systems with the equations of motion), expressed in terms of local coordinates relative to the manifold used. In particular, we estimate the spectrum of the linearized Cahn-Hilliard / Allen-Cahn operator, and specify wide families of $\varepsilon$-dependent weights $\delta(\varepsilon)$, $\mu(\varepsilon)$, acting at each part of the operator, for which the dynamics are stable and rest exponentially small in $\varepsilon$. Our analysis enlightens the role of mass conservation in the classification of the general mixed problem into two main categories where the solution has a profile close to Allen-Cahn, or, when the mass is conserved, close to the Cahn-Hilliard solution.