On a system of coupled Cahn-Hilliard equations

TL;DR

This paper analyzes a coupled system of Cahn-Hilliard equations modeling phase separation in polymers, proving well-posedness, regularization, separation from pure phases, and convergence to equilibrium in two dimensions.

Contribution

It introduces a coupled Cahn-Hilliard system with a novel energy functional, establishing well-posedness, regularization, phase separation, and long-term behavior results.

Findings

Weak solutions exist and are well-posed.

Solutions regularize instantaneously in the conserved case.

In 2D, solutions stay away from pure phases and converge to a stationary state.

Abstract

We consider a system which consists of a Cahn-Hilliard equation coupled with a Cahn-Hilliard-Oono equation in a bounded domain of $\mathbb{R}^d$, $d = 2, 3$. This system accounts for macrophase and microphase separation in a polymer mixture through two order parameters $u$ and $v$. The free energy of this system is a bivariate interaction potential which contains the mixing entropy of the two order parameters and suitable coupling terms. The equations are endowed with initial conditions and homogeneous Neumann boundary conditions both for $u,v$ and for the corresponding chemical potentials. We first prove that the resulting problem is well posed in a weak sense. Then, in the conserved case, we establish that the weak solution regularizes instantaneously. Furthermore, in two spatial dimensions, we show the strict separation property for $u$ and $v$, namely, they both stay uniformly away from the pure phases $\pm 1$ in finite time. Finally, we investigate the long-time behavior of a finite energy solution showing, in particular, that it converges to a single stationary state.