Well-posedness and Global Attractors for Viscous Fractional Cahn-Hilliard Equations with Memory

TL;DR

This paper studies a viscous fractional Cahn-Hilliard model with memory, proving the existence of solutions and the presence of a global attractor, thus advancing understanding of phase separation dynamics with nonlocal operators.

Contribution

It establishes existence, uniqueness, and long-term behavior of solutions for a nonlocal fractional Cahn-Hilliard equation with minimal assumptions on the potential.

Findings

Existence of global weak solutions

Uniqueness of solutions via continuous dependence

Existence of a compact global attractor

Abstract

We examine a viscous Cahn-Hilliard phase-separation model with memory and where the chemical potential possesses a nonlocal fractional Laplacian operator. The existence of global weak solutions is proven using a Galerkin approximation scheme. A continuous dependence estimate provides uniqueness of the weak solutions and also serves to define a precompact pseudometric. This, in addition to the existence of a bounded absorbing set, shows that the associated semigroup of solution operators admits a compact connected global attractor in the weak energy phase space. The minimal assumptions on the nonlinear potential allow for arbitrary polynomial growth.