On a Class of Sixth-order Cahn-Hilliard Type Equations with Logarithmic Potential

TL;DR

This paper analyzes a complex sixth-order Cahn-Hilliard system with a logarithmic potential, proving existence, uniqueness, regularization, and long-term stability of solutions relevant to phase separation models.

Contribution

It introduces a rigorous mathematical analysis of a highly singular sixth-order Cahn-Hilliard equation with logarithmic potential, establishing key properties of solutions.

Findings

Existence and uniqueness of global weak solutions

Solutions exhibit parabolic regularization over time

Existence of a global attractor for the dynamical system

Abstract

We consider a class of six-order Cahn-Hilliard equations with logarithmic type potential. This system is closely connected with some important phase-field models relevant in different applications, for instance, the functionalized Cahn-Hilliard equation that describes phase separation in mixtures of amphiphilic molecules in solvent, and the Willmore regularization of Cahn-Hilliard equation for anisotropic crystal and epitaxial growth. The singularity of the configuration potential guarantees that the solution always stays in the physical relevant domain [-1,1]. Meanwhile, the resulting system is characterized by some highly singular diffusion terms that make the mathematical analysis more involved. We prove existence and uniqueness of global weak solutions and show their parabolic regularization property for any positive time. Besides, we investigate long-time behavior of the system, proving existence of the global attractor for the associated dynamical process in a suitable complete metric space.