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

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 longtime behavior of the system, proving existence of the global attractor for the associated dynamical process in a suitable complete metric space.

Stability of the semi-implicit method for the Cahn-Hilliard equation with logarithmic potentials

“…The approach therein is based on several approximation procedures using smoothed equations and potentials. We note that more recently there has been some new developments on the analysis of the Cahn-Hilliard equation with singular potentials and dynamical boundary conditions [18,17]. Regarding the prior state of the art literature on these topics and more classical theory concerning long time behavior and attractors, we refer the interested readers to [3,18,17,19] and the references therein for more in-depth reviews and discussions.…”