On the Cahn-Hilliard equation with kinetic rate dependent dynamic boundary condition and non-smooth potential: separation property and long-time behavior

TL;DR

This paper studies the Cahn-Hilliard equation with dynamic boundary conditions and non-smooth potentials, establishing separation properties, regularity, and long-term convergence of solutions in 2D and 3D domains.

Contribution

It proves separation properties and regularity results for solutions with singular potentials under dynamic boundary conditions, including instantaneous separation in 2D and eventual separation in 3D.

Findings

Solutions exhibit propagation of regularity over time.

In 2D, solutions stay away from pure phases after any positive time.

Solutions converge to a single equilibrium as time approaches infinity.

Abstract

We consider a class of Cahn-Hilliard equation that characterizes phase separation phenomena of binary mixtures in a bounded domain $\Omega \subset \mathbb{R}^d$ $(d\in \{2,3\})$ with non-permeable boundary. The equations in the bulk are subject to kinetic rate dependent dynamic boundary conditions with possible boundary diffusion acting on the boundary chemical potential. For the initial boundary value problem with singular potentials, we prove that any global weak solution exhibits a propagation of regularity in time. In the two dimensional case, we establish the instantaneous strict separation property by a suitable De Giorgi's iteration scheme, which yields that the weak solution stays uniformly away from the pure phases $\pm 1$ from any positive time on. In particular, when the bulk and boundary chemical potentials are in equilibrium, we obtain the instantaneous separation property with or without possible boundary diffusion acting on the boundary chemical potential. Next, in the three dimensional case, we show the eventual strict separation property that holds after a sufficiently large time. These separation properties are obtained in an unified way with respect to the structural parameters. Moreover, they allow us to achieve higher-order regularity of the global weak solution and prove the convergence to a single equilibrium as $t \rightarrow \infty$.