On the nonlocal Cahn-Hilliard equation with nonlocal dynamic boundary condition and singular potential: well-posedness, regularity and asymptotic limits

TL;DR

This paper studies a complex nonlocal Cahn-Hilliard model with dynamic boundary conditions, establishing existence, uniqueness, regularity, and asymptotic behavior of solutions for various interaction regimes.

Contribution

It introduces a novel analysis of the nonlocal Cahn-Hilliard equation with dynamic boundary conditions and singular potentials, including asymptotic limits and convergence rates.

Findings

Existence and uniqueness of global weak solutions for coupled bulk-boundary system.

Asymptotic limits as kinetic rate approaches zero or infinity are rigorously justified.

Solutions exhibit propagation of regularity and instantaneous strict separation under certain conditions.

Abstract

We consider a class of nonlocal Cahn-Hilliard equations in a bounded domain $\Omega\subset\mathbb{R}^{d}$ $(d\in\{2,3\})$, subject to a nonlocal kinetic rate dependent dynamic boundary condition. This diffuse interface model describes phase separation processes with possible long-range interactions both within the bulk material and on its boundary. The kinetic rate $1/L$, with $L\in[0,+\infty]$, distinguishes different types of bulk-boundary interactions. For the initial boundary value problem endowed with general singular potentials, including the physically relevant logarithmic potential, we first establish the existence and uniqueness of global weak solutions when the bulk and boundary chemical potentials are coupled through a Robin-type boundary condition, i.e., $L\in(0,+\infty)$. The proof of existence is based on a Yosida approximation of singular potentials and a suitable Faedo-Galerkin scheme. Subsequently, we investigate asymptotic limits as the kinetic rate approaches zero or infinity, which yield weak well-posedness for the limiting cases $L=+\infty$ and $L=0$, respectively. Under appropriate assumptions on the interaction kernels, we derive explicit convergence rates for the Yosida approximation as $\varepsilon\to0$ and for the asymptotic limits as $L\to0$ or $L\to+\infty$. Finally, we demonstrate that every global weak solution exhibits a propagation of regularity over time and when $L\in(0,+\infty]$, we establish the instantaneous strict separation property by means of a suitable De Giorgi's iteration scheme.