Long-time dynamics of the Cahn--Hilliard equation with kinetic rate dependent dynamic boundary conditions

TL;DR

This paper studies the long-term behavior of a Cahn--Hilliard model with dynamic boundary conditions, establishing existence of attractors and stability results, and analyzing the convergence as the kinetic rate varies.

Contribution

It provides a comprehensive analysis of the long-time dynamics of the KLLM model, including existence of attractors and their stability, and links to the GMS model as the kinetic rate increases.

Findings

Existence of a global attractor for the KLLM model.

Convergence of solutions to stationary points over time.

Stability of the GMS model's attractor under kinetic rate perturbations.

Abstract

We consider a Cahn--Hilliard model with kinetic rate dependent dynamic boundary conditions that was introduced by Knopf, Lam, Liu and Metzger (ESAIM Math. Model. Numer. Anal., 2021) and will thus be called the KLLM model. In the aforementioned paper, it was shown that solutions of the KLLM model converge to solutions of the GMS model proposed by Goldstein, Miranville and Schimperna (Physica D, 2011) as the kinetic rate tends to infinity. We first collect the weak well-posedness results for both models and we establish some further essential properties of the weak solutions. Afterwards, we investigate the long-time behavior of the KLLM model. We first prove the existence of a global attractor as well as convergence to a single stationary point. Then, we show that the global attractor of the GMS model is stable with respect to perturbations of the kinetic rate. Eventually, we construct exponential attractors for both models, and we show that the exponential attractor associated with the GMS model is robust against kinetic rate perturbations.