Long-time dynamics of the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions. (February 2022)

• Record Type: Journal Article
• Authors: Garcke, Harald; Knopf, Patrik; Yayla, Sema
• Is Part Of: Nonlinear analysis. Volume 215(2022)
• Publication Date: 2022-02
• Subjects: 35B40 -- 35B41 -- 35K35 -- 35K61 -- 35Q92 -- 37L30; Cahn–Hilliard equation -- Dynamic boundary conditions -- Long-time dynamics -- Stability of global attractors -- Robustness of exponential attractors; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112619
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a Cahn–Hilliard model with kinetic rate dependent dynamic boundary conditions that was introduced by Knopf et al. (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 et al. (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.