On the Cauchy problem of a sixth-order Cahn–Hilliard equation arising in oil-water-surfactant mixtures

Article type: Research Article

Authors: Zhao, Xiaopeng^{; *}

Affiliations: College of Sciences, Northeastern University, Shenyang 110004, China. E-mail: zhaoxiaopeng@mail.neu.edu.cn

Correspondence: [*] Corresponding author. E-mail: zhaoxiaopeng@mail.neu.edu.cn.

Abstract: We study the well-posedness and asymptotic behavior of solutions to the Cauchy problem of a three-dimensional sixth-order Cahn–Hilliard equation arising in oil-water-surfactant mixtures. First, by using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the H2-norm of the initial data is sufficiently small. Moreover, we establish suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

Keywords: Sixth-order Cahn–Hilliard equation, global well-posedness, asymptotic behavior, decay estimates

DOI: 10.3233/ASY-201616

Journal: Asymptotic Analysis, vol. 122, no. 3-4, pp. 201-224, 2021