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

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 H 2 -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.