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

TL;DR

This paper investigates the global existence, uniqueness, and decay rates of solutions for a sixth-order Cahn-Hilliard equation modeling oil-water-surfactant mixtures, revealing decay properties similar to lower-order heat equations.

Contribution

It establishes the global well-posedness and decay rates for the sixth-order Cahn-Hilliard equation using energy methods and Sobolev estimates, which is novel for such high-order equations.

Findings

Existence of unique global strong solutions under small initial data.

Decay rate of solutions matches that of a fourth-order heat equation.

Decay estimates are obtained via negative Sobolev norm analysis.

Abstract

We study the global well-posedness and asymptotic behavior of solutions for the Cauchy problem of 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 initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the time decay rate of strong solutions. It is worth pointing out that although the problem we considered is a sixth-order parabolic equation, the time decay rate is equivalent to the decay rate of fourth-order generalized heat equation, which is better than our expect.