A Class of Sixth Order Viscous Cahn-Hilliard Equation with Willmore Regularization in ℝ3

Xiaopeng Zhao,Ning Duan

doi:10.3390/math8111865

Abstract

The main purpose of this paper is to study the Cauchy problem of sixth order viscous Cahn–Hilliard equation with Willmore regularization. Because of the existence of the nonlinear Willmore regularization and complex structures, it is difficult to obtain the suitable a priori estimates in order to prove the well-posedness results, and the large time behavior of solutions cannot be shown using the usual Fourier splitting method. In order to overcome the above two difficulties, we borrow a fourth-order linear term and a second-order linear term from the related term, rewrite the equation in a new form, and introduce the negative Sobolev norm estimates. Subsequently, we investigate the local well-posedness, global well-posedness, and decay rate of strong solutions for the Cauchy problem of such an equation in R3, respectively.

Highlights

The phase-field method is a powerful tool for studying the dynamics of heterogeneous materials, such as phase separations in binary mixtures, crystal faceting, epitaxial thin film growth, and multi-phase fluid flows, just to name a few
The system responds by removing these orientations in the shape of the crystal (Wulff shape) driven by minimizing the total surface energy of the system
In order to overcome loss of smoothness and ill-posedness of the anisotropic Cahn–Hilliard model, a higher order derivative regularization can be added to the surface energy

Summary

Introduction

The phase-field method is a powerful tool for studying the dynamics of heterogeneous materials, such as phase separations in binary mixtures, crystal faceting, epitaxial thin film growth, and multi-phase fluid flows, just to name a few. Wang et al [13], Liu et al [14], and Coclite and di Ruvo [15] considered the global well-posedness and large time behavior of solutions for some typical higher-order evolution equations, their main methods are frequency decomposition method, Green’s function method, and fixed point theorem. In this paper, supposing that the nonlinear function f (u) = 0 F (s)ds = u3 − u and the viscous parameter e = 1, we consider the Cauchy problem of the sixth-order viscous Cahn–Hilliard equation with Willmore regularization in R3. Because there exists the lower-order linear term on the right side of problem (5), it is difficulty to use the Fourier splitting method to study the decay rate of solutions.

Preliminaries

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3

Conclusions