Global well-posedness and decay of solutions to the Cauchy problem of convective Cahn-Hilliard equation

TL;DR

This paper establishes the global existence, uniqueness, and decay rates of solutions to the 3D convective Cahn-Hilliard equation, revealing optimal decay behaviors and preservation of negative Sobolev norms using a refined energy approach.

Contribution

It introduces a refined pure energy method to prove global well-posedness and decay rates for the 3D convective Cahn-Hilliard equation, including optimal decay of higher derivatives.

Findings

Optimal decay rates of higher-order derivatives obtained

Negative Sobolev norms are preserved and enhance decay

Global well-posedness established for the 3D convective Cahn-Hilliard equation

Abstract

In this paper, we consider the global well-posedness and time-decay rates of solution to the Cauchy problem for 3D convective Cahn-Hilliard equation with double-well potential via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the $\dot{H}^{-s}$ ($0<s\leq\frac12$) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.