  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.

Analysis of PDEs