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

In this paper, we consider the global well-posedness of solutions for the Cauchy problem of N-dimensional Cahn-Hilliard equation with convective term. We first construct the local smooth solutions; then, by combining some a priori estimates, continuity argument, the local smooth solutions are extended step by step to all t > 0 provided that the initial data are suitably small and the smooth nonlinear functions satisfy certain local growth conditions at some fixed point (u) over bar is an element of R. In addition, for the N-dimensional Cahn-Hilliard equation without convective term, we also establish the similar result.