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

Abstract

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 $${\bar{u}}\in {\mathbb {R}}$$ . In addition, for the N-dimensional Cahn–Hilliard equation without convective term, we also establish the similar result.