Global well-posedness of the 3D Cahn-Hilliard equations

Abstract

The Cauchy problem of the Cahn-Hilliard equations is studied in three-dimensional space. Firstly, we construct its approximate fourth-order parabolic equation, obtaining the existence of solutions by the Aubin-Lions's compactness lemma. Furthermore, we prove the uniqueness of the solution. Then, the global well-posedness is demonstrated by using energy estimates. At last, we consider a special case and get a better result about it.