On the uniqueness of strong solution to the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system

This paper is mainly concerned with an initial-boundary value problem of the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system with the Landau potential in a two and three dimensions. The existence of strong solutions with bounded and strictly positive density for this system was constructed by Giorgini and Temam \cite{GT}. However, whether uniqueness holds has remained an open question. The present work solves this question and we prove the uniqueness of strong solution. Our method mainly relies on some extra time weighted estimates and the Lagrangian approach.