On the uniqueness of strong solution to the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system (2508.09761v1)

Abstract: 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. Global existence of strong solutions with bounded and strictly positive density for this system was proven by Giorgini and Temam \cite{GT}; however, an additional smoothness assumption on the initial density was needed to prove uniqueness in \cite{GT1}. Whether uniqueness holds without this additional assumption has remained an open question. The present work solves this question and we finally establish uniqueness of the strong solution in the framework in \cite{GT}. Our method relies on some extra time weighted estimates and the Lagrangian approach.