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

TL;DR

This paper proves the uniqueness of strong solutions for the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system with Landau potential in 2D and 3D, addressing an open problem in mathematical fluid dynamics.

Contribution

It establishes the first proof of uniqueness for strong solutions of this complex coupled system using time weighted estimates and Lagrangian methods.

Findings

Uniqueness of strong solutions is proven for the system.

The method involves time weighted estimates and Lagrangian approach.

Addresses an open question in the mathematical analysis of fluid systems.

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. 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.