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

2025/08/13 by Jiahong Wu, Jiang, Lingxin, Wu, Jiahong +2
Mathematics · Computer Science · Materials Science · #Differential Equations and Numerical Methods #Advanced Mathematical Modeling in Engineering #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.2508.09761

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 \citeGT. 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.

Citations

• On the Stationary Nonlocal Cahn-Hilliard-Navier-Stokes System:\n Existence, Uniqueness and Exponential Stability
• The incompressible navier-stokes equations in vacuum
• Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density
• Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density
• DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS

Related

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• Global regularity and asymptotic stabilization for the incompressible Navier-Stokes-Cahn-Hilliard model with unmatched densities
• On the Stationary Nonlocal Cahn-Hilliard-Navier-Stokes System:\n Existence, Uniqueness and Exponential Stability