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This is a Quantitative Study study.

October 2, 2025Open Access

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

Key Points

The study establishes the uniqueness of strong solutions for the nonhomogeneous Navier-Stokes-Cahn-Hilliard system.
Strong solutions were previously shown to exist with bounded density, but uniqueness was uncertain until now.
The methodology employs time weighted estimates and the Lagrangian approach to prove uniqueness in solutions.
This finding resolves a long-standing open question in the field of fluid dynamics and phase separation.

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 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.

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Cite This Study

Jiang et al. (Wed,) studied this question.

synapsesocial.com/papers/68de5d9c83cbc991d0a204d2 https://doi.org/https://doi.org/10.48550/arxiv.2508.09761

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