Inhomogeneous 2D Navier--Stokes equations: Existence, uniqueness, stability, continuity in time and energy conservation of weak solutions

2025/04/23 by Stefan Škondrić, Škondrić, Stefan
#math.AP

paper · pdf · doi:10.48550/arxiv.2504.16638

Abstract

We present a novel and direct proof of the existence and uniqueness of weak solutions of the inhomogeneous incompressible Navier--Stokes equations without vacuum. The analysis we employ to prove the strong convergence of the approximating sequence, which is based on the relative energy method, reveals how to conclude the stability and uniqueness of weak solutions. To the best of our knowledge, these global-in-time stability estimates are completely new. Furthermore, for the first time, we establish energy conservation for weak solutions.

Related

• Relative Energy Method For Weak-Strong Uniqueness Of The Inhomogeneous Navier-Stokes Equations
• Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density
• The Navier-Stokes equations: on the existence of a weak solution enjoying the energy equality
• On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics
• Regularity aspects of Leray-Hopf solutions to the 2D Inhomogeneous Navier-Stokes system and applications to weak-strong uniqueness