What question did this study set out to answer?

To provide a direct proof for the existence, uniqueness, global-in-time stability, and energy conservation of weak solutions to the inhomogeneous incompressible 2D Navier–Stokes equations without vacuum.

August 22, 2026Journal of Mathematical Fluid Mechanics0 citationsOpen Access

Inhomogeneous 2D Navier–Stokes Equations: Existence, Uniqueness, Stability, Continuity in Time and Energy Conservation of Weak Solutions

SŠStefan Škondrić

Key Points

To provide a direct proof for the existence, uniqueness, global-in-time stability, and energy conservation of weak solutions to the inhomogeneous incompressible 2D Navier–Stokes equations without vacuum.
Implemented the relative energy method to analyze and prove the strong convergence of approximating solution sequences.
Derived global-in-time stability estimates for two-dimensional inhomogeneous incompressible fluid systems without vacuum states.
Proved the existence and uniqueness of weak solutions for the two-dimensional inhomogeneous incompressible Navier–Stokes equations.
Formulated novel global-in-time stability estimates and established strict energy conservation properties for weak solutions.

Abstract

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.

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