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Existence and uniqueness of Leray-Hopf weak solution for the inhomogeneous 2D Navier--Stokes equations without vacuum

Stefan Škondrić

TL;DR

We analyze the 2D inhomogeneous incompressible Navier–Stokes equations with density bounded away from vacuum and establish existence and uniqueness of Leray–Hopf weak solutions via a robust relative energy framework. The core method combines a direct approximation by regular data with a novel $W^{-1,4}$-stability analysis that handles the challenging cross-term involving density fluctuations, and it yields an energy-conserving weak solution in addition to an explicit stability bound. A key innovation is the introduction of immediately strong solutions and corresponding energy functionals, which, together with transport flow estimates, enable strong convergence and energy equality in the limit. The results also demonstrate energy conservation for weak solutions and provide a precise, quantitative stability estimate that implies continuous dependence on initial data, with potential impact on broader no-vacuum incompressible flow problems in two dimensions.

Abstract

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

Existence and uniqueness of Leray-Hopf weak solution for the inhomogeneous 2D Navier--Stokes equations without vacuum

TL;DR

We analyze the 2D inhomogeneous incompressible Navier–Stokes equations with density bounded away from vacuum and establish existence and uniqueness of Leray–Hopf weak solutions via a robust relative energy framework. The core method combines a direct approximation by regular data with a novel -stability analysis that handles the challenging cross-term involving density fluctuations, and it yields an energy-conserving weak solution in addition to an explicit stability bound. A key innovation is the introduction of immediately strong solutions and corresponding energy functionals, which, together with transport flow estimates, enable strong convergence and energy equality in the limit. The results also demonstrate energy conservation for weak solutions and provide a precise, quantitative stability estimate that implies continuous dependence on initial data, with potential impact on broader no-vacuum incompressible flow problems in two dimensions.

Abstract

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

Paper Structure

This paper contains 19 sections, 19 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. Notion of solution and main result
  3. Literature overview
  4. Weak vs. strong uniqueness
  5. Lagrangian coordinates
  6. Relative energy method and $W^{-1,4}$-stability
  7. Outline of the paper
  8. Refined main result and strategy of proof
  9. Refined main result
  10. Immediately strong solutions and improved $W^{-1,4}$-stability
  11. New strategy to prove \ref{['thm:mainthmrefined']}
  12. Some transport theory
  13. Weak and renormalized solution of the continuity equation
  14. The flow associated to non-Lipschitz vector fields
  15. More estimates on the velocity
  16. ...and 4 more sections

Key Result

Theorem 1.2

For every $u_0 \in L^2_\sigma(\mathbb{R}^2)$, the two-dimensional homogeneous Navier-Stokes equations admit unique global-in-time Leray--Hopf solution $u$ which satisfies the energy equality and we have

Theorems & Definitions (41)

  • Definition 1.1: Leray--Hopf weak solution
  • Theorem 1.2: Existence and uniqueness for the two-dimensional homogeneous Navier-Stokes system
  • Theorem 1.3: Existence and uniqueness for the two-dimensional inhomogeneous Navier-Stokes system
  • Theorem 2.1
  • Remark 2.2: On the parabolic regularization
  • Remark 2.3: On the no vacuum propagation
  • Definition 2.4: Immediately strong solutions
  • Proposition 2.5
  • Proposition 2.6
  • Definition 3.1: Weak and renormalized solution
  • ...and 31 more