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Wiener Algebras Methods for Liouville Theorems on the Stationary Navier-Stokes System

Nicolas Lerner

TL;DR

This work develops a Wiener-algebra–based framework to obtain Liouville-type theorems for the stationary Navier–Stokes system in $\mathbb{R}^3$, linking vanishing behavior at infinity to rigidity through low-frequency control of the velocity. By exploiting spectral localization and the invariance of Wiener algebras under singular integrals, the authors prove that sufficiently small or localized low-frequency components of the velocity (and certain components of the Bernoulli head pressure $Q$) force the velocity to vanish, under a mild conjectural setting. The paper yields refinements of classic results (Galdi and Chae), showing that projecting onto the near-zero frequency block with $\alpha_0(D)$ and placing it in $L^{9/2}$ or that certain low-frequency parts of $\mathbf{C}^2 v$ or $\nabla Q$ belong to $L^{6/5}$ suffices for triviality. This approach highlights the power of Wiener-algebra regularity and low-frequency analysis in handling nonlinear terms via singular integrals, with implications for uniqueness and structure of stationary incompressible flows.

Abstract

We prove some Liouville theorems for the stationary Navier-Stokes system for incompressible fluids. We provide some sufficient conditions on the low frequency part of the solution, using some properties of classical singular integrals with respect to Wiener algebras.

Wiener Algebras Methods for Liouville Theorems on the Stationary Navier-Stokes System

TL;DR

This work develops a Wiener-algebra–based framework to obtain Liouville-type theorems for the stationary Navier–Stokes system in , linking vanishing behavior at infinity to rigidity through low-frequency control of the velocity. By exploiting spectral localization and the invariance of Wiener algebras under singular integrals, the authors prove that sufficiently small or localized low-frequency components of the velocity (and certain components of the Bernoulli head pressure ) force the velocity to vanish, under a mild conjectural setting. The paper yields refinements of classic results (Galdi and Chae), showing that projecting onto the near-zero frequency block with and placing it in or that certain low-frequency parts of or belong to suffices for triviality. This approach highlights the power of Wiener-algebra regularity and low-frequency analysis in handling nonlinear terms via singular integrals, with implications for uniqueness and structure of stationary incompressible flows.

Abstract

We prove some Liouville theorems for the stationary Navier-Stokes system for incompressible fluids. We provide some sufficient conditions on the low frequency part of the solution, using some properties of classical singular integrals with respect to Wiener algebras.
Paper Structure (25 sections, 60 theorems, 423 equations)

This paper contains 25 sections, 60 theorems, 423 equations.

Table of Contents

  1. Introduction
  2. The classical Liouville theorem
  3. Liouville theorems for the stationary Navier-Stokes system
  4. Known results
  5. A preliminary formulation of our results
  6. Preliminaries
  7. A linear lemma
  8. The stationary Navier-Stokes system
  9. Some first regularity results involving the Wiener algebra
  10. Regularity results, continued
  11. More on the Bernoulli head pressure
  12. First facts
  13. The Bernoulli head pressure $Q$ is a non-positive function
  14. More properties of the Bernoulli head pressure
  15. Assumptions on the Laplacean of the Bernoulli Head Pressure
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $f$ be a bounded harmonic function on $\mathbb R^{d}$. Then $f$ is a constant function.

Theorems & Definitions (156)

  • Theorem 1.1: Joseph Liouville, 1844
  • proof
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • Conjecture 1.5
  • Claim 1.6
  • proof : Proof of the Claim
  • Remark 1.7
  • Remark 1.8
  • ...and 146 more