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Liouville type theorems for the fractional Navier-Stokes equations without the integrability condition of velocity in $\mathbb{R}^3$

Wendong Wang, Guoxu Yang, Jianbo Yu

TL;DR

This work establishes Liouville-type rigidity for the steady fractional Navier–Stokes equations in $\mathbb{R}^3$ with no velocity integrability condition, for $s\in(\tfrac12,1)$. It combines Lizorkin's multiplier theorem to derive $L^p$ estimates for the fractional linear Oseen operator with Coifman–McIntosh–Meyer commutator bounds to handle the dissipation and nonlocal effects, and develops an $L^p$ theory for a generalized 3D stationary Oseen system under a small-perturbation framework. The authors prove that any smooth solution converging to a nonzero constant at infinity must be equal to that constant, first under a regularity class and then via a corollary with stronger bounds; they also treat the nonzero-infinity case by a perturbation/energy method to arrive at the same rigidity. These results extend Liouville-type phenomena to the fractional NS setting and clarify how far-field limits interact with nonlocal dissipation without requiring velocity integrability, highlighting the roles of the fractional Laplacian and nonlocal commutator structure. The findings have implications for the uniqueness and asymptotic behavior of stationary fractional fluid models and contribute to the broader understanding of nonlocal dissipative PDEs in three dimensions.

Abstract

Motivated by the classification of solutions of harmonic functions, we investigate Liouville type theorems for the fractional Navier-Stokes equations in $\mathbb{R}^3$ under some conditions on the boundedness of fractional derivatives. We prove that the smooth solution must be a trivial solution provided that it uniformly converges to a nonzero constant vector at infinity by applying Lizorkin's multiplier theorem to establish \(L^p\) estimates for the fractional linear Oseen system and Coifman-McIntosh-Meyer type commutator estimates for the dissipation term. It is noteworthy that the integrability of velocity is not required here.

Liouville type theorems for the fractional Navier-Stokes equations without the integrability condition of velocity in $\mathbb{R}^3$

TL;DR

This work establishes Liouville-type rigidity for the steady fractional Navier–Stokes equations in with no velocity integrability condition, for . It combines Lizorkin's multiplier theorem to derive estimates for the fractional linear Oseen operator with Coifman–McIntosh–Meyer commutator bounds to handle the dissipation and nonlocal effects, and develops an theory for a generalized 3D stationary Oseen system under a small-perturbation framework. The authors prove that any smooth solution converging to a nonzero constant at infinity must be equal to that constant, first under a regularity class and then via a corollary with stronger bounds; they also treat the nonzero-infinity case by a perturbation/energy method to arrive at the same rigidity. These results extend Liouville-type phenomena to the fractional NS setting and clarify how far-field limits interact with nonlocal dissipation without requiring velocity integrability, highlighting the roles of the fractional Laplacian and nonlocal commutator structure. The findings have implications for the uniqueness and asymptotic behavior of stationary fractional fluid models and contribute to the broader understanding of nonlocal dissipative PDEs in three dimensions.

Abstract

Motivated by the classification of solutions of harmonic functions, we investigate Liouville type theorems for the fractional Navier-Stokes equations in under some conditions on the boundedness of fractional derivatives. We prove that the smooth solution must be a trivial solution provided that it uniformly converges to a nonzero constant vector at infinity by applying Lizorkin's multiplier theorem to establish estimates for the fractional linear Oseen system and Coifman-McIntosh-Meyer type commutator estimates for the dissipation term. It is noteworthy that the integrability of velocity is not required here.
Paper Structure (4 sections, 10 theorems, 86 equations)

This paper contains 4 sections, 10 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:main1']} and Corollary \ref{['cor']}
  4. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $\frac{1}{2}<s<1$. Suppose that $u \in \dot{H}^s\left(\mathbb{R}^3\right) \cap \dot{W}^{1+2s,\infty}\left(\mathbb{R}^3\right)$ is a smooth solution of eq:main with the uniform condition eq:the uniform condition at spatial infinit, then $u\equiv u_\infty$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1: Coifman-McIntosh-Meyer type commutator estimate, Theorem 6.1, LS20
  • Remark 2.2
  • Lemma 2.3: Fractional Leibniz rule
  • Theorem 2.4: Lizorkin
  • Definition 2.5
  • ...and 10 more