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Refined Liouville-Type Theorems for the Stationary Navier--Stokes Equations

Youseung Cho, Minsuk Yang

Abstract

We study smooth solutions to the three-dimensional stationary Navier--Stokes equations and establish new Liouville-type theorems under refined decay assumptions. Building on the work of Cho et al., we introduce a refinement to previously known integrability criteria and analyze the associated averaged quantities. Our main result shows that if the $L^p$ growth rate of a solution remains bounded for some $3/2 < p < 3$, then the solution must be trivial. The proof combines averaged decay estimates, energy inequalities, and an iteration scheme.

Refined Liouville-Type Theorems for the Stationary Navier--Stokes Equations

Abstract

We study smooth solutions to the three-dimensional stationary Navier--Stokes equations and establish new Liouville-type theorems under refined decay assumptions. Building on the work of Cho et al., we introduce a refinement to previously known integrability criteria and analyze the associated averaged quantities. Our main result shows that if the growth rate of a solution remains bounded for some , then the solution must be trivial. The proof combines averaged decay estimates, energy inequalities, and an iteration scheme.
Paper Structure (5 sections, 9 theorems, 69 equations)

This paper contains 5 sections, 9 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Energy estimates
  4. Proof of Theorem \ref{['T1']}
  5. Proof of Theorem \ref{['T2']}

Key Result

Theorem 1

Let $g$ satisfy Assumption A1. Suppose $\mathbf{u}$ is a smooth solution to E11 such that Then $\mathbf{u} = 0$.

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2: Lemma 3 of MR4354995
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 6 more