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Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition

Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi

Abstract

We study the asymptotic behavior of solutions to the steady Navier-Stokes equations outside of an infinite cylinder in $\mathbb{R}^3$. We assume that the flow is periodic in $x_3$-direction and has no swirl. This problem is closely related with two-dimensional exterior problem. Under a condition on the generalized finite Dirichlet integral, we give a pointwise decay estimate of the vorticity at the spatial infinity. Moreover, we prove a Liouville-type theorem only from the condition of the generalized finite Dirichlet integral.

Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition

Abstract

We study the asymptotic behavior of solutions to the steady Navier-Stokes equations outside of an infinite cylinder in . We assume that the flow is periodic in -direction and has no swirl. This problem is closely related with two-dimensional exterior problem. Under a condition on the generalized finite Dirichlet integral, we give a pointwise decay estimate of the vorticity at the spatial infinity. Moreover, we prove a Liouville-type theorem only from the condition of the generalized finite Dirichlet integral.
Paper Structure (7 sections, 6 theorems, 93 equations)

This paper contains 7 sections, 6 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm_asymp']}
  3. Preliminary estimates
  4. $L^q$-energy estimates
  5. Pointwise behavior via maximum principle
  6. Proof of Theorem \ref{['thm_asymp']}
  7. Proof of Theorem \ref{['thm:liouville']}

Key Result

Theorem 1.1

Let $(v,p)$ be a smooth axisymmetric solution of sns with no swirl satisfying eq:Diri with some $q \in [2,\infty)$. Then, we have and

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Remark 2.1
  • ...and 2 more