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Liouville-type theorems for the new Taylor--Couette flow of the stationary Navier--Stokes equations

Hideo Kozono, Yutaka Terasawa, Yuta Wakasugi

TL;DR

The paper investigates stationary incompressible Navier–Stokes flow in the annulus between two rotating cylinders and proves Liouville-type rigidity results. Under small Reynolds number and either axisymmetry with a small $L^{\infty}$-norm or a bounded periodic pressure, any smooth solution must be a generalized Taylor–Couette flow with explicitly determined angular and axial components; when the pressure is bounded or $z$-periodic, this reduces to the canonical Taylor–Couette flow. In the non-axisymmetric setting, a similar smallness regime forces axial symmetry and the generalized Taylor–Couette form; a parallel Stokes-limit remark shows uniqueness of this solution in the bounded case. Overall, the work connects boundary-driven laminar dynamics in the annulus to a unique exact flow structure, highlighting the role of small-Reynolds-number effects in enforcing symmetry and a specific Taylor–Couette-type solution.

Abstract

We study the stationary Navier--Stokes equations in the region between two rotating concentric cylinders. We first prove that, under the small Reynolds number, if the fluid is axisymmetric and if its velocity is sufficiently small in the $L^\infty$-norm, then it is necessarily a generalized Taylor-Couette flow which is a new exact solution of the Navier--Stokes equations. If, in addition, the associated pressure is bounded or periodic in the $z$-axis, then it coincides with the well-known canonical Taylor-Couette flow. Next, we give a certain bound of the Reynolds number and the $L^\infty$-norm of the velocity such as the fluid is indeed, necessarily axisymmetric. It is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the new exact form of the Taylor-Couette flow.

Liouville-type theorems for the new Taylor--Couette flow of the stationary Navier--Stokes equations

TL;DR

The paper investigates stationary incompressible Navier–Stokes flow in the annulus between two rotating cylinders and proves Liouville-type rigidity results. Under small Reynolds number and either axisymmetry with a small -norm or a bounded periodic pressure, any smooth solution must be a generalized Taylor–Couette flow with explicitly determined angular and axial components; when the pressure is bounded or -periodic, this reduces to the canonical Taylor–Couette flow. In the non-axisymmetric setting, a similar smallness regime forces axial symmetry and the generalized Taylor–Couette form; a parallel Stokes-limit remark shows uniqueness of this solution in the bounded case. Overall, the work connects boundary-driven laminar dynamics in the annulus to a unique exact flow structure, highlighting the role of small-Reynolds-number effects in enforcing symmetry and a specific Taylor–Couette-type solution.

Abstract

We study the stationary Navier--Stokes equations in the region between two rotating concentric cylinders. We first prove that, under the small Reynolds number, if the fluid is axisymmetric and if its velocity is sufficiently small in the -norm, then it is necessarily a generalized Taylor-Couette flow which is a new exact solution of the Navier--Stokes equations. If, in addition, the associated pressure is bounded or periodic in the -axis, then it coincides with the well-known canonical Taylor-Couette flow. Next, we give a certain bound of the Reynolds number and the -norm of the velocity such as the fluid is indeed, necessarily axisymmetric. It is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the new exact form of the Taylor-Couette flow.
Paper Structure (6 sections, 5 theorems, 103 equations)

This paper contains 6 sections, 5 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Axially symmetric case
  3. General case
  4. Preliminaries
  5. Proof of Theorem \ref{['thm:liouville']}
  6. Proof of Theorem \ref{['thm:liouville:2']}

Key Result

Theorem 1.1

Let $(v,p)$ be an axially symmetric smooth solution of ns in $\Omega$ with the boundary conditions eq:bc. There exists a constant $C_1(\nu,R_1,R_2)>0$ such that if $\omega_1, \omega_2$ and $\| v \|_{L^{\infty}}$ satisfy and respectively, then $(v,p)$ must be the generalized Taylor--Couette flow with some constants $a, b \in \mathbb R$, where the constants $A$ and $B$ are the same as in eq:mu:et

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof