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Existence and non-uniqueness of classical solutions to the axially symmetric stationary Navier-Stokes equations in an exterior cylinder

Zijin Li, Xinghong Pan

TL;DR

The authors address the existence and non-uniqueness of classical, axially symmetric stationary Navier–Stokes flows in an exterior periodic cylinder subject to small boundary perturbations. They linearize about the scale-invariant base flow $\frac{\nu}{r}\mathbf{e}_r+\frac{\mu}{r}\mathbf{e}_\theta$ and decompose the problem into Fourier modes in the axial variable, solving zero and nonzero modes with Euler-type ODEs and Biot–Savart-type representations that involve modified Bessel functions. A careful nonlinear estimate in a tailored function space leads to a contraction mapping, proving global small-data existence of a decaying perturbation $\boldsymbol{v}$ and thus a solution $\boldsymbol{u}=\frac{\nu}{r}\mathbf{e}_r+\frac{\mu}{r}\mathbf{e}_\theta+\boldsymbol{v}$; when $\nu<-2$, the authors construct infinitely many such solutions by varying the swirl parameter, establishing non-uniqueness. This work connects to the 2D Stokes paradox and an open problem of Yudovich on global stationary, periodic flows, and extends axisymmetric stationary NS analysis in exterior domains with precise decay and regularity. The methods rely on detailed mode-by-mode linear estimates and a robust fixed-point argument in weighted spaces.

Abstract

In this paper, we show existence and non-uniqueness on the axially symmetric stationary Navier-Stokes equations in an exterior periodic cylinder. On the boundary of the cylinder, the horizontally swirl velocity is subject to the perturbation of a rotation, the horizontally radial velocity is subject to the perturbation of an interior sink, while the vertical velocity is the perturbation of zero. At infinity, the flow stays at rest. We construct a solution to such problem, whose principal part admits a critical decay for the horizontal components and a supercritical decay for the vertical component of the velocity. This existence result is related to the 2D Stokes paradox and an open problem raised by V. I. Yudovich in [Eleven great problems of mathematical hydrodynamics, Mosc. Math. J. 3 (2003), no. 2, 711--737], where Problem 2 states that: Show (spatially) global existence theorems for stationary and periodic flows. Moreover, if the horizontally radial-sink velocity is relatively large ($ν<-2$ in our setting), then the solution to this problem is non-unique.

Existence and non-uniqueness of classical solutions to the axially symmetric stationary Navier-Stokes equations in an exterior cylinder

TL;DR

The authors address the existence and non-uniqueness of classical, axially symmetric stationary Navier–Stokes flows in an exterior periodic cylinder subject to small boundary perturbations. They linearize about the scale-invariant base flow and decompose the problem into Fourier modes in the axial variable, solving zero and nonzero modes with Euler-type ODEs and Biot–Savart-type representations that involve modified Bessel functions. A careful nonlinear estimate in a tailored function space leads to a contraction mapping, proving global small-data existence of a decaying perturbation and thus a solution ; when , the authors construct infinitely many such solutions by varying the swirl parameter, establishing non-uniqueness. This work connects to the 2D Stokes paradox and an open problem of Yudovich on global stationary, periodic flows, and extends axisymmetric stationary NS analysis in exterior domains with precise decay and regularity. The methods rely on detailed mode-by-mode linear estimates and a robust fixed-point argument in weighted spaces.

Abstract

In this paper, we show existence and non-uniqueness on the axially symmetric stationary Navier-Stokes equations in an exterior periodic cylinder. On the boundary of the cylinder, the horizontally swirl velocity is subject to the perturbation of a rotation, the horizontally radial velocity is subject to the perturbation of an interior sink, while the vertical velocity is the perturbation of zero. At infinity, the flow stays at rest. We construct a solution to such problem, whose principal part admits a critical decay for the horizontal components and a supercritical decay for the vertical component of the velocity. This existence result is related to the 2D Stokes paradox and an open problem raised by V. I. Yudovich in [Eleven great problems of mathematical hydrodynamics, Mosc. Math. J. 3 (2003), no. 2, 711--737], where Problem 2 states that: Show (spatially) global existence theorems for stationary and periodic flows. Moreover, if the horizontally radial-sink velocity is relatively large ( in our setting), then the solution to this problem is non-unique.

Paper Structure

This paper contains 14 sections, 10 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Functional spaces and the main result
  4. Strategy of proof to the main result
  5. Related works, organization and notations of the paper
  6. Reformulation in the Fourier mode and the zero-mode estimates
  7. The zero mode of $\boldsymbol{v_\theta}$ and its estimate
  8. The zero modes of $\boldsymbol{v_r}$ and $\boldsymbol{v_z}$ and their estimates
  9. The non-zero Fourier modes and their estimates
  10. The non-zero mode of $\boldsymbol{v_\theta}$ and its estimate
  11. The non-zero mode of $\boldsymbol{(v_r,v_z)}$ and their estimates
  12. The nonlinear estimate and proof of the main result
  13. Existence
  14. Non-uniqueness of the case for $\nu<-2$

Key Result

Theorem 1.1

For fixed $\nu<0,\,\mu\in \mathbb R$ and $\lambda_\theta>3$, $\lambda_z>2$, $\lambda>3/2$, there exists $\epsilon,\,\tau>0$ depending on the aforementioned constants, such that if then the system ns0 has a solution $\boldsymbol{u}$ such that where $\boldsymbol{v}\in{\mathcal{B}_{\tau}}\,$ satisfying for some constants $C>0$, depending on $\mu,\,\nu,\,\lambda_\theta,\,\lambda_z,\,\lambda$ mentio

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • ...and 11 more