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Stability of Three-dimensional Oseen Vortices under Helical Perturbations

Quentin Vila

Abstract

We study the long-time behaviour of helically symmetric infinite-energy solutions to the incompressible Navier-Stokes equations in the whole space $\mathbb{R}^3$. Our solutions are $H^1$-perturbations of a Lamb-Oseen vortex whose circulation Reynolds number can be any fixed real number. If $v$ denotes the helical velocity perturbation, no matter how large at initial time in $H^1(\mathbb{R}^3)$, we show that the scale-invariant quantities $\|v(t)\|_{L^2}$ and $\sqrt{t} \|\nabla v(t)\|_{L^2}$ converge to zero as $t \to +\infty$. This proves that the Oseen vortex is globally stable with respect to $H^1$-helical perturbations. Our analysis relies on a logarithmic energy estimate for the perturbation $v$, on the Ladyzhenskaya inequality for helical vector fields, and on Poincaré's inequality which implies an exponential decay in time for the velocity components whose mean value is zero along the symmetry axis.

Stability of Three-dimensional Oseen Vortices under Helical Perturbations

Abstract

We study the long-time behaviour of helically symmetric infinite-energy solutions to the incompressible Navier-Stokes equations in the whole space . Our solutions are -perturbations of a Lamb-Oseen vortex whose circulation Reynolds number can be any fixed real number. If denotes the helical velocity perturbation, no matter how large at initial time in , we show that the scale-invariant quantities and converge to zero as . This proves that the Oseen vortex is globally stable with respect to -helical perturbations. Our analysis relies on a logarithmic energy estimate for the perturbation , on the Ladyzhenskaya inequality for helical vector fields, and on Poincaré's inequality which implies an exponential decay in time for the velocity components whose mean value is zero along the symmetry axis.
Paper Structure (8 sections, 14 theorems, 85 equations, 1 figure)

This paper contains 8 sections, 14 theorems, 85 equations, 1 figure.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Remarks.
  4. Global Well-Posedness of the Helical Navier-Stokes Equations
  5. Estimates for the Zero-Vertical-Mean Component
  6. Remark 3.6.
  7. Asymptotic Behaviour
  8. Additional Results and Comments

Key Result

Theorem 1.1

Let $a\in\mathbb{R}$. Let $u_0\in X$. Equation eqNSdimensionless with initial data $u_0$ has a unique global solution $u(t) = v(t) + aU(t)$ where $v\in\mathscr{C}^0([0,+\infty[\,,H)$. Moreover, the following limit holds

Figures (1)

  • Figure 1: The tangent vector to a helical curve of radius $r$ and pitch $2\pi L$ is the Beltrami vector $e_B = re_\theta + Le_z$, which satisfies $e_B\cdot\nabla = \partial_\theta + L\partial_z$ in cylindrical coordinates. A differential scalar map $f$ is helical if $e_B\cdot\nabla f = 0$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Lemma 2.1: $L^2$ and $L^\infty$ estimates for the Oseen vortex
  • Proposition 2.2: Local existence
  • proof
  • Lemma 2.3: Ladyzhenskaya's inequality for helical maps
  • Theorem 2.4: Global existence
  • proof
  • Proposition 3.1: Logarithmic energy estimate for the $H^1$ component
  • proof
  • Corollary 3.2
  • ...and 12 more