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Long-time behavior of an Arc-shaped Vortex Filament and its Application to the Stability of a Circular Vortex Filament

Masashi Aiki

Abstract

We consider a nonlinear model equation, known as the Localized Induction Equation, describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. We show stability estimates for an arc-shaped vortex filament, which is an exact solution to an initial-boundary value problem for the Localized Induction Equation. An arc-shaped filament travels along an axis at a constant speed without changing its shape, and is oriented in such a way that the arc stays in a plane that is perpendicular to the axis. We prove that an arc-shaped filament is stable in the Lyapunov sense for general perturbations except in the axis-direction, for which the perturbation can grow linearly in time. We also show that this estimate is optimal. We then apply the obtained stability estimates to study the stability of a circular vortex filament under some symmetry assumptions on the initial perturbation. We do this by dividing the circular filament into arcs, apply the stability estimate to each arc-shaped filament, and combine the estimates to obtain estimates for the whole circle. The optimality of the stability estimates for an arc-shaped filament also shows that a circular filament is not stable in the Lyapunov sense, namely, certain perturbations can grow linearly in time.

Long-time behavior of an Arc-shaped Vortex Filament and its Application to the Stability of a Circular Vortex Filament

Abstract

We consider a nonlinear model equation, known as the Localized Induction Equation, describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. We show stability estimates for an arc-shaped vortex filament, which is an exact solution to an initial-boundary value problem for the Localized Induction Equation. An arc-shaped filament travels along an axis at a constant speed without changing its shape, and is oriented in such a way that the arc stays in a plane that is perpendicular to the axis. We prove that an arc-shaped filament is stable in the Lyapunov sense for general perturbations except in the axis-direction, for which the perturbation can grow linearly in time. We also show that this estimate is optimal. We then apply the obtained stability estimates to study the stability of a circular vortex filament under some symmetry assumptions on the initial perturbation. We do this by dividing the circular filament into arcs, apply the stability estimate to each arc-shaped filament, and combine the estimates to obtain estimates for the whole circle. The optimality of the stability estimates for an arc-shaped filament also shows that a circular filament is not stable in the Lyapunov sense, namely, certain perturbations can grow linearly in time.
Paper Structure (12 sections, 11 theorems, 79 equations, 1 figure)

This paper contains 12 sections, 11 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction and Problem Setting
  2. Function Spaces, Notations, and Main Theorem
  3. Proof of Theorem \ref{['th1']}, \ref{['th2']}, and \ref{['thopt']}
  4. Proof of Theorem \ref{['th1']}
  5. Proof of Theorem \ref{['thopt']}
  6. Remark on Theorem \ref{['th2']}
  7. Long-time Behaviour of a Circular Vortex Filament
  8. Concluding Remarks
  9. On Theorem \ref{['thopt']}
  10. On the Stability of Circular Vortex Filaments
  11. Comparison with Known Results on the Euler Equation
  12. Possible Applications

Key Result

Theorem 2.3

For any $R>0$ and $\theta \in (0,\pi )$, there exists $C_{\ast}>0$ such that for any initial perturbation $\hbox{$\varphi$}_{0} \in H^{4}(I_{\theta R})$ satisfying Assumption as, problem (slant3) has a unique time-global solution $\hbox{$x$}(s,t)$ satisfying $|\hbox{$x$}(s,t)|=1$ for all $s\in I_{\t Furthermore, $\hbox{$\varphi$}(s,t)=\hbox{$x$}(s,t)- \hbox{$x$}^{R}(s,t)$ satisfies the following e

Figures (1)

  • Figure 1: A schematic of the problem setting for the initial-boundary value problem (\ref{['slant2']})

Theorems & Definitions (15)

  • Definition 2.1
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 5 more