Lyapunov Unstable Motion Bifurcating from a Circular Vortex Filament

1. Problem Statement

The paper investigates the stability dynamics of closed vortex filaments in $\mathbb{R}^3$ governed by the Localized Induction Equation (LIE):
$$x_t = x_s \times x_{ss}$$
where $x(s,t)$ is the position vector of the filament parametrized by arc length $s$. The specific focus is on the circular vortex filament (a vortex ring) of radius $R$, denoted as $x^R$.

The central problem addresses a subtle but critical distinction in stability theory:

• Orbital Stability: The solution remains close to the manifold of symmetries (rotations and translations) of the circular filament.
• Lyapunov Stability: The solution remains close to the specific representative trajectory $x^R(t)$ without quotienting by symmetries.

Previous work by the authors established that the circular filament is orbitally stable under asymmetric perturbations but identified Lyapunov instability due to the linear growth of translation modes. This paper aims to rigorously construct explicit solutions that demonstrate this instability: solutions that stay uniformly close to the circular orbit (satisfying orbital stability estimates) but drift secularly away from the reference motion (failing Lyapunov stability).

2. Methodology

The authors employ a combination of geometric analysis, bifurcation theory, and functional analysis.

A. Ansatz: Axial Screw Motions

The authors define a specific class of solutions called Axial Screw Motions. A solution $x(s,t)$ is an axial screw motion if it maintains a constant shape while:

1. Rotating around the symmetry axis ($e_3$) with angular velocity $\Omega$.
2. Translating along the symmetry axis with speed $V$.
3. Undergoing a constant tangential slip with speed $c$.

Mathematically, the ansatz is:
$$x(s, t) = Q^{\Omega t}_z y(s + ct) + V t e_3$$
where $Q^{\Omega t}_z$ is the rotation matrix, and $y(s)$ is a closed curve in $H^4(\mathbb{T})^3$ with $|y_s|=1$.

B. Reduction to a Nonlinear Elliptic System

Substituting the ansatz into the LIE reduces the time-dependent PDE to a stationary nonlinear ordinary differential equation (ODE) for the profile $y(s)$:
$$y_s \times y_{ss} = c y_s + \Omega e_3 \times y + V e_3$$
By utilizing the Frenet-Serret frame and the arc-length constraint ($|y_s|=1$), the authors reformulate this into a system involving the normal ($v$) and binormal ($w$) components of the perturbation $z = y - x^R_0$.

C. Bifurcation Analysis

The core of the proof relies on the Crandall-Rabinowitz Local Bifurcation Theorem.

1. Linearization: The system is linearized around the trivial solution (the circular filament $x^R_0$).
2. Spectral Analysis: The authors analyze the linearized operator $L_\Omega$. They identify critical angular velocities $\Omega_k$ where the kernel of the operator becomes non-trivial (dimension 1).
   $$\Omega_k = \frac{\sqrt{k^2 - 1}}{R^2}, \quad k \in \mathbb{N}, k \geq 2$$
3. Transversality: They verify the transversality condition required for bifurcation, ensuring that the derivative of the operator with respect to the parameter $\Omega$ maps the kernel element outside the range of the operator.
4. Implicit Function Theorem: Using the theorem, they prove the existence of a smooth branch of non-trivial solutions $(\Omega_\lambda, y_\lambda, c_\lambda, V_\lambda)$ bifurcating from the circular filament at $\lambda = 0$.

3. Key Contributions

1. Construction of Explicit Unstable Solutions: The paper constructs a family of closed vortex filaments (axial screw motions) that emerge from the circular filament. These are the first explicit examples of solutions that are orbitally stable but Lyapunov unstable in this context.
2. Quantification of Secular Drift: The authors prove that while these new solutions remain within a distance $C|\lambda|$ of the orbital manifold $\Sigma$ for all time, they drift linearly away from the reference circular motion $x^R(t)$ along the symmetry axis.
3. Clarification of Stability Concepts: The work provides a concrete mathematical realization of the "gap" between orbital stability and Lyapunov stability for vortex filaments, distinguishing between staying close to the shape/orbit versus staying close to the specific trajectory.
4. Differentiation from Prior Work: The authors clarify why their results differ from previous constructions by Kida [29] and Garcia and Vega [19]. Unlike those works, which focused on tangent vectors or non-closed filaments, this paper strictly enforces the closed filament constraint with fixed arc length, which is essential for the stability analysis of vortex rings.

4. Main Results

Theorem 1.5 (Main Theorem):
For any integer $k \geq 2$, there exists a smooth family of axial screw motions $\{x^{(\lambda)}\}$ bifurcating from the circular filament $x^R$ at the critical angular velocity $\Omega_k = \frac{\sqrt{k^2-1}}{R^2}$.

The family satisfies two crucial properties:

1. Orbital Stability Estimate: There exists a constant $C > 0$ such that for all $\lambda$ near 0 and all $t \geq 0$:
   $$\text{dist}(x^{(\lambda)}(t), \Sigma) \leq C|\lambda|$$
   This confirms the solutions stay close to the manifold of circular filaments.
2. Lyapunov Instability (Secular Drift): For any $\lambda \neq 0$, there exist $\gamma, t_0 > 0$ such that for $t \geq t_0$:
   $$|x^{(\lambda)}(s, t) - x^R(s, t)| \geq \gamma t$$
   The drift is caused by a difference in the axial translation speed $V_\lambda$. Specifically, the speed of the bifurcating solution is slower than the reference circle:
   $$V_\lambda = \frac{1}{R} - \left( \frac{k^2(k^2-1)}{2R^3} \right) \lambda^2 + o(\lambda^2)$$

5. Significance

• Theoretical Rigor: The paper bridges the gap between formal asymptotic analysis and rigorous mathematical proof regarding the stability of vortex rings. It confirms that linear instability modes (translation) can lead to non-trivial, globally existing nonlinear solutions that exhibit long-term drift.
• Physical Implications: In fluid dynamics, this suggests that while a vortex ring may appear stable in shape (orbital stability), small perturbations can cause the ring to slowly separate from its original trajectory over time. This is crucial for understanding the long-term evolution of vortex rings in inviscid fluids (e.g., in atmospheric physics or engineering applications like jet engines).
• Mathematical Framework: The methodology of reducing the LIE to a bifurcation problem for closed curves with arc-length constraints provides a robust template for analyzing stability in other geometric evolution equations.

In summary, Aiki and Higaki demonstrate that the circular vortex filament is Lyapunov unstable despite being orbitally stable, by explicitly constructing a family of "screw-like" vortex filaments that slowly drift away from the reference circle while maintaining their circular-like shape.