[Paper Review] On the winding number for particle trajectories in a disk-like vortex patch of the Euler equations

Kyudong Choi, In-Jee Jeong|arXiv (Cornell University)|Aug 12, 2020

Navier-Stokes equation solutions15 references4 citations

TL;DR

This paper establishes that for most fluid particles in a disk-like vortex patch of the 2D incompressible Euler equations, the winding number around the origin grows linearly in time at a rate close to $\frac{1}{4\pi}$, provided the initial patch is a small $L^1$-perturbation of the unit disk. The result relies on controlling velocity and angular speed near the origin using log-Lipschitz estimates and time decomposition, showing that the winding rate converges to the circular patch value as the perturbation shrinks.

ABSTRACT

We consider vortex patch solutions of the incompressible Euler equations in the plane. It is shown that the winding number around the origin for most particles in the patch grows linearly in time when the initial patch is close to a disk enough.

Motivation & Objective

• To analyze the long-time winding behavior of fluid particles in disk-like vortex patches of the 2D Euler equations.
• To quantify how close the winding number rate is to that of a circular vortex patch when the initial patch is a small perturbation of the unit disk.
• To establish a refined dynamical stability notion beyond shape preservation, focusing on particle trajectories and angular motion.
• To provide quantitative estimates on the proportion of particles whose winding rate deviates by at most $C\delta^{1/12}$ from $\frac{1}{4\pi}$, where $\delta$ measures the initial perturbation.

Proposed method

• Define the winding number $N_x(T)$ as the time integral of the angular velocity component divided by radial distance along particle trajectories.
• Decompose the time interval $[0,T]$ into regions based on the particle's distance from the origin, using dyadic annuli $A_i^\epsilon$ and the exterior region.
• Use Yudovich's log-Lipschitz estimate and $L^\infty$ bounds on velocity to control the angular speed $u_{\text{tan}}/|x|$ in different spatial regimes.
• Estimate the contribution to the winding integral over regions where the particle is far from the origin and near the origin separately, using decay in time and spatial measure.
• Construct a large set $H_T \subset \Omega_0$ of particles that avoid the origin and stay in annuli where the velocity field is close to the circular case, ensuring uniform control.
• Apply a Borel-Cantelli-type argument over increasing time intervals to extend the result to infinite time, showing liminf control on the long-time average winding rate.

Experimental results

Research questions

• RQ1 How does the winding number of fluid particles in a disk-like vortex patch evolve over time when the initial patch is a small perturbation of the unit disk?
• RQ2 What proportion of particles in such a patch maintain a winding rate close to $\frac{1}{4\pi}$, the value for the circular patch?
• RQ3 Can the winding rate be controlled uniformly in time, and what is the quantitative dependence on the initial perturbation size $\delta = |D \triangle \Omega_0|$?
• RQ4 Does the long-time average winding rate converge to $\frac{1}{4\pi}$ for a positive measure set of initial particles, and if so, how fast?
• RQ5 How do the singularities in the Biot-Savart law (e.g., $u \sim \log(1/|x|)$ near the origin) affect the winding dynamics despite the velocity being bounded?

Key findings

• For any $T > 0$, there exists a set $H_T \subset \Omega_0$ with $|H_T| \geq |\Omega_0| - C\delta^{1/12}$ such that $\left| \frac{N_x(T)}{T} - \frac{1}{4\pi} \right| \leq C\delta^{1/12}$ for all $x \in H_T$, where $\delta = |D \triangle \Omega_0|$.
• There exists a set $H \subset \Omega_0$ with $|H| \geq |\Omega_0| - C\delta^{1/12}$ such that $\liminf_{t \to \infty} \left| \frac{N_x(t)}{t} - \frac{1}{4\pi} \right| \leq C\delta^{1/12}$ for all $x \in H$.
• The winding rate for most particles converges to $\frac{1}{4\pi}$ as $\delta \to 0$, with the error bounded by $C\delta^{1/12}$, indicating stability of the angular motion under small perturbations.
• The analysis shows that the set of particles hitting the origin has measure zero, justifying the definition of winding number for almost all particles.
• The method controls the integral of $u_{\text{tan}}/|x|$ by decomposing time based on radial distance and using uniform bounds on velocity and log-Lipschitz continuity.
• The result provides a dynamical refinement of classical shape stability, showing that not only does the patch remain close to a disk, but particle trajectories also exhibit nearly identical rotational behavior.