# On the estimate of distance traveled by a particle in a disk-like vortex patch

## TL;DR

This paper analyzes how the distance traveled by particles in a 2D vortex patch grows, showing linear growth for most particles when the patch is close to a disk in shape.

## Contribution

It provides a new estimate on particle travel distance in vortex patches that are nearly circular, extending understanding of fluid particle trajectories in Euler flows.

## Key findings

- Travel distance grows linearly for most particles in nearly circular vortex patches.
- The result applies when the initial vortex patch is close to a disk in shape.
- The analysis is based on the characteristic function of a bounded open set as initial vorticity.

## Abstract

We consider the incompressible two-dimensional Euler equation in the plane in the case when its initial vorticity is the characteristic function of a bounded open set. We show that the travel distance grows linearly for most of fluid particles initially placed on the set when the area of the symmetric difference between the set and a disk is small enough.