# On the analyticity of the trajectories of the particles in the patch problem for $2$D Euler and aggregation equations

## TL;DR

This paper proves that particle trajectories in certain 2D fluid models are analytic in time when starting from patch initial data, using detailed analysis of the Beurling transform and associated equations.

## Contribution

It introduces a novel approach to establish time analyticity of particle trajectories for 2D Euler and aggregation equations with patch initial data.

## Key findings

- Particle trajectories are analytic in time for patch solutions.
- The Beurling transform plays a key role in the analysis.
- New equations for the Lagrangian flow are derived and studied.

## Abstract

We give a proof of the analiticity in time for the particle trajectories associated with the solutions of some transport equations when the initial datum is a patch. These results are obtained from a precise study of the Beurling transform, which provides estimates for the solutions of some new equations satisfied by the lagrangian flow.