On the analyticity of the trajectories of the particles in the planar patch problem for some active scalar equations

We prove analyticity in time of the particle trajectories associated with the solutions of some transport equations when the initial condition is the characteristic function of a regular bounded domain. These results are obtained from a detailed study of the Beurling transform, that represents a derivative of the velocity field. The precise estimates obtained for the solutions of an equation satisfied by the Lagrangian flow, are a key point in the development.