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

Abstract

<p style='text-indent:20px;'>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.</p>