Existence and Regularity for Vortex Patch Solutions of the 2D Euler Equations

In “Global regularity for vortex patches” (Commun. Math. Phys. 1993), Bertozzi and Constantin formulate the vortex patch problem in the level-set framework and prove a priori estimates for this active scalar equation. By extending the tools used to prove these estimates, we construct solutions and show propagation of higher Hölder regularity. This constitutes a proof of the regularity of vortex patches, carried out solely in the level-set framework.