Vortex Patches Choreography for Active Scalar Equations

This paper deals with the existence of $N$ vortex patches located at vertex a regular polygon sides that rotate around center constant angular velocity. That is done for Euler and (SQG)$_\beta$ equations, $\beta\in(0,1)$, but may be also extended to more general models. The idea desingularization Thomsom point system, is, vortices sides. proof based on study contour dynamics equation combined application infinite dimensional Implicit Function theorem well--chosen function spaces.