Rigorous derivation of the leapfrogging motion for planar Euler equations

The main goal of this paper is to explore the leapfrogging phenomenon in the inviscid planar flows. We show for 2d Euler equations that under suitable constraints, four concentrated vortex patches leapfrog for all time. When observed from a translating frame of reference, the evolution of these vortex patches can be described as a non-rigid time periodic motion. Our proof hinges upon two key components. First, we desingularize the symmetric four point vortex configuration, which leapfrogs in accordance with Love's result \cite{Love1893}, by concentrated vortex patches. Second, we borrow some tools from KAM theory to effectively tackle the small divisor problem and deal with the degeneracy in the time direction. Our approach is robust and flexible and solves a long-standing problem that has remained unresolved for many decades.