Time-periodic vortices near translating symmetric dipole patches

We prove the existence of time-periodic solutions of the two-dimensional incompressible Euler equations bifurcating from a translating vortex pair.

The reference configuration consists of two symmetric vortex patches of equal strength and opposite sign traveling at constant speed.

In a regime of large separation between the vortices, the dynamics may be viewed as a small perturbation of an integrable system.

Working in a co-moving frame and using the contour dynamics formulation, we reduce the problem to a nonlinear transport equation for the vortex boundaries.

The linearized operator exhibits degeneracies associated with symmetries and transport effects.

By combining a Lyapunov-Schmidt reduction, Nash-Moser scheme, and spectral analysis with sharp asymptotic expansions of the eigenvalues in order to overcome the degeneracy, we construct families of non-rigid, time-periodic vortex patch solutions for a large Cantor set of parameters.

The analysis reveals that translating dipoles possess a surprisingly rich nearby dynamics, far beyond the classical rigid paradigm usually associated with vortex patch motion.

More generally, the approach developed in this work is flexible and robust, and is expected to extend to a broader class of nonlocal PDEs from Fluid Mechanics with degenerate behaviours.