Mathematical Analysis Seminars @ DISIM

Title: Translating dipole patches and non-rigid time-periodic vortices

Abstract: Time-periodic structures are among the most striking coherent phenomena arising in two-dimensional inviscid fluid dynamics. While rigidly rotating or translating vortex configurations have been extensively studied, much less is known about genuinely time-periodic motions that occur near traveling equilibria, especially when the vortices possess internal degrees of freedom.The goal of this talk is to show the existence of time-periodic solutions of the 2D incompressible Euler equations bifurcating from a translating vortex pair. The reference configuration consists of two symmetric vortex patches of size of order 1, and of equal strength with opposite sign, traveling at constant speed. In the regime of large separation between the two vortices, we construct families of non-rigid, time-periodic vortex patch solutions for a large Cantor set of parameters. An important difficulty is due to a degenerate mode associated to symmetries and transport effects. We combine a Lyapunov–Schmidt reduction, a Nash–Moser iteration, and spectral analysis with sharp asymptotic expansions of the eigenvalues to overcome this degeneracy.

This is a joint work with Taoufik Hmidi and Riccardo Montalto.