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di ANALISI MATEMATICA
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Il giorno Venerdì 4 Ottobre 2024, alle ore 12:00 nella Sala Professori I Livello del DMA, il Prof. Massimiliano Berti, della Sissa di Trieste, terrà il seguente
seminario:

Quasi-periodic vortex patches of 2d Euler

Abstract: We prove the existence of time quasi-periodic vortex patch solutions of the 2d-Euler equations in R^2, close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux-Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in an infinite dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself. This is the first time such an idea is implemented in KAM for PDEs.  Joint work with Z. Hassaina and N. Masmoudi.