KAM theory for active scalar equations

Abstract

In this paper, we establish the existence of time quasi-periodic solutions to generalized surface quasi-geostrophic equation $(\mathrm{gSQG}{)}_{\alpha }$ in the patch form close to Rankine vortices. We show that invariant tori survive when the order $\alpha$ of the singular operator belongs to a Cantor set contained in $(0,\frac{1}{2})$ with almost full Lebesgue measure. The proof is based on several techniques from KAM theory, pseudo-differential calculus together with Nash-Moser scheme in the spirit of the recent works \cite{Baldi-Berti2018,Berti-Bolle15}. One key novelty here is a refined Egorov type theorem established through a new approach based on the kernel dynamics together with some hidden Töpliz structures.