KAM Theory for Active Scalar Equations

In this memoir, we establish the existence of time quasi-periodic solutions to generalized surface quasi-geostrophic equation ( gSQG ) α (\operatorname {gSQG})_\alpha in the patch form close to Rankine vortices. We show that invariant tori survive when the order α \alpha of the singular operator belongs to a Cantor set contained in ( 0 , 1 2 ) (0,\frac 12) 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 of Baldi and Berti (2018) and Berti and Bolle (2015). 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.