2021/10/16 by Zineb Hassainia, Taoufik Hmidi, Hassainia, Zineb +3 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.2110.08615
arxiv created 2021/11/16 · arxiv updated 2021/11/17
In this paper, we establish the existence of time quasi-periodic solutions to generalized surface quasi-geostrophic equation (\rm gSQG)α in the patch form close to Rankine vortices. We show that invariant tori survive when the order α of the singular operator belongs to a Cantor set contained in (0,\frac12) 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 \citeBaldi-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.