KAM theory for active scalar equations

Zineb Hassainia; Taoufik Hmidi; Nader Masmoudi

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arXiv · 2110.08615

KAM theory for active scalar equations

Zineb Hassainia·Taoufik Hmidi·Nader Masmoudi

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Abstract

In this paper, we establish the existence of time quasi-periodic solutions to generalized surface quasi-geostrophic equation $({\rm gSQG})_\alpha$ 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 $(0,\frac12)$ 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 \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 T\"opliz structures.

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Zineb Hassainia, Taoufik Hmidi, Nader Masmoudi. 2021-10-16. KAM theory for active scalar equations. https://arxiv.org/abs/2110.08615

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