KAM theory for active scalar equations

TL;DR

This paper proves the existence of quasi-periodic solutions for a class of active scalar equations, extending KAM theory techniques to establish invariant tori near vortex solutions in a mathematical fluid dynamics context.

Contribution

It introduces a novel Egorov type theorem and applies advanced KAM and Nash-Moser methods to generalized surface quasi-geostrophic equations, expanding the understanding of their solution structure.

Findings

Invariant tori exist for the gSQG equations near Rankine vortices.

Survival of invariant tori for a Cantor set of parameter values with full measure.

Development of a refined Egorov theorem using kernel dynamics and T"opliz structures.

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.