KAM Theory for Active Scalar Equations

In this memoir, we establish the existence of time quasi-periodic solutions to generalized surface quasi-geostrophic equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis g upper S upper Q upper G right-parenthesis Subscript alpha"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>gSQG</mml:mi> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>α</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(\operatorname {gSQG})_\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the patch form close to Rankine vortices. We show that invariant tori survive when the order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the singular operator belongs to a Cantor set contained in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma one half right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,\frac 12)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.