KAM theory for active scalar equations

Abstract

In this paper, we establish the existence of time quasi-periodic solutions to generalized surface quasi-geostrophic equation (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, 1 2 ) 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 [5,10]. 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. 8.3.1. Tame estimates of the iterated commutators 8.3.2. Iterated kernel estimates 8.3.3. Iterated Töplitz matrix operators 9. Reducibility of the linearized operator 9.1. Structure on the normal direction 9.2. Asymptotic structure 9.3. Reduction of the transport part 9.3.1. Transport equation with constant coefficients 9.3.2. Straightening of the transport equation 9.4. First conjugation of the linearized operator 9.5. Reduction of the nonlocal part 9.6. Complete reduction up to small errors 9.6.1. Frequency localization of operators 9.6.2. Localization on the normal direction 9.6.3. KAM reduction of the remainder term 9.6.4. Approximate inverse in the normal direction 10. Proof of the main result 10.1. Nash-Moser scheme 10.2. Final Cantor set estimates Appendix A.

Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations

“…Finally, we point out that the use of suitable isotropic tori is a commodity but it is not essential to get the triangular structure up to small errors. This point will be discussed in Section 5, see also [35].…”

“…We point out that this approach has been successfully implemented to generate quasi-periodic solutions to several quasi-linear and fully nonlinear autonomous PDE's, see for instance [4,5,6,14]. Recent progress towards applying KAM theory for the vortex dynamics has been performed for gSQG equation [35] and Euler equation [15]. Finally, we point out that the use of suitable isotropic tori is a commodity but it is not essential to get the triangular structure up to small errors.…”

“…We mainly follow the same approach as in [11] which reduces the search of an approximate right inverse of (5.60) to the search of an approximate right inverse in the normal directions. The main difference with [11] is to be able to bypass the use of the isotropic torus in a similar way to the recent paper [35].…”

“…Therefore to invert this triangular system it suffices to get an approximate right inverse for the linearized operator in the normal direction, denoted in what follows by L ω . We notice that in Section 5.3, and similarly to [35], we can bypass the use of isotropic torus by a slight modification of Berti-Bolle approach. Actually, according to Proposition 5.1, we can conjugate the linearized operator with the transformation described by (5.63) computed at the torus i 0 and get a triangular system with small errors mainly of "type Z".…”

“…At the equilibrium state (corresponding to ε = 0) L ω is diagonal and we shall see that the set of parameters (λ, ω) leading to the existence of a right inverse is almost full. Now remark that even for ε small, the perturbation affects the main part of the operator in a similar way to water waves [4,14] or generalized SQG equation [35] and then we should construct the suitable change of coordinates in order to reduce the positive part of the operator to a diagonal operator. Later we should implement KAM scheme to diagonalize the zero-order part.…”