Quasi-periodic incompressible Euler flows in 3D

Baldi, Pietro; Montalto, Riccardo

doi:10.1016/j.aim.2021.107730

Cited by 35 publications

(36 citation statements)

References 40 publications

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“…Here we need to extend their construction to the framework of of symplectic change of coordinates with C q regularity. We point out that similar results with slight variations have been established in [7,13] in a non-symplectic framework.…”

Section: Reduction Of Ordersupporting

confidence: 85%

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

Hmidi¹,

Roulley²

2021

Preprint

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In this paper, we shall implement KAM theory in order to construct a large class of time quasi-periodic solutions for an active scalar model arising in fluid dynamics. More precisely, the construction of invariant tori is performed for quasi-geostrophic shallow-water equations when the Rossby deformation length belongs to a massive Cantor set. As a consequence, we construct pulsating vortex patches whose boundary is localized in a thin annulus for any time.

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Section: Reduction Of Ordersupporting

confidence: 85%

“…The proof of the following lemma is quite similar to Lemma 5.3. in [7] with the a minor difference in the weighted norms. See also [35,Lem.…”

Section: Triangularization Up To Error Termsmentioning

confidence: 99%

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

Hmidi¹,

Roulley²

2021

Preprint

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“…This is the case for instance in the water-waves equations where several results has been obtained in the past few years on the periodic and quasi-periodic settings [1,4,5,14,15,17,65]. For similar discussion we refer to [6] for the 3D incompressible Euler equations with quasi-periodic forcing and to [16] for the vortex patches solutions to the 2D Euler equations.…”

Section: Introductionmentioning

confidence: 99%

KAM theory for active scalar equations

Hassainia¹,

Hmidi²,

Masmoudi³

2021

Preprint

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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.

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“…See also [23,24] for a linear Schrödinger equation on zoll manifold with unbounded potential. We remark that by implementing the above techniques the KAM-type results of quasi -linear PDEs such as incompressible Euler flows in 3D [2] and forced Kirchhoff equation on T d [14] have been built recently.…”

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confidence: 99%