Steady asymmetric vortex pairs for Euler equations

Hassainia, Zineb; Hmidi, Taoufik

doi:10.3934/dcds.2020348

Cited by 22 publications

(15 citation statements)

References 62 publications

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“…We shall construct small asymmetric co-rotating and counter-rotating pairs of vortex patches near the vortex pairs in (1.5) and (1.6). This extends the desingularizion result of [29], obtained in the Eulerian case α = 0, to gSQG equations (1.1) with α ∈ (0, 1). For the sake of clarity we shall give an elementary statement; for a complete statement see Propositions 3.1 and 3.2.…”

supporting

confidence: 82%

“…In this section we will give a precise statement of Theorem 1.5, which describes the structure of the set of vortex patch solutions in a neighborhood of the point vortex pairs. We shall only treat the case α ∈ (0, 1), since the limiting case α = 0 was addressed in [29].…”

Section: Asymmetric Pairsmentioning

confidence: 99%

“…In [34], however, Hmidi and Mateu gave a direct proof showing the existence of co-rotating and counter-rotating vortex pairs, using an elegant desingularization of the contour dynamics equations and an application of the implicit function theorem. The same technique was implemented for the desingularization of the asymmetric pairs [29], Karman street [21] and, very recently, the vortex polygon [22]. See [6] for related results where point vortices are instead desingularized into doubly-connected patches.…”

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confidence: 99%

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Multipole vortex patch equilibria for active scalar equations

Hassainia¹,

Wheeler²

2021

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We study how a general configuration of finitely-many point vortices, in a state of uniform rotation or translation with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. Using a technique first introduced by Hmidi and Mateu for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular, at which point classical techniques such as Lyapunov-Schmidt reduction can be used. Provided the point vortex equilibrium is non-degenerate in a natural sense, solutions can be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we prove the existence of various families of solutions with patches arranged in asymmetric pairs, regular polygons, body-centered polygons, and nested regular polygons. These configurations are degenerate due to their additional symmetries, but this difficulty can be overcome by integrating the appropriate symmetries into the formulation of the problem.

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supporting

confidence: 82%

Section: Asymmetric Pairsmentioning

confidence: 99%

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confidence: 99%

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Multipole vortex patch equilibria for active scalar equations

Hassainia¹,

Wheeler²

2021

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“…An analytical proof based on the bifurcation theory and complex analysis tools was devised by Burbea in [18] who proved the existence of m-fold (for any m ∈ N * ) symmetric V-states bifurcating from Rankine vortices with angular velocity Ω m := m−1 2m . More investigations on the V-states in different settings like the doubly-connected patches, vortex pairs, boundary effects or for different models has been implemented during the past decade by several authors, for more details we refer to [19,20,27,28,29,30,31,32,33,34,36,37,38,39,40]. Concerning (QGSW) λ there are a few results dealing with relative equilibria.…”

Section: Model Relative Equilibria From Periodic To Quasi-periodic So...mentioning

confidence: 99%

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

Hmidi¹,

Roulley²

2021

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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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“…This approach sounds to be flexible and robust and has been adapted recently by different authors to cover various interesting point vortex configurations associated to multiple models. For instance, it was used by [22] to extend the construction for the case α ∈ [1,2) and in [54] for the desingularization of the asymmetric pairs. The same technique was used to desingularize a spatial periodic distribution called Karman vortex street [44] and a similar study was performed for the Thomson polygon [45].…”

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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Steady asymmetric vortex pairs for Euler equations

Cited by 22 publications

References 62 publications

Multipole vortex patch equilibria for active scalar equations

Multipole vortex patch equilibria for active scalar equations

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

KAM theory for active scalar equations

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