An analytical and numerical study of steady patches in the disc

Hoz, Francisco de la; Hassainia, Zineb; Hmidi, Taoufik; Mateu, Jоan

doi:10.2140/apde.2016.9.1609

Cited by 41 publications

(26 citation statements)

References 36 publications

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“…(ii) We obtain global existence of non trivial solutions in the patch form for (1.1) where only a few results on global existence are known in the periodic case, see [2,23,24,25,36,53] and the references therein. (iii) Some technical problems are behind the limitation of α > 0 and we cannot go to α = 0.…”

Section: Introductionmentioning

confidence: 98%

“…In addition, formation of small loops are discovered when the two branches are close enough. Boundary effects on the emergence of the V-states were analyzed through the disc example in [36], with important numerical experiments putting in evidence the oscillation of the Burbea's curves. Note also that for Euler equations a second bifurcation of countable branches from the ellipses occurs but the shapes have in fact less symmetry and being at most two-folds.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

KAM theory for active scalar equations

Hassainia¹,

Hmidi²,

Masmoudi³

2021

Preprint

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“…It is worth pointing out that Burbua's approach has been intensively exploited in the few last years in different directions. For instance, this was implemented to prove the existence of rotating patches close to Kirchhoff's ellipses [4,21], multiconnected patches [20,22,25], patches in bounded domains [8], non trivial rotating smooth solutions [5] and rotating vortices with non uniform densities [13]. We mention that many of these results apply not only to the 2D Euler equations but also to more singular nonlinear transport equations, but with much more involved computations.…”

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confidence: 99%

Steady asymmetric vortex pairs for Euler equations

Hassainia

Hmidi

2021

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we study the existence of co-rotating and counterrotating unequal-sized pairs of simply connected patches for Euler equations. In particular, we prove the existence of curves of steadily co-rotating and counterrotating asymmetric vortex pairs passing through a point vortex pairs with unequal circulations. We also provide a careful study of the asymptotic behavior for the angular velocity and the translating speed close to the point vortex pairs.

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“…Here we only recall two existence results. In [8], based on bifurcation theory, de la Hoz-Hassainia-Hmidi-Mateu proved that for any b ∈ (0, 1) and m a positive integer, there exists a family of m-fold symmetric rotating patches bifurcating from the steady patch D b , b ∈ (0, 1), with angular velocity Ω m = (m−1+b 2m )/(2m). These rotating patches are simply-connected, moreover, the angular velocity lies in the interval (0, 1/2) just as the whole plane case.…”

Section: Introductionmentioning

confidence: 99%

On radial symmetry of rotating vortex patches in the disc

Wang¹,

Zuo²

2019

Preprint

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In this note, we consider the radial symmetry property of rotating vortex patches for the 2D incompressible Euler equations in the unit disc. By choosing a suitable vector field to deform the patch, we show that each simply-connected rotating vortex patch D with angular velocity Ω, Ω ≥ max{1/2, (2l 2 )/(1 − l 2 ) 2 } or Ω ≤ −(2l 2 )/(1 − l 2 ) 2 , where l = sup x∈D |x|, must be a disc. The main idea of the proof, which has a variational flavor, comes from a very recent paper of Gómez-Serrano-Park-Shi-Yao, arXiv:1908.01722, where radial symmetry of rotating vortex patches in the whole plane was studied.

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An analytical and numerical study of steady patches in the disc

Cited by 41 publications

References 36 publications

KAM theory for active scalar equations

KAM theory for active scalar equations

Steady asymmetric vortex pairs for Euler equations

On radial symmetry of rotating vortex patches in the disc

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