Periodic solutions with prescribed minimal period of vortex type problems in domains

Bartsch, Thomas; Sacchet, Matteo

doi:10.1088/1361-6544/aaaf2d

Cited by 4 publications

(4 citation statements)

References 36 publications

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“…A generalized analysis of Leapfrogging vortex rings have been also discussed in [15,22,23,43,52] and the references therein. In addition to the leapfrogging motion, many periodic non-rigid configuration of vortex point system set in a general domain were also discussed in [4,5,36,55]. Very recently, time quasi-periodic solutions have been explored for Euler equations around the ellipses [12] or around the Rankine vortices in the disc [40].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…The second equation in (1) is nothing but the Biot-Savart law which can be written in the complex form:…”

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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 the first one vortices with possibly different strengths and of arbitrary number are close to a critical point of the so called Robin function h Ω (z) = g Ω (z, z) and the configuration of vortices looks after rescaling like a rigidly rotating solution of the N -vortex system on R 2 , see [4,6]. In the second type of solutions, shown in [10], two identical vortices rotate around their center of vorticity while the center itself follows a level line of h Ω . The third result holds for an arbitrary number of identical vortices, which separated by time shifts follow the same curve close to the boundary of a simply connected bounded domain, [5].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Periodic solutions for the N-vortex problem via a superposition principle

Gebhard¹

2018

Discrete &Amp; Continuous Dynamical Systems - A

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We examine the N -vortex problem on general domains Ω ⊂ R 2 concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the formwith some regular and symmetric, but in general not explicitely known function g : Ω × Ω → R. The investigation relies on the idea to superpose a stationary solution of a system of less than N vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and the configurations, under which multiple T -periodic solutions are shown to exist for every T > 0 small enough. The crucial condition holds in generic bounded domains and is explicitely verified for an example in the unit disc Ω = B 1 (0). In particular we therefore obtain various examples of periodic solutions in B 1 (0) that are not rigidly rotating configurations.MSC 2010: Primary: 37J45; Secondary: 37N10, 76B47

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“…In general domains Ω ⊂ R 2 , the Hamiltonian H Ω does not share the advantages of the cases stated above. Still, using perturbative approaches the existence of some types of periodic solutions could also be established in general domains [3,4,5,8,15]. In recent years, also different kinds of stationary solutions, i.e.…”

Section: Introductionmentioning

confidence: 99%

Stability of Periodic Solutions of the N-vortex Problem in General Domains

Gebhard

Ortega

2019

Regul. Chaot. Dyn.

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We investigate stability properties of a type of periodic solutions of the N -vortex problem on general domains Ω ⊂ R 2 . The solutions in question bifurcate from rigidly rotating configurations of the whole-plane vortex system and a critical point a 0 ∈ Ω of the Robin function associated to the Dirichlet Laplacian of Ω. Under a linear stability condition on the initial rotating configuration, which can be verified for examples consisting of up to 4 vortices, we show that the linear stability of the induced solutions is solely determined by the type of the critical point a 0 . If a 0 is a saddle, they are unstable. Otherwise they are stable in a certain linear sense. The proof uses a criterion for the bifurcation of multiple eigenvalues, which is applied to suitable Poincaré sections. Beyond linear stability, Herman's last geometric theorem allows us to prove the existence of isoenergetically orbitally stable solutions in the case of N = 2 vortices.

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Periodic solutions with prescribed minimal period of vortex type problems in domains

Cited by 4 publications

References 36 publications

Steady asymmetric vortex pairs for Euler equations

Steady asymmetric vortex pairs for Euler equations

Periodic solutions for the N-vortex problem via a superposition principle

Stability of Periodic Solutions of the N-vortex Problem in General Domains

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