Symmetric equilibria for the N-vortex problem

Kuhl, Christian

doi:10.1007/s11784-015-0242-3

J. Fixed Point Theory Appl.

2015

DOI: 10.1007/s11784-015-0242-3

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Symmetric equilibria for the N-vortex problem

Christian Kuhl

Abstract: This paper is concerned with the study of existence and properties of symmetric stationary solutions for the dynamics of N point vortices in an ideal fluid constrained to a dihedrally symmetric twodimensional domain Ω, which is governed by a Hamiltonian systemwhere zi = (xi, yi), i = 1, . . . , N, and whereis the so-called Kirchhoff-Routh path function under some conditions on the vorticities Γi.Mathematics Subject Classification. 37N10, 37C25, 76M30.

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Cited by 6 publications

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Periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Bartsch

2016

Arch. Math.

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We are concerned with the dynamics of N point vortices z1, . . . , zN ∈ Ω ⊂ R 2 in a planar domain. This is described by a Hamiltonian systemwhere Γ1, . . . , ΓN ∈ R \ {0} are the vorticities, J ∈ R 2×2 is the standard symplectic 2 × 2 matrix, and the Hamiltonian H is of N -vortex type:Γj Γ k g(zj, z k ).Here g : Ω × Ω → R is an arbitrary symmetric function of class C 2 , e.g. the regular part of a hydrodynamic Green function. Given a nondegenerate critical point a0 ∈ Ω of h(z) = g(z, z) and a nondegenerate relative equilibrium Z(t) ∈ R 2N of the Hamiltonian system in the plane with g = 0, we prove the existence of a smooth path of periodic solutions z (r) (t) = z (r)k (t) → a0 as r → 0. In the limit r → 0, and after a suitable rescaling, the solutions look like Z(t).Mathematics Subject Classification (2010). Primary 37J45; Secondary 37N10, 76B47.

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Periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Bartsch

2016

Arch. Math.

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Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Bartsch¹,

Gebhard²

2016

Math. Ann.

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The paper deals with singular first order Hamiltonian systems of the formwhere J ∈ R 2×2 defines the standard symplectic structure in R 2 , and the Hamiltonian H is of N -vortex type:This is defined on the configuration space {(z 1 , . . . , z N ) ∈ Ω 2N : z j = z k for j = k} of N different points in the domain Ω ⊂ R 2 . The function F : Ω N → R may have additional singularities near the boundary of Ω N . We prove the existence of a global continuum of periodic solutions z(t) = (z 1 (t), . . . , z N (t)) ∈ Ω N that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium Z(t) ∈ R 2N of the N -vortex problem in the whole plane (where F = 0). Examples for Z include Thomson's vortex configurations, or equilateral triangle solutions. The domain Ω need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of 2π-periodic H 1/2 functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for S 1 -equivariant gradient maps that we adapt to this class of potential operators.MSC 2010: Primary: 37J45; Secondary: 37N10, 76B47

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Equilibria for the N-vortex-problem in a general bounded domain

Kuhl¹

2016

Journal of Mathematical Analysis and Applications

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Symmetric equilibria for the N-vortex problem

Cited by 6 publications

References 9 publications

Periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

Equilibria for the N-vortex-problem in a general bounded domain

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