Periodic solutions of the N point-vortex problem in planar domains

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DaiQianhui_2014_10_31.pdf (17.97 MB)

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2014

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DOI:
http://dx.doi.org/10.22029/jlupub-9676

Abstract

The N point-vortex Hamiltonian system from fluid dynamics is a prominent and representative example of a first order singular Hamiltonian system which appears in Mathematical Physics. The logarithmic singularities appearing in the Hamiltonian are of a very different type compared with those appearing in classical (e.g. celestial) mechanics. Owing to the complexity of the Hamiltonian function and the compactness problems, investigating periodic solutions of the system in planar domains is highly nontrivial. This dissertation addresses two issues related to this topic.The first issue concerns the point vortex system in the disk. The system for two vortices in the disk is completely integrable in the sense of Liouville. Appropriate canonical coordinate transformations are presented to reduce the degrees of freedom. By examining the reduced system, it is revealed that any non-asymptotic motion of two identical vortices in the disk is a superposition of a uniform rotation of the coordinate frame and a periodic motion in this rotating frame. Examples of trajectories both in the rotating frame and in the original system are plotted numerically. The existence of symmetric periodic solutions, so-called ´vortex crystals´, in the disk is also discussed. It is proved that the configurations of an open or centered N-polygon, of symmetric two N-polygons and of alternate two N-polygons with generic choice of radii or strengths exist for the system in the disk.The second part of this dissertation focuses on the N-vortex problem in general planar domains. It is proved that when all the strengths of the N vortices are the same, a stable critical point of the Robin function associated with this domain yields periodic solutions with arbitrarily small minimal period oscillating around this point. These periodic solutions are near a singularity of the phase space. Moreover, they are simple choreographies, i.e. the N vortices travel on the same circle with a phase-shift. The proof is based on a careful investigation of the behavior of the action functional near this singularity. This result applies in particular to generic bounded domains, which may be simply or multiply connected. Then it is generalized to the system of two point vortices with arbitrary non-opposite strengths. A similar result is also obtained when the logarithm function log |z1 - z2|is replaced by |z1 - z2|- alpha with alpha greater than 0.

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