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Symmetric Periodic Solutions in the N-Vortex Problem

Abstract : This thesis focuses on the study of the periodic solutions of the N-vortex problem of positive vorticity. This problem was formulated by Helmholtz more than 160 years ago and remains an active research field. For an undetermined number of vortices and general vorticities the system is not Liouville integrable and periodic solutions cannot be determined explicitly, except for relative equilibria. By using variational methods, we prove the existence of infinitely many non-trivial periodic solutions for arbitrary N and arbitrary positive vorticities. Moreover, when the vorticities are positive rational numbers, we show that there exists only finitely many energy levels on which there might exist a relative equilibrium. Finally, for the identical N-vortex problem, we show that there exists infinitely many simple choreographies.
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Qun Wang. Symmetric Periodic Solutions in the N-Vortex Problem. Fluids mechanics [physics.class-ph]. PSL Research University, 2018. English. ⟨NNT : 2018PSLED069⟩. ⟨tel-02462245⟩

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