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Applications of Arnold's variational principle to the stability of vortices in ideal and viscous flows

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TURW02 - Rigorous analysis of incompressible fluid models and turbulence

We revisit Arnold’s variational approach to the stability of steady-state solutions of the two-dimensional Euler equations. In thecase of planar vortices, we study in detail the quadratic form that represents the second variation of the energy on the isovorticalsurface. We show in particular that, for a large class of radially symmetric vortices with strictly decreasing profile, the secondvariation is negative definite for all perturbations that preserve the total circulation. We use that property to give a new stabilityproof for the Oseen vortex as a self-similar solution of the 2D Navier-Stokes equations, and to investigate the vanishing viscositylimit of axisymmetric vortex rings. This talk is based on joint work with V. Sverak.

This talk is part of the Isaac Newton Institute Seminar Series series.

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