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Nonlinear stability of circular vortex patches. (English) Zbl 0584.76062
This remarkable paper proves that the stationary solution of the two- dimensional homogeneous incompressible Euler equations consisting of a circular vortex patch is Lyapunov stable, global in time, in the \(L^ 1\)-norm on the vorticities. It is shown that a nearly circular vortex patch must evolve in such a way that the area of the region of deviation from circularity is uniformly bounded, globally in time. This paper naturally fits into the programs of establishing a priori stability estimates pioneered by V. I. Arnold [Sov. Math., Dokl. 6, 773–777 (1965); translation from Dokl. Akad. Nauk SSSR 162, 975–978 (1965; Zbl 0141.43901)] and extensively developed recently by D. Holm, J. Marsden, the reviewer and A. Weinstein [Phys. Lett., A 98, 15–21 (1983)]. This method relies heavily on the Poisson geometry of the underlying phase space, and in particular on the kinematic conserved quantities, called Casimirs. This method does not work directly in the present case due to difficulties arising in the analytical description of the coadjoint orbit formed by vortex patches. The paper retains the spirit of this method but combines it with very careful estimates on the second variation of the energy to get the stated result in a large disk enclosing the given vortex patch.
Reviewer: T.Ratiu

MSC:
76E30 Nonlinear effects in hydrodynamic stability
76B47 Vortex flows for incompressible inviscid fluids
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