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Vortex rings in $$\mathbb{R}^3$$ and rearrangements. (English) Zbl 0991.76012
Summary: We study the existence of steady axisymmetric vortex rings in ideal fluid. A functional, comprising a linear combination of kinetic energy and impulse, is to be maximized subject to the constraint that a quantity related to vorticity belongs to a set of rearrangements of a given function. Generalized solutions of a quite specific type are shown to exist, arising as extreme points of a convex extended constraint set. In the case when the given function is the indicator of a set of finite measure, we demonstrate the existence of proper maximizers and local maximizers.

##### MSC:
 76B47 Vortex flows for incompressible inviscid fluids 76M30 Variational methods applied to problems in fluid mechanics 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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