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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.

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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