Vortex rings in \(\mathbb{R}^3\) and rearrangements. (English) Zbl 0991.76012

Badiani, T. V.; Burton, G. R.

Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 457, No. 2009, 1115-1135 (2001).

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

Keywords:

variational problem; existence of steady axisymmetric vortex rings; ideal fluid; rearrangements; convex extended constraint set; maximizers; local maximizers