an:01658890
Zbl 0991.76012
Badiani, T. V.; Burton, G. R.
Vortex rings in \(\mathbb{R}^3\) and rearrangements
EN
Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 457, No. 2009, 1115-1135 (2001).
00075056
2001
j
76B47 76M30 76B03
variational problem; existence of steady axisymmetric vortex rings; ideal fluid; rearrangements; convex extended constraint set; maximizers; local maximizers
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.