A global branch of steady vortex rings. (English) Zbl 0628.76032

Amick, C. J.; Turner, R. E. L.

J. Reine Angew. Math. 384, 1-23 (1988).

A steady vortex ring of prescribed strength and propagation speed can be described in terms of a Stokes stream function \(\psi\). A flux constant k measures the flow through the center of the axisymmetric vortex ring. For \(k=0\), Hill in 1894 found an explicit solution for the semi-linear elliptic equation satisfied by \(\psi\). In this paper it is shown that there is an unbounded, closed, connected branch of solutions emanating from Hill’s vortex in the space of pairs (k,\(\psi)\).

MSC:

76B47	Vortex flows for incompressible inviscid fluids
35B32	Bifurcations in context of PDEs
35R05	PDEs with low regular coefficients and/or low regular data
35R35	Free boundary problems for PDEs

Keywords:

degree theory; global bifurcation; steady vortex ring; Stokes stream function; semi-linear elliptic equation; unbounded, closed, connected branch of solutions; Hill’s vortex