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Global vortex rings and asymptotic behaviour. (English) Zbl 0845.76017
The existence of solution is investigated in the case of steady axisymmetric vortex rings in an ideal fluid. The dependence between stream-function and vorticity is supposed to be known. The vortex-strength parameter and the propagation velocity of the vortex ring are prescribed as well. A direct variational method is used; it is proved that the corresponding functional is differentiable, and the minimizing sequences of a constrained variational problem are precompact. The main result on the asymptotic behaviour is that the cross-section of a steady vortex ring shrinks to a point, and the vortex ring degenerates into a singular circle as the vortex-strength parameter tends to infinity.

MSC:
76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
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