Long Time Evolution of Concentrated Vortex Rings with Large Radius: Long Time Evolution of Concentrated Vortex Rings

We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on N annuli of radii of the order of r 0 and thickness ε . We prove that when r 0 = | log ε | α , α > 1 , the vorticity field of the fluid converges for ε → 0 to the point vortex model, in an interval of time which diverges as log | log ε | . This generalizes previous result by Cavallaro and Marchioro in (J Math Phys 62:053102, 2021), that assumed α > 2 and in which the convergence was proved for short times only.