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arXiv · 2405.06518

Long time evolution of concentrated vortex rings with large radius

Paolo Buttà·Guido Cavallaro·Carlo Marchioro
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Abstract

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 $\varepsilon$. We prove that when $r_0= |\log \varepsilon|^\alpha$, $\alpha>1$, the vorticity field of the fluid converges for $\varepsilon \to 0$ to the point vortex model, in an interval of time which diverges as $\log|\log\varepsilon|$. This generalizes previous result by Cavallaro and Marchioro in [J. Math. Phys. 62, 053102, (2021)], that assumed $\alpha>2$ and in which the convergence was proved for short times only.

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Paolo Buttà, Guido Cavallaro, Carlo Marchioro. 2024-05-10. Long time evolution of concentrated vortex rings with large radius. https://doi.org/10.1007/s10955-024-03381-x

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