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A confinement result for axisymmetric fluids. (English) Zbl 1165.76318

Summary: We consider an incompressible, inviscid, axisymmetric fluid moving in \(\mathbb{R}^3\) without swirl (a vortex ring). We study its time evolution via the Euler equation, assuming that initially the vorticity is concentrated at a finite distance, say \(r_0\), from the symmetry axis. We prove a bound on the growth of the support of the vorticity. Namely we show that the fluid is confined in a cylinder with radius \(r\leq r_0+\text{constant}\,t^{1/4}\log(e+t)\).

MSC:

76B75 Flow control and optimization for incompressible inviscid fluids
76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
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References:

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