Filamentation near Hill's vortex

Abstract

For the axisymmetric incompressible Euler equations, we prove linear in time filamentation near Hill's vortex: there exists an arbitrary small outward perturbation growing linearly for all times. This is based on combining recent nonlinear orbital stability statement with a dynamical bootstrapping scheme for particle trajectories. These results rigorously confirm numerical simulations by Pozrikidis [45] in 1986.

On vortex stretching for anti-parallel axisymmetric flows

“…Even in the case of T 3 , we are not aware of any results giving infinite vorticity growth for smooth vorticity, not relying on the 2+ 1 2 dimensional geometry (however see [42]). In our previous work, we obtained arbitrarily large but finite growth of ω(t, •) L ∞ in R 3 using perturbations of the Hill's vortex [9]. Very recent numerical computations by Hou suggests finite time singularity for axisymmetric Euler with swirl in the interior of the domain [25].…”

“…(for solvability, e.g. see Section 2.4 of [9]). Then, along the flow, we have the important Cauchy formula…”

Stability and instability of Kelvin waves

“…We remark that the improved stability statements, Theorem 1.2 and Corollary 1.3, are essential in the proof of above instability result. For other works on perimeter growth of perturbations, we refer to [16] for a disk patch, [17] for the Lamb dipole, [15] for the Hill's spherical vortex.…”

Infinite growth in vorticity gradient of compactly supported planar vorticity near Lamb dipole

“…see [35]), there are only a few results on the norm growth for smooth solutions; see references in [9,17]. In our companion work [9] (based on orbital stability [7]), we consider perturbations of the Hill's vortex, which is a traveling-wave solution to the axisymmetric Euler equation supported on the unit ball in R 3 . The Hill's vortex could be seen as the 3D analogue of the Lamb dipole.…”

“…The Hill's vortex could be seen as the 3D analogue of the Lamb dipole. In [9], stability and instability of the Hill's vortex is established using a largely parallel argument as in this work. However, let us point out the main differences: the gradient growth in the current paper is strictly harder to achieve since we face the restriction that the vorticity must vanish on the symmetry axis {x 2 = 0}, for the vorticity to be smooth.…”