On Vorticity Gradient Growth for the Axisymmetric 3D Euler Equations Without Swirl

Finite-Time Singularity Formation for Strong Solutions to the Axi-symmetric 3D Euler Equations

“…We mention without details the results of Yudovich [57], Nadirashvilli [46], Denisov ([16], [17]), Kiselev and Šverák [38], and Zlatoš [58]. There are also a few results on infinite time singularity formation in the 3D Euler equations such as [57], [25], and [18]. To the author's knowledge, the only result on finite-time singularity formation for finite-energy solutions to the 3D Euler equation prior to the present one is that of the author and I. Jeong [24] on hour-glass shaped axi-symmetric domains with a sharp corner.…”

Filamentation near Hill's vortex

“…Infinite time linear growth of ∇ 2 ω 2 L ∞ and arbitrarily long but finite time exponential growth of ω L ∞ are achieved in the current work for C ∞ c (R 3 )-vorticity. In the presence of a physical boundary, obtaining growth is strictly simpler since the vorticity may not vanish on the boundary, leading to a stable growth mechanism; see [20,19,17,9] and the references therein. When the boundary is absent but if the domain is periodic with respect to an axis, then it is well-known that one can use the 2 1 2 -dimensional flow construction to obtain linear in time growth of the 3D vorticity (see [21,39,4,32]).…”

“…For an explicit representation of the axi-symmetric Biot-Savart law u = K[ξ], we refer to [23, Section 1], [56], [17].…”

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

“…While global regularity for smooth initial data is known (e.g. 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 .…”