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

Abstract

We prove linear in time filamentation for perturbations of the Lamb dipole, which is a traveling wave solution to the incompressible Euler equations in R 2 . The main ingredient is a recent nonlinear orbital stability result by Abe-Choi. As a consequence, we obtain linear in time growth for the vorticity gradient for all times, for certain smooth and compactly supported initial vorticity in R 2 . The construction carries over to some generalized SQG equations.

Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field

“…In the periodic setting, an example of Zlatos shows that the vorticity gradient can grow at least exponentially [88]. Choi and Jeong obtain linear in time growth for the vorticity gradient for certain smooth and compactly supported initial vorticity in R 2 [16]. Classical solutions to the 3D Euler equations could develop finite-time singularities ( [14,28]).…”

Singularity formation in the incompressible Euler equation in finite and infinite time

“…There has been subsequent work exhibiting this type of behavior (often stated for the gradient of the vorticity) on domains with the boundary [13,114,218,219,166,134,181], on the torus [58,221] near stationary solutions and the plane near a vortex dipole [35]. The works [218,219,166] provide linear-in-time growth rates whereas the others provide superlinear growth rates, with [134] on the domain of a disk standing out as the sole example of double-exponential growth from smooth data of the vorticity gradient, saturating the bound (36).…”

“…We are interested in the long time behavior of this dynamical system. The formula (35) shows that the vorticity at any later time is just some area preserving rearrangement of its initial conditions. As discussed in §2.1, any perfect fluid flow can be thought of as a path in the group of such transformations D µ pM q (the configuration space of the flow).…”

“…Let ω 0 P L 8 pM q. There exists a unique weak Euler solution of class ω P L 8 pp´8, 8q ˆM q with the following properties (a) ωptq takes the form (35) for a unique Hölder continuous flow of homeomorphisms Φ t P C αptq pM q for C :" CpM q and αptq :" expp´C|t|}ω 0 } L 8 q. Consequently, }ωptq} L 8 pM q " }ω 0 } L 8 pM q for all t P R. (b) ω P CpR; L p pM qq X C w pR; L 8 pM qq for all p P r1, 8q, where C w pR; L 8 pM qq denotes the space of functions which are continuous in time with values in the weak-˚topology of L 8 pM q. (c) The solution map S t is continuous in the weak-˚topology of L 8 pM q, i.e.…”