Global Bifurcation of Rotating Vortex Patches

Abstract

We rigorously construct continuous curves of rotating vortex patch solutions to the twodimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [WOZ84, Ove86] of singular limiting patches with 90 • corners at which the relative fluid velocity vanishes. For solutions close to the disk, we prove that there are "Cat's eyes"-type structures in the flow, and provide numerical evidence that these structures persist along the entire solution curves and are related to the formation of corners. We also show, for any rotating vortex patch, that the boundary is analytic as soon as it is sufficiently regular.

Steady asymmetric vortex pairs for Euler equations

“…Regularity of the nonlinear functional. This section is devoted to the regularity study of the nonlinear functional F introduced in 11and (17) and which defines the V-states equations. We proceed first with the Banach spaces X and Y of Hölder type to which the Implicit Function Theorem will be applied.…”

“…More precisely, the rotating patches appear as a collection of one dimensional branches bifurcating from Rankine vortices at the discrete angular velocities set m−1 2m , m ≥ 2 . These local branches were extended very recently to global ones in [17], where the minimum value on the patch boundary of the angular fluid velocity becomes arbitrarily small near the end of each branch. The regularity of the V-states boundary has been conducted in a series of papers [3,4,17,24].…”

“…These local branches were extended very recently to global ones in [17], where the minimum value on the patch boundary of the angular fluid velocity becomes arbitrarily small near the end of each branch. The regularity of the V-states boundary has been conducted in a series of papers [3,4,17,24]. The existence of small loops in the bifurcation diagram has been proved recently in [19].…”

Kármán vortex street in incompressible fluid models

“…fulfills V (0, f ) = V 0 , where V 0 is defined in (18). The parameters satisfy: ε 0 ∈ (0, min{1, l 4 }), σ < 1, α ∈ (0, 1), and X is defined in (31).…”

Imperfect Bifurcation for the Quasi-Geostrophic Shallow-Water Equations

“…These corners coincide with hyperbolic stagnation points in the rotating reference frame in which the patch is steady. From an analytical point of view, this problem is still open and some progress has been recently made in [13].…”