Vorticity confinement for 2D incompressible flows in an infinite cylinder

Abstract

We study the confinement of vorticity for two-dimensional incompressible flows in an infinite cylinder. For Navier-Stokes solutions with non-negative and compactly supported initial vorticity, we derive quantitative decay estimates showing that the vorticity mass outside regions whose distance from the initial support grows like $\sqrt{t\log^\alpha t}$ (with $\alpha>1$) or like $t^\beta$ (with $\beta>1/2$) becomes, respectively, super-polynomially or stretched-exponentially small. The analysis combines an iterative scheme with an antisymmetry property of the Biot-Savart kernel. In the Euler case, by coupling this approach with a first-moment estimate from [Commun. Math. Phys., 367, 1077-1093, 2019], we recover the confinement bound of [Commun. Math. Phys., 367, 1077-1093, 2019] and refine it slightly: the diameter of the vorticity support grows at most like $(t\log t)^{1/3}$, rather than $t^{1/3}\log^2 t$.