Vorticity confinement for 2D incompressible flows in an infinite cylinder

TL;DR

This paper investigates how vorticity remains confined in 2D incompressible flows within an infinite cylinder, providing decay estimates and bounds on the growth of vorticity support over time for Navier-Stokes and Euler solutions.

Contribution

It introduces new quantitative decay estimates for vorticity outside expanding regions and refines existing confinement bounds for Euler flows.

Findings

Vorticity outside certain regions decays super-polynomially or stretched-exponentially.

The vorticity support diameter grows at most like (t log t)^{1/3}.

The analysis combines iterative schemes with kernel antisymmetry and moment estimates.

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 $t^{1/3}\log^2 t$.