On global regular axially-symmetric solutions to the Navier-Stokes equations in a cylinder

TL;DR

This paper investigates the behavior of axially symmetric solutions to the Navier-Stokes equations in a finite cylinder, deriving specific energy estimates for the vorticity components under certain boundary conditions.

Contribution

The authors establish new energy estimates for the vorticity in axially symmetric Navier-Stokes solutions with specific boundary conditions, advancing understanding of solution regularity.

Findings

Derived bounds for vorticity components in cylindrical domains.

Identified limitations in obtaining global estimates for nonslip boundary conditions.

Provided a framework for future analysis of axisymmetric Navier-Stokes solutions.

Abstract

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $\Omega\subset\mathbb{R}^3$. We assume that $v_r$, $v_\varphi$, $\omega_\varphi$ vanish on the lateral part of boundary $\partial\Omega$ of the cylinder, and that $v_z$, $\omega_\varphi$, $\partial_zv_\varphi$ vanish on the top and bottom parts of the boundary $\partial\Omega$, where we used standard cylindrical coordinates, and we denoted by $\omega= {\rm curl}\, v$ the vorticity field. Our aim is to derive the estimate $$ \left\|\frac{\omega_{r}}{r}\right\|_{V\left(\Omega\times (0,t)\right)}+\left\|\frac{\omega_{\varphi}}{r}\right\|_{V\left(\Omega\times (0,t)\right)} \leq \phi(\operatorname{data}),$$ where $\phi$ is an increasing positive function and $\|\ \|_{V\left(\Omega\times (0,t)\right)}$ is the energy norm. We are not able to derive any global type estimate for nonslip boundary conditions.