On global regular axially-symmetric solutions to the Navier-Stokes equations in a cylinder
Wiesław J. Grygierzec, Wojciech M. Zajączkowski
TL;DR
This work investigates global regularity for axisymmetric solutions of the 3D Navier–Stokes equations in a finite cylinder under mixed boundary conditions that enforce decay of certain vorticity components on the lateral boundary and endcaps. The authors develop a priori bounds for the vorticity components through a swirl-controlled energy framework, stream-function formalism, and order-reduction estimates, leveraging Hardy and interpolation inequalities to handle axis singularities. They present two complementary global-estimate routes (OZ and GZ), showing that the key quantity $X(t)=\|Φ\|_{V(Ω^t)}+\|Γ\|_{V(Ω^t)}$ remains bounded by data under suitable assumptions, thereby proving the existence of global regular axisymmetric solutions in the cylinder. The results advance understanding of global behavior for axisymmetric Navier–Stokes with cylindrical geometry and highlight remaining challenges for nonslip boundary conditions.
Abstract
We consider the axisymmetric Navier-Stokes equations in a finite cylinder $Ω\subset\mathbb{R}^3$. We assume that $v_r$, $v_\varphi$, $ω_\varphi$ vanish on the lateral part of boundary $\partialΩ$ of the cylinder, and that $v_z$, $ω_\varphi$, $\partial_zv_\varphi$ vanish on the top and bottom parts of the boundary $\partialΩ$, where we used standard cylindrical coordinates, and we denoted by $ω= {\rm curl}\, v$ the vorticity field. Our aim is to derive the estimate $$ \left\|\frac{ω_{r}}{r}\right\|_{V\left(Ω\times (0,t)\right)}+\left\|\frac{ω_{\varphi}}{r}\right\|_{V\left(Ω\times (0,t)\right)} \leq φ(\operatorname{data}),$$ where $φ$ is an increasing positive function and $\|\ \|_{V\left(Ω\times (0,t)\right)}$ is the energy norm. We are not able to derive any global type estimate for nonslip boundary conditions.