On the regularity of axially-symmetric solutions to the incompressible Navier-Stokes equations in a cylinder

TL;DR

This paper investigates the regularity of axially-symmetric solutions to the incompressible Navier-Stokes equations in a cylinder, introducing weighted estimates and order-reduction techniques to establish conditions for solution regularity.

Contribution

The authors develop new weighted and Sobolev estimates that reduce the nonlinear complexity, demonstrating solutions remain regular under specific boundedness conditions of the azimuthal velocity component.

Findings

Solutions are 'almost regular' under derived estimates.

Regularity is maintained if a certain ratio involving $v_ heta$ remains bounded.

Order-reduction estimates facilitate analysis of nonlinear terms.

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 boundary $\partial \Omega$ of the cylinder, and that $v_z$, $\omega_\varphi$, $\partial_z v_\varphi$ vanish on the top and bottom parts of the boundary $\partial \Omega$, where we used standard cylindrical coordinates, and we denoted by $\omega =\mathrm{curl}\, v$ the vorticity field. We use weighted estimates and $H^3$ Sobolev estimate on the modified stream function to derive three order-reduction estimates. These enable one to reduce the order of the nonlinear estimates of the equations, and help observe that the solutions to the equations are ``almost regular''. We use the order-reduction estimates to show that the solution to the equations remains regular as long as, for any $p\in (6,\infty)$, $\| v_\varphi \|_{L^\infty_t L^p_x}/\| v_\varphi \|_{L^\infty_t L^\infty_x}$ remains bounded below by a positive number.