On the regularity of axially-symmetric solutions to the incompressible Navier-Stokes equations in a cylinder
W. S. Ożański, W. Zajączkowski
TL;DR
We study axisymmetric solutions to the 3D incompressible Navier–Stokes equations in a finite cylinder with boundary conditions that enforce no-slip-like constraints on the swirl and vorticity components. The authors develop weighted and elliptic estimates for the modified stream function $\psi_1=\psi/r$ and derive three order-reduction estimates that reduce nonlinear complexity, enabling a conditional regularity result. They show that a regular solution remains regular as long as the swirl satisfies $|v_\varphi|_{L^\infty_t L^p_x}/|v_\varphi|_{L^\infty_t L^\infty_x}$ is bounded away from zero for any $p>6$, with explicit bounds expressed in terms of an energy-like quantity $X(t)$. The work provides a quantitative framework tying swirl control to global regularity in axisymmetric cylinder domains, contributing to the understanding of partial regularity and near-axi-symmetric dynamics in Navier–Stokes flows.
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 boundary $\partial Ω$ of the cylinder, and that $v_z$, $ω_\varphi$, $\partial_z v_\varphi$ vanish on the top and bottom parts of the boundary $\partial Ω$, where we used standard cylindrical coordinates, and we denoted by $ω=\mathrm{curl}\, v$ 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.