Global Regularity of Axisymmetric Navier-Stokes Equations with NHL Boundary Conditions under a Critical Smallness Condition

TL;DR

This paper proves global regularity for axisymmetric Navier-Stokes equations with NHL boundary conditions under a specific smallness condition on initial data, extending criticality theory to physically relevant boundary settings.

Contribution

It establishes a new global regularity result for axisymmetric flows with NHL boundary conditions using a critical smallness criterion, involving refined inequalities and boundary analysis.

Findings

Global regularity holds under a specific smallness condition on initial data.

The proof uses maximum principles, refined inequalities, and boundary analysis.

The result extends criticality theory to flows with NHL boundary conditions.

Abstract

We investigate the global regularity problem for the three-dimensional incompressible Navier-Stokes equations restricted to axisymmetric flows in a finite cylinder $D = \{(r,\theta,x_3): 0 \le r \le 1, 0 \le \theta < 2\pi, 0 \le x_3 \le 1\}$, subject to the Navier-Hodge-Lions (NHL) boundary condition. While global existence of smooth solutions is known in the swirl-free case, the presence of swirl ($v_\theta \neq 0$) introduces vortex stretching that may potentially lead to finite-time singularity formation. In this work, we prove that if the initial data satisfy a scaling-invariant smallness condition of the form \[ \frac{9C_1C_3^{1/2}}{4}\left(\frac{1}{2}\|V_0\|_{L^4}^4 + \|\Omega_0\|_{L^2}^2\right)^{1/4}\|\Gamma_0\|_{L^4} \le \frac{1}{4}, \] where $V = v_\theta/\sqrt{r}$, $\Omega = \omega_\theta/r$, $\Gamma = r v_\theta$, and $C_1, C_3$ are explicit constants given in this paper, then the solution remains globally regular for all time. The proof proceeds via a transformed system for $\Omega$ and $V$, leveraging a maximum principle for $\Gamma$, refined Agmon-type inequalities to control $\|v_r/r\|_{L^\infty}$, and delicate boundary analysis of the finite cylinder geometry. Key energy estimates yield $L^\infty_T L^4_x$ bounds for all velocity components, which fall within the regularity class, thereby precluding finite-time blow-up. The result extends the known criticality theory for axisymmetric Navier-Stokes flows to the setting of NHL boundary conditions, which are physically relevant for flows with stress-free or slip-type constraints on lateral and horizontal boundaries.