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

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.