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 ($v_{\theta }\mathrm{\ne }0$) 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 $$\frac{9C_1{C}_3^{1\mathrm{/}2}}{4}{(\frac{1}{2}\mathrm{\parallel }{V}_0{\mathrm{\parallel }}_{L^4}^4+\mathrm{\parallel }{\mathrm{\Omega }}_0{\mathrm{\parallel }}_{L^2}^2)}^{1\mathrm{/}4}\mathrm{\parallel }{\mathrm{\Gamma }}_0{\mathrm{\parallel }}_{L^4}\le \frac{1}{4},$$ where $V=v_{\theta }\mathrm{/}\sqrt{r}$, $\mathrm{\Omega }={\omega }_{\theta }\mathrm{/}r$, $\mathrm{\Gamma }=r{v}_{\theta }$, 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 $\mathrm{\Omega }$ and $V$, leveraging a maximum principle for $\mathrm{\Gamma }$, refined Agmon-type inequalities to control $\mathrm{\parallel }{v}_r\mathrm{/}r{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$, and delicate boundary analysis of the finite cylinder geometry. Key energy estimates yield $L_T^{\mathrm{\infty }}{L}_x^4$ 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.