Global Regularity of Axisymmetric Navier-Stokes Equations with NHL Boundary Conditions under a Critical Smallness Condition
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866909053996761088 |
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| author | Chan, Tsz-Lik |
| author_facet | Chan, Tsz-Lik |
| contents | We investigate the global regularity problem for the three-dimensional incompressible Navier-Stokes equations restricted to axisymmetric flows in a finite cylinder $D = \{(r,θ,x_3): 0 \le r \le 1, 0 \le θ< 2π, 0 \le x_3 \le 1\}$, 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 ($v_θ\neq 0$) 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 \[ \frac{9C_1C_3^{1/2}}{4}\left(\frac{1}{2}\|V_0\|_{L^4}^4 + \|Ω_0\|_{L^2}^2\right)^{1/4}\|Γ_0\|_{L^4} \le \frac{1}{4}, \] where $V = v_θ/\sqrt{r}$, $Ω= ω_θ/r$, $Γ= r v_θ$, 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 $Ω$ and $V$, leveraging a maximum principle for $Γ$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18011 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global Regularity of Axisymmetric Navier-Stokes Equations with NHL Boundary Conditions under a Critical Smallness Condition Chan, Tsz-Lik Analysis of PDEs We investigate the global regularity problem for the three-dimensional incompressible Navier-Stokes equations restricted to axisymmetric flows in a finite cylinder $D = \{(r,θ,x_3): 0 \le r \le 1, 0 \le θ< 2π, 0 \le x_3 \le 1\}$, 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 ($v_θ\neq 0$) 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 \[ \frac{9C_1C_3^{1/2}}{4}\left(\frac{1}{2}\|V_0\|_{L^4}^4 + \|Ω_0\|_{L^2}^2\right)^{1/4}\|Γ_0\|_{L^4} \le \frac{1}{4}, \] where $V = v_θ/\sqrt{r}$, $Ω= ω_θ/r$, $Γ= r v_θ$, 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 $Ω$ and $V$, leveraging a maximum principle for $Γ$, 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. |
| title | Global Regularity of Axisymmetric Navier-Stokes Equations with NHL Boundary Conditions under a Critical Smallness Condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.18011 |