On global regular axially-symmetric solutions to the Navier-Stokes equations in a cylinder
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| Format: | Preprint |
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2025
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| author | Grygierzec, Wiesław J. Zajączkowski, Wojciech M. |
| author_facet | Grygierzec, Wiesław J. Zajączkowski, Wojciech M. |
| contents | We consider the axisymmetric Navier-Stokes equations in a finite cylinder $Ω\subset\mathbb{R}^3$. We assume that $v_r$, $v_φ$, $ω_φ$ vanish on the lateral part of boundary $\partialΩ$ of the cylinder, and that $v_z$, $ω_φ$, $\partial_zv_φ$ vanish on the top and bottom parts of the boundary $\partialΩ$, where we used standard cylindrical coordinates, and we denoted by $ω= {\rm curl}\, v$ the vorticity field. Our aim is to derive the estimate $$ \left\|\frac{ω_{r}}{r}\right\|_{V\left(Ω\times (0,t)\right)}+\left\|\frac{ω_φ}{r}\right\|_{V\left(Ω\times (0,t)\right)} \leq ϕ(\operatorname{data}),$$ where $ϕ$ is an increasing positive function and $\|\ \|_{V\left(Ω\times (0,t)\right)}$ is the energy norm. We are not able to derive any global type estimate for nonslip boundary conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_05098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On global regular axially-symmetric solutions to the Navier-Stokes equations in a cylinder Grygierzec, Wiesław J. Zajączkowski, Wojciech M. Analysis of PDEs 35A01, 35B01, 35B65, 35Q30, 76D03, 76D05 We consider the axisymmetric Navier-Stokes equations in a finite cylinder $Ω\subset\mathbb{R}^3$. We assume that $v_r$, $v_φ$, $ω_φ$ vanish on the lateral part of boundary $\partialΩ$ of the cylinder, and that $v_z$, $ω_φ$, $\partial_zv_φ$ vanish on the top and bottom parts of the boundary $\partialΩ$, where we used standard cylindrical coordinates, and we denoted by $ω= {\rm curl}\, v$ the vorticity field. Our aim is to derive the estimate $$ \left\|\frac{ω_{r}}{r}\right\|_{V\left(Ω\times (0,t)\right)}+\left\|\frac{ω_φ}{r}\right\|_{V\left(Ω\times (0,t)\right)} \leq ϕ(\operatorname{data}),$$ where $ϕ$ is an increasing positive function and $\|\ \|_{V\left(Ω\times (0,t)\right)}$ is the energy norm. We are not able to derive any global type estimate for nonslip boundary conditions. |
| title | On global regular axially-symmetric solutions to the Navier-Stokes equations in a cylinder |
| topic | Analysis of PDEs 35A01, 35B01, 35B65, 35Q30, 76D03, 76D05 |
| url | https://arxiv.org/abs/2511.05098 |