A regularity criterion for the angular component of velocity in the norm $L_\infty(0,T;L_p(Ω)),\;\frac 3 p <1$ in axisymmetric 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 $Ω\subset\R^3$. 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 $ω=\curl v$ the vorticity field. We use $H^3$ Sobolev estimates for the modified stream function (stream function divided by radius) and energy type estimates for gradient of swirl to derive two order reduction estimates. Using the estimate \[ \|v_φ\|_{L_\infty(0,T;L_p)Ω)}\les A, \] where A is a given number and $p>3$ we prove the existence of global regular axially-symmetric solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_14964 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A regularity criterion for the angular component of velocity in the norm $L_\infty(0,T;L_p(Ω)),\;\frac 3 p <1$ in axisymmetric 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 $Ω\subset\R^3$. 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 $ω=\curl v$ the vorticity field. We use $H^3$ Sobolev estimates for the modified stream function (stream function divided by radius) and energy type estimates for gradient of swirl to derive two order reduction estimates. Using the estimate \[ \|v_φ\|_{L_\infty(0,T;L_p)Ω)}\les A, \] where A is a given number and $p>3$ we prove the existence of global regular axially-symmetric solutions. |
| title | A regularity criterion for the angular component of velocity in the norm $L_\infty(0,T;L_p(Ω)),\;\frac 3 p <1$ in axisymmetric Navier Stokes equations in a cylinder |
| topic | Analysis of PDEs 35A01, 35B01, 35B65, 35Q30, 76D03, 76D05 |
| url | https://arxiv.org/abs/2507.14964 |