On the regularity of axially-symmetric solutions to the incompressible Navier-Stokes equations in a cylinder
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Ożański, W. S. Zajączkowski, W. |
| author_facet | Ożański, W. S. Zajączkowski, W. |
| 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 boundary $\partial Ω$ of the cylinder, and that $v_z$, $ω_φ$, $\partial_z v_φ$ vanish on the top and bottom parts of the boundary $\partial Ω$, where we used standard cylindrical coordinates, and we denoted by $ω=\mathrm{curl}\, v$ the vorticity field. We use weighted estimates and $H^3$ Sobolev estimate on the modified stream function to derive three order-reduction estimates. These enable one to reduce the order of the nonlinear estimates of the equations, and help observe that the solutions to the equations are ``almost regular''. We use the order-reduction estimates to show that the solution to the equations remains regular as long as, for any $p\in (6,\infty)$, $\| v_φ\|_{L^\infty_t L^p_x}/\| v_φ\|_{L^\infty_t L^\infty_x}$ remains bounded below by a positive number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16670 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the regularity of axially-symmetric solutions to the incompressible Navier-Stokes equations in a cylinder Ożański, W. S. Zajączkowski, W. Analysis of PDEs 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 boundary $\partial Ω$ of the cylinder, and that $v_z$, $ω_φ$, $\partial_z v_φ$ vanish on the top and bottom parts of the boundary $\partial Ω$, where we used standard cylindrical coordinates, and we denoted by $ω=\mathrm{curl}\, v$ the vorticity field. We use weighted estimates and $H^3$ Sobolev estimate on the modified stream function to derive three order-reduction estimates. These enable one to reduce the order of the nonlinear estimates of the equations, and help observe that the solutions to the equations are ``almost regular''. We use the order-reduction estimates to show that the solution to the equations remains regular as long as, for any $p\in (6,\infty)$, $\| v_φ\|_{L^\infty_t L^p_x}/\| v_φ\|_{L^\infty_t L^\infty_x}$ remains bounded below by a positive number. |
| title | On the regularity of axially-symmetric solutions to the incompressible Navier-Stokes equations in a cylinder |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.16670 |