On global regular axially-symmetric solutions to the Navier-Stokes equations in a cylinder

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.

💡 Research Summary

The paper studies the three‑dimensional incompressible Navier–Stokes system in a bounded cylindrical domain
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