### Global regular solutions to the Navier-Stokes equations in an axially symmetric domain

#### Abstract

We prove the existence of global regular solutions to the Navier-Stokes

equations in an axially symmetric domain in $\mathbb R^3$ and with boundary slip

conditions. We assume that initial angular component of velocity and angular

component of the external force and angular derivatives of the cylindrical

components of initial velocity and of the external force are sufficiently

small in corresponding norms. Then there exists a solution such that velocity

belongs to $W_{5/2}^{2,1}(\Omega^T)$ and gradient of pressure to

$L_{5/2}(\Omega^T)$, and we do not have restrictions on $T$.

equations in an axially symmetric domain in $\mathbb R^3$ and with boundary slip

conditions. We assume that initial angular component of velocity and angular

component of the external force and angular derivatives of the cylindrical

components of initial velocity and of the external force are sufficiently

small in corresponding norms. Then there exists a solution such that velocity

belongs to $W_{5/2}^{2,1}(\Omega^T)$ and gradient of pressure to

$L_{5/2}(\Omega^T)$, and we do not have restrictions on $T$.

#### Keywords

Navier-Stokes equations; axially symmetric domain; global regular solutions; slip boundary conditions

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