### On global regular solutions to the Navier-Stokes equations in cylindrical domains

#### Abstract

We consider the incompressible fluid motion described by the

Navier-Stokes equations in a cylindrical domain $\Omega\subset\R^3$ under

the slip boundary conditions. First we prove long time existence of regular

solutions such that $v\in W_2^{2,1}(\Omega\times(0,T))$,

$\nabla p\in L_2(\Omega\times(0,T))$, where $v$ is the velocity of the fluid

and $p$ the pressure. To show this we need smallness of

$\|v_{,x_3}(0)\|_{L_2(\Omega)}$ and $\|f_{,x_3}\|_{L_2(\Omega\times(0,T))}$,

where $f$ is the external force and $x_3$ is the axis along the cylinder.

The above smallness restrictions mean that the considered solution remains

close to the two-dimensional solution, which, as is well known, is regular.

Having $T$ sufficiently large and imposing some decay estimates on

$\|f(t)\|_{L_2(\Omega)}$ we continue the local solution step by step up to

the global one.

Navier-Stokes equations in a cylindrical domain $\Omega\subset\R^3$ under

the slip boundary conditions. First we prove long time existence of regular

solutions such that $v\in W_2^{2,1}(\Omega\times(0,T))$,

$\nabla p\in L_2(\Omega\times(0,T))$, where $v$ is the velocity of the fluid

and $p$ the pressure. To show this we need smallness of

$\|v_{,x_3}(0)\|_{L_2(\Omega)}$ and $\|f_{,x_3}\|_{L_2(\Omega\times(0,T))}$,

where $f$ is the external force and $x_3$ is the axis along the cylinder.

The above smallness restrictions mean that the considered solution remains

close to the two-dimensional solution, which, as is well known, is regular.

Having $T$ sufficiently large and imposing some decay estimates on

$\|f(t)\|_{L_2(\Omega)}$ we continue the local solution step by step up to

the global one.

#### Keywords

Navier-Stokes equations; existence of regular solutions; global existence; slip boundary conditions

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