### Global axially symmetric solutions with large swirl to the Navier-Stokes equations

DOI: http://dx.doi.org/10.12775/TMNA.2007.013

#### Abstract

Long time existence of axially symmetric solutions to the

Navier-Stokes equations in a bounded cylinder and with boundary slip

conditions is proved. The axially symmetric solutions with nonvanishing

azimuthal component of velocity (swirl) are examined. The solutions are such

that swirl is small in a neighbourhood close to the axis of symmetry but it is

large in some positive distance from it. There is a great difference between

the proofs of global axially symmetric solutions with vanishing and

nonvanishing swirl. In the first case global estimate follows at once but

in the second case we need a lot of considerations in weighted spaces to show it.

The existence is proved by the Leray-Schauder fixed point theorem.

Navier-Stokes equations in a bounded cylinder and with boundary slip

conditions is proved. The axially symmetric solutions with nonvanishing

azimuthal component of velocity (swirl) are examined. The solutions are such

that swirl is small in a neighbourhood close to the axis of symmetry but it is

large in some positive distance from it. There is a great difference between

the proofs of global axially symmetric solutions with vanishing and

nonvanishing swirl. In the first case global estimate follows at once but

in the second case we need a lot of considerations in weighted spaces to show it.

The existence is proved by the Leray-Schauder fixed point theorem.

#### Keywords

Navier-Stokes equations; slip boundary conditions; axially symmetric solutions; large swirl

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