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

Wojciech M. Zajączkowski


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; slip boundary conditions; axially symmetric solutions; large swirl

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