TY - JOUR AU - ZajÄ…czkowski, Wojciech M. PY - 2007/06/01 Y2 - 2024/03/28 TI - Global axially symmetric solutions with large swirl to the Navier-Stokes equations JF - Topological Methods in Nonlinear Analysis JA - TMNA VL - 29 IS - 2 SE - DO - UR - https://apcz.umk.pl/TMNA/article/view/TMNA.2007.013 SP - 295 - 331 AB - Long time existence of axially symmetric solutions to theNavier-Stokes equations in a bounded cylinder and with boundary slipconditions is proved. The axially symmetric solutions with nonvanishingazimuthal component of velocity (swirl) are examined. The solutions are suchthat swirl is small in a neighbourhood close to the axis of symmetry but it islarge in some positive distance from it. There is a great difference betweenthe 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. ER -