Global existence of axially symmetric solutions to Navier–Stokes equations with large angular component of velocity

Wojciech M. Zaj/aczkowski¹

¹ Institute of Mathematics and Cryptology Military University of Technology Kaliskiego 2 00-908 Warszawa, Poland and Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland

Colloquium Mathematicum, Tome 100 (2004) no. 2, pp. 243-263

Résumé

Global existence of axially symmetric solutions to the Navier–Stokes equations in a cylinder with the axis of symmetry removed is proved. The solutions satisfy the ideal slip conditions on the boundary. We underline that there is no restriction on the angular component of velocity. We obtain two kinds of existence results. First, under assumptions necessary for the existence of weak solutions, we prove that the velocity belongs to $W_{4/3}^{2,1}({\mit \Omega }\times (0,T))$, so it satisfies the Serrin condition. Next, increasing regularity of the external force and initial data we prove existence of solutions (by the Leray–Schauder fixed point theorem) such that $v\in W_r^{2,1}({\mit \Omega }\times (0,T))$ with $r>4/3$, and we prove their uniqueness.

Détail

DOI : 10.4064/cm100-2-7

Keywords: global existence axially symmetric solutions navier stokes equations cylinder axis symmetry removed proved solutions satisfy ideal slip conditions boundary underline there restriction angular component velocity obtain kinds existence results first under assumptions necessary existence weak solutions prove velocity belongs mit omega times satisfies serrin condition increasing regularity external force initial prove existence solutions leray schauder fixed point theorem mit omega times prove their uniqueness