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

Volume 100 / 2004

Wojciech M. Zaj/aczkowski Colloquium Mathematicum 100 (2004), 243-263 MSC: 35Q35, 76D03, 76D05. DOI: 10.4064/cm100-2-7


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.


  • Wojciech M. Zaj/aczkowskiInstitute of Mathematics and Cryptology
    Military University of Technology
    Kaliskiego 2
    00-908 Warszawa, Poland
    Institute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland

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