Global special regular solutions to the Navier–Stokes equations in axially symmetric domains under boundary slip conditions

Volume 432 / 2005

Wojciech M. Zaj/aczkowski Dissertationes Mathematicae 432 (2005), 1-138 MSC: 35Q30, 35K20, 76D05, 76D03. DOI: 10.4064/dm432-0-1


Existence and uniqueness of global regular special solutions to Navier–Stokes equations with boundary slip conditions in axially symmetric domains is proved. The proof of global existence relies on the global existence results for axially symmetric solutions which were obtained by Ladyzhenskaya and Yudovich–Ukhovski{\u\i} in 1968 who employed the problem for vorticity. In this paper the equations for vorticity also play a crucial role. Moreover, the boundary slip conditions imply appropriate boundary conditions for vorticity, which is of crucial importance to this paper. Finally, we prove the existence of solutions which remain close to the axially symmetric solutions for all time. The existence of axially symmetric solutions was proved in weighted Sobolev spaces with the weight equal to a power of the distance to the axis of symmetry because in such spaces a global estimate for vorticity could be obtained. Therefore in this paper similar weighted Sobolev spaces are also used.


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

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