Existence of solutions to the nonstationary Stokes system in $H_{-\mu }^{2,1}$, $\mu \in (0,1)$, in a domain with a distinguished axis. Part 2. Estimate in the 3d case

Volume 34 / 2007

W. M. Zaj/aczkowski Applicationes Mathematicae 34 (2007), 143-167 MSC: 35Q30, 76D03, 76D99. DOI: 10.4064/am34-2-2

Abstract

We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity $v_0$ and the external force $f$ belong to some weighted Sobolev spaces. It is assumed that the weight is the $(-\mu )$th power of the distance to the axis. Let $f\in L_{2,-\mu } $, $v_0\in H_{-\mu }^1$, $\mu \in (0,1)$. We prove an estimate of the velocity in the $H_{-\mu }^{2,1}$ norm and of the gradient of the pressure in the norm of $L_{2,-\mu }$. We apply the Fourier transform with respect to the variable along the axis and the Laplace transform with respect to time. Then we obtain two-dimensional problems with parameters. Deriving an appropriate estimate with a constant independent of the parameters and using estimates in the two-dimensional case yields the result. The existence and regularity in a bounded domain will be shown in another paper.

Authors

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

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