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## Applicationes Mathematicae

2007 | 34 | 2 | 143-167
Tytuł artykułu

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

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We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity v₀ and the external force f belong to some weighted Sobolev spaces. It is assumed that the weight is the (-μ )th power of the distance to the axis. Let $f∈ L_{2,-μ}$, $v₀ ∈ H_{-μ}¹$, μ ∈ (0,1). We prove an estimate of the velocity in the $H_{-μ}^{2,1}$ norm and of the gradient of the pressure in the norm of $L_{2,-μ}$. 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.
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Tom
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143-167
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wydano
2007
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• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland
• Institute of Mathematics and Cryptology, Military University of Technology, Kaliskiego 2, 00-908 Warszawa, Poland
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