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 , , μ ∈ (0,1). We prove an estimate of the velocity in the norm and of the gradient of the pressure in the norm of . 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-am34-2-2,
author = {W. M. Zaj\k aczkowski},
title = {Existence of solutions to the nonstationary Stokes system in $H\_{-m}^{2,1}$, m [?] (0,1), in a domain with a distinguished axis. Part 2. Estimate in the 3d case},
journal = {Applicationes Mathematicae},
volume = {34},
year = {2007},
pages = {143-167},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am34-2-2}
}
W. M. Zajączkowski. 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. Applicationes Mathematicae, Tome 34 (2007) pp. 143-167. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am34-2-2/