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

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 34.2 (2007): 143-167. <http://eudml.org/doc/279202>.

@article{W2007,
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 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.},
author = {W. M. Zajączkowski},
journal = {Applicationes Mathematicae},
language = {eng},
number = {2},
pages = {143-167},
title = {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},
url = {http://eudml.org/doc/279202},
volume = {34},
year = {2007},
}

TY - JOUR
AU - W. M. Zajączkowski
TI - 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
JO - Applicationes Mathematicae
PY - 2007
VL - 34
IS - 2
SP - 143
EP - 167
AB - 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.
LA - eng
UR - http://eudml.org/doc/279202
ER -