Long time existence of regular solutions to Navier-Stokes equations in cylindrical domains under boundary slip conditions

W. M. Zajączkowski. "Long time existence of regular solutions to Navier-Stokes equations in cylindrical domains under boundary slip conditions." Studia Mathematica 169.3 (2005): 243-285. <http://eudml.org/doc/284500>.

@article{W2005,
abstract = {Long time existence of solutions to the Navier-Stokes equations in cylindrical domains under boundary slip conditions is proved. Moreover, the existence of solutions with no restrictions on the magnitude of the initial velocity and the external force is shown. However, we have to assume that the quantity $I = ∑_\{i=1\}^\{2\} (||∂_\{x₃\}^\{i\} v(0)||_\{L₂(Ω)\} + ||∂_\{x₃\}^\{i\}f||_\{L₂(Ω×(0,T))\})$ is sufficiently small, where x₃ is the coordinate along the axis parallel to the cylinder. The time of existence is inversely proportional to I. Existence of solutions is proved by the Leray-Schauder fixed point theorem applied to problems for $h^\{(i)\} = ∂_\{x₃\}^\{i\}v$, $q^\{(i)\} = ∂_\{x₃\}^\{i\}p$, i = 1,2, which follow from the Navier-Stokes equations and corresponding boundary conditions. Existence is proved in Sobolev-Slobodetskiĭ spaces: $h^\{(i)\} ∈ W_\{δ\}^\{2+β,1+β/2\}(Ω×(0,T))$, where i = 1,2, β ∈ (0,1), δ ∈ (1,2), 5/δ < 3 + β, 3/δ < 2 + β.},
author = {W. M. Zajączkowski},
journal = {Studia Mathematica},
keywords = {Navier-Stokes system; global existence of regular solutions; large data},
language = {eng},
number = {3},
pages = {243-285},
title = {Long time existence of regular solutions to Navier-Stokes equations in cylindrical domains under boundary slip conditions},
url = {http://eudml.org/doc/284500},
volume = {169},
year = {2005},
}

TY - JOUR
AU - W. M. Zajączkowski
TI - Long time existence of regular solutions to Navier-Stokes equations in cylindrical domains under boundary slip conditions
JO - Studia Mathematica
PY - 2005
VL - 169
IS - 3
SP - 243
EP - 285
AB - Long time existence of solutions to the Navier-Stokes equations in cylindrical domains under boundary slip conditions is proved. Moreover, the existence of solutions with no restrictions on the magnitude of the initial velocity and the external force is shown. However, we have to assume that the quantity $I = ∑_{i=1}^{2} (||∂_{x₃}^{i} v(0)||_{L₂(Ω)} + ||∂_{x₃}^{i}f||_{L₂(Ω×(0,T))})$ is sufficiently small, where x₃ is the coordinate along the axis parallel to the cylinder. The time of existence is inversely proportional to I. Existence of solutions is proved by the Leray-Schauder fixed point theorem applied to problems for $h^{(i)} = ∂_{x₃}^{i}v$, $q^{(i)} = ∂_{x₃}^{i}p$, i = 1,2, which follow from the Navier-Stokes equations and corresponding boundary conditions. Existence is proved in Sobolev-Slobodetskiĭ spaces: $h^{(i)} ∈ W_{δ}^{2+β,1+β/2}(Ω×(0,T))$, where i = 1,2, β ∈ (0,1), δ ∈ (1,2), 5/δ < 3 + β, 3/δ < 2 + β.
LA - eng
KW - Navier-Stokes system; global existence of regular solutions; large data
UR - http://eudml.org/doc/284500
ER -