Topological Methods in Nonlinear Analysis

On global regular solutions to the Navier-Stokes equations in cylindrical domains

Wojciech M. Zajączkowski

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Abstract

We consider the incompressible fluid motion described by the Navier-Stokes equations in a cylindrical domain $\Omega\subset\mathbb R^3$ under the slip boundary conditions. First we prove long time existence of regular solutions such that $v\in W_2^{2,1}(\Omega\times(0,T))$, $\nabla p\in L_2(\Omega\times(0,T))$, where $v$ is the velocity of the fluid and $p$ the pressure. To show this we need smallness of $\|v_{,x_3}(0)\|_{L_2(\Omega)}$ and $\|f_{,x_3}\|_{L_2(\Omega\times(0,T))}$, where $f$ is the external force and $x_3$ is the axis along the cylinder. The above smallness restrictions mean that the considered solution remains close to the two-dimensional solution, which, as is well known, is regular.

Having $T$ sufficiently large and imposing some decay estimates on $\|f(t)\|_{L_2(\Omega)}$ we continue the local solution step by step up to the global one.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 37, Number 1 (2011), 55-85.

Dates
First available in Project Euclid: 20 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461181488

Mathematical Reviews number (MathSciNet)
MR2839517

Zentralblatt MATH identifier
1337.76012

Citation

Zajączkowski, Wojciech M. On global regular solutions to the Navier-Stokes equations in cylindrical domains. Topol. Methods Nonlinear Anal. 37 (2011), no. 1, 55--85. https://projecteuclid.org/euclid.tmna/1461181488


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