## Topological Methods in Nonlinear Analysis

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

Wojciech M. Zajączkowski

#### 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

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

#### References

• R. Adams, Sobolev Spaces, Academic Press, New York, San Francisco, London (1975) \ref\key 2
• O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representation of Functions and Imbedding Theorems, Nauka, Moscow (1975), (in Russian) \ref\key 3
• M. Cannone, Y. Meyer and F. Planchon, Solutions autosimilaires des équations de Navier–Stokes , Séminaire “Équations aux Dérivées Partielles” de l'École polytechnique, Exposé VIII, 1993–1994 \ref\key 4
• J. Y. Chemin and I. Gallagher, On the global wellposedness of the $3d$ Navier–Stokes equations with large initial data .v1 [math. AP] 19 Aug 2005 ;, Ann. Sci. École Norm. Sup., 39 (2006), 679–698 \ref\key 5 ––––, Large, global solutions to the Navier–Stokes equations slowly varying in one direction , arXiv: 0710.5408v2 [math. Ap] 31 Oct 2007 ;, Trans. Amer. Math. Soc., 362 (2010), 2859–2873 \ref\key 6
• J. Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier–Stokes equations , arXiv: 0807.1265v1 [math. Ap] 8 Jul 2008 \ref\key 7
• H. Fujita and T. Kato, On the Navier–Stokes initial value problem \romI, Arch. Rational Mech. Anal., 16 (1964), 269–315 \ref\key 8
• H. Koch and D. Tataru, Well-posedness for the Navier–Stokes equations , Adv. Math., 157 (2001), 22–35 \ref\key 9
• O. A. Ladyzhenskaya, Solutions “in the large” of the nonstationary boundary value problem for the Navier–Stokes system with two space variables , Comm. Pure Appl. Math., 12 (1959), 427–433 \ref\key 10 ––––, On unique solvability of three-dimensional Cauchy problem for the Navier–Stokes equations under the axial symmetry , Zap. Nauchn. Sem. LOMI, 7 (1968), 155–177, (in Russian) \ref\key 11 ––––, Mathematical Theory of Viscous Incompressible Flow, Nauka, Moscow (1970), (in Russian) \ref\key 12
• O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow (1967), (in Russian) \ref\key 13
• J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problémes que pose l'hydrodynamique , J. Math. Pures Appl., 12 (1933), 1–82 \ref\key 14
• P. Maremonti and V. A. Solonnikov, On the estimates of solutions of the nonstationary Stokes problem in S. L. Sobolev spaces with a mixed norm , Zap. Nauchn. Sem. POMI, 222 (1995), 124–150 \ref\key 15
• B. Nowakowski. personal communication \ref\key 16
• V. A. Solonnikov, Solvability of the Stokes equations in S. L. Sobolev spaces with a mixed norm , Zap. Nauchn. Sem. POMI, 288 (2002), 204–231 \ref\key 19
• V. A. Solonnikov and V. E. Schadilov, On a boundary value problem for a stationary system of the Navier–Stokes equations , Trudy Mat. Inst. Steklov, 125 (1973), 196–210, (in Russian)\moreref\nofrills ; English transl., Proc. Steklov Inst. Math., 125 (1973), 186–199 \ref\key 20
• F. Weissler, The Navier–Stokes initial value problem in $L^p$ , Arch. Rational Mech. Anal., 74 (1980), 219–230 \ref\key 21
• W. von Wahl, The equations of Navier–Stokes equations and abstract parabolic equations , Braunschweig (1985) \ref\key 22