Global existence of solutions to Navier–Stokes equations in cylindrical domains

Bernard Nowakowski¹ ; Wojciech M. Zaj/aczkowski²

¹ Institute of Mathematics Polish Academy of Sciences /Sniadeckich 8 00-956 Warszawa, Poland
² Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland and Institute of Mathematics and Cryptology Military University of Technology Kaliskiego 2 00-908 Warszawa, Poland

Applicationes Mathematicae, Tome 36 (2009) no. 2, pp. 169-182.

Résumé

We prove the existence of global and regular solutions to the Navier–Stokes equations in cylindrical type domains under boundary slip conditions, where coordinates are chosen so that the $x_3$-axis is parallel to the axis of the cylinder. Regular solutions have already been obtained on the interval $[0,T]$, where $T>0$ is large, on the assumption that the $L_2$-norms of the third component of the force field, of derivatives of the force field, and of the velocity field with respect to the direction of the axis of the cylinder are small. In this paper we continue the solution to all times.

Détail

DOI : 10.4064/am36-2-5

Keywords: prove existence global regular solutions navier stokes equations cylindrical type domains under boundary slip conditions where coordinates chosen axis parallel axis cylinder regular solutions have already obtained interval where large assumption norms third component force field derivatives force field velocity field respect direction axis cylinder small paper continue solution times