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

W. M. Zajączkowski ¹

¹ 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

Studia Mathematica, Tome 169 (2005) no. 3, pp. 243-285

Résumé

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 $$ I=\sum_{i=1}^2(\|\partial_{x_3}^iv(0)\|_{L_2({\mit\Omega})}+ \|\partial_{x_3}^if\|_{L_2({\mit\Omega}\times(0,T))}) $$ is sufficiently small, where $x_3$ 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 $h^{(i)}=\partial_{x_3}^iv$, $q^{(i)}=\partial_{x_3}^ip$, $i=1,2$, which follow from the Navier–Stokes equations and corresponding boundary conditions. Existence is proved in Sobolev–Slobodetskiĭ spaces: $h^{(i)}\in W_\delta^{2+\beta,1+\beta/2}({\mit\Omega}\times(0,T))$, where $i=1,2$, $\beta\in(0,1)$, $\delta\in(1,2)$, $5/\delta3+\beta$, $3/\delta2+\beta$.

Détail

DOI : 10.4064/sm169-3-3

Keywords: long time existence solutions navier stokes equations cylindrical domains under boundary slip conditions proved moreover existence solutions restrictions magnitude initial velocity external force shown however have assume quantity sum partial mit omega partial mit omega times sufficiently small where coordinate along axis parallel cylinder time existence inversely proportional existence solutions proved leray schauder fixed point theorem applied problems partial partial which follow navier stokes equations corresponding boundary conditions existence proved sobolev slobodetski spaces delta beta beta mit omega times where beta delta delta beta delta beta