Long time existence of regular solutions to Navier–Stokes equations in cylindrical domains under boundary slip conditions
Volume 169 / 2005
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 $$ I=\sum_{i=1}^2(\|\partial_{x_3}^iv(0)\|_{L_2({\mit\Omega})}+ \|\partial_{x_3}^if\|_{L_2({\mit\Omega}\times(0,T))}) $$ 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 $i=1,2$, $\beta\in(0,1)$, $\delta\in(1,2)$, $5/\delta<3+\beta$, $3/\delta<2+\beta$.
