On the Stokes system in cylindrical domains

The existence of solutions to some initial-boundary value problem for the Stokes system is proved. The result is shown in Sobolev-Slobodetskii spaces such that the velocity belongs to $W_r^{2+\sigma,1+\sigma/2}(\Omega^T)$ and gradient of pressure to $W_r^{\sigma,\sigma/2}(\Omega^T)$, where $r\in(1,\infty)$, $\sigma\in(0,1)$, $\Omega^T=\Omega\times(0,T)$. These are special Besov spaces: $B_{r,r}^{2+\sigma,1+\sigma/2}(\Omega^T)$ and $B_{r,r}^{\sigma,\sigma/2}(\Omega^T)$, respectively. The existence is proved by the technique of regularizer.