On the Stokes System in Cylindrical Domains

Abstract

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 the 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.