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Abstract

We prove the existence of solutions to the evolutionary Stokes system in a bounded domain $\Omega\subset\mathbb R^3$. The main result shows that the velocity belongs either to $W_p^{2s+2,s+1}(\Omega^T)$ or to $B_{p,q}^{2s+2,s+1}(\Omega^T)$ with $p>3$ and $s\in\mathbb R_+\cup\{0\}$. The proof is divided into two steps. First the existence in $W_p^{2k+2,k+1}$ for $k\in{\mathbb N}$ is proved. Next applying interpolation theory the existence in Besov spaces in a half space is shown. Finally the technique of regularizers implies the existence in a bounded domain. The result is generalized to the spaces $W_p^{2s,s}(\Omega^T)$ and $B_{p,q}^{2s,s}$ with $p>2$ and $s\in(1/2,1)$.