On the Stokes system in cylindrical domains

TL;DR

This paper proves the existence of solutions to the Stokes system in cylindrical domains within specific Sobolev-Slobodetskii and Besov spaces, using regularizer techniques to handle initial-boundary value problems.

Contribution

It establishes existence results for the Stokes system in cylindrical domains in advanced function spaces, extending previous work with a novel regularizer approach.

Findings

Solutions exist in Sobolev-Slobodetskii spaces

Solutions also exist in corresponding Besov spaces

The regularizer technique is effective for these problems

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 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.