## Existence of solutions to the nonstationary Stokes system in $H_{-\mu}^{2,1}$, $\mu\in (0,1)$, in a domain with a distinguished axis. Part 1. Existence near the axis in 2d

### Volume 34 / 2007

Applicationes Mathematicae 34 (2007), 121-141
MSC: 35Q30, 76D03, 76D07, 76D99.
DOI: 10.4064/am34-2-1

#### Abstract

We consider the nonstationary Stokes system with slip boundary conditions in a bounded domain which contains some distinguished axis. We assume that the data functions belong to weighted Sobolev spaces with the weight equal to some power function of the distance to the axis. The aim is to prove the existence of solutions in corresponding weighted Sobolev spaces. The proof is divided into three parts. In the first, the existence in 2d in weighted spaces near the axis is shown. In the second, we show an estimate in 3d in weighted spaces near the axis. Finally, in the third, the existence in a bounded domain is proved. This paper contains the first part of the proof