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

W. M. Zaj/aczkowski¹

¹ Institute of Mathematics Polish Academy of Sciences /Sniadeckich 8 00-956 Warszawa, Poland and Institute of Mathematics and Cryptology Military University of Technology Kaliskiego 2 00-908 Warszawa, Poland

Applicationes Mathematicae, Tome 34 (2007) no. 2, pp. 121-141.

Résumé

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

Détail

DOI : 10.4064/am34-2-1

Keywords: consider nonstationary stokes system slip boundary conditions bounded domain which contains distinguished axis assume functions belong weighted sobolev spaces weight equal power function distance axis prove existence solutions corresponding weighted sobolev spaces proof divided three parts first existence weighted spaces near axis shown second estimate weighted spaces near axis finally third existence bounded domain proved paper contains first part proof