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 2. Estimate in the 3d case

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. 143-167.

Résumé

We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity $v_0$ and the external force $f$ belong to some weighted Sobolev spaces. It is assumed that the weight is the $(-\mu )$th power of the distance to the axis. Let $f\in L_{2,-\mu } $, $v_0\in H_{-\mu }^1$, $\mu \in (0,1)$. We prove an estimate of the velocity in the $H_{-\mu }^{2,1}$ norm and of the gradient of the pressure in the norm of $L_{2,-\mu }$. We apply the Fourier transform with respect to the variable along the axis and the Laplace transform with respect to time. Then we obtain two-dimensional problems with parameters. Deriving an appropriate estimate with a constant independent of the parameters and using estimates in the two-dimensional case yields the result. The existence and regularity in a bounded domain will be shown in another paper.

Détail

DOI : 10.4064/am34-2-2

Keywords: examine regularity solutions stokes system neighbourhood distinguished axis under assumptions initial velocity external force belong weighted sobolev spaces assumed weight power distance axis prove estimate velocity norm gradient pressure norm apply fourier transform respect variable along axis laplace transform respect time obtain two dimensional problems parameters deriving appropriate estimate constant independent parameters using estimates two dimensional yields result existence regularity bounded domain shown another paper