Nonstationary Stokes System in Cylindrical Domains Under Boundary Slip Conditions

Abstract

Existence and uniqueness of solutions to the nonstationary Stokes system in a cylindrical domain \({\Omega\subset\mathbb{R}^3}\) and under boundary slip conditions are proved in anisotropic Sobolev spaces. Assuming that the external force belong to \({L_r(\Omega\times(0,T))}\) and initial velocity to \({W_r^{2-2/r}(\Omega)}\) there exists a solution such that velocity belongs to \({W_r^{2,1}(\Omega\times(0,T))}\) and gradient of pressure to \({L_r(\Omega\times(0,T))}\), \({r\in(1,\infty)}\), \({T > 0}\). Thanks to the slip boundary conditions and a partition of unity the Stokes system is transformed to the Poisson equation for pressure and the heat equation for velocity. The existence of solutions to these equations is proved by applying local considerations. In this case we have to consider neighborhoods near the edges which by local mapping can be transformed to dihedral angle \({\pi/2}\). Hence solvability of the problem bases on construction local Green functions (near an interior point, near a point of a smooth part of the boundary, near a point of the edge) and their appropriate estimates. The technique presented in this paper can also work in other functional spaces: Sobolev-Slobodetskii, Besov, Nikolskii, Holder and so on.