Nonstationary Stokes system in anisotropic Sobolev spaces

• Published in: Mathematical methods in the applied sciences Vol. 38; no. 12; pp. 2466 - 2478
• Main Author: Zajaczkowski, Wojciech M
• Format: Journal Article
• Language: English
• Published: Freiburg Blackwell Publishing Ltd 01.08.2015; Wiley Subscription Services, Inc
• Subjects: anisotropic Sobolev spaces; Anisotropy; Boundary conditions; Dirichlet problem; Eulers equations; existence and uniqueness; Fluid flow; heat and Poissons equations; Heat equations; Helmholtz-Weyl decomposition; Mathematical analysis; nonstationary Stokes system; Slip; Sobolev space
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.3234

The nonstationary Stokes system with slip boundary conditions is considered in a bounded domain Ω⊂R3. We prove the existence and uniqueness of solutions to the problem in anisotropic Sobolev spaces Wr2,1(Ω×(0,T)),r∈(1,∞). Thanks to the slip boundary conditions, the Stokes problem is transformed to the Poisson and the heat equation. In this way, difficult calculations that must be performed in considerations of boundary value problems for the Stokes system are avoided. This approach does not work for the Dirichlet and the Neumann boundary conditions. Because solvability of the Poisson and the heat equation is carried out by the regularizer technique, we have that S=∂Ω∈C1+α∩Wr3−1/r∩Wσ2−1/σ,σ > 3,α > 0. Copyright © 2014 John Wiley & Sons, Ltd.