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Nonstationary Stokes system in anisotropic Sobolev spaces
Author(s) -
Zaja̧czkowski Wojciech M.
Publication year - 2015
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3234
Subject(s) - sobolev space , mathematics , uniqueness , boundary value problem , mathematical analysis , bounded function , stokes problem , neumann boundary condition , dirichlet boundary condition , poisson's equation , mixed boundary condition , dirichlet problem , dirichlet distribution , homogenization (climate) , domain (mathematical analysis) , stokes flow , anisotropy , elliptic boundary value problem , robin boundary condition , geometry , physics , finite element method , biodiversity , ecology , flow (mathematics) , biology , thermodynamics , quantum mechanics
The nonstationary Stokes system with slip boundary conditions is considered in a bounded domain Ω ⊂ R 3 . We prove the existence and uniqueness of solutions to the problem in anisotropic Sobolev spacesW r 2 , 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 ∈ C 1 + α ∩ W r 3 − 1 / r ∩ W σ 2 − 1 / σ , σ > 3, α > 0. Copyright © 2014 John Wiley & Sons, Ltd.