Stationary Anisotropic Stokes, Oseen, and Navier-Stokes Systems: Periodic Solutions in $\R^n$

Abstract

First, the solution uniqueness, existence and regularity for stationary anisotropic (linear) Stokes and generalised Oseen systems with constant viscosity coefficients in a compressible framework are analysed in a range of periodic Sobolev (Bessel-potential) spaces on $n$-dimensional flat torus. By the Galerkin algorithm, the linear results are employed to show existence of solution to the stationary anisotropic (non-linear) Navier-Stokes incompressible system on torus in a periodic Sobolev space for any $n\ge 2$. Then the solution uniqueness and regularity results for stationary anisotropic Navier-Stokes system on torus are established for $n\in\{2,3,4\}$.