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

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions for stationary anisotropic Stokes, Oseen, and Navier-Stokes systems on an n-dimensional torus, extending classical results to anisotropic and periodic settings.

Contribution

It establishes existence, uniqueness, and regularity results for stationary anisotropic Stokes, Oseen, and Navier-Stokes systems in periodic Sobolev spaces, including nonlinear Navier-Stokes solutions.

Findings

Proved solution existence for anisotropic Stokes and Oseen systems in periodic Sobolev spaces.

Established uniqueness and regularity for anisotropic Navier-Stokes in dimensions 2, 3, and 4.

Demonstrated the applicability of Galerkin methods in this anisotropic periodic framework.

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 $n\in\{2,3,4\}$.