Periodic Solutions in $\mathbb R^n$ for Stationary Anisotropic Stokes and Navier-Stokes Systems

TL;DR

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

Contribution

It establishes the existence and uniqueness of solutions for linear and nonlinear anisotropic systems using fixed point theorems in a periodic Sobolev space framework.

Findings

Proved solution existence for anisotropic Stokes system.

Extended results to nonlinear Navier-Stokes system.

Established regularity results for solutions.

Abstract

First, the solution uniqueness and existence of a stationary anisotropic (linear) Stokes system with constant viscosity coefficients in a compressible framework on $n$-dimensional flat torus are analysed in a range of periodic Sobolev (Bessel-potential) spaces. By employing the Leray-Schauder fixed point theorem, 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. Then the solution regularity results for stationary anisotropic Navier-Stokes system on torus are established.