Spatially-Periodic Solutions for Evolution Anisotropic Variable-Coefficient Navier-Stokes Equations: II. Serrin-Type Solutions

TL;DR

This paper proves the existence of weak, space-periodic solutions to anisotropic Navier-Stokes equations with variable viscosity, extending the understanding of non-stationary fluid flows with complex viscosity structures.

Contribution

It introduces a Galerkin-based method to establish the existence of Serrin-type solutions for anisotropic, variable-coefficient Navier-Stokes equations with relaxed ellipticity conditions.

Findings

Existence of weak solutions with velocity in specified function spaces

Discussion of solution uniqueness and regularity

Application of Galerkin algorithm to complex anisotropic fluids

Abstract

We consider evolution (non-stationary) space-periodic solutions to the $n$-dimensional non-linear Navier-Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in space and time and satisfying the relaxed ellipticity condition. Employing the Galerkin algorithm, we prove the existence of Serrin-type solutions, that is, weak solutions with the velocity in the periodic space $L_2(0,T;\dot{\mathbf H}^{n/2}_{\#\sigma})$, $n\ge 2$. The solution uniqueness and regularity results are also discussed.