A class of exact solutions of the Navier-Stokes equations in three and four dimensions

A few basic, intuitive, properties of the Navier-Stokes system of equations for incompressible fluid flows are discussed in this paper. We present a rephrased interpretation of the Navier-Stokes equation in a space having an arbitrary number of dimensions. We then derive spatially periodic solutions for the velocity and pressure fields that span an unbounded domain in three and four dimensions, given a smooth solenoidal initial velocity vector field. In these solutions all velocity components depend non-trivially on all coordinate directions. & COPY; 2023 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

This paper discusses a few basic, intuitive properties of the Navier-Stokes equations for incompressible fluid flows. The authors propose a rephrased interpretation of the Navier-Stokes equation in a space with arbitrary dimensions. They then derive spatially periodic solutions for the velocity and pressure fields that span an unbounded domain in three and four dimensions, given a smooth solenoidal initial velocity vector field. In these solutions, all velocity components depend non-trivially on all coordinate directions.