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

TL;DR

This paper derives explicit, spatially periodic solutions to the Navier-Stokes equations in three and four dimensions, providing insights into the behavior of incompressible fluid flows in higher-dimensional spaces.

Contribution

It introduces a new class of exact solutions for the Navier-Stokes equations in multiple dimensions, expanding understanding of fluid dynamics beyond traditional three-dimensional analysis.

Findings

Derived spatially periodic solutions in 3D and 4D

Solutions depend non-trivially on all spatial coordinates

Applicable to smooth solenoidal initial velocity fields

Abstract

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.