On Landau's Solutions of the Navier-Stokes Equations

TL;DR

This paper explores Landau's explicit solutions to the steady-state 3D Navier-Stokes equations, examining the effects of relaxing symmetry assumptions and extending considerations to higher dimensions.

Contribution

It investigates the impact of reducing symmetry constraints on Landau's solutions and discusses implications for broader classes of solutions and higher-dimensional cases.

Findings

Symmetry assumptions simplify Navier-Stokes to ODEs.

Dropping symmetry leads to more complex PDE analysis.

Implications for general solutions and higher dimensions are discussed.

Abstract

In 1944 L.D.Landau calculated a very interesting family of explicit solutions of the steady-state 3d Navier-Stokes equations. The solutions are derived under certain assumptions of symmetry, which reduce the Navier-Stokes equations to a system of ODEs. We investigate what happens when some of the symmetry conditions are dropped (and we have to deal with PDEs). Implications of these calculations for more general classes of solutions are also discussed. We also discuss the situation for general dimension.