A type D breakdown of the Navier Stokes equation in d=3 spatial dimensions

TL;DR

This paper demonstrates a specific type D breakdown of the Navier-Stokes equations in three dimensions, highlighting limitations for analytical solutions and emphasizing the need for computational approaches in hydrology applications.

Contribution

It introduces a novel type D breakdown of the Navier-Stokes equations in three dimensions, extending previous findings and discussing implications for solving hydrology problems.

Findings

Type D breakdown occurs in 3D Navier-Stokes equations.

Analytical solutions are limited; computational methods are necessary.

Breakdown also observed in Euler equations.

Abstract

In this paper a type D breakdown of the Navier Stokes (NS) in d=3 is demonstrated. The element of the breakdown also occurs in the Euler equation. We consider the fact that in d=2 Ladyzhenskaya found a generalized type B solution. The discussion revolves around the notion, also found in quantum spin theory, that in the behavior of a system can be quite different from the behavior in d=2 dimensions. Concerning applications, our resolution of the problem implies that e.g. hydrology problems formulated as a NS equation can only be solved in computational approximation.