No Existence and Smoothness of Solution of the Navier-Stokes Equation

TL;DR

This paper argues that smooth, physically reasonable solutions to the Navier-Stokes equation do not exist for transitional flow and turbulence because of singularities at inflection points where the source term vanishes.

Contribution

It presents a novel argument that the Navier-Stokes equation lacks smooth solutions in transitional and turbulent regimes due to inherent singularities.

Findings

Singular points occur at inflection points in velocity profiles.

The Poisson form of Navier-Stokes becomes singular at these points.

No smooth solutions exist for transitional and turbulent flows.

Abstract

The Navier-Stokes equation can be written in a form of Poisson equation. For laminar flow in a channel (plane Poiseuille flow), the Navier-Stokes equation has a non-zero source term and a non-zero solution within the domain. For transitional flow, the velocity profile is distorted, and an inflection point or kink appears on the velocity profile, at a sufficiently high Reynolds number and large disturbance. In the vicinity of the inflection point or kink on the distorted velocity profile, we can always find a point where the source term is zero. At this point, the Poisson equation is singular, due to the zero source term, and has no solution at this point due to singularity. It is concluded that there exists no smooth and physically reasonable solutions of the Navier-Stokes equation for transitional flow and turbulence in the global domain due to singularity.