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

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 ( backward difference (2)u(x, y, z) = F-x (x, y, z, t) 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 backward difference (2)u(x, y, z) = 0. 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 orphysically reasonable solutions of the Navier-Stokes equation for transitional flow and turbulence in the global domain due to singularity.

Automated summary

For laminar flow in a channel, the Navier-Stokes equation has a non-zero source term and solution; for transitional flow, the velocity profile becomes distorted with inflection points where backward difference is zero; due to singularity, there are no smooth or physically reasonable solutions of the Navier-Stokes equation for transitional flow and turbulence in the global domain.