  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.

Fluid Dynamics;Analysis of PDEs;Dynamical Systems;Chaotic Dynamics