Symplectic Representation and Turbulent Global Solutions of Incompressible Navier-Stokes Equations in ${\mathbb{R}}^3$

The incompressible Navier-Stokes equations are considered. We find that there exist infinite non-trivial solutions of static Euler equations. Moreover there exist random solutions of static Euler equations. Provided Reynolds number is large enough and time variable $t$ goes to infinity, these random solutions of static Euler equations are the path limits of corresponding Navier-Stokes flows. But the double limit of these Navier-Stokes flows do not exist. Therefore these solutions are called turbulent solutions.