Terence Tao on the Navier-Stokes Millennium Problem

This is a continuation a previous post on the Navier-Stokes Clay Mathematics Institute Millennium Problem stimulated by a recent article uploaded by Terence Tao: 

• Quantitative bounds for critically bounded solutions to the Navier-Stokes equations

arguing that:

• If a certain norm of the solution stays bounded (or grows at a controlled rate), then the solution stays regular.
• Taken in the contrapositive, they assert that if a solution blows up at a certain finite time T, then certain norms of the solution must also go to infinity.

The certain norm can be the $L^3$ norm over the spatial domain of the velocity $u(t)$ for $t<T$, which thus would serve as an indicator for solution of the Navier-Stokes Millennium Problem. 

I have in sequence of posts expressed criticism of the formulation of the Millennium Problem which can be condensed into: 

• The normalisation to unit viscosity means that the physics of turbulence, which appears for small viscosity and bounded flow velocities, is missing. This makes the Millennium Problem into a purely academic mathematical problem without significance to the real world of fluid flow (and so probably violating the intention of Mr Clay). 
• More precisely, a turbulent solution would not be a classical regular solution nor a solution with unbounded velocities, thus a solution outside the present formulation of the Millennium Problem.

It is thus natural to ask for a reformulation of the Navier-Stokes Problem, in particular since no advance towards a solution in the original formulation has been made since 2000. Compare with this article on Knowino. 

PS This post was triggered by a new comment to the original post by Terence from 2019. See also my next post here.