Dear Terry,

No, this is not necessary. The regularity problem for NS (with small viscosity) is irrelevant. It is trivially solved by any of the many regularization techniques available. Since we are concerned with turbulent flow, the very notion of a constant (small) viscosity is an illusion without any form of mathematical or physical rationale. Since this is conceived to be the major difficulty of the Prize Problem, the formulation misses the point. Nobody knows what viscosity is, because it is a solution dependent quantity, and assuming that it is a small constant is without any reasonable foundation.

The essence are the stability properties as expressed in the linearized equations, which is a convection-reaction problem with oscillating reaction coefficient. Computatationally you discover a lot of cancellation making mean value outputs stable. This is remarkable and wonderful!!

Of course, a fully rigorous mathematical analysis of the lineraized problem would require an analytical solution formula for turbulent solutions, which is not available. But you can set up and study model problems.

This from of weak output stability is the essence of the NS problem with its turbulent solutions. Not the regularity problem, which is a trivial problem.

What you say is that you see no way to progress. Why insist on such a fruitless attitude?

You can get more information (in particular in the book Computational Turbulent Incompressible Flow discuusing the Clay Problem) from my home page at www.nada.kth.se/~cgjoh

I hope you will take a look. I am fully convinced that I have something interesting to contribute. And the alternative you outline seems to be without hope.

Best,

Claes