## onsdag 14 maj 2014

### Towards a Solution of the Clay Navier-Stokes Problem 2

The Clay Millennium Navier-Stokes problem concerns properties of solutions of the incompressible Navier-Stokes equations as the basic model of fluid mechanics of fundamental importance in both science and mathematics.

The Official Formulation Description by Charles Fefferman poses the following alternatives:
1. Existence of smooth solutions for all time from smooth initial data?
2. Cease of existence ("break-down" or "blow-up") of a solution from smooth initial data?
No progress towards a solution has been made since the formulation in 2000. Existence of smooth solutions for all time seems impossible since the viscosity term is not strong enough. All efforts to construct a solution with blow-up have failed because the viscosity term is too strong. No answer thus seems to be possible and a scientific dead-lock is reached.

Over the years I have, without success, tried to convey the message that the reason for the dead-lock is  that Fefferman's problem formulation is both mathematically and physically meaningless, because the fundamental aspect of (Hadamard) wellposedness or stability of solutions to perturbations is not included.

Including well-posedness leads to the following possible answer which is neither 1 nor 2 and which deals with case of small viscosity (compared to initial data):
• Turbulent solutions always develop in finite time from smooth initial data.
• A turbulent (non-smooth) solution is characterized by having a Navier-Stokes residual which is small in a weak $H^{-1}$-norm and large in a strong $H^1$-norm.
• Turbulent solutions are weakly wellposed by having stable mean-value outputs.
I have tried to get some comment from Terence Tao, Charles Fefferman and Peter Constantin, who are in charge of the problem formulation and serve as referees to evaluate proposed solutions. The response I get is that the problem formulation without wellposedness by Fefferman is fine as a mathematical problem, even if it does not make sense from physics point of view. The response is that it may well be that a solution will never be reached, but if so let it be.

But why not include wellposedness and make the Clay Navier-Stokes problem meaningful from a physics point of view and then meaningful as a challenge to development of mathematics? Why not open to possibility instead of impossibility? Why spend major efforts on a meaningless question without answer?

I pose this question to Fefferman, Constantin and Tao, with the hope of getting some response, to be reported.

PS1 We may compare with the lack of global warming since 2000: No progress of the temperature whatsoever. With this evidence one may ask if there may be some fundamental flaw in the idea of global warming.

PS2 Terence Tao sets out to "construct" a selfreplicating solution of the Navier-Stokes equations which "blows up" in a 70 page paper and pen excercise, which shows to be impossible. We let instead the computer construct solutions, which turns out to be possible, and we observe that the constructed solutions become turbulent and thus show a form of blow-up.

PS3 It does not seem that Fefferman et al are interested in communicating outside their own group and so they respond by silence, whatever it means.  Is this a sign of healthy strong science, which Mr. Clay presumably would prefer to support? The consequences are far reaching: If the Clay problem formulation is wrong, then something bigger is wrong.