torsdag 21 februari 2019

Hamming and Tartar on Clay Navier Stokes Problem

Richard Hamming (1915-98)

The mathematician Richard Hamming said:
  • Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.
An example is given in the official formal formulation of the Clay Navier-Stokes Problem by Fefferman, which does not mention the world turbulencewhich in the informal presentation is central:
  • Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.
Everybody, except Fefferman, understands that turbulence is the secret hidden in Navier-Stokes and that the Clay problem, to be more than an intellectual sport standing in the way for sensible information, should ask about a mathematical theory unlocking the secret of turbulence. 

The informal presentation is sensible, while the formal presentation is nothing but an intellectual sport, which neither has practitioners since no progress towards a resolution has been made since 2000, or rather since 1932 when Leray proved existence of weak solutions with uniqueness or wellposedness left completely open.  

We have presented a resolution to a reformulated Clay Problem offering sensible information about the physical process of turbulence, by computation. We hope there are some sensible people that can show a reaction to our resolution. We show by computation that weak solutions exist, are non-smooth/turbulent and have wellposed mean-values such as drag and lift. 

Hamming also said:
  • The purpose of computing is insight, not numbers. … 
  • [But] sometimes … the purpose of computing numbers is not yet in sight.
Yes, we find that being able to compute (turbulent) solutions to Navier-Stokes equations opens to gain insight into the nature and manifestation of turbulence.  DFS is in sight and gives insight! 

So, what insight has DFS brought? Here is one major revelation:
  • bluff body flow = potential flow + turbulent 3d rotational slip separation.
Bluff body flow is thus computable by DFS, which offers a revolutionary new capacity to CFD with a vast field of applications for all sorts of vehicles or life moving through air and water, and the fluid mechanics is understandable!

Also note what the mathematician Luc Tartar says in the presentation of his book on Navier-Stokes:
  • To an uninformed observer, it may seem that there is more interest in the Navier-Stokes equation nowadays, but many who claim to be interested show such a lack of knowledge about continuum mechanics that one may wonder about such a superficial attraction. 
  • Could one of the Clay Millennium Prizes be the reason behind this renewed  interest?
  • Reading the text of the conjectures to be solved for winning that particular prize leaves the impression that the subject was not chosen by people interested in continuum mechanics, as the selected questions have almost no physical content.
  • The problems seem to have been chosen in the hope that they will be solved by specialists of harmonic analysis...
  • I  hope that this particular set of lecture notes...may help the readers understand a little more about the physical content of the equation, and also its limitations, which many do not seem to be aware of.
And as before: the pure mathematicians Fefferman, Constantin and Tao in charge of the problem formulation refuse to participate in any form of discussion.  Why? Lack of knowledge about continuum mechanics, with focus instead on harmonic analysis?

And remember:
  • What is computable is understandable. (Pythagoras)
Luc Tartar


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