fredag 16 maj 2014

Towards Computational Solution of Clay Navier-Stokes Problem 3

The formulation of the Clay Navier-Stokes Prize problem is unfortunate, or more precisely both mathematically and physically meaningless, because the following two completely fundamental aspects are not included:
1. wellposedness
2. turbulence.
To see the effect consider exterior flow with a slip boundary condition, which allows a unique stationary smooth near-solution as potential flow with a Navier-Stokes residual, which scales with the viscosity $\epsilon$. Smooth potential flow thus offers a solution to the NS equations with a
vanishingly small residual under vanishingly small viscosity. But potential flow is not stable since it  under small perturbation develops into a completely different turbulent solution. In other words, potential flow is not wellposed in any sense and thus not a physical solution.

The present problem formulation without 1 and 2 does not allow unphysical smooth potential flow to be distinguished from physical turbulent flow. The result is that the Clay NS problem has no meaningful solution and does not serve the purpose of a Prize problem.

Note that the Clay NS problem is introduced with the following description of the essence of the problem and its importance to humanity:
• 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.
But turbulence is not an issue in the official formulation. The secret to unlock is turbulence, but that is not part of the problem formulation. Something is weird here. I have pointed that out to the President of Clay Mathematics Institute and will report the reaction. Here is the letter:

President
Clay Mathematics Institute

I want to convey the information that the formulation of the Clay Navier-Stokes problem is incorrect both mathematically and physically, because the fundamental aspects of (i) wellposedness and (ii) turbulence, are not included, as exposed in detail in the following sequence of blog posts:

The result is that the problem cannot be given a meaningful solution and thus does not serve well as a Prize problem. Evidence is given by the fact that no progress towards a solution has been made.

I have tried to engage Charles Fefferman, who has formulated the problem, Peter Constantin, who acts as a referee, and Terence Tao, who is working on the problem, into a discussion, but I get no response.

I hope this way to stimulate discussion, which I think would be more constructive than no discussion.

Sincerely, Claes Johnson