tisdag 13 maj 2014

Towards Solution of the Clay Navier-Stokes Problem?

              Watch movie of turbulent flow as solution of the Navier-Stokes equations. 

Quanta Magazine reports in A Fluid New Path in Grand Math Challenge (Febr 24):
  • In a paper posted online on February 3, Terence Tao of the University of California, Los Angeles, a winner of the Fields Medal, mathematics’ highest honor, offers a possible way to break the impasse. 
  • He has shown that in an alternative abstract universe closely related to the one described by the Navier-Stokes equations, it is possible for a body of fluid to form a sort of computer, which can build a self-replicating fluid robot that, like the Cat in the Hat, keeps transferring its energy to smaller and smaller copies of itself until the fluid “blows up.” As strange as it sounds, it may be possible, Tao proposes, to construct the same kind of self-replicator in the case of the true Navier-Stokes equations. 
  • If so, this fluid computer would settle a question that the Clay Mathematics Institute in 2000 dubbed one of the seven most important problems in modern mathematics, and for which it offered a million-dollar prize. 
  • Is a fluid governed by the Navier-Stokes equations guaranteed to flow smoothly for all time, the problem asks, or could it eventually hit a “blowup” in which something physically impossible happens, such as a non-zero amount of energy concentrated into a single point in space?
  • Tao’s proposal is “a tall order,” said Charles Fefferman of Princeton University.
We read that the Grand Math Challenge of the Clay Navier-Stokes Problem is taken on by one of the world's sharpest mathematicians with the plan to construct a solution with smooth initial data which "blows up" in finite time, thus giving a negative answer to the Clay problem. 

Tao thus seeks to construct a "fluid computer" capable of answering a mathematical question concerning the Navier-Stokes equations.

Let us compare with our own approach to the Clay problem based on using a digital computer to solve the Navier-Stokes equations computationally, which offers the following answer for the case of small viscosity as presented in New Theory of Flight (see also blogpost):
  1. Computations produce from smooth initial data functions with Navier-Stokes residuals small in $H^{-1}$ and large in $H^1$, which are non-smooth solutions showing to have stable mean-value outputs and thus represent physical turbulent states.  
  2. Smooth solutions are unstable and thus do not represent physical states.       
In this analysis the aspect of stability is fundamental as identified by Hadamard as well-posedness. Unfortunately, the Clay problem formulation does not include the aspect of well-posedness, and thus is meaningless. Including well-posedness gives a new Clay problem, which can be answered in a meaningful way and this is what we seek to do.

Computations thus produce non-smooth approximate solutions which are well-posed in mean-value sense and thus physical solutions, while smooth solutions show to be unstable and thus are not physical solutions. Our answer is different from Tao's in that computed solutions initiated from smooth initial data do not "blow up" but instead turn turbulent with residuals becoming large in $H^1$ but with stable mean-value outputs.

I have asked Tao for a comment to the message of this post and will report.

More on the Clay problem here and here.

PS1 The fact that there has been no advance towards a solution of the Clay problem as formulated by Charles Fefferman in 2000, without reference to well-posedness, can be seen as evidence that the Clay question is ill posed and thus cannot be answered. The problem thus requires reformulation but mathematicians in charge of the problem formulation do not seem to be open to such a thing.

Hadamard's 1933 paper on the necessity of well-posedned seems to be forgotten. Strange. Very strange. The Navier-Stokes solution does not "blow up" but becomes non-smooth (turbulent), but this is not contained in the present formulation.

PS2 Quanta reports:
  • The real ocean doesn’t spontaneously blow up, of course, and perhaps for that reason, most mathematicians have concentrated their energy on trying to prove that the solutions to the Navier-Stokes equations remain smooth and well-behaved forever, a property called global regularity. 
  • Purported proofs of global regularity surface every few months, but so far each one has had a fatal flaw. (The most recent attempt to garner serious attention, by Mukhtarbay Otelbaev of the Eurasian National University in Astana, Kazakhstan, is still under review, but mathematicians have already uncovered significant problems with the proof, which Otelbaev is trying to solve.)
Amazing: It is observed that the ocean does not blow up spontaneously, but ocean motion is partly turbulent and thus is not smooth and well-behaved and thus falls outside the allowed categories in the Clay problem, as either staying smooth or blowing up. No wonder that the problem as formulated has no solution. See also following post.

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