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What is the New Look of Terence Tao at the Navier Stokes equations?

+ 7 like - 0 dislike
5141 views

In a recent paper by Terence Tao (http://arxiv.org/abs/1402.0290), "Finite time blowup for an averaged three-dimensional Navier-Stokes equation," seems to suggest a new look on this problem.

I have briefly read through this commentator report: https://www.simonsfoundation.org/quanta/20140224-a-fluid-new-path-in-grand-math-challenge/.

I would like to probe to potential mathematical physcisits in this PO site,

What exactly is the new result or new thinking that Terence Tao provides in his paper? How relevant/important is this to solve the Millenium problem for Navier-Stokes equation?

add: There is also a very nice comment from Scott Aaronson http://www.scottaaronson.com/blog/?p=1697, who comments about the papers from L Susskind and T Tao, how their observations shade new lights on (classical or quantum) computations.

asked Apr 6, 2014 in Mathematics by wonderich (1,400 points) [ revision history ]
edited Apr 10, 2014 by wonderich

1 Answer

+ 9 like - 0 dislike

It's an approach for finding a counterexample, in case that smoothness fails. The idea is to find scaling-down blowups for Navier-Stokes type problems, so that a solution near his kind at scale A reproduces a very similar solution at scale A/2, and so on to A/4 until you get a singularity at a finite time. He has given a rather beautiful and nifty way of doing this construction in systems which are very similar formally to Navier Stokes, but modified so that he is able to control the reproduction process at each stage, and prove stability, so that little changes don't wreck his constructions.

The main thing about his method is that it requires that the fluid is actually capable of storing information stably for long enough to construct the next smaller stage, and this is likely false in the highly chaotic world of the turbulent systems, but who knows, it needs to be investigated thoroughly. Unlike other problems, Riemann Hypothesis, or Yang-Mills mass gap, this Clay problem we don't know the answer for sure, there might be a surprise.

If the solutions to Navier Stokes are actually smooth, which is more likely given present knowledge (although not certain), then his method will not solve the problem, rather they will work to exclude certain avenues of proof.

answered Apr 6, 2014 by Ron Maimon (7,535 points) [ no revision ]

Hm just from reading this nice answer, this systematic downscaling remotely reminds me of some kind of applying successive inverse RG transformations ... 

A very nice answer. Thank you, I hear your words. +1.

Ok, but please don't upvote too much, it's not a difficult answer.

There is also a very nice comment from Scott Aaronson: http://www.scottaaronson.com/blog/?p=1697, who comments about the papers from L Susskind and T Tao, how their observations shaded new lights on (classical or quantum) computation.

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