

Counter-Example to the Navier-Stokes Millennium Problem Found - jperras
http://ejde.math.txstate.edu/Volumes/2010/93/abstr.html

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slackenerny
I spent half an hour composing meaningful description of insignificance of the
work (without judging if it is indeed a result, attacked "uniqueness" is more
of a technicality than anything much of a big deal) only to find out author
also published stabs at some other Clay problems:
<http://arxiv.org/abs/0809.4935> (PNP), <http://arxiv.org/abs/0806.2361> (RH)…
at which point I have nothing more to say.

Flagged.

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lolipop1
So first, the equations seems to be mostly used to approximate and model a lot
of stuff. Very precise, I know.

Second, here's wikipedia take on the Millenium problem: Somewhat surprisingly,
given their wide range of practical uses, mathematicians have not yet proven
that in three dimensions solutions always exist (existence), or that if they
do exist, then they do not contain any singularity (smoothness). These are
called the Navier–Stokes existence and smoothness problems. The Clay
Mathematics Institute has called this one of the seven most important open
problems in mathematics and has offered a US$1,000,000 prize for a solution or
a counter-example.

Does the article directly relate to the Millenium Prize problems? I can't
conclude anything but it seems like just a small part of the whole. It think
people are excited because there was a lot of talk about the possible solution
to the P equals or not NP problem.

Maths at those levels are esoteric to most people and as such, presenting
papers like that is pretty useless in mainstream media. If one could find a
good analysis of the conclusions and possibilities, it might be a lot more
useful.

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golwengaud
It looks like the author is attacking statement C from the official problem
description ([http://www.claymath.org/millennium/Navier-
Stokes_Equations/n...](http://www.claymath.org/millennium/Navier-
Stokes_Equations/navierstokes.pdf)). That statement asks for "a smooth,
divergence-free vector ﬁeld u0 (x) on R3 and a smooth f(x, t) on R3 x [0, ∞),
satisfying [some physical-reasonableness conditions] for which there exist no
solutions (p, u) of the [the Navier-Stokes equation, a physical reasonableness
condition, and a sort of sanity condition]." In essence, you're finding a
counterexample for uniqueness.

The author claims to have found exactly such a combination of initial-value
field u0(x) and force function f(x), solving the millenium problem. I can't
really judge his claim yet, as I haven't read the whole paper (and I'm
probably not qualified to say even once I have), but slackenemy presents some
good reasons to be rather dubious elsewhere in the thread. Also, applying
Scott Aaronson's "Eight Signs A Claimed P≠NP Proof Is Wrong"
(<http://scottaaronson.com/blog/?p=458>) isn't pretty.

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seles
what

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seles
oh, this might help

"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."

<http://www.claymath.org/millennium/Navier-Stokes_Equations/>

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exit
that doesn't seem to setup an assertion for which a counter-example could be
shown.

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thurn
My understanding is that the Navier-Stokes problem has to do with showing that
smooth solutions always exist for the three dimensional system of equations,
so a counter-example would be a set of initial conditions for which no smooth
solution exists. On Wikipedia, the problem is described as:

"Prove or give a counter-example of the following statement:

In three space dimensions and time, given an initial velocity field, there
exists a vector velocity and a scalar pressure field, which are both smooth
and globally defined, that solve the Navier–Stokes equations."

[http://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smo...](http://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness)

