Re: solving this impasse could result in solving the Navier-Stokes

David Purvance wrote:
On Aug 23, 4:35 pm, David Purvance <d.purva...@xxxxxxx> wrote:
On Aug 23, 4:08 pm, Stephen Montgomery-Smith

<step...@xxxxxxxxxxxxxxxxx> wrote:
David Purvance wrote:
On Aug 23, 12:57 pm, Stephen Montgomery-Smith
<step...@xxxxxxxxxxxxxxxxx> wrote:
David Purvance wrote:
On Aug 23, 10:44 am, Robert Israel <isr...@xxxxxxxxxxx> wrote:
On Aug 23, 5:06 am, David Purvance <d.purva...@xxxxxxx> wrote:
On Aug 22, 4:09 pm, Stephen Montgomery-Smith
<step...@xxxxxxxxxxxxxxxxx> wrote:
David Purvance wrote:
On Aug 21, 8:37 pm, Robert Israel
<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Stephen Montgomery-Smith <step...@xxxxxxxxxxxxxxxxx> writes:
David Purvance wrote:
This problem is for math professionals.
Euler's equation can be written as a rather simple looking matrix
differential equation
du/dt=A(u) u (1)
where u is an incompressible flow field in wavenumber space and A(u)
is a matrix that is a function of u. It is easy to add viscous shear
to (1) to obtain the Navier-Stokes equations.
The Taylor time expansion of u in Euler's equation results in a second
equivalent matrix differential equation
du/dt = sum{A_n(c_n) t^n} u (2)
where A_n are matrices that are a function of wavenumber alone and c_n
are the Taylor expansion coefficients of u. A_n diagonalize nicely,
i.e., their eigenvectors form unitary matrices and their eigenvalues
are zero or purely imaginary, suggesting that if A_n commute and when
viscous shear is added, the Navier-stokes equations are stable for all
time. Proving that A_n commute boils down to proving if the
eigenvectors of A(u) in (1) can be a function of time. If they cannot,
then the Navier-Stokes Millennium problem is solved,
The proposed solution to (1) and relevant discussion can be found at:
"http://purvanced.wordpress.com/2007/05/09/by-david-purvance/";.
Please chime in and help us resolve this impasse.
Come on guys, and help us out. I am trying to convince David that the
step (7-23) => (7.24) is incorrect, because he doesn't have any reason
to suppose that V is not a function of t. You can more or less read
(7-18) through (7-24) in isolation, without having to read the rest of
the paper.
I am rather convinced that his assertion that the matrices in question
diagonalize nicely are completely correct, and also that his starting
equation (7-1) is correct. Indeed my sense is that he did rather well
to get as far as he did, but that his mistake is essentially an
elementary mistake in finite dimensional linear algebra, and sci.math
regulars like David Ullrich or Robert Israel definitely have sufficient
expertise to help me to point out to David the flaw in his argument.
I think you said it well enough: what do you need me for? Well, perhaps
to find a simple example. Consider
[ t 1 ] [ 0 1 ] [ 1 0 ] [ 0 0 ]
A = [ t^2 t ] = [ 0 0 ] + t [ 0 1 ] + t^2 [ 1 0 ] = A0 + A1 t + A2 t^2.
We diagonalize: A = V L V^(-1) with
[ 1 -1 ]
V = [ t t ]
[ t/2 t/2 ] [ 1/(2t) -1/(2t) ]
but V^(-1) A0 V = [ -t/2 -t/2 ] and V^(-1) A2 V = [ 1/(2t) -1/(2t) ]
are not diagonal.
--
Robert Israel isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
In your example where is the time-invariant vector v such that v^T A =
0 for all t? v is required for incompressibility. I'm all for counter
examples, but pleeeease make them realistic, friend.
But you can make Robert's example include a time invariant vector of the
form you describe. Simply embed his example into a 3 by 3 matrix, in
which all the other entries are 0. Then v=(0,0,1) does what you describe.
I would think that at a minimum a realistic counter example would have
to be at least a 9x9 with v=(k1,k2,k3,0,0,0,-k3,-k2,-k1) (v_perp in my
paper) plus a 3D u_0 from which you could verify v was an eigenvector
of A_0.
I really don't see how Robert Israel's example will lead to anything,
i.e., you get A_1 from knowledge of A_0, A_2 from A_0 and A_1, etc.,
and assuming a matrix exponential solution, which, in turn , depends
on assuming what you are trying to disprove. (You misled the reader a
little when you said he/she only needs to look at (7-18) thru (7-24)
and it led to his rather childish example.)
My paper's Comment 62 at "http://purvanced.wordpress.com/2007/05/09/by-
david-purvance/" pretty much shows why the idea of multiple flow bases
is contradictory.
My "rather childish example" was not meant to be realistic in terms of
your problem. It's simply to illustrate that there's a gap in one
step of your proof. It may well be that the matrices V^(-1) A_k V are
diagonal in the case you're considering, but as far as I can see you
haven't proven it.
Robert Israel isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
And just how does your "Winnie the Pooh" example illustrate that
there's a gap in one step in my proof? Be very specific, please. Eq.
(7-21) states that V^(-1) A_n V must be diagonal.
David, you asked if colleagues of mine could chime in. Robert Israel
chimed in, and he completely agrees with me that you haven't proved your
assertion. Please don't insult Robert by calling his examples "Winnie
the Pooh."
Look, instead of getting defensive when people try to find flaws in your
ideas, why don't you go back and think very hard about either way they
are right or why you are right. I think you are answering back too
quickly, and without enough thought.
And honestly, if you really have cracked the problem (which I think you
haven't), your solution is on the internet for all to see. You have
priority on your solution - no-one can steal it!
I think that you did well to get as far as you did. But even the best
of us make mistakes. I myself have worked on this problem for over 12
years, and I cannot tell you how many times I have thought I have
cracked it! But, based on years of hard won experience, my first
thought is that I have made a mistake. And sometimes I spend weeks
looking for my mistake, which sometimes is quite subtle. In fact, one
time I even made exactly the same mistake you made - assuming that the
derivative of e^{A(t)} is A'(t) e^{A(t)} when A(t) is a time dependent
matrix.
So if I, a professional mathematician, can take weeks to see my mistake,
think that maybe you also might be subject to the same flaw, and may
also take a long time to see where you went wrong.
I liked the way you thought about the problem. You tried to think "out
of the box." But actually nearly all mathematicians try to think "out
of the box." I have tried to throw really crazy ideas at this problem -
not only do I find that they don't work, but then later I find that
other people have also tried exactly the same idea many years before me!
All of us working on this problem are trying really, really hard, and
we are desperately looking for that sleek simple solution that somehow
everyone else missed, as well as those complicated and sophisticated
solutions that also don't work.
So please, allow for the fact that you might have made a mistake. We
are not involved in some conspiracy where only professional
mathematicians are allowed in the "math club." If your solution really
was correct, I assure you that I and Terry Tao would have recognized it,
and would be loudly and gladly trumpeting your great discovery, giving
you full credit.
Best regards, Stephen
Dear Stephen,
What conspiracy?
I have a right to be frustrated with both you and Robert. You both
have given little 2x2 counter examples that I show don't apply to the
NSE and then you back off and say your examples mean something more
general, which I fail to see. This is engaging in "Winnie the
Poohism" as far as I am concerned. And to prove it, rationally like
professional mathematicians, let Robert successfully reply to my
request that he specifically explain how his "now generalized" example
demonstrates that there is a gap in my reasoning. Otherwise you're
both copping out.
"
Dear Robert,
Two comments up there is a typo in my challenge to you. V^(-1) A_n V
should read sum(V^(-1) A_n V). I apologize.
You might be frustrated with us, but I am also getting frustrated with
you. I don't understand why you don't understand that our small 2 by 2
examples are completely sufficient to show that your method of proof is
flawed. If you want larger dimensional examples, perhaps with a few
more properties, simply embed them into larger numbers of dimensions, or
see if you can add more properties. I have identified a very particular
step in your reasoning where I think a proof is missing, and you have
not provided the missing proof. I and Robert have examples to show
this. That they are simple 2 by 2 examples only strengthens our
assertions, it does not weaken them!
As for your remark "Why didn't Terence Tao's comment persuade you quit
conversing with me long ago?" - what does that mean?
Stephen
You are talking in general terms. Be specific, i.e., use mathematical
equations to show how your "now generalized" 2x2 examples mean
anything relevant to the impasse. (You'll have to pull your examples
from my blog.) You hold my feet to the fire and now I am asking
politely that you take some of your own medicine.

Terence essentially said that, as my solution was presented, "B"
cannot be deduced from "A". I didn't interpret what he said to mean
that it was impossible to deduce "B" from "A" and that my solution was
fatally and forever in error. However, if his argument was so
convincing to you as you implied today, why didn't you blow me off
months ago? I didn't solicit your comments on my blog. Most of them
were very helpful, but, why did you bother?

Actually, enough of this already. Please just be mathematically
specific on how your examples further your criticism. This is all I
ask .

For some reason my last comment didn't display, so I'll resubmit:

You are talking in general terms. Be specific, i.e., use mathematical
equations to show how your "now generalized" 2x2 examples mean
anything relevant to the impasse. (You'll have to pull your examples
from my blog.) You hold my feet to the fire and now I am asking
politely that you take some of your own medicine.

Terence essentially said that, as my solution was presented, "B"
cannot be deduced from "A". I didn't interpret what he said to mean
that it was impossible to deduce "B" from "A" and that my solution was
fatally and forever in error. However, if his argument was so
convincing to you as you implied today, why didn't you blow me off
months ago? I didn't solicit your comments on my blog. Most of them
were very helpful, but, why did you bother?

Actually, enough of this already. Please just be mathematically
specific on how your examples further your criticism. This is all I
ask.


I have stated that I don't see how (7-23) => (7-24). Robert has provided a 2x2 example that shows that in general this step does not follow. You reply that his example is "not realistic." But his example is a counterexample to this assertion.

Now what you have to do is provide a proof that (7-23) => (7-24). It will have to use extra properties. You have stated that the existence of v_perp, a vector in all of the matrices kernels, is enough to provide this implication. But you haven't provided a proof.

I have stated that I don't believe that (7-24) is true in this instance, but I haven't provided a proof of this. So it is still open for you to prove this.

I have suggested that you do some numerical calculations and see if (7-24) is true. Personally I think that would settle the issue.

Stephen
.

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