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

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.

.

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