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

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.

Thanks, Stephen
.

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