Resolving this impasse could result in solving the Navier-Stokes Millennium Problem
- From: David Purvance <d.purvance@xxxxxxx>
- Date: Tue, 21 Aug 2007 12:41:06 -0700
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.
.
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