# The Statistics of Navier Stokes Equations

1. Aug 20, 2009

### John Creighto

I am curious about what insight people might have as to the statistics of Navier Stokes equation. I thought of the following way someone might try to calculate these.

1) Choose a bais (Basis A)
2) Pick a discrete number of points to constrain the solution of stokes equation.
3) Find the solution space of this basis when constrained by these points (This can be done by row reduction and will give you a new basis) (Basis B)
4) Relate the derivative of stokes equation, to this new basis via a matrix equation.
5) Set the derivative equal to zero at the constraint points chosen above.
6) Find the new solution space given this derivative constraint. (Basis C)
7) Orthogonalize this basis (use singular value decomposition) to get a new basis again. (Basis D)
8) Find the energy of each basis function from Basis D.
9) Assign the probability of each basis function based upon the energy of each basis function and the temperature of the region.
10) Knowing the probability of each basis function average over space and time to calculate the statistics.

One problem I see is that the energy of each basis function will likely depend upon which basis functions are occupied. The reason is that for instance the energy of wind is related to the cube of the velocity. However, at each estimate of the statistics the energy of each basis function could possibly be re-evaluated. Thus maybe any nonlinearities could be taken care of iteratively. Also note that the basis functions are chosen to be orthogonal, so perhaps this would simplify these nonlinarities somewhat.