"For the requirement that only real solutions are allowed to express themselves, then these complex parts (ghost equations) must necessarily add up to zero. This in essence supplies an additional equation within the solution space."

Look at the book on pdes by Garabedian. There's a discussion of pdes in the complex plane with application to the solution of some linear and nonlinear problems. If this was doable for the Navier-Stokes equations it would have already been done! Many of the techniques used to solve the Euler equations are related to such tricks (eg. conformal mappings and hodograph transforms).

There has been some research lately, for approximate forms of the Euler equations, using quaternions & Kahler algebras. However this will not work for the NSE because of the dissipation terms (i.e. the equations are not Hamiltonian - although it may work for the Euler equations which are).

Thanks for those thoughts & links, Tom. (You have the most amazingly broad literature review of anyone I've ever come across. Truly amazing. You have been so incredibly helpful over these past few years).

I have had my distant sights set on quaternions as a way of more tightly coupling temporal-spatial changes - this is an interesting development. I do feel that this coupling would really be of use in equations where the time-derivatives are of higher order.

Actually, during the past few days, I've been able to extend the Hui/Cox work even further now, into better understanding an arbitrary bulk + fluctuating split solution - for incompressible flows. I'm working on simulations to test the concept. Thus far, the closure equation on the bulk+fluctuating split seems to indicate an interesting simple coupling between the bulk & fluctuating solutions - in terms of wave-number components - different for the bulk & fluctuating portions - slow-fast waves/oscillations. I'm going to try & find the envelope of applicability for this theory.

Thanks so much for your thoughts.

mw

Link added

<http://www.atm.damtp.cam.ac.uk/peopl...m-myth-scanned>