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## Exact solution for fractional Navier-Stokes

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

3:52 PM (39 minutes ago)

 to David, bcc: aimee
Dear Professor Donoho,

For applications to financial volatility, one does not hope for an exact partial differential equation like fractional Navier-Stokes but rather a fractional diffusion with some convection term that can produce a phase transition to turbulence.  I examined the feature that allows Chaurasia-Kumar to solve the fractional Navier-Stokes exactly, and it is that in appropriate coordinates, the space terms become

d^2 u/dr^2 + (1/r) du/dr

and one can obtain an exact formula for the finite Hankel transform for this (see snapshot).  Assuming that phase transition to turbulence persists in the fractional Navier-Stokes as well, I would seek comparison theorems for phase transitions:

if fNS has phase transition to turbulence and the convection term (1/r) du/dr is appended by some F(u) > 1/r du/dr, then the fNS-type equation with F as convection term is also turbulent.  But perhaps more precise results are possible.  I will simply queue these questions for my period of study the next several months with some relief from housing woes in San Francisco.

2 Attachments

Preview attachment Screenshot 2015-07-08 15.34.49.png

Screenshot 2015-07-08 15.34.49.png

Preview attachment chaurasia-solution-fractional-navier-stokes.pdf

chaurasia-solution-fractional-navier-stokes.pdf

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

4:31 PM (0 minutes ago)

 to David, bcc: aimee
There is a fractional Hankel transform for which (1/x df/dx) has a nice formula which could apply to the exact fractional Navier-Stokes.  Unfortunately, it is still unclear whether the form of the convection term that we could determine from empirical volatility data is in such a nice parametric form.  Therefore comparison theorems are necessary.  In general linear parabolic equations of ‘Schroedinger type’ where instead of a convection term one has multiplier potential then one can use Feynman-Kac formula. I hope to clear up this issue when clear numers are available for convection terms.

2 Attachments

Preview attachment Screenshot 2015-07-08 15.34.49.png

Screenshot 2015-07-08 15.34.49.png

Preview attachment 2012-fractional-hankel-transforms.pdf

2012-fractional-hankel-transforms.pdf
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