Exact solution for fractional Navier-Stokes
July 8, 2015 by zulfahmed
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.
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.