onsdag 21 maj 2014

Tao on Clay Navier-Stokes and Turbulence?

Terence Tao is working on the Clay Navier-Stokes Prize Problem and in a recent post considers  Kolmogorov's power law for turbulence. A heuristic derivation goes as follows: The smallest spatial scale $\epsilon$ of a fluctuation $u$ of turbulent incompressible flow of small viscosity $\nu >0$ is determined by a local Reynolds number condition
  • $\frac{u\epsilon}{\nu}\sim 1$.
Assuming the smallest scale carries a substantial part of the total dissipation gives 
  • $\nu(\frac{u}{\epsilon})^2\sim 1$.
Combination gives 
  • $u\sim \nu^{\frac{1}{4}}$
  • $\epsilon\sim\nu^{\frac{3}{4}}$ 
suggesting that the turbulent solution is Lipschitz continuous with exponent $\frac{1}{3}$. 

My question to Tao posed as a post comment is if according to the Clay problem formulation, such a $Lip^\frac{1}{3}$ turbulent solution with smallest scale $\nu^\frac{3}{4}$ is to be viewed as a smooth solution for any small $\nu >0$?

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