- Let me end with a few words about the significance of the problems posed here.
- Fluids are important and hard to understand....our understanding is at a very primitive level.
- Standard methods from PDE appear inadequate to settle the problem.
- Instead, we probably need some deep, new ideas.
onsdag 27 april 2016
Reformulation of Clay Navier-Stokes Problem Needed 5
Fefferman concludes the official formulation of the Clay Navier-Stokes Problem with:
Yes, fluids are hard to understand for a pure mathematician and the understanding appears to be on a very primitive level, and this has led to an unfortunate formulation of the problem leading into a fruitless search for either (i) blowup into infinite fluid velocities in finite time as non-smoothness, or (ii) not blowup as smoothness of solutions.
But for a fluid the distinction between smoothness and non-smoothness concerns the size of velocity gradients. It is a well-known fact since long that compressible flow may exhibit non-smoothness in the form of shocks with large velocity gradients but without large velocities.
The Clay Problem concerns incompressible flow (at unbounded Reynolds numbers), which does not form shocks but instead becomes turbulent for large Reynolds number with again large velocity gradients as expression of non-smoothness, and (most likely) without large velocities.
The clue to solve the Clay Problem offered in the official formulation by searching for infinite velocities in fluid flow, which Tao has picked up in recent attempts to solve the problem, thus appears to be misleading and as such is not helpful to mathematics as science.
Fefferman is asking for some new ideas, but closes the door to any form of communication with computational turbulence as a new idea towards understanding and resolution of the problem.