tisdag 26 april 2016

Reformulation of the Clay Navier-Stokes Problem 4

The official formulation of the Clay Navier-Stokes Problem by Fefferman includes the following statements (with (A) and (B) global existence+regularity and (1)-(3) Navier-Stokes equations):
  • For initial data $u^0(x)$ not assumed to be small, it is known that (A) and (B) hold (also for $\nu = 0$) if the time interval $[0,∞)$ is replaced by a small time interval $[0,T)$, with $T$ depending on the initial data. 
  • For a given initial $u^0(x)$, the maximum allowable $T$ is called the “blowup time.” Either (A) and (B) hold, or else there is a smooth, divergence-free $u^0(x)$ for which (1), (2), (3) have a solution with a finite blowup time. 
  • For the Navier–Stokes equations ($ν > 0$), if there is a solution with a finite blowup time $T$, then the velocity $u_i(x,t)),1≤i≤3$ becomes unbounded near the blowup time.
We read that Fefferman claims that the distinction between (i) YES or (ii) NO to the question of existence+regularity for the Navier-Stokes equations, is between (i) bounded flow velocity for all time and (ii)  unbounded velocity for some "blowup time" $T$.

Fefferman here uses the same distinction as in the classical theory of ordinary differential equations (odes) based on a (correct) mathematical analysis showing that the only way a solution trajectory can cease to exist, is to tend to infinity in finite time.  

But this argument cannot be generalised to partial differential equations (pdes), because a smooth solution to a pde can cease to exist as a smooth solution because of unbounded derivatives of the solution, without the solution itself becoming infinite (as required in the ode case).  

The basic distinction for Navier-Stokes is instead between (i) laminar/smooth flow and (ii) turbulent/non-smooth for all time without blowup to infinity of the velocity, where non-smooth means large velocity gradients.

The official formulation of the problem is unfortunate by (incorrectly) claiming that the question can be reduced to a question of infinite velocities at finite blowup time. The Clay problem thus needs to be reformulated, since an incorrectly formulated problem can only lead in a wrong direction.

In a lecture about the problem, Cafarelli falls in the trap of Fefferman.  

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