torsdag 14 november 2019

Solving the Clay Navier-Stokes Problem with Meaningless Mathematics?

The Clay 2000 Millennium Navier-Stokes problem concerns solutions to the incompressible Navier-Stokes equations:
  • $\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p -\nu\Delta u =0$,
  • $\nabla\cdot u=0$,
where $u(x,t)$ is velocity and $p(x,t)$ pressure depending on a space coordinate $x\in R^3$ and time coordinate $t\ge 0$, $\nu$ is a positive (constant) viscosity, and an initial velocity is given at $t=0$.

The question posed in the official formulation of the problem is:
  • Do smooth solutions exist for all time (global in time)?
  • Or do solutions cease to exist at some finite time (finite time break down)?
Mathematician have been struggling with this problem since the equations were formulated in the 1830s, however with little progress, in particular after 2000. 

The present main assault to solve the problem is led by Terence Tao as the most able mathematician on Earth. Tao approaches the problem along a well traced path based on a theorem stating that if velocities are (suitably) bounded, then smooth solutions existing for small time if initial data are smooth, will not cease to exist and thus exist for all time.

In short: Bounded solutions will stay smooth. And the other way around: The only way smooth solutions may cease to exist is through velocities blowing up to infinity.

In a recent article Tao seeks to give this purely heavily qualitative result (with very little information) a quantitive form (with hopefully more information). The basic result is stated as Theorem 1.2 taking the basic form: If velocities by are bounded by some positive constant A, then first derivatives of velocity and vorticity are bounded by constants of size:
  •    exp exp exp A $= e^{e^{e^A}}$.
In short, if velocities are bounded, then so are gradients (and similarly higher derivates) and so a solution initialised as smooth will stay smooth. 

But the bound on the derivatives with the triple exponent makes no sense. From any reasonable point of view the bound is infinite and thus says nothing about smoothness. 

In this approach to the Clay problem made by mathematicians it appears that reason is gone: If a smooth solution can have basically infinitely large derivatives, then the concept of smoothness is twisted away from any reasonable meaning. Is the idea to solve the Clay problem with meaningless mathematics, to report that it has been solved, once and for all?

In several previous posts I have indicated a different approach to resolve the Clay problem in a meaningful way. Take a look.  The basic insight is that smooth solutions of Navier-Stokes equations in general develop into turbulent solutions which are not smooth. But this does not appear to be something a (pure) mathematician can accept, and then not the Clay Prize committee, even if this is the truth. Is this as an expression of crisis of modern mathematics? Or not at all?

So when is fluid flow turbulent non-smooth? The answer is: When viscous dissipation is of the same size as kinetic energy. More precisely, the basic energy estimate for Navier-Stokes equations reads:
  • $\int \vert u(x,T)\vert^2 dx +2\int_0^T\int\nu\vert\nabla u(x,t)\vert^2dxdt =\int\vert u(x,0)\vert^2 dx$
for $T\gt 0$ with on the left side kinetic energy at time $T$ plus total dissipated viscous energy balancing kinetic energy at initial time $t=0$. Here $u$ is normalised to be of size 1 and the viscosity $\nu $ is smaller than $10^{-6}$, as a typical case when solutions turn turbulent. With a smooth initial solution the viscous dissipation starts out as very small and then grows as turbulence develops with kinetic energy transformed into viscous dissipation with large velocity gradients (of size $\nu^{-1/2}\ge 10^3)$. This is reality very far from the triple exponential world of Tao, but mathematicians do not seem to be willing to listen to reason...I have asked Tao for comment...

On top of the triple exponentials Tao scales the equations so that viscosity is 1 which means that fluid
velocity is boosted with another big factor making the argument even more unphysical and then also unmathematical if meaning is intended.

A Navier-Stokes solution initialised as smooth does not turn non-smooth from velocities blowing up to infinity, but from gradients of velocities becoming large as expression of turbulence which is non-smooth flow. It is very difficult to understand why this not something that Tao understands very well.

PS The Navier-Stokes problem was formulated by pure mathematicians to be solved by pure mathematicians by methods of pure mathematics. Since no progress has been made and none is in sight, my expressed view is that the  problem should be reformulated to make sense for a wider scientific community including applied and computational mathematics. 

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