torsdag 22 maj 2014

Mr Clay and a Meaningless Navier-Stokes Prize Problem

Turbulent flow around a landing gear as non-smooth solution of the 3d incompressible Navier-Stokes equations, by CTLab KTH. Watch also turbulent flow around an airplane in landing configuration. To argue that these flows are smooth would be a meaningless abuse of mathematical language.

The Clay Institute of Mathematics (CMI) founded by Landon T. Clay celebrated the new Millennium by setting up 7 Prize Problems each worth $1 million,  presented in beautiful words:
  • The Clay Mathematics Institute (CMI) grew out of the longstanding belief of its founder, Mr. Landon T. Clay, in the value of mathematical knowledge and its centrality to human progress, culture, and intellectual life....
  • further the beauty, power and universality of mathematical thinking...deepest, most difficult problems... achievement in mathematics of historical dimension
  • elevate in the consciousness of the general public the fact that, in mathematics, the frontier is still open and abounds in important unsolved problems...
  • Problems have long been regarded as the life of mathematics.  A good problem is one that defies existing methods...whose solution promises a real advance in our knowledge. 
I have long argued that since the Navier-Stokes Prize Problem is formulated without including the fundamental aspects of wellposedness and turbulence, it misses these values and thus is not a good Prize Problem. Here is my argument again:

Consider the incompressible Navier-Stokes equations with viscosity $\nu >0$ in the case of (very) large Reynolds number $Re =\frac{UL}{\nu}$ with $U$ global flow speed and $L$ global length scale. Assume $U=L=1$ and thus $\nu$ (very) small. Such flows are observed physically and computationally to be turbulent with substantial velocity fluctuations $u\sim \nu^\frac{1}{4}$ on a smallest spatial scale $\epsilon\sim\nu^\frac{3}{4}$ with corresponding substantial viscous dissipation $\sim 1$.  For the jumbojet in the above simulation $Re\approx 10^8$ and the smallest scale a fraction of a millimeter. The heuristic argument to this effect goes as follows:

A: Breakdown to smaller scales only takes place for sufficiently large local Reynolds number (of size 100 or more), which gives the following relation for the fluctuations $u$ on the smallest scale $\epsilon$:
  • $\frac{u\epsilon}{\nu}\sim 1$.
B: Substantial dissipation on smallest scale $\epsilon$ means 
  • $\nu (\frac{u}{\epsilon})^2\sim 1$.
Combination of A and B gives $u\sim \nu^\frac{1}{4}$ and $\epsilon\sim\nu^\frac{3}{4}$ as stated. This can be viewed to express Lipschitz-Hölder continuity with exponent $\frac{1}{3}$ and thus that turbulent solutions for (very) small $\nu$ are non-smooth, because they are $Lip^{\frac{1}{3}}$ on (very) small scales. 

The existence of such turbulent solutions can mathematically be proved by standard methods by regularization on scales much smaller than $\epsilon$, which does not change the solution but the NS equation. 

For smooth data such solutions to regularized NS could formally be proved to be smooth in the sense of the formulation of the NS Prize Problem by Fefferman, but this would be in conflict with the observation that solutions are non-smooth  ($Lip^{\frac{1}{3}}$) on (very) small scales $\sim\nu^\frac{3}{4}$. 

The only mathematically and physically reasonable way to resolve this conflict of definitions, would be to view turbulent solutions to be non-smooth ($Lip^{\frac{1}{3}}$ on very small scales), and thus as weak solutions, with weakly small but strongly large Euler residuals, and the aspect of wellposedness would then be of focal interest.

Computational sensitivity (stability) analysis shows that turbulent weak solutions, are weakly wellposed in the sense that solution mean-values are not highly sensitive to perturbations of data (while point-values are).

Stability analysis further shows that globally smooth solutions with derivatives of unit size for smooth data of unit size, are unstable and thus are not physical solutions. 

The net result is that the present formulation of the NS Prize Problem is meaningless from both mathematical and physical point of view. A meaningful formulation must include wellposedness and turbulence as key issues, with existence settled by standard techniques, and a meaningful resolution would have to offer mathematical evidence of weak wellposedness and features of turbulence.

I have asked Terence Tao, as a world leading mathematician working on the Prize Problem, about his views on the aspects I have brought up, and will report his response. I have earlier many times asked Fefferman the same thing but the only response I get is "To me my formulation is meaningful". 

What would then Mr Clay say if he understood that the NS Prize Problem is not meaningful     
outside a small group of mathematicians (which may contain just one person), when comparing to the mission to which he donated his Prize:
  •  the value of mathematical knowledge and its centrality to human progress, culture, and intellectual life....
  • further the beauty, power and universality of mathematical thinking...deepest, most difficult problems... achievement in mathematics of historical dimension. 
PS It is remarkable (or deplorable) that my repeated request to start a discussion about the formulation of the Prize problem is met with complete silence from those in charge of the problem. If my view-points are silly, that could be said by those who know better. If they are not silly, maybe even relevant, then it would be silly (or deplorable) to not say anything.  In either case, silence is not reasonable and it is tiresome to keep silent under increasing pressure from the outside world to say something... 

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