torsdag 12 december 2013

The Truth about the Clay Navier-Stokes Problem

The resolution of d'Alembert's paradox (article here) and the new theory of flight developed together with Johan Hoffman, reveals the following truth about the Clay Millennium Problem on existence and smoothness of solutions of incompressible Navier-Stokes equations, in the case of (vanishingly) small viscosity and exterior bluff body flow:
  • Smooth solutions of the Navier-Stokes equations are unstable and thus not wellposed physical solutions.
  • Wellposed physical solutions are partly turbulent and as such are non-smooth with weakly small but strongly not-small residuals.   
Potential solutions are smooth solutions to the Navier-Stokes equations with vanishing viscosity, but are unstable and thus are not wellposed physical solutions.  Computations show that wellposed physical  solutions do exist and can be described as potential flow modified by 3d rotational slip separation into turbulent wake flow. 

The truth about the Clay Navier-Stokes problem in the case of small viscosity exterior flow with smooth data, is thus:
  1. Smooth solutions are not wellposed, because they are all unstable. 
  2. Wellposed physical solutions have to be non-smooth, as the only way to avoid illposedness according to 1. 
  3. The conclusion is that smooth wellposed solutions do not exist and computation shows that non-smooth well-posed solutions do exist. 
In the formulation of the Clay problem by Charles Fefferman, well-posedness is not an issue, which according to French mathematician Hadamard makes the formulation meaningless, since only well-posed solutions are meaningful.

Fefferman's answer to my question why his problem formulation does not include wellposedness and thus is meaningless, is that the formulation is meaningful to Fefferman and that is enough for him. Why mr Clay accepts a meaningless problem formulation is unknown: A meaningless problem cannot have a meaningful solution and the $1 million prize will never be given out, which cannot be the meaning of mr Clay. 

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