tisdag 10 december 2013

Computational Solution of the Clay Navier-Stokes Problem

The computations underlying the New Theory of Flight to soon be presented to the world in Journal of Mathematical Fluid Mechanics, suggest the following resolution of the Clay Millenium Problem on existence or non-existence of smooth solutions to the incompressible Navier-Stokes equations with smooth data:
  • Consider exterior flow, governed by the incompressible Navier-Stokes equations with viscosity $\nu >0$, around a given solid body with smooth boundary with given smooth flow at infinity as given smooth data. Let $N$ be any given finite amount of computation as number of flops. Then there is a choice of small viscosity $\nu$ such that it is impossible to compute an approximate solution of the Navier-Stokes equations by time-stepping with a Navier-Stokes residual which is small (e.g smaller than 1) in the entire fluid domain within the limit of computation $N$. 
This suggests a resolution in the negative: For given smooth data there is no smooth solution to the Navier-Stokes equations if $\nu$ is chosen small enough. 

The argument is that a smooth solution should be computable by time-stepping an approximate solution with small residual within a given amount of computation $N$.  An observed impossiblity of time-stepping an approximate solution with small pointwise residual within the given limit $N$, then gives evidence of non-existence of a smooth solution.

Note that interpreting a smooth solution as possible to compute by time-stepping with small residual, introduces an aspect of stability into the notion of smoothness. This is necessary since a potential solution has a small Navier-Stokes residual for $\nu$ small, yet is not computable by time-stepping because of instability. In the formulation of the Clay problem this point is missed, which makes the problem ill-posed and without good answer. The result is that the Clay $1million Prize will never be given out, against the intention of Clay.

Another resolution based the theory of flight goes as follows: Existence of smooth solutions of the Navier-Stokes equations with small viscosity would make flight impossible. Since flight is observed to be possible, smooth solutions do not exist. 

2 kommentarer:

  1. is your paper that you link finally accepted in a journal? Were the referee comments as devastating as in AIAA?

  2. Of course it was rejected by mathematicians because it questioned the problem formulation by Fefferman as lacking the aspect of wellposedness identified by Hadamard in the 1930s. But Hadamard had a good point which is valid today also for mathematicians. And yes, many referee reports are off the point and my experience is that the better my article is the less will it be understood by ignorant referees.