onsdag 20 april 2016

Turbulent Euler Solutions and the Clay Navier-Stokes Problem 2

This is a continuation from the previous post:

We compute an approximate turbulent solution $U_h$ to the Euler equations using G2 on a given mesh with mesh size $h$ characterised by substantial turbulent dissipation. We ask if with $U_h$ given, it is possible to construct a function $\hat U_h$ which solves the Navier-Stokes equations for some viscosity $\nu_h$?

 This can be answered by pointwise computing the Euler residual $E(\hat U_h)$ of a regularisation $\hat U_h$ of $U_h$ together with the Laplacian $\Delta\hat U_h$ and then defining
  • $\hat h =\frac{E(\hat U_h)}{\Delta U_h}$.   
If it turns out that $\hat h >0$, then we have a function $\hat U_h$ which exactly solves the Navier-Stokes equations with a viscosity $\hat h$, and if $U_h$ is turbulent, so will $\hat U_h$ be.

Depending on the variation of $\hat h$, we could argue that we have constructed an exact solution to a modified Navier-Stokes equation (with constant viscosity), with the modification depending on the variation of the computed $\hat h$, a solution which is turbulent and thus non-smooth. 

This argument has a connection to that presented by Terence Tao in a setting of modified Euler/Navier-Stokes equations.  The difference is that we use a computed solution of great complexity instead the analytical solution of less complexity constructed by hand by Tao.

There is strong evidence from experimental observation and computing that solutions to the Navier-Stokes  equations with small viscosity, are always turbulent and thus that the Clay problem about global existence of smooth solutions has a negative answer. Thus it seems pretty clear that computational evidence can settle the Clay problem, but this may not be accepted by a jury of mathematicians trained in analytical mathematics developed before the computer.  It may be that without computational evidence the problem may stay unsolved for ever, or that an answer by analytical mathematics becomes so particular that the very meaning of the problem is lost.

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