Millennium Prize Problem: Navier-Stokes

/ Paul Pukite (@whut)

Watched the hokey movie Gifted on a plane ride. Turns out that the Millennium Prize for mathematically solving the Navier-Stokes problem plays into the plot.

I am interested in variations of the Navier–Stokes equations that describe hydrodynamical flow on the surface of a sphere.  The premise is that such a formulation can be used to perhaps model ENSO and QBO.

The so-called primitive equations are the starting point, as these create constraints for the volume geometry (i.e. vertical motion much smaller than horizontal motion and fluid layer depth small compared to Earth’s radius). From that, we go to Laplace’s tidal equations, which are a linearization of the primitive equations.

I give a solution here, which was originally motivated by QBO.

Of course the equations are under-determined, so the only hope I had of solving them is to provide this simplifying assumption:

{frac{partialzeta}{partialvarphi} = frac{partialzeta}{partial t}frac{partial t}{partialvarphi}}

If you don’t believe that this partial differential coupling of a latitudinal forcing to a tidal response occurs, then don’t go further. But if you do, then:

 

 

 

 

5 thoughts on “Millennium Prize Problem: Navier-Stokes”

  1. This might be of use in winning the prize:

    Singularity of Navier-Stokes Equations

    “From the present study, it is clear that turbulence cannot be duplicated due to the
    singularity of Navier-Stokes equations at a velocity inflection point and that the turbulence is
    composed of numerous singularities. This may be useful information for the question in the
    millennium prize problems “

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  2. There is no singularity in the Navier-Stokes equation as stated in the Official Problem Description (OPD) of the NS Millennium Problem. In the OPD, the incompressible fluid form of the NS equation (where the divergence of the fluid velocity is constrained to zero) is what is used, and I have shown existence, smoothness, and uniqueness of solutions to the NS equation of this form, thereby solving this Millennium Problem (at least unofficially). This need not be the case, however, for other forms of the NS equation. For example, if the fluid has a finite acoustic velocity, then supersonic flow would be possible which would result in sonic shock waves.

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