On Sunday, November 8, 2020 at 10:18:06 AM UTC-6, Jos Bergervoet wrote:
> The Navier-Stokes equations can be simplified in two ways:=20
> by putting to zero the compressibility and/or the viscosity,=20
> which then leaves us with 4 cases..=20
I won't repeat what was said in an earlier reply (about surveying the
field before jumping in), but will note a few things. The best way to
address the problem is to remove the constraints and broaden it back out
to the simple and elegant form
d_t(rho) + del . (rho u) =3D 0
d_t(rho u) + del . (rho u u + P) =3D rho g
with constitutive laws
(d_t + u.del) rho =3D 0 - non-compressibility
P =3D (p - lambda del.V) I - mu (del u + (del u)^+) - the stress model
where I is the identity dyad, and P the stress tensor dyad
... and to broaden it to include the *other* transport equations for the
other Noether 4-currents of the kinematic group. The 2 equations above
are the transport equations for mass and momentum. The kinematic group -
the Bargmann group - also has kinetic energy, and *especially* angular
momentum and moment. These transport equations should also be included
and the whole system dealt with in its entirety ... especially the
equations for angular momentum, because this figures prominently in the
actual fluid dynamics that come out of the Navier-Stokes equation!
You want to make money on this, and that's your motivation? Rather than
just that of advancing science and mathematical physics? Well, then you
had better hurry. Because if we solve it first, we're *refusing* the
prize and nobody's going to get anything.
Moneyed interests have no place in science and mathematics and
Perelman's precedent will be honored and continued.