The Navier-Stokes equations can be simplified in two ways:Assume all of the simple ways have been tried, look them up, and
by putting to zero the compressibility and/or the viscosity,
which then leaves us with 4 cases..
Does the "millennium problem" of proving or disproving the
smoothness of the solution require the full case, or would
solving it for a simplified case already be enough? (I'm
asking because we don't want to do the work and then still
not get one million dollar, of course!)
I would expect that the doubly simplified case is too
trivial.. but is the solution in that case actually known
already? That would be the question:
"Are there solutions for a non-compressible, non-viscous
fluid that start with smooth initial conditions and then
develop a singularity?"
Since non-viscosity means the equations are time reversal
invariant, the question could also be: can you start with
a singular solution and have the time-evolution smooth it
out? (To me the answer seems likely to be yes, but as said,
I don't know whether it has been proven. It might be both
simple and difficult, like the Goldbach conjecture..)
--
Jos
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
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:
by putting to zero the compressibility and/or the viscosity,
which then leaves us with 4 cases..
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?
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