Navier-Stokes questions..

[Jos Bergervoet]

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..

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..)

[George Hrabovsky]

Post by Jos Bergervoet
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
"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

Assume all of the simple ways have been tried, look them up, and
make sure you get the same results. If you don't, then figure out
where you made your mistake. Assume your results are wrong until
you prove that they are correct. Only then proceed to make a claim.
Good luck.

[Jos Bergervoet]

In <https://en.wikipedia.org/wiki/Quantum_decoherence> we read:
"Decoherence has been developed into a complete framework, but it
does not solve the measurement problem, ..."
Somewhat later, however, <https://en.wikipedia.org/wiki/Quantum_decoherence#Phase-space_picture> we read:
"... the system appears to have irreversibly collapsed onto a state
with a precise value for the measured attributes, relative to that
element. And this, provided one explains how the Born rule coefficients
effectively act as probabilities as per the measurement postulate,
constitutes a solution to the quantum measurement problem."

So, is this measurement problem solved or not?! Wikipedia seems to give
conflicting views!

Perhaps they mean that one part of the problem is solved, namely the
*appearance* to an observer of just one single outcome, even though
the system still is in a big mixed state. And another part is not yet
solved: that the probability of appearance is given by the Born rule, i.e.
by the squared amplitudes of the components of the mixed state..

But is the latter really not proven? If you describe the system and its
observer both by quantum mechanics, where this "observer" is some kind
of counter of outcomes, wouldn't that lead to the counter result state
being strongly centered around the Born rule prediction?

And if so, what more could there possible be to prove about this
measurement problem?

[Rock Brentwood]

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.

[Jos Bergervoet]

Post by Rock Brentwood

Post by Jos Bergervoet
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),

Yes, but surveying the field was exactly my aim! By posting in s.p.r.
I was hoping to find the experts' opinion about the state of affairs..
In particular: which of the 4 cases has been, or has not been solved?!

Post by Rock Brentwood
... 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

The additional equations will be added as constraints, like angular
momentum conservation is a useful constraint in solving for elliptical
planet orbits?

Post by Rock Brentwood
... especially the
equations for angular momentum, because this figures prominently in the
actual fluid dynamics that come out of the Navier-Stokes equation!

To 'broaden it' as you write, seems like the opposite of what I was
looking for (looking at simplified cases) although adding constraints
of course does simplify things.. Still, I'm curious about the simple
question: which ones of the simplified cases have been solved already?

Post by Rock Brentwood
You want to make money on this, and that's your motivation?

I was planning to solve one of the open millennium problems each year,
which would give me a decent income. :-) But OK, if it's not appreciated
I'll just have to predict the stock market. QM is well-suited for it:
https://phys.org/news/2018-02-stock-quantum-harmonic-oscillator.html
https://arxiv.org/abs/1009.4843