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Nov 8, 2020, 11:18:06 AM11/8/20

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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..)

--

Jos

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

Nov 10, 2020, 3:09:58 PM11/10/20

to

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.

Nov 20, 2020, 3:18:59 PM11/20/20

to

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?

--

Jos

"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?

--

Jos

Nov 22, 2020, 4:08:15 PM11/22/20

to

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.

Nov 28, 2020, 6:35:08 AM11/28/20

to

On 20/11/22 10:08 PM, Rock Brentwood wrote:

> 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:

> 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),

Yes, but surveying the field was exactly my aim! By posting in s.p.r.
> field before jumping in),

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?!

> ... 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
> 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

momentum conservation is a useful constraint in solving for elliptical

planet orbits?

> ... especially the

> equations for angular momentum, because this figures prominently in the

> actual fluid dynamics that come out of the Navier-Stokes equation!

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?

> You want to make money on this, and that's your motivation?

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

--

Jos

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