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  • Navier-Stokes questions..

    From Jos Bergervoet@21:1/5 to All on Sun Nov 8 16:18:02 2020
    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

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  • From George Hrabovsky@21:1/5 to Jos Bergervoet on Tue Nov 10 12:09:54 2020
    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..

    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
    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.

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  • From Jos Bergervoet@21:1/5 to All on Fri Nov 20 20:18:54 2020
    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

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  • From Rock Brentwood@21:1/5 to Jos Bergervoet on Sun Nov 22 21:08:12 2020
    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.

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  • From Jos Bergervoet@21:1/5 to Rock Brentwood on Sat Nov 28 11:35:03 2020
    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:
    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?!

    ... 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?

    ... 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?

    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

    --
    Jos

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