Comments on: Penny Smith’s Proof on the Navier-Stokes Equations

By: Navier-Stokes Equations for the Layperson « Continuous Everywhere but Differentiable Nowhere

Tue, 10 Nov 2009 02:51:53 +0000

[...] I finally found a really nice explanation of the problem on this blog which attempts to explain a proposed (but now shown wrong) solution. Read up to but [...]

By: Analytical Solution to the Navier Stokes Equation Reported « Unruled Notebook

Thu, 19 Mar 2009 04:54:53 +0000

[...] rather than a constructive one but nevertheless is a great effort. Brooke Moses has written an explanatory blog post on this and an analysis of greater rigour is also available. A nice discussion is also available [...]

By: Philosophia Naturalis Part Deux « Unruled Notebook

Philosophia Naturalis Part Deux « Unruled Notebook — Sun, 02 Nov 2008 05:30:08 +0000

[...] Instead of directing us to the more sensational blog posts on this topic, we shall direct us to Penny Smith�s Proof on the Navier-Stokes Equations, a post by Brooks Moses at his Notes on Divergent Simulations blog, which explains the technical [...]

By: Nonoscience / Philosophia Naturalis Part Deux

Nonoscience / Philosophia Naturalis Part Deux — Mon, 28 Jul 2008 17:17:13 +0000

By: a-kozachok1

a-kozachok1 — Sat, 15 Mar 2008 10:59:25 +0000

Dear Brooks !
I very hope, that you and your colleagues from US universities will formulate the comment in this occasion:
MILLENNIUM PRIZE PROBLEM (NAVIER– STOKES EQUATIONS) IS SOLVABLE BY CLASSICAL METHODS.
The Navier-Stokes equations can be reduced correctly to more simple classical equations of mathematical physics . The problem of an existence proof of solutions of such equations is not so actual.
1. It is known, divu and divv are identical to infinitesimal magnitude and velocity of the relative modification of the infinitesimal volume. Therefore divergency of acceleration divw – acceleration of the relative modification of the same volume. In that case for incompressible liquid alongside with requirements divu=0, divv=0 is necessary to accept divw=0.
2.The requirement divw=0 for incompressible liquid is formulated by analogy and proved. Operation div will convert the Navier – Stokes equations to the three-dimensional Laplace equation for pressure p=p(x,y,z,t) at some limitations of a mass force vector. The time t enters into Laplace equation as parameter.
Read more on http://a-kozachok1.narod.ru ,“Sixth Millennium Problems (NAVIER-STOKES equations) is solvable by classical methods” (Russian).

Yours faithfully, Alexandr Kozachok

By: Alexandr Kozachok

Alexandr Kozachok — Sun, 17 Feb 2008 03:36:45 +0000

MILLENNIUM PRIZE PROBLEM (NAVIER– STOKES EQUATIONS) IS SOLVABLE BY CLASSICAL METHODS

Kozachok A.A., Kiev, Ukraine

Formulated by Clay Mathematics Institute the sixth Millennium Problems about existence and smoothness of solutions of the Navier – Stokes equations periodically was discussed at numerous forums (http://grani.ru/Society/Science/m.112524.html). On recognition of some commentators the complete presentation of problem’s solution can demand about thousand pages for mathematical formulas (http://lib.mexmat.ru/forum/viewtopic.php?t=4289). The author of Official Problem Description–Charles Fefferman has set the task about demonstration of existence and smoothness of the solution, instead of solution’s obtaining. However, the Navier-Stokes equations can be reduced correctly to more simple classical equations of mathematical physics . The problem of an existence proof of solutions of such equations is not so actual.
It is known, divu and divv are identical to infinitesimal magnitude and velocity of the relative modification of a volume element of the strained medium. Therefore divergency of acceleration divw, probably, there is a magnitude, identical to a acceleration of the relative modification of the same volume. In that case for incompressible liquid alongside with requirements divu=0, divv=0 it is necessary to accept divw=0.
The requirement divw=0 for incompressible liquid is formulated by analogy and proved. Operation div will convert the Navier – Stokes equations to the three-dimensional Laplace equation for pressure p=p(x,y,z,t) at some limitations of a mass force vector. The time enters into Laplace equation as parameter.
Laplacian of the Navier – Stokes equations (pressure p=p(x,y,z,t) – harmonic function) and a change of a variable (velocity on acceleration) allow to gain system of conventionally independent integro-differential equations with acceleration’s components w. In that case components of the acceleration for ideal incompressible liquid are harmonic functions too. The change of a variable allows to use boundary conditions of an adhesion of a fluid absolutely correctly. According to this requirement vectors of acceleration on firm immobile boundary line are equal to null.
Conversion of the Navier-Stokes equations to more simple equations has actually removed a problem of an existence proof and smoothness of their solution. It is possible to use known effects about properties of harmonic functions or representation of the common decision of the Laplace equation (http://continuum-paradoxes.narod.ru, the link « Manual, p.1 », p. 58). More in detail on a site http://continuum-paradoxes.narod.ru, the link “Russian pages”, “Sixth Millennium Problems (NAVIER-STOKES equations) is solvable by classical methods (in Russian)”.
Yours faithfully, Alexandr Kozachok

By: Alexandr Kozachok

Alexandr Kozachok — Sat, 15 Dec 2007 09:22:34 +0000

Dear Dmitri Gorskin!
Where your “exact analytical solution” is published?
Alexandr Kozachok

By: Dmitri Gorskin

Dmitri Gorskin — Tue, 04 Dec 2007 10:53:54 +0000

I found exact analytical solution for 3 D viscous nonsteady flow(without cappilar effect) many years ago.But I never thought,that it is nesessary to prove existence this smooth solution for small period of time.It is clear, like a day.
But when I start research it, I was surprised.The water has two properties, like living object(like cell).Competition and reproductions some of his parameters.It was amazing.

By: Recovering my youth

Recovering my youth — Sun, 22 Apr 2007 22:09:29 +0000

[...] UPDATE: I finally found a really nice explanation of the problem on this blog which attempts to explain a proposed (but now shown wrong) solution. Read up to but not including the paragraph beginning “So, how does Penny Smith’s analysis approximate this by a set of hyperbolic equations?” [...]

By: Alexandr Kozachok

Alexandr Kozachok — Tue, 24 Oct 2006 08:26:00 +0000

Dear participants of discussion !
You can detect unreasonableness in Penny Smith’s calculations if you will look the reference ” NAVIER -STOKES Equations Paradoxes ” on a site http://continuum-paradoxes.narod.ru . In detail look “Russian Pages”.
Alexandr Kozachok

By: gs

gs — Sun, 15 Oct 2006 14:07:05 +0000

Brooks, thanks for your clarification on October 7. Good luck in your search for a postdoc.

By: Nonoscience / Philosophia Naturalis Part Deux

Nonoscience / Philosophia Naturalis Part Deux — Thu, 12 Oct 2006 17:00:35 +0000

[...] The other one is more recent and is about the withdrawn paper of Dr. Penny Smith on the proposed existence solution for the Navier Stokes equations. Instead of directing us to the more sensational blog posts on this topic, we shall direct us to Penny Smith’s Proof on the Navier-Stokes Equations, a post by Brooks Moses at his Notes on Divergent Simulations blog, which exlpains the technical possibilities of such a scheme of transforming the NS equations into a hyperbolic equivalent and the implications. [...]

By: Sun

Sun — Mon, 09 Oct 2006 18:58:25 +0000

Brooks: You are right. In reality, there is disturbance all the time.

By: Brooks

Brooks — Mon, 09 Oct 2006 17:31:56 +0000

Sun: As with any chaotic system, unique solutions can be extremely sensitive to initial conditions, and still be unique for a given initial condition. Thus, given a particular set of initial conditions, there is only one set of turbulent eddies that will result — but, if that set of initial conditions is perturbed by a tiny amount, the perturbation will grow, and by the time turbulence develops, the eddies will be quite different.

You won’t see this in experiments, of course, since it’s impossible to get exactly the same initial conditions twice. But it does show up in computations, unless you introduce randomness to avoid it.

By: Sun

Sun — Mon, 09 Oct 2006 16:47:57 +0000

Brooks: Thanks for the nice explanation of the N-S equation. Just wonder how a unique solution of N-S equation describe a turbulent flow…

By: Arunn

Arunn — Mon, 09 Oct 2006 06:23:14 +0000

Brooks: Yes LBM uses always this artificial compressibility. I am just into it recently to simulate porous medium flows, my area of research interest wherein I encountered this. But I am yet to get a hands on on this; just yet fumbling with the LBM code…

By: Brooks

Brooks — Mon, 09 Oct 2006 05:39:50 +0000

Arunn: Artificial compressiblity is, indeed, how all of the Lattice-Boltzmann methods that I have seen work — as of yet, nobody has developed a LBM that can deal with anything other than hyperbolic problems.

There seem to be a number of variants on LBMs that have clever enhancements to the artificial-compressiblity equations to do a better job of modeling incompressible flows, but as far as I can tell from a quick look through the literature, they’re all still essentially solving hyperbolic equations with a finite rate of information propogation — there aren’t any methods that I’ve seen that do a complete many-iteration relaxation at each timestep (which is how you’d do a non-hyperbolic problem with a system like that).

Smoothed-particle hydrodynamics models also use approximations like this, and with those I find it rather more surprising that I haven’t seen a relaxation-at-each-timestep incompressible method. But maybe there is such a thing and I just haven’t seen it.

By: Arunn

Arunn — Mon, 09 Oct 2006 05:04:40 +0000

Brooks (and gs and Arun): Firstly, just to make it clear, if at all there was any slight doubt; I am Arunn, different from Arun who also seems to be discussing our pet problems and share the same field… ;)

It is unfortunate that the paper is taken off. But I agree with Brooks that nevertheless what is discussed here is worthwhile. Also, isn’t this pseudo-comprressibility method the bread and butter of LBM?

By: Brooks

Brooks — Mon, 09 Oct 2006 03:20:06 +0000

Arun: In the general case, there's certainly no guarantee of an upper bound -- in the nondimensionalization, pressure gets scaled by the length- and time-scales used to do the nondimensionalizing, and so for a solution that gives a pressure increase P, one can easily produce a problem with the same nondimensionalized form but faster velocities and a different lengthscale, which will have a pressure increase of n*P for arbitrary n. (Or, at least, I think that's true.) Thus, there's no general guarantee. In the more interesting case, I guess at this point it's an open question whether a proof of this sort will be able to impose a clear bound on the pressure as a function of the initial conditions. Penny's didn't, I believe.

By: Arun

Arun — Sun, 08 Oct 2006 15:15:30 +0000

What I think one would want to know, physically speaking, is whether over time a flow of a physical fluid that is initially well within in the incompressible regime require a change to the compressible regime? i.e., do the immortal solutions imply a upper bound on the pressure?