Talk:Navier–Stokes existence and smoothness

Possible links to examples of simpler solutions?

It says the analog for R2 is solved but doesn't point to the article or reference stating examples, solutions, or proofs.

JWhiteheadcc 09:24, 14 May 2007 (UTC)

Notation clue?

What if, where it identifies $\Delta$, we add parenthetically the notation I am more used to seeing, e.g:

(also written $\nabla ^{2}$)

One of the biggest barriers I have in reading math is that there are so many different notations for the same concept, and so many different usages for the same symbols. $\Delta$ is used in quite a few different ways, for example. In this case, the change in notation really had me scratching my head, since this is a familiar equation. For the casual reader, it could be a bigger stumbling block.

Note that $\Delta$ is the form in the original problem statement, so I don't think we should rewrite it, just provide the conceptual link. But many of the linked references use the $\nabla ^{2}$ notation.

Bob Kerns (talk) 21:39, 23 December 2007 (UTC)

$\Delta$ is the notation most used by mathematicians (especially PDE theorists), so it makes sense to use it here. I see your point, though. In fact, this article needs a lot of rewriting, and more background and history. As it is, it is based too closely on the Clay Institute problem description. The Clay description is ultraprecise and narrow for a good reason: there should be room no dispute on whether a solution warrants the million dollar prize. However, that's not necessarily right for Wikipedia. The Riemann hypothesis and Hodge conjecture articles, for example, are not mere writeups of the official Clay Institute problem descriptions. What this article should be, I think, is an overview of rigorous, mathematical approaches to the Navier-Stokes equations. The Clay Institute problem should be there, but in context. It should be pointed out that there are other mathematical problems under the same heading which are also very important even if there is no million dollar prize, one example being the compressible Navier-Stokes equations. Perturbationist (talk) 03:04, 31 December 2007 (UTC)

Bob, reading the above comment it looks like I went on a rant about this article and sort of ignored your suggestion. I didn't mean to do that. My point was, I think we should do what you said, but it wouldn't really flow into the text the way it's written now, so we should rewrite the article first. Perturbationist (talk) 02:37, 1 January 2008 (UTC)

Not to worry, I didn't take it as a rant at all! You raise a good point -- should this be about the Clay Institute problem, or about the mathematical problem? While the former may be more exciting to read about, the latter is probably more useful.

I also like that you identify that this notation is favored by PDE theorists. I think that while the math world has all these notational camps, Wikipedia should try to cross-reference them. I don't care which notation is used, just that notation differences should be minimized as a stumbling block for readers familiar with one or the other. Placing it in context as "$\Delta$ is the notation most used by mathematicians (especially PDE theorists)" is really helpful to learn when to expect which notation. Despite my ancient background with Macsyma and PDE theorists, I've had more to do with engineers and physicists overall, so your phrase lights a little LED over my head...

It almost seems like a "Math notation" Wikipedia project would be helpful, to catalog the various systems of notation and variants, and which articles using a particular notation could simply reference.

I find keeping the notation systems straight for, say, category theory, to be harder than the theory itself! (At the level I grasp it, anyway!) Bob Kerns (talk) 06:18, 4 January 2008 (UTC)

It sounds like we have similar ideas about the direction this article should go in. I want to run by you the changes I have in mind and see what you think. The Clay problem should be mentioned, but alongside more general problems. For example, perhaps the most natural problem, and one to which a lot of effort has been devoted, is to solve the Navier-Stokes equations on a compact region $\Omega$ in $\mathbb {R} ^{3}$ with smooth boundary, with the boundary condition $u=0$ on $\partial \Omega$. The formulation of the Clay problem avoids boundary conditions -- this is meant to make the problem easier -- but that doesn't mean we should. As for the notation, the text is way too dense. The quantifier symbols ($\forall$ and $\exists$) should be replaced by English, and there is no need to repeat expressions like $f{\mathcal {2}}(C^{\infty }(\mathbb {R} ^{3}))^{3}$ all over the place. It would be better to say something like "let $f$ be a smooth time-dependent vector field on $\mathbb {R} ^{3}$".

The math notation project certainly sounds like a good idea, though it may be a lot of work to get off the ground. Perturbationist (talk) 04:09, 6 January 2008 (UTC)

Euler vs Navier-Stokes

Crowsnest argues that there is a clear distinction between the Euler and Navier-Stokes equations, but this is not so: Euler solutions are defined as viscosity solutions, which are solutions of the Navier-Stokes equations as the viscosity tends to zero. Since the viscosity in the Clay problem formulation can be arbitrarily small, the limit as the viscosity tends to zero effectively is included, which means that the Euler equations effectively are included as a (generally acknowleged most interesting) limit case. Why advocate on purely formal grounds that the Euler equations should be excluded? —Preceding unsigned comment added by Egbertus (talk • contribs) 10:57, 5 August 2008 (UTC)

The Euler equations are inviscid equations, see e.g. Landau and Lifshitz, Fluid Mechanics, 1987, p.3 or any other good textbook on fluid dynamics. Whether solutions to the Euler equations are the limit of solution to the Navier-Stokes equations as the viscosity tends to zero, is still an open question and yet another outstanding problem (even more in connection with boundary conditions at a solid wall). Even stronger: solutions to the Euler equations, of which potential flow solutions are a special case, behave in many cases in a very different way as the solution to Navier-Stokes equations does, for the same problem. So any reference to the Euler equations with respect to the Clay millennium prize is off-topic. -- Crowsnest (talk) 12:43, 5 August 2008 (UTC)

The Euler solutions considered in the article are mathematically defined as vanishing viscosity solutions of the Navier-Stokes equations. Hence there is a strong connection between Euler and Navier-Stokes solutions. This is also acknowledged in the official problem formulation, which mentions the Euler equations as a "open and very important" limit case. —Preceding unsigned comment added by Egbertus (talk • contribs) 13:40, 5 August 2008 (UTC)

The problem formulation is clear: the Euler equations are the N-S equations with the viscosity set equal to zero. This does not imply that the Euler equations are the asymptotic limit for vanishing viscosity in the N-S equations, which is questionable. Further the problem statement clearly distinguishes between the Euler eqs. (ν=0) and N-S eqs. (ν>0), and all the requested proofs are for (ν>0), which are the N-S eqs. and not he Euler eqs. So the reference to Hoffman & Johnson is off-topic. Crowsnest (talk) 13:59, 5 August 2008 (UTC)Egbertus (talk) 08:56, 6 August 2008 (UTC)

Removed section "Proposed resolutions"

I removed the section "Proposed resolutions" talking about "...smooth laminar potential flow around a circular cylinder (with zero drag)..." which is non-sensical: potential flow solutions do not exist for laminar flow around a cylinder, and laminar flow around a cylinder gives non-zero drag. Crowsnest (talk) 14:06, 5 August 2008 (UTC)

Crowsnest is probably confused by the term laminar (here a synonym for smooth) and this term is therefore removed. Any book on fluid mechanics will present potentlal flow (stationary inviscid incompressible irrotational flow).

—Preceding unsigned comment added by Egbertus (talk • contribs) 14:58, 5 August 2008 (UTC)

Laminar flow, as commonly used in fluid dynamics, is not a synonym for smooth flow. It is always associated with viscosity (through the Reynolds number), see e.g. Lamb (1932), Hydrodynamics, p. 32, or: Landau & Lifshitz (1987), Fluid Mechanics, p. 110. Crowsnest (talk) 18:38, 6 August 2008 (UTC)

The wikipedia article on laminar flow reads: "Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers.... It is the opposite of turbulent flow. In nonscientific terms laminar flow is "smooth," while turbulent flow is "rough." " Evidently, according to wikipedia, laminar is associated with smooth and turbulent with non-smooth or rough, which makes sense. To associate laminar with very viscous flow is possible, since very viscous flow is smooth, but for slightly viscous flow it is not natural, since slightly viscous flow can be both laminar and turbulent.Egbertus (talk) 19:02, 6 August 2008 (UTC)

This section is of little relevance. The authors themselves already state in their paper that their work does not comply with the requests made in the description of the Millenium prize problem. The paper does not even solve the Navier-Stokes equations. Crowsnest (talk) 23:44, 6 August 2008 (UTC)

Crowsnest takes the role of the judge of the prize and declares that "this proposed resolution does not comply with the requirements in the statement of the prize problem". Is Crowsnest the judge? Next Crowsnest gives his/her own suggestive (negative) interpretation of the article. Again, is really Crowsnest the judge? What are in fact the scientific credentials of Crowsnest? The added section should be deleted since it expresses the personal opinions of Crowsnest. Right? Crowsnest states that regularized Euler equations are solved, which are forms of Navier-Stokes equations, and thus seems to admit that the article concerns Navier-Stokes. Right? Egbertus (talk) 06:00, 7 August 2008 (UTC)

As said, Hoffman & Johnson themselves declare in the BIT paper that they do not comply with the conditions as stated in the problem formulation for the Clay Millenium Problem Prize. I propose to completely remove this material, as being off-topic since Hoffman & Johnson do not solve the Navier-Stokes equations. See also the discussion on Talk:D'Alembert's paradox#Vague Reference?. -- Crowsnest (talk) 09:47, 7 August 2008 (UTC)

I suggest Crowsnest let HJ speak for themselves: The article clearly addresses the Clay problem and proposes a resolution, and Crowsnest is not the judge. What are in fact the scientific credentials of Crowsnest? Mathematician, fluid dynamisist, academician?Egbertus (talk) 09:53, 7 August 2008 (UTC)

Please cite where it says in the paper that they propose a resolution for the Navier-Stokes problem. -- Jitse Niesen (talk) 10:32, 7 August 2008 (UTC)

It is stated "we present evidence of (II)", where (II) is defined by "The Clay Mathematics Institute millennium problem on the incompressible Navier–Stokes equations formulated by Fefferman [7] asks for a proof of (I) global existence of smooth solutions for all smooth data, or a proof of the converse (II) non global existence of a smooth solution for some smooth data, referred to as breakdown or blowup." It is also stated that "since the viscosity in the Navier–Stokes equations is allowed to be arbitrarily small and solutions of the Euler equations are defined as viscosity solutions of the Navier–Stokes equations under vanishing viscosity, with slip boundary conditions modeling vanishing skin friction, the Euler equations effectively are included in the millennium problem as a most difficult limit case."Egbertus (talk) 13:27, 8 August 2008 (UTC)

The presentation by Crowsnest of the resolution by HJ was removed, since it expresses an evaluation of the proposed resolution which is not based on published scientific work, only on the personal opinion of Crowsnest. The headline with Hoffman-Johnson in big letters was also removed. Egbertus (talk) 14:03, 8 August 2008 (UTC)

Hoffman & Johnson (H&J) start, directly in the introduction of the BIT paper with title "Blowup of incompressible Euler solutions", and directly after giving their description of the Clay Millennium Problem with:

"... In this note we address the analogous problem for the inviscid incompressible Euler equations, which for some reason is not explicitly a Millennium Problem, although mentioned briefly in [7] and in [6] referred to as "a major open problem in PDE theory, of far greater physical importance than the blowup problem for Navier-Stokes equations, which of course is known to the nonspecialists because it is a Clay Millennium Problem". In fact, since the viscosity in the Navier-Stokes equations is allowed to be arbitrarily small and solutions of the Euler equations are defined as viscosity solutions of the Navier-Stokes equations under vanishing viscosity, the Euler equations effectively are included in the Millennium Problem. We present evidence that a specific initially smooth solution of the Euler equations, potential flow around a circular cylinder, in finite time exhibits blowup into a turbulent non-smooth solution, that is we present evidence of (II). ..."

Where reference [7] is the statement of the Clay Prize Problem, clearly stating:

"... These problems are also open and very important for the Euler equations (ν = 0), although the Euler equation is not on the Clay Institute's list of prize problems. ..."

and reference [6] is a paper by P. Constantin.

So H&J admit that they do not comply with the conditions of the prize, which is (as is in its name) on the Navier-Stokes equations, asking for proofs of existence and smoothness (I), or the opposite, i.e. proofs regarding their breakdown (II). Thereafter, see the quote above, H&J claim to provide evidence with respect to (II), but in the context of the Euler equations (or more precise: the "regularized Euler equations" as defined by H&J), which is not the context of the Clay Millennium Prize.

On these grounds, this material is off-topic, and in accordance with WP:OFFTOPIC, I removed it.

If you have good arguments why this material should be in this article, discuss this here until consensus is reached, before re-instating the material again.

If you re-instate the material before consensus is reached, than that is (to my opinion) disruptive editing, see WP:DIS.

Crowsnest (talk) 17:48, 8 August 2008 (UTC)

Dear Crowsnest: Yes, there are very good reasons for putting up a reference to the proposed resolution by HJ: It is published in a refereed journal of high standard and HJ are leading reseachers. No other resolution has ben proposed or published. The problem is of highest scientific interest. It is controversial, and that adds to the interest. Crowsnest takes the role of a prize judge and claims that the resolution is no good, and therefore removes the reference. But Crowsnest is not the judge, and does not support his/her personal opinions by published references where the work of HJ is criticised or shown to be wrong. I ask Crowsnest: (1) Are you the judge of the prize? (2) What published scientific work is the basis for your removal of the reference? I may be that wikipedia has to enter and settle this dispute.Egbertus (talk) 07:36, 9 August 2008 (UTC)

With regard to your questions:

1. I am not involved in judging the prize, nor have I any other interest with respect to this prize.
2. The basis for the removal is the work itself, stating it does not comply with the conditions of the Clay prize, which explicitly exclude the Euler equations, as explained several times above. So please comment on that, instead of asking whether I or other editors are judging this prize. You do not provide any evidence to support your view that this material should be included in the article, you only stay asking (the same) questions.

Controversiality in itself does not make this a subject of scientific interest. It only does when it is backed up with evidence. And whether something is of scientific interest does not make it automatically of encyclopedic interest. Hoffman & Johnson boost forward several claims regarding resolving five key issues in physics in their book and (draft) papers, which are not backed up by the material they present, and without self-reflection on the validity of their arguments in relation to the enormous claims made. For myself, I tend in general to avoid spending much work on such discussions, since most times it is just a waist of time arguing with people who by their incapacity of self-reflection are not open-minded on a scientific subject.

Crowsnest (talk) 08:37, 9 August 2008 (UTC)

Regarding controversial subjects, also see WP:FRINGE#Evaluating scientific and non-scientific claims. But as said, the material is removed because it itself states not to be on the subject of the Clay prize. -- Crowsnest (talk) 08:48, 9 August 2008 (UTC)

Crowsnest expresses personal opinions about the work by HJ, not backed in scientific writing, and thus irrelevant as concerns wikipedia. Again: Viscosity solutions of the Euler equations are considered, which effectively are solutions of the Navier-Stokes equations. The article claims resolution. Since Crowsnest admits to not be a prize judge, please refrain from claims that the proposed resolutions does not comply with the rules. This is something for the judge. Controversy itself is not any reason to remove material from wikipedia. Better to back the controversy by scientific writing. Science is sometimes controversial. I ask wikipedia to enter the dispute. It seems that Crowsnest is locked in position.Egbertus (talk) 09:02, 9 August 2008 (UTC)

What do you mean with:

• "... Viscosity solutions of the Euler equations ...", since the Euler equations are without viscosity by definition, see e.g. Landau & Lifshitz, Fluid Mechanics, 1987, p. 3.
• And what is meant with your statement that these solutions "... effectively are solutions of the Navier-Stokes equations ...".
• Does the paper solve the Navier-Stokes equations, as requested for in the Clay Millennium Prize?
• Does the paper solve the equations in the whole of the unbounded domain $\mathbb {R} ^{3}$, or in the whole of the periodic domain $\mathbb {R} ^{3}/\mathbb {Z} ^{3}$, as requested in the problem statement for the prize?

Crowsnest (talk) 17:18, 9 August 2008 (UTC)

Where does the article claim resolution with respect to the Clay prize? Then Lord Kelvin and von Helmholtz are more likely candidates, showing the instability of the Euler equations for a sufficient strong tangential jump in the velocity. They at least considered an unbounded domain, as demanded by the Clay prize problem statement.

Since you are so persistent in trying to get this reference place here or on d'Alembert's paradox, let me ask you two questions which seem relevant, regarding WP:COI, since you avoided responding on this[1] when I asked about this on your talk page:

1. Are you one of the authors of the BIT article?
2. Are you directly connected to the research groups of the authors of the BIT article?

Crowsnest (talk) 09:33, 9 August 2008 (UTC)

The claim is stated in the above citation from the article. Your questions are irrelevant for the scientific discussion. The scientific source is the published work by HJ. My opinion is irrelevant. I have asked you about your scientific credentials for your opinions about the work by HJ, which you insist should influence wikipedia.Egbertus (talk) 09:46, 9 August 2008 (UTC)

One of the pillars of Wikipedia is verifiability, which is based on reliable sources, and whether I have blue eyes, or good scientific credentials is irrelevant. Neither makes me a reliable source, whatever I say about this or whatever my scientific credentials are.

What is relevant, is whether you have a conflict of interest, as stated in my questions above, which you have not answered yet. -- Crowsnest (talk) 10:09, 9 August 2008 (UTC)

Crowsnest gives no scientific sources nor credentials for his personal evaluation of the work by HJ. Wikipedia is not intended for such contributions. The published work by HJ concerns a major open problem, for which by definition there can be no general consensus. Wikipedia must enter and settle the dispute. The arguments are presented on this talk page, and do not have to be repeated,Egbertus (talk) 11:47, 9 August 2008 (UTC)

I do not give a personal evaluation of the work, I just repeat what the authors themselves state: that the work is not within the context of the Clay prize.

Consensus in the WP sense, WP:CONS, in this respect has nothing to do with general consensus on the scientific content of the paper. The WP community has to reach consensus on whether or not this material should be included on WP, and if included: in which formulations.

You express as your opinion that this is an important paper. That may be, but is not relevant, since the paper is not on the subject of this article. So it does not belong in this article.

Please give arguments, and please answer the two questions regarding whether you have a conflict of interest or not. Crowsnest (talk) 12:08, 9 August 2008 (UTC)

I have asked for editor assistance since we are getting nowhere, just repeating arguments.Egbertus (talk) 13:35, 9 August 2008 (UTC)

I added reference to the Euler blowup article by HJ under partial results and also ref to d'Alembert's paradox, as problems related to the Clay problem. Hope this can satisfy Crowsnest, since claims of resolution are not posted, only claims of related results as partial results.Egbertus (talk) 18:07, 9 August 2008 (UTC)

Why do you re-instate the disputed material, with the reason for the dispute being primary the question whether it should be included or not? Why do you not discuss this here, give arguments, and (if there comes consensus on containing this material in the article) put a proposed text here? This is rather in conflict with your asking for editor assistance.

How can work on a "related problem", not on -- or converging towards -- the Navier-Stokes equations, be a partial result with respect to the Clay Millennium Prize Problem. Hoffman & Johnson claim that the Navier-Stokes equations converge towards the Euler equations (and next they numerically solve yet another set of equations, called the "regularized Euler equations). The Navier-Stokes equations also converge towards the Stokes equations, which is a linear set of equations and much more amendable to proofs on uniqueness and smoothness. But nobody would dare to say that results regarding the Stokes equations are partial results with respect to the solution of Millennium Problem.

What has d'Alembert's paradox (on the drag experienced by bodies in a fluid flow) to do with the Millenium prize (on the N-S equations in an unbounded domain without an object to interact with)?

So I strongly oppose to this re-insertion of this material, instead of discussing it here. -- Crowsnest (talk) 18:48, 9 August 2008 (UTC)

Awaiting editor assistance I tried a different presentation under partial results, in order to not get stuck on a technical discussion whether viscosity solutions of the Euler equations are Navier-Stokes solutions. In the presentation of the problem WP speaks about turbulence as the essence of the problem: "...to the first person providing a hint on the phenomenon of turbulence. In that spirit of ideas, the Clay Institute set a concrete mathematical problem". The HJ article shows how an intially smooth solution of the Euler equations breaks down into a turbulent viscosity solution of the Euler equations, and thus "provides a hint on the phenomenon of turbulence". What is remarkable is the lack of other "hints" and the math stalemate evidenced by Tao. Why not give a reference to the only published hint there is? Or can Crowsnest give reference to some other "hint" worth of mentioning?Egbertus (talk) 23:47, 9 August 2008 (UTC)

The problem is not a different representation of the material, the problem is that this material is off-topic. What WP says on the Clay prize is irrelevant, that is explicitly excluded as a primary source, see WP:PRIMARY. Then your presentation of the H&J article is incorrect: the article does not solve the Euler equations (without viscosity), but the H&J "regularized Euler equations" (with some form of dissipation). Solutions of the regularized Euler equations are assumed to converge towards solutions of the Euler equations, which is not proven in the paper or the book of H&J.

But let us assume for the moment they solve the Euler equations. It is your opinion, and apparently also H&J's, that results obtained for the limit of one equation, in which certain highest-derivative terms vanish, gives clues about the uniqueness and smoothness of the N-S equations. It is like saying that for the Burgers equation $\partial _{t}u+u\partial _{x}u-\nu \partial _{x}^{2}u=0\,$, the inviscid limit ($\nu =0$) having discontinuous shock solutions, predicts something about uniqueness and smoothness of the viscous Burgers equation ($\nu >0$). Which is not true, since the viscous Burgers equations has smooth solutions, see e.g. Taku Yanagisawa (2007), "Asymptotic behavior of solutions to the viscous Burgers equation", Osaka J. Math., 44 (1): 99–119 {{citation}}: Cite has empty unknown parameter: |1= (help). So this way of reasoning is incorrect, as shown before by Jitse Niesen[2].

Direct numerical simulations of turbulence, which are numerical solutions to the N-S equations in an periodic 3D domain, see e.g. Moin, Parviz (1998), "DIRECT NUMERICAL SIMULATION: a Tool in Turbulence Research", Annual Review of Fluid Mechanics, 30: 539, doi:10.1146/annurev.fluid.30.1.539, are much closer to providing hints with respect to the Clay problem, than H&J solution of non N-S equations in a bounded region. Yet they do not advocate themselves as partial results for the Clay prize.

Further you stay asking questions, and do not answer questions. In this way, this is not a discussion, only a difference of opinion. I stay answering your questions, and giving arguments why your reasoning is incorrect in my opinion. -- Crowsnest (talk) 08:38, 10 August 2008 (UTC)

To Jitse Niesen: I am perfectly willing to discuss. The inviscid Euler equations do not have pointwise solutions, like inviscid Burgers eq, and some form of regularized solutions have to be sought. This is what HJ do, and regularized Euler solutions are Navier-Stokes solutions and thus not off-topic. HJ take one further important step and consider wellposedness of regularized Euler solutions, which is the basic problem of turbulence, since regularized Euler solutions show to be turbulent. HJ show wellposedness of mean-vlaues such as drag and lift of turbulent Euler solutions, and thus provide an important step towards uncovering the enigma of turbulence. HJ show that potential solutions are unstable and illposed with respect to all outputs even mean-values and thus cannot be observed in real flow, which resolves d'Alembert's paradox. Concerning Burgers (and Euler) it is amisconception to claim that solutions with small positive viscosity are smooth since derivatives are very large, and functions with very large derivatives are not smooth, by definition of smoothness. You repeatedly bring up the mathematically erronous idea that you can distinguish an infinitesimally small number from zero. You are not alone, but that does not make this idea more correct. Finally, to speak about the inviscid Euler equations without telling what kind of solutions are considered (e.g regularized solutions), is meaningless. Egbertus (talk) 09:09, 10 August 2008 (UTC)

The Burgers equation was just introduced as an example for which it is not possible to transfer properties from the limiting case without viscosity to the case with viscosity. -- Crowsnest (talk) 12:55, 10 August 2008 (UTC)

To Crowsnest: I answer all questions of relevance. A question to you: In what sense are Moins DNS providing hints to the resolution of the Clay problem? If they do it is of great interest. Concerning verifiability: WP is not seeking to supply the truth but more modestly information based on verifiable scientific sources such as articles in refereed journals, right?Egbertus (talk) 09:18, 10 August 2008 (UTC)

You only answer the questions you like to answer.

There are many DNS solutions of the N-S equations in a 3D periodic domain, so in that respect in accordance with the problem statement. They provide hints, but no proof, that in general the solutions are smooth.

Concerning verifiability and truth, see WP:V and WP:TRUTH. But first of all, WP is an encyclopedia, so being true and verifiable are in itself not enough reason for inclusion, see WP:ENC. -- Crowsnest (talk) 12:04, 10 August 2008 (UTC)

To Crowsnest: What is your motivation of removing the reference to HJ under partial results? Off-topic?? I repeat HJ consider regularized Euler solutions which are Navier-Stokes solutions, and show wellposedness of meanvalue outputs of under small viscosity. How can this be off-topic? The central problem of turbulence is wellposedness of meanvalue outputs. How can you claim that this is off-topic? If it is not off-topic it should be cited since it is published in refereed journals. Right? Please be reasonable.Egbertus (talk) 09:34, 10 August 2008 (UTC)

My motivation is that the discussion should be here, instead of edit-warring in the article.

You know the rules, you even have been blocked for it, I requested you to discuss on the talk page and only re-insert material after consensus is reached. On which you did not object. And now, since you seem not to be able to provide arguments why this material is not off-topic, you start again with re-inserting and reverting in the article.

So I have the following questions to you, on which I am (still) waiting for an answer by you:

1. Does the H&J paper solve the Navier-Stokes equations, as requested in the problem statement for the Clay Millennium Prize?
2. Does the H&J paper solve equations in an unbound $\mathbb {R} ^{3}$ or periodic $\mathbb {R} ^{3}/\mathbb {Z} ^{3}$ domain, as requested in the problem statement for the Clay Millennium Prize?
3. Are solutions of the inviscid Euler equations also solutions of the Navier-Stokes equations?
4. Are solutions of the regularized Euler equations also solutions of the Navier-Stokes equations?
5. Do proofs of existence and smoothness, made for the inviscid Euler equations, transfer to the Navier-Stokes equations?
6. Are you one of the editor's of the H&J paper? Or are you involved in their research groups?

I will not answer more of your questions before getting answer with respect to the above questions. -- Crowsnest (talk) 12:04, 10 August 2008 (UTC)

I posted a request for contributions by other editors on Wikipedia Talk:WikiProject Mathematics and Wikipedia Talk:WikiProject Physics. -- Crowsnest (talk) 12:12, 10 August 2008 (UTC)

Dear Crowsnest: It is good that also you ask for editor assistance. Your questions: 1. HJ claim to do that in the article. It is up to the scientific community and prize committee to judge, not me. 3. What do you mean by inviscid Euler solutions, pointwise, regularized? 4. Yes, answers also 3. 5. Again what is meant by exist and smoothness of inviscid Euler equtions? cf 3. 6. Irrelevant. This is a basic principle of Wikipedia of keeping the identity of the editor out of the discussion. Do you understand my questions concerning your use of the notion of solution of the Euler equations? If not, I will explain in more detail, why you have to define what you mean by solution. This may not be well understood in fluid mechanics circles but is completely crucial as concerns the Clay problem. Best regards.Egbertus (talk) 12:50, 10 August 2008 (UTC)

A further question to Crowsnest: Since you block references to HJ work on the Clay page, why don´t you do the same thing on the related d'Alembert page?Egbertus (talk) 13:15, 10 August 2008 (UTC)

At that moment, see this diff, I removed the text referring to the Clay prize, as being off-topic. I did not remove the other material or the reference to the BIT paper (containing also claims regarding d'Alembert's paradox) because there was an ongoing discussion on that topic on the talk page. -- Crowsnest (talk) 08:51, 11 August 2008 (UTC)

Ad 1. I don't think Hoffman & Johnson claim to do so. Your evidence for that is two sentences in the paper: "we present evidence of (II)" (from the summary) and "since the viscosity in the Navier–Stokes equations is allowed to be arbitrarily small and solutions of the Euler equations are defined as viscosity solutions of the Navier–Stokes equations under vanishing viscosity, with slip boundary conditions modeling vanishing skin friction, the Euler equations effectively are included in the millennium problem as a most difficult limit case" (at the end of Section 1). However, (II) refers to "non global existence of a smooth solution for some smooth data, referred to as breakdown or blowup." and I think that (II) in the sentence you quote refers to non-existence in the Euler equations. And the second sentence is at best ambiguous, does "effectively included" mean they're included or not? Fortunately, Hoffman & Johnson are more explicit in Section 7: "In particular, there may be strong reason to include the Euler equations as a most useful limit case of the Navier–Stokes millennium problem, instead of discarding it on formal reasons." That implies that the Euler equation are not included in the millennium problem.

You left out the answer to 2. I'd like to see some proof for your answers for 3 and 4. For "existence of Euler", you can use the definition given by Fefferman for existence of Navier–Stokes except that ν = 0; I think you would call this pointwise and non-regularized. -- Jitse Niesen (talk) 13:23, 10 August 2008 (UTC)

In addition to Jitse Niessen:

Ad 2. You do not answer question 2.

Ad 3. I mean the inviscid Euler equations, not the "regularized". For example: in a 2D domain with a cartesian (x,y) coordinate system, and (u,v) the associated velocity components, v=p=0 everywhere and u=1 for y>0 and u=0 elsewhere, is a solution to the unforced inviscid Euler equations. I do not consider this to be a solution to the N-S equations for ν>0.

Ad 5. If you like, answer for both the inviscid and regularized Euler equations. Please provide reliable references.

Ad 6. Relevant according to WP:COI. I do not ask you to disclose your identity. I just want to know whether you answer both questions with "no", or at least one with "yes".

-- Crowsnest (talk) 10:05, 11 August 2008 (UTC)

Ad 3. A better example, not requiring weak solutions: in 3D, with cartesian coordinate system (x,y,z) and velocity components (u,v,w) in these respective directions, v=w=p=0 and u=f(y,z) is a solution of the inviscid Euler equations without exterior forcing. In particular, all smooth functions f are a solution. Whileonly the subset Δf=0 is a solution of the N-S equation. -- Crowsnest (talk) 14:02, 11 August 2008 (UTC)

Dear Jitse: You don´t think HJ claim resolution, but you are evidently not sure. Why not write to HJ and ask? Or should I do that? The issue of the whole space/periodic is just to keep it simple, but if you can do something with boundary condition, the better. If you take into account wellposedness, which is necessary to have a meaningful discussion, you see that the essence is the size (in various norms) of the residual vs Euler and the viscous term in NS, and that the precise nature of the viscous term is irrelevant for mean-value outputs. This makes regularized Euler solutions also solutions to Navier-Stokes with small viscosity. Right? Egbertus (talk) 13:54, 10 August 2008 (UTC)

To C S: Please motivate why you undid revision by Egbertus (talk) 14:48, 10 August 2008 (UTC)

I am sure that Hoffman & Johnson do not make a clear claim of resolution in their paper. They may claim resolution privately but that's irrelevant here. Any personal communication is also irrelevant because it's not verifiable for Wikipedia's purposes.

I assume that with "wellposedness" you mean the same concept as Hoffman & Johnson in their paper, and not the standard meaning of the word. That's not mentioned in the millennium problem and thus not relevant here. -- Jitse Niesen (talk) 14:54, 10 August 2008 (UTC)

What makes you so sure? The question is if what they present is a resolution or not, regardless of whatvere explicit claims are made. Is it a resolution? Even a step towards a resolution is better than nothing, right? It is well understood that wellposedness is central to any discussion of differential equations. What makes you think it is not? Why not focus on science instead of just formality without science?Egbertus (talk) 20:59, 10 August 2008 (UTC)

It is not to WP to decide whether something is a resolution to the prize problem or not. A core issue for inclusion is verifiability by reliable sources, and it is verifiable by reading this reference itself that it solves equations different from the N-S equations, and in a different geometrical setting as asked for in the problem statement. Apart from the fact that I read several times in the article that they realise their work it outside the scope of the Clay Millennium problem. And H&J giving arguments why they think the scope of the problem statement, as it is, should be extended to also include their work. -- Crowsnest (talk) 08:40, 11 August 2008 (UTC)

Crowsnest's concern about COI is valid. Claes Johnson (the J in HJ) has said (regarding the D'Alembert paradox) that "We have rewritten the Wikipedia article, to include the new resolution, and nobody is protesting, so it represents the new truth."[3] I would say this is relevant to Crowsnest's question. Claes Johnson also says that the BIT article "proposes a resolution"[4] of the Clay Navier-Stokes problem, if that matters. 76.197.56.242 (talk) 23:41, 10 August 2008 (UTC)

The page [5] has some correspondence between J and Tao (an expert in this area). Tao appears to be rather skeptical of the claims made by J. The work by H&J should not be included on wikipedia unless it is confirmed by multiple reliable sources. The account Egbertus has no edits except for attempts to promote the work of H&J. R.e.b. (talk) 14:50, 11 August 2008 (UTC)

I added link to book home page connecting to Tao´s blog.Egbertus (talk) 09:43, 12 August 2008 (UTC)

To Crowsnest: Do you claim that all external links to various professional blogs and home pages should be deleted? Then there is a lot to delete for the other Clay problems.Egbertus (talk) 14:35, 12 August 2008 (UTC)

I do not know what is on the articles on the other Clay millennium problems, and they have not my interest, so I cannot comment on that. -- Crowsnest (talk) 07:12, 13 August 2008 (UTC)

Dear Jitse: I already discussed, see above, but I don´t see that you or Crowsnest do that. What was your reason to remove the ref? Why do you consistently seek to remove any ref to HJ? Don´t you see that this is violation of NPOV by one-sidedly suppressing information?Egbertus (talk) 16:16, 12 August 2008 (UTC)

When your edit is reverted, you should allow some time for discussion. You should not re-revert immediately.

I don't think all external links to various professional blogs and home pages should be deleted. I do think most are inappriopriate. It depends on the expertise of the person and the contents of the page being linked to.

I'm removing any reference to Hoffman and Johnson from this article because I think their work is not relevant in this context. It's not an violation of NPOV, because NPOV does not say that all information about the topic should be included. -- Jitse Niesen (talk) 12:34, 13 August 2008 (UTC)

You write that you "think" that HJ work is not relevant. Is that an argument? What do you "think"? Why is it not relevant? Don't you understand that violation of NPOV is serious? There is no proposal of resolution what I know of except that by HJ. If you speak of "all information" this is it. Why delete all information? How can that not be against NPOV?Egbertus (talk) 05:35, 14 August 2008 (UTC)

NPOV is not the issue, since the material is not relevant for the article, as being off-topic, as the article itself states. Saying it is relevant and important is not enough, see WP:RELNOT. -- Crowsnest (talk) 21:26, 14 August 2008 (UTC)

OK, so you admit that the ref. is relevant and important, but yet it is deleted. This not NOPV. To advocate that it is off-topic is just your personal hang-up which should not control WP, right?Egbertus (talk) 07:37, 15 August 2008 (UTC)

I don't see such an admission from Crowsnest. The point of RELNOT is that saying something is relevant and important doesn't establish that it is relevant and important. You have to document the HJ paper's relevance and importance through citations of verifiable reliable sources per the verifiability policy--we are not going to accept its relevance and importance purely by your say-so.

As for NPOV, that policy says (in its "Undue weight" section), "[v]iews that are held by a tiny minority should not be represented except in articles devoted to those views." The set of people known to claim HJ is relevant is {Johnson, Egbertus}, a set with cardinality at most two and possibly less. So, the NPOV policy says to leave the HJ reference out of the article. 76.197.56.242 (talk) 08:18, 15 August 2008 (UTC)

You are not well informed. HJ are leading researchers in mathematical/computational fluid mechanics. Study the literature and http://www.isihighlycited.com/ listing the most cited researchers in the world. I think WP should build on knowledge not ignorance.Egbertus (talk) 08:52, 15 August 2008 (UTC)

And I think Wikipedia is not a forum for you to promote your work or your friends' works. It is true Claes Johnson is a distinguished professor of applied mathematics. No one has disputed this. It is also clear that he has used his reputation to push his paper through the editorial process at Journal of Mathematical Fluid Mechanics. This is clear from the published correspondence on his wiki. The referee reports and initial editor's reports are all overwhelmingly negative, just as they were from the previous rejections at other journals (the much more prestigious ones). As for the initial acceptance letter from the chief editor [6], it only corrobates what everyone here has been saying. The "resolution" is overblown. In order to be published (and that letter is in no way a final acceptance), various conditions that tone down the language of the paper must be obeyed. For example, "The title should sound less emphatic, for example: "A new approach to resolve d'Alembert's paradox" or simply (what I would prefer) "A note on d'Alembert's paradox"." and "The overall attitude of the paper should be less provocative and its tone less pompous." This is hardly compelling evidence for us to believe that this work has gained any degree of acceptance from other distinguished mathematicians. Indeed the editor comments, " I think that JMFM could publish your paper as a note for stimulating discussion of an important problem...I should look like a note pointing out some observations which are hoped to shed new light at an old problem. This should initiate new discussion and investigation of the problem." As I have thought from the beginning, this paper was accepted (at least tentatively) on the basis of being "interesting" even if it goes nowhere close to actually resolving D'Alembert's paradox. Many such papers exist in many fields. Only time can tell if they have any value. It is certainly not your decision to make that it is either a done deal and should be included in an encyclopedia or turn Wikipedia into an endless debate forum where you can have a discussion that you are unable to have in real life. --C S (talk) 10:12, 15 August 2008 (UTC)

Neither CS is well informed. Science is about making progress in an open discussion. CJ is one of the most cited researchers in the world with wide influence and auditorium and does not need WP. But I think WP needs input from the frontiers of research, not just second or third hand information, which often is incorrect because of misunderstanding and misinterpretation. CS is consistently deleting any input from one of the leading research groups in the world in their field. Isn´t there anybody reading this who can help CS to a more reasonable and for WP productive standpoint? An what are the scientific credentials of CS? What kind of playground is WP?Egbertus (talk) 10:39, 15 August 2008 (UTC)

Who's not well informed? This makes about the 14 time you've been told that Wikipedia is not about explaining "frontiers of research" but is precisely about explaining "second or third hand information". It seems almost useless to point you to a previously pointed at policy page, but here it is: Wikipedia:reliable source. I understand you don't like how Wikipedia works. But Wikipedia is not going to change its way just because you don't like it. Wikipedia is not a playground. It's not a place where you can just do whatever you want. At this point, you are simply flooding the discussion pages with no new information. --C S (talk) 11:17, 15 August 2008 (UTC)

You are wrong CS: Ignorance is not a virtue, not on WP in particular. You do not seem to understand the role of scientists to supply first hand information by publishing in refereed journals. Second or third hand information can never replace first hand information. In particular not on WP. I hope you can understand and stop relentless blocking of new information on WP.Egbertus (talk) 11:27, 15 August 2008 (UTC)

You evidently continue and now delete all information on hydrodynamic stability I put up under Navier-Stokes equations. Don´t you understand that this is a completely fundamental aspect, maybe the most important there is, which has been studied by many prominent researchers including Nobel Prize winners. Why do you delete this information from WP? Is there noboy who can help CS out of this destructiveness.Egbertus (talk) 11:38, 15 August 2008 (UTC)

Maybe WP is not for scientists, as it is run by some of its editors. I change to Knol.Egbertus (talk) 13:37, 15 August 2008 (UTC)

Hydrodynamic stability

Hello, Whilst Hydrodynamic stability, henceforth referred to as HS, may well be important, I don't think it is something that sits squarely between turbulence and the limitations of the NS equations. My suggestion is to create on sentence somewhere and link it to an article about HS. It doesn't really work where it was placed. User A1 (talk) 13:30, 15 August 2008 (UTC)

Assumptions on the Navier–Stokes equations

This section was rather imprecise:

1. It states that the Navier–Stokes equations assume conservation of energy, which they do not. There is viscous energy dissipation, and the equations do not state where this energy goes (which is also not important within this context).
2. Temperature effects on density are not mentioned.
3. As correctly pointed out by 86.80.203.194 the Mach condition on incompressibility is not the condition to decide whether a flow can be approximated as incompressible. This Mach condition is appropriate for instance for steady flow around airfoils, with a distinctive (fixed) object relative to which velocities are measured. In this Clay Millennium Prize problem there is no such object. Further, for unsteady flows, conditions under which the flow can be assumed incompressible are more complicated and different from the Mach condition. For instance, in acoustics problems the flow is always compressible.

Since this discussion on the assumptions under which the incompressible and homogeneous Navier–Stokes equations can be assumed to hold is not so relevant in the present context (the conditions are given in the problem statement), I deleted this section (excluding the last paragraph which is an introduction to the next two sections). -- Crowsnest (talk) 08:31, 2 November 2008 (UTC)

Advices for person who want to find exact analytical solution of Navier-Stokes equation

1. This problem is extremely difficult.

If you will work fast or for money (for 1 million dollars) you will destroy your brain yourself.It is very dangerous problem for health. Our brain cannot to solve it fast.

2. There were 12 attempts to solve it in USA and Europe till 2008 year. But they had faults. One person wrote, that he lost 8 hours during 6 years every day. And he made mistakes. He could not solve it.

3. Mathematicians created very complex notation, which stop them. I think it interferes in of you to find this solution.

4. This problems useful not for mathematicians, but for all scientists. The Navier-Stokes equation - it is very complex mathematical model. But mathematicians think today, that it is simple mathematical model. Look please on lecture ( Luis Cafarelli). http://claymath.msri.org/navierstokes.mov He thinks so and many others.

5. If your organization needs for this solution, I can tell you for 10 minutes. But you have to know the theory of differential equation and physics. my phone is 514-5267971 gorskin @ hotmail . com Without mathematics you cannot create a mathematical model, for example for radioactive decay.If you don't know radioactivity, you don't understand the solution of this equation, if you are mathematician only.

6. I don't want to publish all my results, because it will be use for design of weapon. Canadian research consul (he worked for Government) didn't want to support my work with using exact solution of Navier-Stokes equation, for application. It is not interesting for any Canadian organizations. But when people developed math theory for radio, they had the same problems many years ago.

8. Exact solution of Navier-Stokes equation for turbulent flow is nonlinear waves. Navier-Stokes equation can describe many physical and biological process, for example: turbulence, mitosis. I try to use solution for describing finance process with the stocks on tsx.com. There is a chaos in the stock exchange, likes in the fluid.It is possible to describe it with Navier-Stokes equation sometimes.

9. I found the solution of parts of this problem. There is not global regularity for Navier-Stokes for some cases. With Navier-Stokes equation we can determine the constant Feigenbaum. We can find this constant, if we see on the peninsula in Nova Scotia in Canada.

Main properties of exact solutions on Navier-Stokes equations are:

1. More than one function is solution. It is group.

2. Very sensible for initial data.

3. You cannot predict future or past on some period time, if you know some solution for turbulent flow for time t=0 c.

4. The solution, for example, can be a nonlinear wave, solutions and many others.

5. We can find such initial data, that there is not any turbulence for high number of Re, we can destroy turbulence for high number of Re.It is theoretical result of course.

6. There are solutions, like vortex. It can compete with each other for momentum (impulse), matter (substance).They can bisect or join, connect in the one object (like new "chemical elements").Tornado is example.

These properties have any bacteria and cancer cells. But there is not genes. We can not to use the theory of Charles Darwin (inherit properties) for water.Water have some properties of life object only.

I had written part of my article, but it is stolen from my room or from my bag. Dmitri Gorskin (talk) 21:30, 27 January 2009 (UTC)

On the 2d case

About"Partial results

The Navier–Stokes problem in two dimension has already been solved positively since the 60's: there exist smooth and globally defined solutions.[2]"

I scanned the book [2]

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows", 2nd edition, Gordon and Breach, 1969.

and did not find this result. Anyone can specify the pages of this result. Thank you!

The only paper I know about the 2d case is one of Weigant, VA and Kazhikhov, AV On the existence of global solutions to two-dimensional Navier-Stokes equations of compressible viscous fluids

Siberian Math. J,36 1108--1141,1995

zhangvszhang 18:22 27th July 2009(UTC)

Jorma Jormakka's announcement

Since we've just been through this with P vs. NP, I'll simply make a note of http://ejde.math.txstate.edu/Volumes/2010/93/jormakka.pdf, which is asserted by its author to be a solution satisfying the problem definition. It appears that, by the author's own admission, this exploits a defect in the way that "external forces" are defined in the official problem. There's no coverage in third-party sources as of yet, so I would tend to oppose mentioning it in the article. However, if the Clay Mathematics Institute comments on the technicality being exploited in this solution, that would likely be enough for it to get mentioned in the article. — Gavia immer (talk) 02:05, 22 August 2010 (UTC)

Move from article footer

- 4. The Clay Math Institute Navier-Stokes problem, as it is stated in the official problem statement, is proved in the peer-reviewed journal paper Jorma Jormakka: Solutions to three-dimensional Navier-Stokes equations, Electron. J. Diff. Equ. Vol 2010(201), No. 93., pp. 1-14. http://ejde.math.txstate.edu. To the stated problem there exists a counterexample both in the space-periodic and nonperiodic cases (Statements D and B are true). Whether the Clay Math accepts this solution or reformulates the problem is up to them. The requires changes are not small and show that the problem was not well-understood at the time the problem was posed. Notably, the claim that a solution can be uniquely continued from t=0 to some finite time is wrong under the initial conditions given in the problem statement. The same error is repeared in this Wiki page.

- - (Comment to the Wiki page moderator: The validity of the above mentioned EJDE article can easily be checked by any undergraduate student and it has been accepted to be correct by the mathematical community for about two years, and it is checked by many competent mathematicians. As the result is a bit embarassing to the PDE community, no verication of this peer-reviewed journal result has been given in American newspapers, that so well verified e.g. that Irak has nuclear weapons in 2003 and made big news to verify the WTC dust analysis showing the spectrum of thermite. Quite strangely, the strongest supporters of the false theorem of uniqueness have also not made a public statement that they were wrong.)

This comment originally by User:88.114.55.128 NOT by (shoo Sinebot) User A1 (talk) 11:09, 28 August 2010 (UTC)

Just an additional note: the IP editor who posted this has identified himself on my talk page as being this same Jorma Jormakka. My earlier comments in the preceding section still apply in the absence of independent coverage. — Gavia immer (talk) 15:56, 28 August 2010 (UTC)

Dear Gavia immer

I answered to Robert Coulter below, but my answer cannot be seen in the talk page. Could you do something to it. If Robert Coulter is making false claims against my article, then my response should be visible. Just to mention again. I was not the person first announcing my work to Wiki. I simply put the reference to such a form that it is certianly correct and cannot be irritating anybody. I think, if Wiki is open to many contributors, this reference should be there. The article has not been refuted by anybody and the fact that neither Clay nor anybody makes any announcements that it is wrong is indication that it is correct. Especially, Terence Tao has not shown the article to be wrong. For my part personally it is fully irrelevant whether there is a reference to my article in Wiki os not, but I do not like incorrect claims to be made against the paper on Wiki Talk-pages. If such are made, then I must respond to them. Please, make my answer to Robert Coulter visible. Sincerely, Jorma Jormakka —Preceding unsigned comment added by 88.114.55.128 (talk) 05:07, 7 September 2010 (UTC)

If your comment below wasn't visible, it was likely a transient problem with this page being cached. If you can see this response, you should be able to see your own posting in the section below. I had nothing to do with any such problem and have no control over how this talk page functions. — Gavia immer (talk) 05:23, 7 September 2010 (UTC)

Discussion on Jormakka's Claim

To summarise this lengthy discussion, Robert Coulter has demanded that there should not be any comment on the Wiki page to a published peer-reviewed article in a reputable mathematical journal though it is clearly relevant to the Wiki page. Coulter has presented no scientific arguments to claim that there is any error in the published article. All his arguments were already answered on Tao's blog and they can be seen there. His arguments are not correct. They are based on physical intuition that energy should be finite, however the Clay problem setting is not physical in this sense. The article has been checked by many reputable mathematicians and is easy to check again for anybody with even very minimal mathematical basic knowledge. The result in the article is mathematically undisputable and the exact Navier-Stokes problem as posed by the Clay Math Institute is solved in the article. Clay Math may or may not grant a prize for this article, or it may decide to correct the problem and grant some other prize for this article. The dispute seems to be whether this information should be available to Wiki readers or not. I do not know how to remove this dispute since I do not know what and why Robert Coulter opposes, only that he opposes mentioning the article in any way. Jorma Jormakka 88.114.55.128 (talk) 09:18, 20 September 2010 (UTC)

I refute Jormakka's claim. See Posts Refuting Jormakka Claim Note also, in the same blog, that Terence Tao has noted problems with the alleged proof.

Recommend all references to Jormakka's alleged proof removed from article.

~~Robert Coulter~~ —Preceding unsigned comment added by 76.123.120.172 (talk) 21:32, 3 September 2010 (UTC)

The EJDE article has been checked by referees and many competent mathematicians. The footnote in the Wiki article that the solutions are not unique in the situation specified by Fefferman has been confirmed also by Terence Tao. The footnote and the mention of nonuniqueness is useful for readers in this context and it is not disputed. Terence Tao's comments do not show any error in the article. He makes the following statement. 1) Tao states that the solutions are indeed not unique as stated by Fefferman. This is to be understood that solutions are not unique under the conditions stated by Fefferman. 2) Tao says that if the pressure would have been required bounded then the solutions would have been unique. This is true, and the EJDE article states the same: growth conditions are needed to pressure. 3) Tao states without any basis that the external force has to be given as a point function and not as a feedback function using the velocity. The EJDE article states that Fefferman's problem formulation should have stated so but it does not. 4) Tao makes one error in his fast written response. He claims that using feedback forces solutions are not unique. They certainly are. The nonuniqueness is only in defining the initial conditions. When the initial conditions are fully defined, uniqueness of the solutions follows from the local-in-time existence and uniqueness theorem. Physically, when there are several solutions for zero external force, application of external force in the beginning can steer the solution to any of these solutions and the solution stays in the selected solution for all time after the force is stopped. Try steering a car to a given direction, it indeed turns where you want and and after that, if you simply allow the car to go where it goes, you do not need to turn the steering wheel. These are all comments by Tao. Three of them are irrelevant, and the last of them is false. There was also a student cowgod42 trying to show the proof incorrect in reddit.com. While he posted a claim that "Supposed proof... is utterly wrong", all his arguments were shown incorrect, which he finally admitted. Concerning the footnote in Wiki. It is correct. Tao has not claimed it is incorrect, neither has anybody else. —Preceding unsigned comment added by Jorma Jormakka88.114.55.128 (talk) 04:06, 7 September 2010 (UTC)

Robert Coulter's refutation is what is called "original research" that must not be done in the Wikipedia. The citation to the EJDE article is a normal citation to scientific literature. The information that is citated is not in any way in dispute. Coulter's claim is unfounded and based on his own unpublished conclusions. The reference to Terence Tao's answer in a discussion part of his blog is not a scientific reference, indeed Tao did not want to discuss the article but wrote an opinion in order to forbid all discussion of the article. If Coulter (or Tao) wants to refute the claim, it should be done by publishing an article in a peer-reviewed forum, or in some other way make a scientifically respectable refutation of the EJDE article. As there is no such reference, Coulter's opinion is original research and has no place in Wiki. it is also necessary to use critizism of sources when considering the value of a blog opinion by another scientist, who himself is probably trying to solve the same problem. Jorma Jormakka —Preceding unsigned comment added by 88.114.55.128 (talk) 09:28, 7 September 2010 (UTC)

Absence of comment on a proof does not indicate correctness. It is not the burden of the scientific/mathematical community to prove the negation of your proof. Your proof must be confirmed by reputable scholars in the field. I could find none. In fact, nearly all comments I could find about this paper on the internet are highly critical of the paper. Wikipedia should remove the reference to this alledged proof for reasons I just stated. 76.123.120.172 (talk) 22:54, 7 September 2010 (UTC)Robert Coulter

I do not know why the reference to this article has been attacked so heavily by Robert Coulter. The comment in Wiki is that the solutions are not unique. This fact is not in dispute in any way. Even Tao has acknowledged it. The EJDE article is a peer-reviewed journal paper, which has been checked by reputable third party referees before publishing. The editor of EJDE who accepted the article is a reputable mathematician in his field and the people thanked in the article are also reputable scholars in mathematics and physics. If a published paper is to be refuted, it should be done in respectable scientific forums, not in Web comments or Wiki Talk pages. It is indeed the task of the mathematical community to show that a result published in a peer-reviewed journal of good repution is wrong, if it is wrong. So far the article is accepted as it has not been refuted. Lack of concensus in a scientific community is in general no argument against a scientif result, however in mathematics there is usually a concensus: proven results are assumed to be correct unless the proofs are found incorrect. If Robert Coulter wants to refute the article, Wiki Talk page is not a place for it. He should publish an article, and the correct journal to submit such an article is EJDE. For what Internet discussions matter in mathematics, there are also no valid refutations in the Internet against the article, if there were I would withdraw the article personally. There is the post of cowgod42, the single comment by Tao, and the incorrect claims by Robert Coulter. That is all, and they are all refuted. Comments from Clay concerning this article may come in two years, most probably not earlier. Jorma Jormakka —Preceding unsigned comment added by 88.114.55.128 (talk) 17:54, 12 September 2010 (UTC)

Before this gets too out of hand, Wikipedia is not a suitable place for research discussions. If there is not consensus within the scientific community, then outside communities (such as wikipedia editors) cannot comment one way or the other on the topic. I am certain that there must be scientific venues for such discussions. Robert is quite right in suggesting that third party citations would greatly enhance the ease with which us simple WP editors could digest something of this nature. 23:22, 7 September 2010 (UTC)

There is consensus concerning the Wiki comment. It is correct. The relevant results in the EJDE article (Lemmas/Theorems 2.1-2.4) are all correct and confirmed by the third party referees and editor. The conclusion of the article is that unless Theorem 2.4 is accepted as a proof of Statement D in the official problem statement, the official problem statement must be corrected. Theorem 2.4 uses a feedback steering force as the external force. If this is allowed, then there is a finite-time blowup. It it is not allowed, then the problem statement needs to be modified so that it adds restrictions to pressure (at least exclude feedback forces. There is nothing in the official problem statement that presently excludes feedback forces.) However, this can be the discussion that needs third party citations. This part is not addressed in the Wiki comment and either way (accept the proof or correct the problem) the EJDE article is a partial result to the millennium prize problem. I hope this answer satisfies both Gavia immer and Robert Coulter. This discussion has gone far too far for a simple mathematical result and a minor comment to Wiki with a citation to a journal paper. I am sure that the vast majority of citations in Wiki are not as well confirmed as this elementary proven mathematical result of non-uniqueness. Else there could be no discussion on non-mathematical topics at all. Jorma Jormakka

The issue here is if mentioning of the supposed proof is warranted in this article. The only person that I know promoting this is the author. There is no known verification of this proof. Non-action of others to disprove does not translate to verification of the proof. If Wikipedia took that policy with every supposed proof of a famous conjecture, then the article would be filled with contradictory statements since at any given time there are numerous supposed proofs taking differing positions on a conjecture. Wikipedia should remove the reference to this supposed proof or risk losing credibility on this subject 76.123.120.172 (talk) 16:53, 13 September 2010 (UTC)Robert Coulter

I will re-iterate that wikipedia is not the discussion area for this, and these discussions would be best had offline. Until there is a clear, digestable account of this by a third party, us plebs can't comment on this one way or the other. All that can be said here is "this is not the place". Including this into WP should only be done by a non-author of the article. With no ability to comprehend the contents of the proposed article, which is highly specialised, this will not be added. Misinterpreting the article would be a greater error than omission. WP can afford to be "behind the times" on this one. User A1 (talk) 17:24, 13 September 2010 (UTC)

I entered a POV on the section being discussed. The reasons for this have been mentioned above. The non-neutral section is the last two sentences of Item 3. Rbcoulter (talk) 18:53, 13 September 2010 (UTC)Robert Coulter

Dear Gavia immer

I changed the title of Robert Coulter's subtitle "Jormakka Claim Disputed" to the subtitle "Discussion of Jormakka's Claim" since Robert Coulter has not in any way scientifically disputed any claim of mine, neither the comment in Wiki, nor the EJDE article. This is similar manipulation of media as the character cowgod42 in reddit.com, who posted to many places a message "Supposed proof by Jormakka ... is utterly wrong" while he had not even read the article and when he read it he admitted that all his claims were wrong. This can be seen by opening the message. As this is Wiki Talk pages and I am not even suggesting that at this point anything more than the short and certainly correct comment that already is there should be added to Wiki, I think this campaign by Robert Coulter against my so called claim is totally out of place in Wiki. This really should stop and I suggest Robert Coulter will not any more insert unsubstantiated claims against my published peer-reviewed article in this forum. The reference in Wiki is to the nonuniqueness and it is not disputed. There are very seldom any public announcements of mathematical papers made by some eminent mathematicians in the Internet. Usually, a paper is accepted, published, and that is it. It is already verified by the journal. There is no claimin the Wiki page that the Navier-Stokes problem is solved. I wrote there that the nonuniqueness is proven in the article and therefore it is claimed that the Navier-Stokes can be solved. Surely, this is correct. The suggestion to refer to my article come from a non-author, not know to me. The article is verified by the peer-review process. Jorma Jormakka —Preceding unsigned comment added by 88.114.55.128 (talk) 13:15, 14 September 2010 (UTC)

My understanding concerning this article is that it must be neutral. As Gavia Immer has stated, we are not to discuss the technical merits of the supposed proof here. I would be more than glad to discuss its technical merits elsewhere. The only issue here is the neutrality status of the section I noted above. This clearly violates the standards for the article. I do not see how and author entering a statement referring to his/her own work in the article text can be considered neutral.Rbcoulter (talk) 13:44, 14 September 2010 (UTC)

It is a published peer-reviewed journal article in a reputable journal. It is a part of the common mathematical body of knowledge. It is not an unverified manuscript. I did not originally enter the statement but modified it to satisfy the requirements set by Gavia immer and to be certainly correct. Your effort is similar to as if you would state 1+1=2 and I would say that this is not verified because no reputable scientist in arithmetics has made a public statement that Robert Coulter is correct, 1+1 is indeed 2. Why should any expert make any statements concerning a published paper if it is correct. They make statements usually when papers are incorrect. The article is fully neutral. It does not claim that the problem is solved. The reference to my paper only highlightens a correct and relevant fact. What do you exactly oppose in the small comment in Wiki and why? Mathematical papers are all neutral. Jorma Jormakka —Preceding unsigned comment added by 88.114.55.128 (talk) 17:39, 14 September 2010 (UTC)

The article says "..Therefore it is claimed that the Statement D (and also B) of the Clay Math. official problem statement can be proved.." See the following wikipedia guidelines concerning neutrality. Especially note the section about using the word "claimed". http://en.wikipedia.org/wiki/Wikipedia:Neutral_point_of_view The paper at EJDE is simply a "claim" of a proof. If the Clay Institute decides that this proof is correct. Then a statement can be inserted to that fact at that time. Meanwhile, it is just speculation and opinion. This, by the guidelines in the link, are not allowed because it is non-neutral.Rbcoulter (talk) 19:17, 14 September 2010 (UTC)Robert Coulter

Alternatively, if Wikipedia takes the position that the EJDE article IS sufficient to represent proof of one of the parts of the Clay NS problem, then this should be listed under a new section maybe called "Full Results". The word "claimed" should not be used since this implies controversy. So there are really only two choices:

1. The EJDE article is a "claim" of a proof. If this is the case, then it has no business in the article at all since it represents an opinion per the neutrality guidelines of Wikipedia.

2. The EJDE article is a proof. Then it should be listed under a new section "Full Results".

In summary, Wikipedia either accepts the EJDE article as current fact or not. It should not post claims or opinions on this matter. Rbcoulter (talk) 20:01, 14 September 2010 (UTC)Robert Coulter

Fine. I will formulate the last statement as it is proved in the EJDE article. "The article concludes: Unless Theorem 2.4 (in the article) is accepted as a proof of Statement D, the official problem statement must be corrected." Then it is not a claim in Wiki, it is also not committing Wiki to support the claim. It is a verbatim reference to the article without interpretation. This statement is exactly what is proven in the checked article. It is not a claim (in your sense of the word claim) that something is proven. The claim is also proven in the article.

Let me explain the argument of the EJDE article very clearly. 1) There is no dispute of the fact that the solutions are not unique (Lemma 2.1, Theorem 2.2) if the external force is a point force defined before and any solution is (u,p) is searched for. This fact is verified even by Terence Tao, while there is no need for Tao or anybody to verify a mathematical fact that is proven in the article. 2) Theorem 2.3 is obtained by selecting function g(t) which has a singularity at finite time. This theorem is not in dispute. Thus, there exist blow-up solutions. So far we must agree. 3) Theorem 2.4 is not referring to the Clay setting at all, so no arguments can be made against it that feedback forces are not accepted in the Clay problem (as Tao argues). They are in the assumption of Thorem 2.4. We only ask if Theorem 2.4 is correct as it is stated. 4) The solutions in Theorem 2.4 are unique since inserting the feedback force the time derivative of the velocity is cancelled, thus there is no possibility of nonuniqueness that was caused by undefined time derivatives of the velocity. It also follows directly from the local-in-time existence and uniqueness theorem, and it follows from the general property of feedback forces: they are steering forces and steer the solution to a unique force. Theorem 2.4 has been checked by EJDE and is correct. 5) The conclusions in Section 4 are: Unless Theorem 2.4 is accepted as a proof of Statement D, the official problem statement must be corrected. This conclusion is not in dispute. Unless Theorem 2.4 is a solution, it is necessary to exclude feedback forces that currently are not excluded in the official problem statement. 6) The question is whether Clay accepts the solution or modifies the problem statement. One or the other must be done. In either case it is news and relevant to the Wiki article.

There is no need for the Clay institute to "verify" mathematically proven statements that have already been through the scientific procedure. This is the only peer-reviewed journal article in a respectable journal stating a result that is so directly relevant to the Clay math problem statement. Thus, there is no reason to refer to any proof attempt in arxiv. (But I do not object, if you want to include them, there are not many. There was Penny Smith's attempt 2006 but it was withdrawn). You are the only person who is still trying to argue against this article but you have not presented any valid mathematical argument to support your claim. I do not know of any valid argument against this proof and it has been read by many experts of the field. Mathematicians mainly comment if they see errors, not if they do not see errors. There is no custom of asserting that mathematical papers are correct by posting to Internet that also this article is correct. This is the role of journals. When an article is published by a mathematical journal, it is an assertation that this is believed to be correct, show it wrong if you can.

The question of neutrality does not apply to mathematically proven results. In many other fields the research results are not proven in a strict sense and it is possible to present the results in a certain light. In mathematics this is not so. All results must be proven. When there is a proof, it is either correct or wrong. If it is correct, it is neutral: it presents facts as they are. Some times, though rarely, there are errors in proofs published in respectable mathematical journals. The scientific practice is to assume that published proofs are correct until they are shown incorrect. In mathematics, the author is expected to withdraw the paper if it is not yet published or to admit (publish or distribute) a correction in case an error is found. This practise is followed and I also follow it. If you have a valid argument agaist the EJDE article I am fully ready to admit that there is an error. I do not know if you are a mathematician but if you are, you should know that mathematics is not a field where authors present results that have issues of neutrality. The Wiki page statement of a claim does not refer to a mathematically proven claim but to other more hazy fields. In mathematics there are assumptions, claim and a proof, and a claim with a proof is not merely a claim. This is the sense and context the word claimed was used in the Wiki comment. The result is partial, not Full result, since we do not know yet whether Clay will choose to fix the problem or to conclude that it is solved. This is more of a legal matter, not a mathematical issue. Mathematically, the problem is solved. The solution in the EJDE article answers exactly the problem that was posed.

As for this that I formulated the comment to my article personally, let me say the following. There was some other person who tried to insert a comment: It is proven (ref to my paper). This comment is in fact correct. Gavia immer removed it and stated that it should be verified by third party sources. As Gavia immer's statement sounds fully absurd to any mathematician (peer-reviewed journal articles are not verified by third-party sources? So what is the peer-review for?) I reformulated the comment and inserted it to Wiki. I am by no means the only one who considers this article correct but there are no cheergroups in mathematics who consider it their job to defend articles of their favorite author. The EJDE article has a very serious claim to the millennium prize and therefore it must be given some publicity so that other mathematicians notice it and can argue against it. Why the American newspapers have not made news of it is not known to me. However, that is an issue outside mathematics. This article has been for over two years available in arxiv and when it first appeared in arxiv, it was sent to many bulleting boards by a person under the name ansobol, it also was published by ResearchGate and other sites. It has been read by many. Clay's Carlson has got it, Fefferman also. Why there is not public announcement that it is approved? The public announcement only needs to be made after two years from the publication of an article. At the moment enough publicity should be given to the article so that those who can break the proof will do so before the two years period is at the end. In mathematics, no news means that the article has not been broken and is assumed correct. If you want to make an Internet discussion of the EJDE article, let me know, I am happy to defend the article, as I already did against your comments on Tao's blog. Jorma Jormakka —Preceding unsigned comment added by 88.114.55.128 (talk) 05:54, 15 September 2010 (UTC)

Robert Coulter, if you try to refute the EJDE article, then the correct mathematical way to do it is that you write 1) either a mathematical paper and submit it to EJDE or 2) you write a short letter to the editor of EJDE explaining your mathematical arguments. Assuming that the editor finds your arguments convincing he will then ask from me a response to your arguments and both are published in EJDE as conversation related to the EJDE article. But your arguments must be much better than the ones you wrote on Tao's blog of the velocity being only apparently nonunique or force being embedded in pressure. They must be sound mathematical arguments, else the editor will not consider them. I home the revised comment in Wiki now satisfies you. It is not only a claim, that is an opinion. The Wiki page does not take a stand whether the quoted phrase is correct or not. It is not original research since original research in an encyclopedia means presenting in an encyclopedia results that are not published elsewhere (primary sources). This article is published elsewhere and is a secondary source to Wiki. (Notice, Gavia immer, this is what the requirement that Wiki information must be based on secondary sources actually means. It does not mean that an author cannot cite his peer-reviewed published work, which is checked.) Instead, Coulter's claim that the EJDE article is wrong is original research since it is not published elsewhere. Such a claim has no place in Wiki, nor on Wiki talk pages. A mathematically proven fact is neutral knowledge no matter who wrote it to Wiki, as correct mathematical results represent facts as they are. The relevance of the comment in the article is high, since the official problem statement is misleading. In case this answer does not satisfy you two, Robert and Gavia, do as you wish. I have mainly continued this discussion as long as this because I have wanted to verify if there is indeed control of Wiki content by people who are not at all experts on the subject, against those who actually have scientific competence. Jorma Jormakka —Preceding unsigned comment added by 88.114.55.128 (talk) 08:28, 15 September 2010 (UTC) ¨¨¨¨ —Preceding unsigned comment added by 88.114.55.128 (talk) 08:31, 15 September 2010 (UTC)

Jormakka, there is a simple solution. Provide a citation to your work that says "It has ebeen shown by Jormakka et al that"... by a third party, and this whole thing goes away. Wikipedia relies on scientists to work together to generate a body of work in a fashion digestable to others.

THere are two issues here (1) if your work has a large impact, naturally others will simply cite your work, and in turn someone will add it here. Scientists using wikipedia as a place to promote their work is problematic for us -- it is a scientific publication, true, but again it is difficult to fully establish its notability and the limitations of the work in its proper context without peer discussion (peer review is hidden to wikipedia) (2) Wikipedia is not "run" by anyone, and I for one am attempting to do my level best, yes in a subject area that is not in my area of expertise, to attempt to make heads or tails of it. It is difficult for me to interpret the article, as it is indeed over my head in terms of technical content. By making claims such as "I have wanted to verify if there is indeed control of Wiki content by people who are not at all experts on the subject, against those who actually have scientific competence" you are shooting yourself in the foot by showing that you are not willing to enter a proper discussion. This kind of pointed behaviour I assume is not true for scientists in general. User A1 (talk) 09:50, 15 September 2010 (UTC)

Dear Gavia immer, Accept my apologies. I am indeed a scientist. I have experienced this as an effort by two nonexperts to try to find arguments why my article should not be referred in Wiki. If I have wrongly thought it is you, I most sincerely apologize. It is stressing and tiresome to make an announcement of a solution to a famous problem. I do not try to promote my work for selfish reasons. If I had wanted to do so, why is there no personal page in Wiki about me? As a professor, I could have easily arranged it. Why I have no home page, nor keep no blog. No, I am a person who does not like publicity and only makes a publicity effort when it is really required. In this special case I am morally oblidged to make the EJDE article publicly known since the article is already published and the deadline for granting the prize is in two years. If there is an error in my article, I should not get a prize. If the article is not made publicly known, somebody who could find a flaw in it will not even know about it. So, I will get the prize undeservedly. Thus, I should and will give the others all the chance to break the proof. The legal and official claim for the prize is already made in the EJDE article. My goal is to let people to break it if they can (they cannot for sure, so I happily give them a chance. Coulter thinks they can, so let them try. I ask them to try all they want. You think it is not sure I am correct, so give the people a chance to break the proof. Now many do not know it.) I am willing to enter any discussion, also concering if there is some control in Wiki, which you say there is not. In my opinion judging from this, there seems to be a very hard effort to get this reference to my work removed and the reasons seem to change when one reason is removed. As said, the reference to my work was inserted by somebody else, not me, only Gavia immer removed it. So, I did put it back. The reference to this article is very useful for the readers since the official problem statement is wrongly posed (if my proof is not accepted) and my article is a published paper that has a very serious claim to the prize. There has been peer discussion of the article. As for this that somebody will cite my work anyway, could you please stop the nonsense and start thinking what the case is with famous mathematical problems. This is a paper proving a famous problem. No mathematical author will refer to the article before there is an official announcement, i.e., the prize is given. As with any encyclopaedia. it is for the editors of the encyclopaedia to decide what information they include, what not. Whether you want to have a reference to this article or not is up to you, I do not mind it either way. As you are the editors, you decide. If there is no control and Wiki has contributions from anybody, let the comment stay. It is correct, relevant, not self-promotion, important, according to me and I know the article very well, also what feedback it has got. But for my part, do as you wish. I have no more time for this discussion. Jorma Jormakka

There have been some changes to the disputed section. The word "claimed" has been removed. Unfortunately, this does not remove the controversy from the article. The last two sentences of item #3 imply a disagreement and/or proof of the Clay NS problem. It is either controversy, or it is not. If it is controversy, it should not be mentioned at all until reliable sources have verified the results. If the EJDE paper is accepted as reliable (This is Jormakka's argument.) by the Wikipedia's editors then it should have prominent and non-controversial placement in the article. After all, it would represent significant progress on the problem (per Jormakka). I recommend a new section "Full Results" describing the EJDE paper. The article cannot simultaneously claim that the NS problem is unresolved but also mention Jormakka's EJDE paper. It must either accept the EJDE paper showing explicitly which sections of the NS problem have been resolved or make no mention of the EJDE paper. Rbcoulter (talk) 11:35, 15 September 2010 (UTC)Robert Coulter

OK, my proof is correct. It is not only my opinion and claim. It is a mathematical fact. Do what ever you want. However, as it is now stated in Wiki, Wiki does not in any way take a stand whether the comment is correct or not. Therefore, the quote from my article is just a statement what the article says. There is no need at this point to rearrage the whole Wiki page, i.e., the quote does not imply that Wiki thinks the quote is true but only states it. It is true however. Jorma Jormakka

Jorma Jormakka -- Your argument is valid (concerning the reference to your proof in the article) if the credibility and reliability of the EJDE can be established. My personal opinion is that EJDE is not a reliable source on this matter because a doubt the quality of the peer review process on your proof. However, if User A1 / Gavia Immer believe that EJDE is a reliable source then I would have no basis for a non-neutrality claim as long as the article explicitly backs the proof without using words implying that it needs secondary verification. Once the article is made neutral, I will remove the flag. Note that I may still argue against including the EJDE paper in the article, but I will only do that on the Talk page. I also would most likely continue to argue against the technical findings in the EJDE paper, but per the guidelines above, I must do that elsewhere.Rbcoulter (talk) 14:41, 15 September 2010 (UTC)Robert Coulter

Dear Robert Coulter,

The EJDE article got a specially hard and long peer-review in EJDE. It was first checked by a group of highly competent Finnish experts in the Summer of 2009, and was sent to several foreign top experts, some of whom commented. EJDE is an American journal of good reputation and this article, being a proof of a famous problem, was naturally looked at very carefully. EJDE promises 3 months referee time, this article was in review for 10 months. I do not know your qualifications on mathematics, but this EJDE article has been checked by mathematicians and physicists (not all are mentioned as they did not want to be thanked) whose published record even in google scholar is very good. I think that your attack on the level of EJDE as a respectabe journal is totally unfounded. The article was passed on ordinary peer-review, not in any special arranged way. I submitted it, it was received, reviewed and accepted. It got a more careful peer-review (much longer) since the claim in the article was exceptional. You have not shown any errors in the EJDE article, neither has Tao or anybody else. On what basis do you do your attack on this article? If you are a mathematician, just read the article. It is intentionally written to be so easy that any second year student can check it. Maybe Gavia immer (classical history?) cannot check it but practically anybody who knows how to derivate sinus and cosinus can check it. It is really, really easy to check. It is exaclty like stating 1+1=2. No mathematician can have any problems with it. Please, ask somebody reliable if you cannot read the article, before making statements of the review process. Still of the review process of this article. In Summer 2009 several American experts, including Terence Tao, were sent this article while it was being checked by the Finnish mathematicians. Tao did not respond. If he had some comments, or could have shown an error, he should have responded, but he did not. Later he also refused to discuss the article. Just today I got a email from a person in the USA who told that he had posted to Tao's blog a post that I should morally and legally get the prize since the Clay math has not excluded feedback forces, so they are allowed and the article is correct. After four hours this this post was removed from the blog. This is the level of discussion done by some experts. However, the review in EJDE and before it in many other forums concering this article has filled all the requirements that are imposed to mathematical articles. You have no basis on the claim that the quality of the peer-review has not been good. Jorma Jormakka

Dear Gavia immer,

Robert Coulter should not be allowed to put a flag that makes my responses to him invisible. I commented on this to you and you said it is not you, Robert Coulter admits it is he. He has done it again, the only comment visible in Wiki is that he has refuted my proof, which he certainly has not done. A mathematical proof is only refuted by showing errors. This he he has not done. I ask Gavia immer to remove the flag and to forbid Robert Coulter from sabotasing Wiki talk pages. Jorma Jormakka

Dear Gavia immer,

Let us agree the following. If you are the editor, then you decide the text and I agree with it. For my part there never was any special need to have my work referred by Wiki, have it or remove it. If you are not the editor, then I guarantee that the reference is correct, relevant, useful and should be there and I take the full responsibility of the comment I wrote, as I already have in the EJDE article. There are no citations to the article yet and I do not expect any to appear soon, Clay has been informed but has not commented anything yet, Americal newspapers have not been informed nor have made annuncements, email newspapers have been informed but have not responded, scientifically there is no dispute: the article is correct. I leave it up to you to decide but I really do not have more time on this discussion. Jorma Jormakka —Preceding unsigned comment added by 88.114.55.128 (talk) 09:34, 16 September 2010 (UTC)

Dear Gavia immer and Robert Coulter,

It is not true, as Coulter claims, that the comments on the EJDE are critical. I have never received any critical comment of the article after it was published. Instead I have received the following "I want to read your NS paper in more detail. I asked an expert about it and he said it's probably right." and "I finally got around to reading your paper on the Navier Stokes equations. I made a comment on Terence Tao's website saying that you deserve the Clay Math prize for it this afternoon. I essentially said that even though the framers of the problem overlooked the possibility of a feedback force, the problem statement did not prohibit this type of force. Four hours later, it was deleted. I guess it violated his blog policy. Oh well. The truth is that you legally and morally deserve the prize. It said "external forces". According to http://www.physicsclassroom.com/class/energy/u5l2a.cfm that could mean "applied force, normal force, tension force, friction force, and air resistance force." These forces are not independent of the properties of the object that the force acts upon. So there is no reason why the Clay Math Prize should exclude feedback forces as you gave in your solution. It's their fault that they overlooked this. I realize that the way they phrased the problem, a mathematician is trained to immediately think the f is independent of the u. But still, it never says this explicitly in the problem statement." Jorma Jormakka —Preceding unsigned comment added by 88.114.55.128 (talk) 10:59, 16 September 2010 (UTC)

Dear Jorma Jormakka:

The peer review process is currently hidden from public view on this EJDE article. If otherwise, please let me know. What was the degree of endorsement from these peers? Did they simply check the math? For example, ensure that the derivatives are correct. For example, there could indeed be blow-up solutions for NS equations but not satisfy the Clay conditions for solving the Clay NS problem. The peers, for example, may have only checked that the blow-up solutions are valid but made no endorsement that they meet the Clay requirements. If a qualified mathematician would come forward and backup up your claim, this would go a long ways in removing the controversy of including a reference to your paper in this article.

I would be more than glad to discuss the technical merits of your paper. Unfortunately, we are not allowed to do it here or at Terence Tao's blog. I have avoided discussing the technical aspects of your paper here. I would appreciate that you do the same. This Talk page, per Wikipedia's guidelines, should only discuss things like neutrality of the article and the reliability of sources, (e.g. the EJDE paper).76.123.120.172 (talk) 16:15, 16 September 2010 (UTC)Robert Coulter

Dear Robert Coulter. The peer review checked the article. The end of the article states of the Clay assumptions. Notice that there have been several versions of the Clay Navier-Stokes problem statement. Some have the titme Existence & Smoothness, some Existence and Smoothness. There has also been slight changes in the text during the years, though now one finds only two virtually identical texts with these two titles. This was not the case, there was e.g. a claim of uniqueness at some point (2008). The EJDE article refers to the text that was the official problem statement in Summer 2007-2008 when this article was written. Still, even in the present text there is the same main problem: feedback forces are not excluded. In other detailed comments from me, try to find the exact problem statement from 2008 if you find some statements from me that seem to refer to text that the present problem statement does not have. The issue of neutrality is the issue of correctness for mathematical papers. However, I do not have more time to this discussion. Do as you wish to do with the comment in Wiki. Jorma Jormakka

I've looked at Jorma Jormakka's paper. It's good in terms of its logic. It was subject to peer review. The only real dispute is "Does it solve the Navier Stokes problem?" If you ask mathematicians like Terence Tao on his blog, they'll say no. Why? Because the external force given in Jormakka's proof is a function of the solution; it is a feedback control function and there is an understanding amongst mathematicians that the external force cannot be a function of the solution. However, when looking at the Clay Mathematics Institute problem description of the Navier Stokes equations, at least at the present time, there is nothing explictly stated in there which prohibits the external force from being a function of the solution. Furthermore, most external forces are functions of properties of the objects in which they act upon, for instance normal force, friction, air resistence, so from a physicist's point of view Jormakka's solution makes sense. Therefore, it looks like legally and morally Jormakka deserves the Clay Mathematics Prize for his solution. The fact that the mathematics community does not consider these types of solutions to be solutions is irrelevant. His solution satisfies the problem description that the Clay Mathematics Institute wrote up. A deal is a deal. Congrats to Jorma Jormakka on his solution. —Preceding unsigned comment added by Logicker (talk • contribs) 19:11, 16 September 2010 (UTC)

If you want to change the Wikipedia article you need to change every remark that implies that the Clay NS problem is still unresolved. For example, a statement needs to be entered in the first paragraph showing that the parts of the Clay problem have been proved. References to the unsolved NS problem need to be done in the past tense. All ambiguity in the article needs to be cleaned up. You can't have one part of the article saying the matter is unresolved, and another part saying it has been solved. I assume that user Logicker supports your position. Gavia Immer seems to be neutral on this. So I guess there is probably no one stopping you from changing the article since I would be outvoted. I will remove the non-neutral flag after you have removed all ambiguities in the article. 76.123.120.172 (talk) 22:41, 16 September 2010 (UTC)Robert Coulter

Dear Robert Coulter, You asked if he peer-reviewers checked the claims on the Clay problem. I can tell one comment from a Finnish professor of mathematics, a top expert on his field, who was one of the readers of the article in 2009. I thanked all readers when the article was published July 2010 and wrote that the EJDE referee finally allowed me to keep the phrases (originally it seemed that he will not)"Unless Theorem 2.4 is accepted as a proof of Statement D, the official problem statement must be corrected." and "The changes needed so that the presented proof is no longer valid are not a small straightforward modification.". The professor wrote: "I did not think that the referee would allow you to keep those statements, though both are completely correct of course." In general, the referees were not asked, nor did comment, on the issue whether Clay should accept the proof. They and me are not at all sure that Clay will give any prize, I even doubt they have any money. One adjunct professor thought that feedback forces are probably forbidden but could not argue why this would be so. One adjunct professor said that "Yes, you solved the problem that was stated but it is not what they want and you will not get the prize." One professor wrote that if "somebody poses a problem for million bugs, he should have checked the formulation and pay the prize if the formulated problem is solved." So, it has been a mixed reception. We cannot say what Clay will decide, so let us not make any major changes to the Wiki page. Yes, this is a serious candidate and mathematically it is correct, but will Clay accept it on not is up to their board. It is not any major headline news that is needed, only enough publicity so that those who want to attack the proof will notice it. Thank you Robert for discussing this issue, feedback is always useful and I give publicity to this paper not for self-promotion but exactly in order to give the doubters a chance to express their arguments against the paper. This is not a review by Web discussion, the paper already was reviewed and accepted. This is in my opinion not checking the correctness, that was already checked. This is not for deciding if Clay should give the prize, that will be up to Clay. This is only for those who have not read the article and are sceptical about it to express and formulate their counterarguments and to see if they hold. I do not think they can possibly hold. The solutions are nonunique (Lemma 2.1/Th 2.2), there is a blow-up solution for zero force (Th 2.3), a feedback force selects one solution and can select the blowup solution (Th 2.4). So, the only question to ask is if feedback forces are allowed in the Clay problem and why should they not be when they are not excluded. I do not think this logic can be broken. Jorma Jormakka —Preceding unsigned comment added by 88.114.55.128 (talk) 08:43, 17 September 2010 (UTC)

At BEST you may have a very weak argument on the semantics of the problem statement. The peer reviewers you mentioned appeared to realize this. An analogy would be like having the Great Race (see http://en.wikipedia.org/wiki/1908_New_York_to_Paris_Race) from New York to Paris, driving to Paris, Texas and attempting to claim the prize. Since the paper makes no attempt to resolve the mathematical mysteries of Navier Stokes, the only hope here is to make a LEGAL case out of it. But the legal case seems weak also. It would involve allowing terms like external force and pressure to be interchangeable. Of course, you never know what the American legal system could do. Look what the Indiana legislature did in 1897 concerning geometry and pi -- http://en.wikipedia.org/wiki/Indiana_Pi_Bill 76.123.120.172 (talk) 15:09, 17 September 2010 (UTC)Robert Coulter

Dear Robert Coulter. Your analogy with the Paris New York race is very good. The PDE community is small and they propose a prize competition to the whole world. The PDE people live in Paris Texas and have never heard of Paris France. Likewise, the PDE community thinks that an external force is naturally independent of the object (I took lots of math and do not remember that this should be necessarily so on any field) and have never heard of feedback forces that the rest of the world considers as one typical type of external forces e.g. in systems theory and control theory. The PDE community never attended such classes, nor though that anybody who is not from Paris Texas counts in this race anyway. So, what do you imagine an expert from the bigger world would do, would he naturally think that this Paris must be Paris Texas? Jorma Jormakka88.114.55.128 (talk) 19:25, 17 September 2010 (UTC)

Dear Robert Coulter or Gavia immer, Could you finally remove the tag that Robert put so that this discussion is private while what is visible is Robert's claim that he has disputed my proof and Tao has found errors. Both claims are wrong. I make no comment on Indiana_Pi_Bill. It will be for the Clay Math to decide. The EJDE article solves exactly the problem that was posed. This is what is done in mathematics: if a problem is posed, it is exactly that problem that should be solved. Thus, if Clay Math fulfill its promise, they should grant the prize. If they cheat, then they cheat. Jorma Jormakka¨88.114.55.128 (talk) 17:01, 17 September 2010 (UTC)

To Robert Coulter. I do not recall having stated anything more than I have proven. The message I sent to Chris Neukirchen (reddit) was that the NSE problem posed by CMI is solved exactly as they formulated, but only because of the problems or errors in their problem formulation. The EJDE article also states that unless Theorem 2.4 is a proof, the problem must be corrected. I have never claimed that the article solves the whole NSE problem, only the exact problem posed by CMI. The point is not actually semantic, mathematical problems MUST be posed correctly. I think your arguments against this proof have been based on misunderstanding the proof. It is not a great mathematical result for NSE research for which I claim undeserved self-promotion and fame, it is a very simple, totally elementary, paper that solves a millennium problem as it is stated. It may have two outcomes: either Clay pays the prize as one should in such a case, or the people in the field of PDEs will be a bit less arrogant since they apparently cannot even formulate a millennium prize problem correctly. Outsiders, especially professors, are not always cranks.88.114.55.128 (talk) 17:22, 17 September 2010 (UTC)

To the Editors:

There are really only two cases here:

1. Jormakka's paper MAY solve certain sections of the CLAY NS problem.

   If this is the case, then the article is non-neutral by allowing Jormakka's paper to be mentioned here.  The article, in effect, would
   be advertising his position, which by my reading of the Wikipedia neutrality guidelines is not allowed.  The article is not meant to be
   a forum where one can assert there opinions about a subject.

2. Jormakka's paper DOES solve certain sections of the CLAY NS problem.

   The entire article needs toe be re-written to reflect this fact.  It cannot have contradictory statements  -- for example, claiming the problem
   is unsolved earlier in the article and later in the article purport that it has been solved.

Rbcoulter (talk) 12:35, 18 September 2010 (UTC)Robert Coulter

Dear Gavia immer, I see that Robert Coulter has disputed the neutrality of the small comment to my article. The article does solve the official problem statement as it is stated, and as a mathematical result it is fully neutral, as mathematical facts are neutral. The comment in Wiki does not increase my chances of getting the Clay prize, the rules for the prize do not mention Wikipedia but publication in a peer-reviewed reputable mathematical journal, which has already been made. EJDE is such a journal. Why this comment is useful is that if somebody is interested in the millennium prize problem, s/he would like to know that there already is a serious claim for the prize. If somebody does not think I deserve the prize, s/he can refute the article mathematically.

As for the case of non-neutrality in the case that the article only MAY solve the Clay problem, I do not see where would be the case of non-neutrality. If the editors want, they may add more discussion that the Clay Math Institute may or may not decide in two years that the article solves the problem. (The other case is that the official problem statement is modified and I will be given some other prize from other funds. This is still news.) I cannot consider mentioning any scientific published article in Wiki as a source of facts as adverticing the position of the author. Relevant articles should be mentioned in the Wiki page and this is relevant to the millennium problem description. It clarifies a little understood issue in the millennium problem formulation.

If somebody else wants to rewrite the Wiki article as Robert Coulter suggests, it can be done by him/her. I will not rewrite the article since even this small and quite correct comment has resulted in so many claims against me arguing that adding the comment is not neutral. See how many comments I have had to write to defend a very minor comment in Wiki. It is already absurd. I originally inserted that type of a comment that Robert Coulter seems to think there should be, but it was deleted. I do not think the editors of this Wiki page should take any stand to whether the article will be granted the prize by Clay or not, and no rewriting of the article is needed at this stage. The mention to my article suffices.

What actually happened with this article was simply the following. I noticed (after quite much work, it is not at all easy to notice) that the solutions to the millenium prize problem are not unique in the formulation as Clay stated, and for zero force there exists a blow-up solution. This had not been noticed before in 10 years since there was an old incorrect theorem stating that the solutions are unique. The theorem is "proven" e.g. in Temam (see my paper) and was used as an argument against my article by two journals until one of my proof checkers noticed the error in Temam's proof. (One journal AASF directly picked up a sentence of the official Clay problem formulation in the Spring 2008, which had the statement of unique solutions. It is not any more in the official problem statement since somebody (probably Fefferman) corrected the statement of uniqueness at some time in 2008-2009.) As the solutions were not unique and there was a blowup solution, I found a feedback force that selects the blowup solution. Thus it gives exactly the solution that the millennium problem asks for. Feedback forces were not considered in the problem statement since the solutions were though to be locally unique and then it would not matter if the force is a feedback force or a point force. There is nothing wrong with feedback forces. The PDE community has naturally always known about feedback forces and they have been accepted as external forces in the NS problem, since they appear in many practical applications of NSE.

After I presented my proof, the PDE community has tried to ignore it, and if it is not possible, to refute it, stating e.g. like Tao that feedback forces are not allowed. Mathematically, as they are not forbidden in the problem statement, they are allowed. The PDE community has tried to ignore the article completely, and like Robert Coulter here tries to do, to ridicule the article, claim that it is disputed, claim that even mentioning it improves my chances in getting the prize etc. Why is it so important that this article should not be known to anybody?

If my counterexample is incorrect, giving it publicity does not improve my chances but only causes it to be broken faster. Then it should disappear in some days. Indeed, if my paper is so obviously wrong not to warrant any discussion, why my widely published claim has not been debunked yet, it is almost a month since the Wiki comment, two years in arxiv, over two months after being published in a journal - journal publications must be refuted, they cannot be simply ignored in the scientific method? Penny Smith's article accepted to a journal in 2006 only lasted for some days and was withdrawn before publication. In general, wrong published proofs of millennium problems are usually fast shown incorrect. If my counterexample is correct, it has a clear place in Wiki as the reader may just as well be informed that there is a very serious candidate for this prize. Mathematically and legally my article is a solution to the Clay problem.

As a conclusion, I ask to remove the note that there is a dispute of the neutrality of the article. I do not see any neutrality dispute. Robert Coulter has not given any argument against my article, he has not shown any argument that my comment is not neutral. Is it correct in Wiki to dispute mathematical facts wihout any reason, simply since you do not like them? I have discussed with Robert Coulter very long and answered to all possible arguments of neutrality. If this discussion does not satisfy him, he does not argue in a rational way. I ask you Gavia immer now as the page moderator to look at the discussion and to think of your rules. Thus, there is a journal article checked through a normal peer-review process. Its content is not scientfically disputed at least so far. It does solve the Clay problem. The PDE community tries to ignore the article for some reason and it is given no publicity in media. They do not try to present mathematical arguments, they try to claim that a published article is so wrong that they do not even need to comment it. However, they do not seem to be able to show what is wrong with the article, and it is not at all so obviously wrong - just the opposite, it is clearly correct. Why my article should not have even a small mention in Wiki? Its correctness is checked, its relevance to the Wiki page is clear. Why is it so important that it is not mentioned anywhere?

In case the issue is now of the internal logic of the Wiki article, then the comment I inserted does not commit Wiki to supporting the correctness of my article. I think this is what you, Gavia immer, think is wisest at this point. It simply mentions that there exists such an article making such a statement. If it will be confirmed after two years, the article can be rewritten to state clearly that the millennium problem is solved. I do not think Wiki editors should decide now whether it solves the full problem or not, nor is it necessary to do so. If Robert Coulter or somebody else wants to rewrite the Wiki article, it is their business. I will not, I do not usually comment my own work, and have described many times why I choose to do so in this special case. I do not find the Wiki page too illogical at the moment. It states that the problem is open and later mentions that there is an article that makes such a claim. If somebody wants to describe my result better, they can do it but it will not be me writing it. If the editors of the page can find somebody who can check the article (i.e. can derivate sinus and cosinus and knows what a partial derivative means), then they can verify the relevant theorems. They have been verified by some 20-30 people so far. It takes 2-3 hours maximum. Jorma Jormakka88.114.55.128 (talk) 14:44, 19 September 2010 (UTC)

It appears that Jorma Jormakka insists on talking about the technical merits of his article -- so I must respond in kind. Feedback Forces -- This term has no meaning in the context of NS. Forces are either internal or external. The right side of the NS equations are the sum of the forces on a fluid element. You are exploiting the the fact the forces are added on the right side to obtain the resultant force that causes the acceleration depicted on the left side of the NS equations. So if this resultant force happens to become infinite the left side will blow-up. This is obvious. So if you have a blowup solution (some part of the flow field becomes infinite), is this the result of having applied an infinite external force or did this force evolve internally from the initial conditions of the flow field? There is an easy way to determine this -- energy balance calculation. If only internal forces are active, then the global energy must decrease at every moment. Your solution achieves global infinite energy in finite time which is impossible without an external force. A good analogy here is that there are craters on the earth that are known to be of volcanic or meteoric in origin. Volcanic origin means that they were only created via energy sources within the earth. As it turns out, the earth is only able to create forces within a certain bound. Some minerals found at certain craters (e.g. the meteor crater near Flagstaff, AZ) require forces exceeded those known to exist internally on the earth. The forces needed require a meteor impact. This way scientists can determine the meteor craters. The initial flow field of a NS scenario has bounded energy. The REAL mystery of the NS problem is if sum or all of this initial energy can concentrate itself in a infinitesimal part of the flow field at some later time. NS problem ALLOWS the application of an external force that is finitely constrained. Your external force is clearly not finite via the energy balance calculation. This means that it is a trivial blowup solution.Rbcoulter (talk) 19:56, 19 September 2010 (UTC)Robert Coulter

Dear Robert Coulter. Mathematically your first argument is totally wrong. The force in Theorem 2.4 has always value zero. It does not become infinite. It is defined as a feedback force, i.e., it is an external force, up to the singularity and then smoothly continued as zero to the whole time axis. It is not a trivial blow-up case where the external force has a singularity. Notice, the example can only be made for the external force having zero value in a neighborhood of the singularity because otherwise the force also has a singularity. It is in no way trivial but a very special case. Then you present again your old energy argument from the Tao discussion. There is no demand that the solutions to the Clay Math Navier-Stokes problem should be physical, only the initial conditions that they specify must be physical, and they are. The initial conditions that Clay specifies are not complete. It is usually necessary in differential equations to specify also the time derivatives of the unknowns at t=0. Clay has not done so because a faulty theorem claimed that it is not necessary to do so in NSE. However, it is necessary. Therefore the time derivatives are undefined and thus the physical initial conditions that Clay specificies are in fact not physical. Physical fluid has all time derivatives of velocity defined at t=0. This Clay fluid does not have. The feedback force select a solution. The feedback force has zero value and thus it has zero energy, but as the initial values of u(x,t) are not completely specified, the force sets the initial time derivatives of u(x,t) at t=0 to values that develop infinite energy when time goes on. It is not the force that has the infinite energy but the initial time derivatives of u(x,t) at t=0. You should not use faulty physical reasoning in a mathematical problem. Such reasoning is not valid here since the Clay problem is not defined in a correct physical manner. No space-periodic solution in R^3 x time is ever physical as it has infinite energy. It is absurd to demand that a counterexample that is especially created to be non-physical (blow-up) should be in some other sense physical. Simply, take the assumptions as Clay gave them and check them against my solution. Theorem 2.4 has all conditions and the only difference is that the force is a feedback force. Thus, the only question is if feedback force is forbidden in the Clay problem and it is not. If you think this is so easy and obvious counterexample to find, why did you not present it yourself. It would have clarified the problem statement. In reality this example was not easy to find, and the only people who would call it easy are either envious that somebody presented a solution, mathematically so poor that they cannot even understand what is done in the simple article - it is easy to check, it was not easy to do, or indeed so good that for them solving a three-dimensional nonlinear partial differential equation of three functions is trivially easy.

Can you remove the tags that you again added so that the last comments are not visible: you are sabotaging Wiki talk pages. Your arguments are not mathematically sound and I have already answered them on Tao's blog. If you want to present mathematical arguments, send a letter to the editor of EJDE. What is your motivation for doing this? Why do you dispute a proven fact? Jorma Jormakka88.114.55.128 (talk) 09:18, 20 September 2010 (UTC)

Send a letter to EJDE? So the only way to refute your argument is send a letter to your publisher? Very amusing. Also, you said "..dispute a proven fact". This sums up the whole problem I have with the Wikipedia article. If your proof is fact, then why isn't the entire article re-written to highlight this fact? The reason is because the editors have some doubt about the proof. They are waiting for some confirmation from an independent reliable source. IMO, this will never come. Unfortunately, the editors have mistakenly made the article non-neutral in the meantime by allowing the article to refer to your yet-to-be confirmed proof. what is my interest in this? For quite some time I have been interested in fluid flow, I am sure much longer than you have been. For the past several months I have posted and read comments on NS at Terence Tao's blog. A couple of months ago you posted a link to your "proof". It didn't take to long to see that it was meaningless because of the infinite global energy. Since you were recalcitrant in not admitting the flaws in your paper, Terence Tao made the wise decision to forbid further discussion on it at his blog. Unfortunately, Wikipedia has not been as wise, and is still under the delusion that your proof means something. -- I cannot speak for your papers on the other Clay problems. I only know you have flaws with the NS problem. Wny don't you focus your energies on these other proofs?76.123.120.172 (talk) 10:16, 20 September 2010 (UTC)Robert Coulter

Dear Robert Coulter, You have not located any errors in the EJDE article, nor has anybody else. If you had had a valid argument I would have accepted it, but you have none. I first studied these equations in 1978, and you studied them much before that, so you must be very old then. It is not good to be too old as a mathematician. You should not manipulate the discussion by adding tags to make only your comments visible, it is not correct on Wiki Talk pages. Jorma Jormakka 88.114.55.128 (talk) 15:16, 20 September 2010 (UTC)

First, I apologize for erring concerning your age. My first exposure to Navier Stokes was also in 1978. Your argument (which I have acknowledged above) is based on sneaking energy and/or forces into to the Clay problem assumptions. The crux of your argument is that if it is not forbidden then it must be allowed. Specifically, you have constructed what appear to be valid blowup solutions to the NS equations. I could have checked the derivatives, but I assumed they were correct. So in that since there may not be "errors" in your paper. The problem comes about when you say that your paper solves the Clay NS problem. There are bounds on external forces in the Clay problem statement that you have exceeded. You can't disguise that fact by calling the right side of your equation external force-free. The energy growth indicates that the external force is there. 76.123.120.172 (talk) 17:41, 20 September 2010 (UTC)Robert Coulter

Dear Robert Coulter. The bounds for the external force are not exceeded. The force has value zero in the whole space-time. The external force needs only to be a feedback force for a few milliseconds and can after that be smoothly continued as zero. The situation is that when you insert the force in Th 2.4 into the equation, it cancels the time derivative of the velocity. Then there is only a unique solution. After that there is no force, the velocity follows the trajectory it has. The force at the beginning sets the time derivatives of the velocity to the values of the selected blow-up solution and after that the solution is unique by the local-in-time existence and uniqueness theorem. The force is zero and therefore in the transform in the beginning of section 3 the force remains as zero and will not have singularity, while u(x,t) and p(x,t) do have a singularity if g(t) has a singularity. The problem in the Clay problem statement is that the initial conditions are not physically correctly set. Therefore these solutions are to certain extent unphysical. There is no energy giving external force. You can see it in Th 2.3, that has force zero, not as a feedback force. It has infinitely many solutions. All of them are mathematically perfectly valid solutions, yet they are not all physically acceptable. But they all satisfy the conditions that Clay wants. Thus, the error is in the Clay problem setting, not in the solutions in my paper. The Clay problem setting does not uniquely define the solutions and some are for you and other physicists unacceptable (I assume you must be a physicist sine you argue as a physicist). My article, as the comment in Wiki, says that either you accept Th 2.4 as a proof of Statement D, or you fix the problem setting of the official problem description. This is a correct and fair statement.

Let us try to resolve the neutrality dispute. The source that is referred to is for Wiki a reliable source (peer-reviewed journals are the best according to Wiki). It does not represent a too large fraction of the article, it is simply a small comment. It does not claim that the problem is solved in the article, it only says that the article states so. This is not refering to an opinion, the article is technically correct, mathematically it is a solution, for physicist it is not. The main Wikiarticle does not in my opinion need to give this article a more prominent place, but if somebody wants to edit themain article, one can mention that there is a published paper which contains a result that may be considered by Clay for the prize. I have informed Fefferman of this article several times. He (or maybe somebody else) did fix the incorrect sentence that the official problem statement had Spring 2008 that the solutions are for a long time known to be unique up to some T>0. Now you do not find this statement. I wrote aboout it to Fefferman in Spring 2008 and later. Fefferman has not made any statement that feedback forces are not allowed. Morally he should have, if they are not allowed as if somebody approaches the author of the problem statement and presents a proof that does not fill the required conditions, why should the expert not comment on this. So I assume they are allowed. So, this is the state. Let us see if I claim too much in the Wiki comment. I think not, the Wiki comment only states what the "reliable source" states, and the source is indeed correct. Let us see if all views have been represented. We could of course refer to Robert Coulter's view, but it is not published. We cannot use Tao's blog since it is a discussion and the reader will not get any clear view. There already is a reference to Tao's blog (the main well written part) in the article. I think referring to the discussion part is not good, it is not a reliable source. So, do you want a fringe view to your comments? I think not. There is no suitable reliable source to refer. We could add some text that the result is not yet citated by other authors. About this that it is no verified, it is verified by being in a reputable journal. If you have some suggestion, not dropping the comment to the article, I am ready to consider it. So far you have only proposed removing all reference to this article. Do you have any constructive suggestion? Jorma Jormakka 88.114.55.128 (talk) 18:09, 20 September 2010 (UTC)

Dear Robert Coulter. I have not checked this carefully but I was contacted by one person who had calculated the momentum in the solutions of Lemma 2.1. You can do the transform in the beginning of Section 3 to any solution for zero force and get a similar result, so what follows is what holds to these my solutions. The momentum is not conserved. There apparently is a well known theorem saying that it should be but this theorem most certainly uses the older well known but incorrect result that the solutions are unique. Assuming that this is correct what the person told, then the solutions with g'(t) not zero are not momentum conserving and therefore unphysical. But they fill all conditions set by Clay. Thus, it seems that the way Clay set the problem, there are unphysical solutions. The correct way to fix the issue is to see what additional physical condition should be included. The Clay problem does not have such an additional condition, so my solution is correct for their problem. But for physicists, decide what you want from these solutions and add the appropriate additional physical condition. For instance, physical fluid has all time derivatives defined at t=0, while Clay fluid does not. I think this is some help to PDE research from another field. I do not understand why the reception of the article has been so negative. Besides, the simple easy approaches in sections 2 and 3 may contain some interesting results if you look at them more carefully. Jorma Jormakka88.114.55.128 (talk) 18:32, 20 September 2010 (UTC)

At this point, we must agree to disagree. I offer the following compromise. Remove all references to the article under the Partial results section. Find another mathematician who will go on record to back up your claim (Mathematician is a person with a PhD in mathematics where such degree is generally recognized as such in the US). Once this is done add a new section at the bottom that refers to your proof. Stick to the facts. Don't speculate about it solving the Clay NS problem. Simply state that a proof was submitted to Clay . If Clay reports that it has some significance it can be added at that time.76.123.120.172 (talk) 21:20, 20 September 2010 (UTC)Robert Coulter

Dear Robert Coulter. Your first condition is already filled. You find several names thanked in the article, the editor and the referees of EJDE. It is a bit absurd to go on asking professors of mathematics to confirm a peer-reviewed journal article to some person on Wiki Talk pages, but as you wish, so I will ask a person who is professor, has a Ph.D. in math and has read the article and back it up. These were your conditions. Please, do not complain any more of this point. You gave your conditions, so they will be filled to the letter. I do not want you any more to return to your complaints. After that I will make a new section removing the comment from the partial results. the text in the new section is correct, as everything I have written so far. Not based on misunderstanding and unsubstantiated claims like many things you have written so far. Jorma Jormakka88.114.55.128 (talk) 05:34, 21 September 2010 (UTC)

The dispute is about the correctness of Lemma 2.1, Theorems 2.2, 2.3 and 2.4 and the remarks in Section 4. I have checked these parts of the EJDE article and found them correct. Henryka Siejka-Jormakka, Ph.D. math