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)

Jormakka Claim Disputed

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 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) Concerning that Fefferman clearly states that the solutions are unique, 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, indeed if we assume that Fefferman's problem formulation does not have errors, we must conclude that Fefferman meant that the force must be a feedback force since otherwise the solutions are not unique. 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 gven 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 te 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. All his arguments were shown incorrect, which he finally admitted. Concerning the footnote. 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 argument is that a Fields medalist Terence Tao claims in his blog that the checked and peer-reviewed EJDE article is wrong. Let us notice that Terence Tao states that the solutions are not unique while another and more famous Fields medalist Charles Fefferman states in the Clay problem formulation that they are unique. Thus, one Fields medalist claims in his blog that another Field medalist was wrong in formulating a million dollar problem. The EJDE article assumes that Fefferman did not make any errors but stated the problem exactly as he wanted. Then the solution to the stated problem is in the EJDE article. If there is a case of selecting which one is wrong: a Fields medalist writing a fast blog, or whether a more famous Fields medalist carefully formulating an important problem, then I guess the error is more likely to be in the first alternative. I think the only problem of accepting the EJDE article is that the solution is so simple, but it was not so easy to find it. Jorma Jormakka

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 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 to 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. Comments from Clay concerning this article may come in two years, most probably not earlier. Jorma Jormakka

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)