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This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.PhysicsWikipedia:WikiProject PhysicsTemplate:WikiProject Physicsphysics C This article has been rated as C-class on Misplaced Pages's content assessment scale. Mid This article has been rated as Mid-importance on the project's importance scale. This article is supported by Fluid Dynamics Taskforce.

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."

I scanned the book

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 th--94.27.64.130 (talk) 05:11, 31 October 2010 (UTC)is 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 Misplaced Pages. 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. Misplaced Pages 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, Misplaced Pages 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 Misplaced Pages 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. Misplaced Pages 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/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 Misplaced Pages 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 Misplaced Pages.

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

In summary, Misplaced Pages 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. Misplaced Pages 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) Misplaced Pages 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 Misplaced Pages'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 Misplaced Pages'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 Misplaced Pages 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/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/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 Misplaced Pages 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 Misplaced Pages 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 Misplaced Pages 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, Misplaced Pages 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 quite 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 backs it up. These were your conditions. Please, do not complain any more on 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 130.188.8.11 (talk) 06:26, 21 September 2010 (UTC)

Dear Robert Coulter and Gavia immer. I hope the revised comment in Wiki satisfies both of you. The lengthy discussion with Coulter and checking of the EJDE article elsewhere makes it very believable that all that is stated in the comment is true. it is not original research for Wiki but based on a reliable source. The correctness of the pertinent theorems lemma 2.1, Th 2.2, Th 2.3 and Th 2.4 are not in doubt. Th 2.4 has exactly the assumptions that Statement D has, with the exception that the external force is stated to be a feedback force. A feedback force can be considered an external force, thus Th 2.4 has the conditions of Statement D. Arguments against physicality of the solution are irrelevant. The solutions are not physical, neither do they need to be. I could get more back up of the result but you must understand that mathematicians are not fond of taking part into any quarrel in Wiki Talk pages. I do not think it is correct to ask for such statements from anybody. I asked for one filling Coulters explicit conditions, there was another anonymous commentor. This must be enough. Let this discussion now end. Jorma Jormakka88.114.55.128 (talk) 06:59, 21 September 2010 (UTC)

The new section reads more like an advertisement for Jormakka's proof. It should simply state that a proof was submitted to Clay. Title should be something like "Jormakka Proof", not "Published Solutions". If Clay accepts it as a solution, then the title of the new section can be changed. Jormakka - "Lengthy discussion" does not make your proof believable. It is statements like these that hurt your credibility on this issue. Only matter here is that you did publish a paper concerning the Clay problem. The above discussion was about if/how this "proof" may be mentioned in this article. To be fair to others, if someone else submits a published proof to Clay, it should also be mentioned in a similar section at the bottom of the article.76.123.120.172 (talk) 10:04, 21 September 2010 (UTC)Robert Coulter

Dear Jorma Jormakka! Lemma 2.5. in your article repeats well-known results ( http://lib.prometey.org/?id=15227 Heinbockel J.H. (2001) Introduction to Tensor Calculus and Continuum Mechanics. Trafford Publishing , ISBN: 978-1553691334 , p. 294). I hope on your reply. Best regard --94.27.64.130 (talk) 05:11, 31 October 2010 (UTC)

EJDE paper

Gavia Immer mentioned the P vs NP problem further up the page. I notice per item #45 that besides this recent work on Navier-Stokes, Jorma Jormakka also solved P vs NP in 2008:

": In September 2008, Jorma Jormakka proved that P is not equal to NP by showing that the subset sum problem cannot be solved in polynomial time. His paper "On the existence of polynomial-time algorithms to the subset sum problem" is available at http://arxiv.org/abs/0809.4935."

The arxiv paper about subset-sum was last revised on August 4, 2010 and is, per the author, "still believed to be correct". These two Clay Millenium problems are in very different areas of mathematics, so Jorma Jormakka is obviously a highly versatile researcher. However, a proposed solution to a famous problem like NS has to be taken as an extraordinary claim (WP:REDFLAG). It would be helpful to have secondary sources regarding the EJDE paper before we include it in the article, at least with the current description. Is there an entry in Math Reviews about it? Has Fefferman said anything? 69.111.192.233 (talk) 06:57, 22 November 2010 (UTC)

I rewrote the section for neutrality and length, but would support removing it unless secondary sources appear. 69.111.192.233 (talk) 10:02, 22 November 2010 (UTC)

Dear Gavia immer, The text that was inserted by the previous person was neither correct and nor neutral. Especially the claims that Terence Tao would have discussed the problem and shown errors in it are false. Tao neither discussed nor found errors. Also, incorrect was the claim that the proof is based on ambiguous language in the problem statement. The EJDE article is based on clear statements, actually lack of necessary conditions, in the problem statement. The EJDE article has been checked by many competent mathematicians. You are wellcome to show it incorrect if you can. It is in Wiki so that those who can show it incorrect will do so before the two years from publication of the article have passed. It is a published peer.reviewed journal article, and the highest type of references used in Wiki are published peer-reviewed journal articles. It can be referred to without any statements by Fefferman. There are other versatile researchers, no reason to make suggestive hints. Maybe the commentor would like to take a look also e.g. at arxiv:1011.3962 and the articles in google.scholar before concluding anything of my competence. I rewrote the text to be neutral and correct, and made the argument clearer. Jorma Jormakka —Preceding unsigned comment added by 88.114.52.63 (talk) 07:48, 23 November 2010 (UTC)

Prof. Jormakka, I'd ask you to stop editing the article, per Misplaced Pages's guideline on conflict of interest, WP:COI. Let uninvolved editors write the section. I felt that I gave a neutral explanation of the ambiguous language issue, but I'm open to suggestions about alternate phrasing. That Tao discussed the issue with you is an obvious fact. It's true that he didn't continue the discussion for as long as you'd have liked him to, and I didn't mention that in the paragraph, but I can add something about it if you want. It's also obviously true that he described what he saw as problems in your paper. Of course his opinion that there are problems is not guaranteed to be correct, but I think my wording made it sufficiently clear that it was his opinion. My judgment is that the opinion of a specialist like Tao on a subject like this is a significant point of view which should therefore be included under our neutrality policy. I felt ok using his blog as source since it was first introduced into the article by you yourself.

Also,it's not accurate to say that published papers are the highest form of reference in Misplaced Pages. They are acceptable for most things but extraordinary claims (which this is) require extraordinary proof, which in this case means secondary sources. We just went through something like this in the P vs NP article, where someone was trying to add a reference to a P=NP "proof" that got into an Indian journal somehow. You (or someone) also mentioned Penny Smith. You might remember that her Navier-Stokes paper had to be withdrawn because one of the earlier theorems it relied on turned out to be wrong despite having been in a journal (the error got past the referees somehow). And to not belabor the obvious, if getting a paper published settled an issue, the Clay Institute would issue its million dollar prizes immediately on publication instead of having a two year waiting period.

Of course I'm still in favor of removing the section completely. The version I wrote was intended as a compromise, mentioning the paper in what I hoped a neutral way with due weight. But really, the significance of the paper to the topic can only be established by secondary sources, and the closest thing we have to that is Tao's comments, which are quite unfavorable. 69.111.192.233 (talk) 08:43, 23 November 2010 (UTC)

I removed the section. I'm ok with the idea of putting something back, but it should be shorter and more neutral than what I removed. I'm mostly ok with its description of Tao's comment but would want to adjust the phrasing a bit more. 69.111.192.233 (talk) 08:49, 23 November 2010 (UTC)

Fine, as you removed it, put there the text you want but write it correctly. The reasons there should be a reference are 1) Somebody else, not me, wanted to put it there originally. I think if somebody wanted it there, there should be good reasons not to include it. 2) It is highly relevant to the millennium prize problem description because it contains the only solution that have been published in a peer-reviewed reputable journal and has not be found incorrect, unlike Deolalikar's unpublished and refuted paper, or Smith's withdrawn paper. The paper as it is has now lasted over 2.5 years while many mathematicians have read it. Tao's fast written comment does not change anything since his comments are wrong. 3) The article shows two important problems in the problem statement. At least previously, have not checked now, there was an error also on the Wiki page. It claimed that the periodic problem is stated in torus. It is not, it is stated in R^4 with periodic velocity and force. Jorma Jormakka —Preceding unsigned comment added by 88.114.52.63 (talk) 10:49, 23 November 2010 (UTC)

I see, the first mention of Tao was in this edit which I guess was by someone other than you. I'm sorry about the confusion. If you want to suggest some short neutral wording to reference the paper, please do so here on the talk page. In particular it should state somehow that the problem solved was not the expected one (I can accept that my own attempt to explain that wasn't so good). Alternatively I can ask at Wikiproject Math for someone from there to suggest wording.

If you are seeking technical discussion of the paper's contents, Misplaced Pages is not the place for that. You might instead open a Math Overflow thread (www.mathoverflow.net). There are a lot of good mathematicians there (including Tao) and you are sure to get well-informed comments. 69.111.192.233 (talk) 01:07, 24 November 2010 (UTC)

eq

Long discussion this page has. Isn`t a 32kb limit for main article pages? W.org should apply some trimm-article-bots. Hello, allow me to point my problem <quote> 4. There exists a constant ${\displaystyle E\in (0,\infty )}$ such that ${\displaystyle \int _{\mathbb {T} ^{3}}\vert \mathbf {v} (x,t)\vert ^{2}dx<E}$ for all ${\displaystyle t\geq 0\,.}$</quote>

allow e be yours constant, then find modulus of integrative.

v=340m/s
dx=0.1m, where upperscript in integrantor is 10m;

if you do the integration then you get

ky=integ.(340^2*0.1)|<inf>1</inf> = 1 156 000 - 115 600 = 1 040 400 m^3/s^2.

if a e constant equal ky then, unit value which is? value?

e_l = ky/e = 1 040 400/2.78 m^3/s^2 = 374 244.6 m^3/s^2

if arguement

ky < e

isnot true, then

ky > e it is true.

my question is next: may it be out there both inequality substraction in T-space, meaning:

374 244 m^3/s^2 - 2.78 = ?

the problem may have been copied to the article page with a mistake, or it is inappropiatly at w.org. 86.121.67.182 (talk) 18:42, 22 September 2011 (UTC)Paul

solution is

${\displaystyle E\in (0,\infty )\cdot }$metre

About the problem

Considering that

• to solve the Navier–Stokes equations is equivalent to approximately solve the problem using the classical molecular dynamics

and that

• to solve the problem using the classical molecular dynamics is equivalent to approximately solve the problem using the first principle molecular dynamics,

is not it trivial that the problem has only weak solutions, if we accept the uncertainty principle?

CES1596 (talk) 17:01, 10 February 2011 (UTC)

Weak solution, better formulation?

The article states:

"The mathematician Jean Leray in 1934 proved the existence of so called weak solutions to the Navier–Stokes equations, satisfying the equations in mean value, not pointwise."

The phrase "in mean value" is misleading to the layperson.

Some physicists for example might not even consider that a "solution" which tests correctly against all (rapidly falling) smooth functions, in particular smooth L2-approximations of Dirac delta distributions, could be less than a strong solution.

Explaining the meaning and also the shortcomings of the 'weak solution' concept isn't easy. If someone can do this concisely, please change that part of the article. Otherwise I would rather remove the last part of the sentence and instead leave it at something like:

"The mathematician Jean Leray in 1934 proved the existence of so called weak solutions to the Navier–Stokes equations. See 'weak solution' for further explanation." — Preceding unsigned comment added by 85.179.12.135 (talk) 18:18, 26 July 2011 (UTC)

broken foot note

There is a note "" but there is no reference number 6 for this to refer to. RJFJR (talk) 19:00, 9 September 2011 (UTC) ( Heinbockel J.H. (2001) Introduction to Tensor Calculus and Continuum Mechanics. Trafford Publishing , ISBN: 978-1553691334 ). — Preceding unsigned comment added by 188.163.17.146 (talk) 17:40, 16 September 2011 (UTC)

NS Existence and Smoothness: An Algebraic Topologic Proof

Navier-Stokes problem has been completely solved in the following paper:

A. Prástaro, Geometry of PDE's. IV: Navier-Stokes equation and integral bordism groups, J. Math. Anal. Appl. 338(2)(2008), 1140-1151. DOI: 10.1016/j.jmaa.2007.06.009. MR2386488(2009j:58028); Zbl 1135.35064]

For complementary results, see also the following References.

A. Prástaro, Extended crystal PDE's, Mathematics Without Boundaries: Surveys in Pure Mathematics. (Eds. P. M. Pardalos and Th. M. Rassias.) Springer-Heidelberg New York Dordrecht London (to appear). arXiv: 0811.3693.

The classification of global space-time weak, singular and smooth solutions, for any initial smooth conditions (vector-field, isobaric-pressure, temperature), for compact, 3-dimensional smooth compact domains, is given by means of suitable integral bordism groups of the Navier-Stokes equation, in the above quoted works. It may be useful to emphasize that global smooth solutions do not necessitate to be (average) asymptotic stable ones. (They are always stable at finite times.) A general geometric criterion to study such stability is also given in and and in the papers quoted below.

A. Prástaro, (Un)stability and bordism groups in PDE's, Banach J. Math. Anal. 1(1)(2007), 139-147. MR2350203(2009e:58036); Zbl 1130.58014.

A. Prástaro, Extended crystal PDE's stability.I: The general theory, Math. Comput. Modelling 49(9-10)(2009), 1759-1780. DOI: 10.1016/j.mcm.2008.07.020. MR2532085(2011b:58041); Zbl 1171.35322.

A. Prástaro, Extended crystal PDE's stability.II: The extended crystal MHD-PDE's, Math. Comput. Modelling 49(9-10)(2009), 1781-1801. DOI: 10.1016/j.mcm.2008.07.021. MR2532086(2011b:58042); Zbl 1171.35323

A. Prástaro, On the extended crystal PDE's stability.I: The n-d'Alembert extended crystal PDE's, Appl. Math. Comput. 204(1)(2008), 63-69. DOI: 10.1016/j.amc.2008.05.141. MR2458340(2010h:58058); Zbl 1161.35054.

A. Prástaro, On the extended crystal PDE's stability.II: Entropy-regular-solutions in MHD-PDE's, Appl. Math. Comput. 204(1)(2008), 82-89. DOI: 10.1016/j.amc.2008.05.142. MR2458342(2010h:58059); Zbl 1161.35462.

In order to introduce the interested reader, let us shorter resume in some steps the proceeding to follow when one aims to know whether a global smooth solution exists in correspondence of some smooth Cauchy data N0 on a fixed compact 3-dimensional smooth space-like manifold.

1) Verify that N0 is contained in the regular submanifold (NS) of the manifold (NS). ((NS) is the sub-equation that is formally integrable and completely integrable.)

2) Characterize the singular integral bordism class of N0, i.e., characterize the singular integral bordism group of (NS).

3) Then N1, another compact 3-dimensional smooth space-like manifold, belongs to the same singular integral bordism class of N0 iff they have the same integral characteristic numbers (on their boundaries).

4) N1 belongs to the same smooth integral bordism class of N0 iff N1 is diffeomorphic to N0. (For example between N0=D^3 and N1=T^3 cannot exist a smooth global solution...)

5) Then, in order to know if a global smooth solution is average asymptotic stable one can apply the new Prástaro's general geometric criterion. Such a stable solution has not turbulence in the sense here requested.

Of course in order to understand the technicalities it is necessary to have the patient to carefully read Prástaro's works ! (Agostino.prastaro (talk) 11:19, 4 July 2013 (UTC))

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