Talk:Navier–Stokes existence and smoothness

message: 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)
thread_title:  Jorma Jormakka's  announcement 

message: The type D breakdown of the d=3 Navier Stokes equation, as defined by the paper of C.Fefferman, Clay Mathematics 2000, is given in my paper, 
http://www.tandfonline.com/doi/abs/10.1080/23311835.2017.1284293
Perhaps this adds to our understanding of the problem or to the confusion.   — Preceding unsigned comment added by 188.206.101.161 (talk) 02:59, 9 October 2017 (UTC)
replies: text: 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)

text: 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)
thread_title:  Move from article footer 

message: 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:  
:"[Not equal]: 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)
replies: text: 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)
replies: text: Prof. Jormakka, I'd ask you to stop editing the article, per Wikipedia'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 Wikipedia.  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)

text: 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, Wikipedia 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)
thread_title:  EJDE paper 

message: Considering that
 
* to solve a Navier–Stokes equation is equivalent with to approximately solve the problem using classical molecular dynamics
and that
* to solve the problem using classical molecular dynamics is equivalent with to approximately solve the problem using 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)
thread_title:  About the problem 

message: 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.[3]"
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)
thread_title:  Weak solution, better formulation? 

message: There is a note "[6, p. 294]" but there is no reference number 6 for this to refer to.  RJFJR (talk) 19:00, 9 September 2011 (UTC)
thread_title:  broken foot note 

message: Navier-Stokes problem has been completely solved in the following paper:
:[1] 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.
:[2] 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 (2014), 415-481. DOI: 10.1007/978-1-4939-1106-6. arXiv: 0811.3693[math.AT].
: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 [1] and [2] and in the papers quoted below.
:[3] 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.
:[4] 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.
:[5] 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
:[6] 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.
:[7] 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.
: More recently a new proof on the existence of smooth global solutions, defined on all R^3 is given in the following paper:
:[8]  A. Prástaro, The Maslov index in PDEs geometry.  arXiv: 1503.07851.
(The geometric methods used are the same ones focused on the Prástaro's PDEs Algebraic Topology.)
: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)
thread_title:  NS Existence and Smoothness: An Algebraic Topologic Proof 

message: 46.203.12.80 wrote an paper as an extern link. But there is no explanation about it and I do not know whether paper has been reviewed, then I'll copy it the bellow in this notes field.
: Navier –Stokes First Exact Transformation.
--Enyokoyama (talk) 15:12, 8 November 2013 (UTC)
replies: text: Year!　Thanks! I have already comfirm that Universal Journal of Applied Mathematics is an peer-reviewed journal. I've read it and seen that you don't solve the Millennium problem but you claim that the NSF Exact Transformation of the classical method would give a nice hint for it. I might expect more explanations in another article on wikipedia or so on.--Enyokoyama (talk) 08:32, 9 November 2013 (UTC)

text: Where's the commentary on these papers? These should be treated as good primary sources but we need secondary sources to validate and weigh. --NeilN talk to me 20:14, 1 August 2014 (UTC)
replies: text:  It should also be noted that the publisher (Horizon Research Publishing) is listed on Jeffrey Beall's list of 'Potential, possible, or probable predatory scholarly open-access publishers':  http://scholarlyoa.com/publishers/ AnonymousMath (talk) 22:07, 1 August 2014 (UTC)
thread_title:  Yet another solution proposed? 

message: Classical solutions
In 2013, Mukhtarbay Otelbaev of the Eurasian National University in Astana, Kazakhstan, proposed a solution. As a serious attempt to solve an important open problem, the proof was immediately inspected by others in the field, who found at least one serious flaw. Otelbaev is attempting to fix the proof, but other mathematicians are skeptical.
Alternative solutions
 Terence Tao in 18 March, 2007announced three possible strategies of an alternative solutions if one wants to solve the full Millennium Prize problem for the 3-dimensional Navier-Stokes equation. Strategy 1 “Solve the Navier-Stokes equation exactly and explicitly (or at least transform this equation exactly and explicitly to a simpler equation)” is used in works:
*Navier –Stokes First Exact Transformation
*Navier –Stokes Second Exact Transformation
The author of these works Alexandr Kozachok has offered (in February 2008 –       Internet , in 2008, 2010, 2012 – INTERNATIONAL CONFERENCE reports, in November 2013 and February 2014 - INTERNATIONAL journal) two exact transformations to the simpler equations. These transformations are executed by well-known classical methods of mathematical physics.   Therefore not only some professionals, but  also  educational,  social and many other sites have published or paid attention to these works  .
--93.74.76.101 (talk) 20:15, 8 December 2014 (UTC)
replies: text: A solution to such an important problem as this needs to be submitted to a top-notch journal, and the result reported.  If the argument is correct, even unknown authors can succeed through this path, and the journal will be happy to publish.  For example, Yitang Zhangs work on the twin-prime conjecture was submitted, accepted, and now has many cites, and lots of follow-up work (showing that others believe the result).  Here the journal is of lower reputation (surely less than such an important result would warrant), and the papers have *no* independent cites (the first is cited only by the second, and the second has no cites, according to google scholar).  On such an important problem as this, independent verification by experts in the field is needed.  Without that it's not notable.   LouScheffer (talk) 20:52, 22 December 2014 (UTC)

text: Anonymous IP is a well-known mathematics crank on Wikipedia.  See, for instance, here, where he claims that the chain rule is wrong and that the Helmholtz decomposition is wrong.  Discussion is pointless.  Just revert his edits.  Sławomir Biały (talk) 01:57, 30 December 2014 (UTC)
replies: text:  Indeed it seems for a number of years he has been applying his miscomprehension of the chain rule in order to come up with a number of absurd results. He interprets any feedback as positive confirmation of his work, including rejection letters from referees, discussions where others are trying to correct him, and silence, after others get fed up discussing basic concepts of mathematics. So discussion is certainly pointless. Hopefully, he will eventually grow tired of vandalising the wikis he edits. However, since this has been going on for years there doesn't seem much chance of that happening. AnonymousMath (talk) 15:12, 30 December 2014 (UTC)

text: Nope.  We are not a forum.  It is not our mandate to find your errors for you.  One way to get your errors pointed out is to offer a bounty for anyone finding an error.  I think $500 should do the trick, but you should actually be prepared to pay it out pretty quickly.  Sławomir Biały (talk) 14:29, 24 January 2015 (UTC)
thread_title: Attempt at solution[edit]

message: I am not exactly sure of Wikipedia's policy on protecting wikis, however it may be advantageous to restrict the number of so called 'solutions' being constantly added/removed from the wiki (see above). Unfortunately, the prestige attached to solving this problem seems to attract numerous well meaning but ultimately incorrect attempts -- some of which are published in journals lacking proper peer review (see above). Serious attempts are normally quickly retracted, however there are also a vocal class of not so serious attempts from individuals who relentlessly push their 'solutions' on well known members of the field and places like Wikipedia. A plausible solution should be easily identifiable by acceptance from experts in the field -- none of the above 'solutions' meet this basic criteria. — Preceding unsigned comment added by 85.181.119.143 (talk • contribs) 10:17, 6 July 2014 (UTC)
replies: text: Not done: requests for increases to the page protection level should be made at Wikipedia:Requests for page protection. Sam Sailor Sing 11:12, 6 July 2014 (UTC)
thread_title:  Protection from self-promotion 

message: I notice that this Talk page is littered with arguments from and against what is almost certainly crackpottery.  Note that WP:TPO specifically permits—I'd say encourages—the reversion of WP:PROMO from Talk pages.  It doesn't matter if the math is good or bad, significant or trivial.  WP relies on third-party reliable sources to evaluate material.  Talk pages are for discussing ways to improve the article, and since unsupported claims, even in refereed journals, is completely off-limits, there is nothing to discuss in the first place.  Such material is simply disruptive.
I'd be BOLD and simply delete most of this Talk page, both the nonsense and the trollfood responses.  However, I defer to long-term readers/editors of this page.  Other options are simply hiding the discussions, using  and , or deeply hiding by tweaking the archive parameters, or manually archiving to a special purpose archive (by creating Talk:Navier–Stokes existence and smoothness/Nevermind say). Choor monster (talk) 11:14, 7 June 2015 (UTC)
replies: text: I agree. Someone else hid a small part with . Maybe that's what should be done to most of the discussion here? --AndyBloch (talk) 19:50, 10 June 2015 (UTC)
replies: text: I've been aggressively reverting crackpots peddling their nonsense in the few math/physics article talk pages I follow.  Most editors don't seem to be aware that this is permitted/encouraged.  Most Talk page edits are considered inviolate.  But not all.  A few other math/physics editors know this, but very few it seems.  When I felt Prastaro's spam-level needed admin-level fixing and went to ANI, one math topic regular chimed in to the discussion with the idea that maybe I ought to be blocked. Choor monster (talk) 16:24, 11 June 2015 (UTC)
thread_title:  Clean up is allowed 

message: There is clearly a difference of opinion as to whether Otalbaev's unsuccessful (so far) proof should be included.  On the one hand, it did not prove the desired results, and most mathematicians think the general approach cannot do so.  On the other hand it is definitely notable, being covered in Nature (journal).
I would argue it should be left in as an example of how the 'proof' process works.  An expert proposes a proof, it's inspected by other experts, and if and only if the other experts in the field believe it, the proof becomes accepted knowledge.  This is of course completely obvious to any mathematician, but many Wikipedia readers are (presumably) not all that familiar with how mathematics works.  So I think this is helpful.
This general idea appears in lots of other mathematics articles.  The four color theorem reports on original, incorrect proofs (which did not particularly touch on the methods finally used).  P = NP reports on a famous but futile attempt.  Fermat's last theorem shows this process in detail - Wiles submitted a proof, a flaw was found, he spent a year repairing it, and it was finally accepted.
So I would argue this should be left in, as an example of the mathematical process, for the benefit of any casual reader.  Other opinions are of course welcome, LouScheffer (talk) 12:12, 8 October 2016 (UTC)
thread_title:  Otelbaev attempt at proof and sociology of math. 

message: This would easily be solved by counting all the points of the sphere from which its circle has been defined according to three static singular points defining the perfect curve of that circle.
The (non-)compressible fluid-theories could then be computed if those points have been expressed.
Conjecture: there are three successive points on the curve of the geometric circle, what is their mathematical expression?
zionion blogpot com 102.176.162.48 (talk) 14:02, 18 September 2021 (UTC)
thread_title:  Spherical points 

message: In response to  and recent edits. The book "The Navier-Stokes problem in the 21st century" by Pierre Lemarié-Rieusset is a very qualified and useful source. Tao's work takes up about one page total (out of 700), and is clearly contextualized as being important for showing that the particular nonlinear structure is important, and that function space bounds and energy estimates are insufficient for regularity. That information, i.e. that certain kinds of approaches will not work, is certainly valuable for some practitioners, but there doesn't seem to be any reason to think it compares to many more significant works on Navier-Stokes (Hopf, Serrin, Caffarelli−Kohn−Nirenberg, Buckmaster−Vicol, Sverak etc). More to the point, I have not seen any source which suggests that Tao's paper is very important and not just an interesting paper. Gumshoe2 (talk) 02:23, 6 December 2021 (UTC)
replies: text: Although citations are an imperfect metric, in some form they indicate what other mathematicians believe important.  In 5 years, Tao's work has accumulated about half the cites (173) that the "classic" papers in the field (those by Scheffer (no relation)) and Shnirelman (300-400 cites each) that are cited in the Clay official problem description.  Looking at these citing documents briefly, it looks like most references to Tao's paper are about the Navier-Stokes problem, as opposed to applying his techniques to other problems (another typical source of cites).  On the other hand, another review of the current situation (by Robinson) also only mentions Tao's work in passing.  My preference is to leave it in, if for no other reason than to indicate the field is not dead (the other cites are from 1934 and 1969).  LouScheffer (talk) 00:53, 7 December 2021 (UTC)
replies: text: If such citations are the metric, then there's at least 100 articles that warrant mention. But wikipedia is not the place for an up-to-date academic review, so I think there's no need to resort to such imperfect criteria anyway. There's no reason to mention Tao's work when there are several high-quality papers from the last 20 years whose importance is widely acknowledged in the literature. Gumshoe2 (talk) 04:05, 7 December 2021 (UTC)

text: I'll write a version of the section, then maybe we can discuss specifics in a more useful way. Gumshoe2 (talk) 04:24, 7 December 2021 (UTC)
thread_title:  Importance of Tao's work 

message: https://doi.org/10.3390/e24030339
Entropy | Free Full-Text | No Existence and Smoothness of Solution of the Navier-Stokes Equation (mdpi.com)  — Preceding unsigned comment added by Un11imig (talk • contribs) 12:22, 30 November 2023 (UTC)
thread_title:  A new solution proposed by Hua-Shu Dou 

message: There are 4 equations in the text. Eq1 and Eq2 are the components x and y of the NS equations, Eq3 is the continuity equation... and what is the Eq 4?? JosuAguirrebeitia (talk) 14:01, 16 May 2024 (UTC)
thread_title:  on the equations of the 2D case