Talk:Navier–Stokes existence and smoothness/Archive 5

Archive 1

←

Archive 3

Archive 4

Archive 5

NS Existence and Smoothness: An Algebraic Topologic Proof

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))

94.153.74.179 (talk) 14:21, 15 February 2014 (UTC)Agostino.prastaro (talk) 19:56, 13 March 2014 (UTC) (Agostino.prastaro (talk) 16:51, 9 April 2015 (UTC))

Clean up is allowed

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 {{collapsetop}} and {{collapsebottom}}, 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)

I agree. Someone else hid a small part with {{hat|reason=[[WP:NOTHERE]]}}. Maybe that's what should be done to most of the discussion here? --AndyBloch (talk) 19:50, 10 June 2015 (UTC)

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)

Otelbaev attempt at proof and sociology of math.

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)

Move from article footer

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)

- 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)