Navier-Stokes Equation Progress?

Penny Smith, a mathematician at Lehigh University, has posted a paper on the arXiv that purports to solve one of the Clay Foundation Millenium problems, the one about the Navier-Stokes Equation. The paper is here, and Christina Sormani has set up a web-page giving some background and exposition of Smith’s work. I should emphasize that I know just about nothing about this kind of mathematics, but I’m reporting on this here for two reasons:

1. It looks plausible that this really is important.

2. Penny Smith tells me that she is a regular reader of this weblog.

Update: There’s an informative news article about this on the Nature web-site.

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176 Responses to Navier-Stokes Equation Progress?

  1. ttt says:

    I was wondering if Penny was interested in writing a textbook?
    You seemed to have garnered alot of genuine interest in your work!

    I would be interest in buying that book!

  2. csrster says:

    SteveM re: Kolomogorov: I would think the answer is “not at all” in the sense that the challenge is to prove that solutions to NSE remain finite precisely on the smallest, dissipative, scales where Kolmogorov breaks down.

    Of course we computational fluid dynamicists all know the result, but we’d still like to see a proof :-)

  3. oub says:


    The paper is withdrawn, but can somebody point out to me the
    *regularity* of the solutions, as stated in her original work?
    Are they supposed to be classical solutions?


  4. MathPhys says:

    I just looked up Sormani’s web site, and there is a statement there that reads

    “Regretably the journal referee did not catch the error or Smith would have corrected it before publication, either restating the result or rewriting the theorem.

    If the restatement wasn’t strong enough to obtain the immortal smooth solution to the Navier Stokes, she never would have made a claim to have found one.

    It is possible it was a very subtle error that could only be picked up due to the combined efforts of all the mathematicians who looked at Smith’s work”.

    So it’s the fault of the referee and/or all the mathematicians who read the paper. I didn’t know that!!!

  5. MathPhys says:

    Seriously, Smith had an idea, and it turned out to be wrong. No big deal, we all make mistake. Just don’t blame the poor referee.

  6. penny smith says:

    It was indeed a VERY subtle error. And, I can mostly fix the paper
    (except for Navier StoKES) if people would stop beating me up for makeing a subtle error. It is not as if I have been dishonest.

    Moreover to answer some people nasty comments. I have been working on nonlinear PDE my whole 30 year working career–and have scored big sucesses. My first paper solved a major problem in compressible gas dynamics including three dimensions. My second Paper gave a smooth family of transonic flows in a plane exterior problem disproving
    a famous and incorrect theorem.
    I didn’t get to Institute for Advanced Study by being a moron,
    even if today I do feel like a moron.

    Second, not everyone has experts on tap to check their work on NS.
    I submitted to a preprint file, and was glad that an error was found.

    I am not dishonest. I am not arrogant. I do not seek publicity.
    I just try to do math the best I can, and have endured considerable sacrifices to do that.

    I have some big successes. I now have some failures. Please don’t beat the shit out of me. Ok?

    Moreover, working on hard problems makes errors more likely.
    Newton, Gauss, Poincare, Einstein, Hilbert all have published false
    theorems. Were they morons?
    It’s hard enough as it is.

  7. MathPhys says:


    No one says you were dishonest or arrogant. But when Christina Sormani implies that it’s basically the referee’s fault, I find that that’s unfair on her part.

  8. jeremy says:

    I don’t know how many of us have gone through all Penny’s relevant papers, but suddenly some of us decided to give advices to Penny on how to conduct her research. I don’t know if that is necessary. Every researcher has his/her own style of doing research, hardly anyone can claim one is better than the other. Remember Perelman, if he has waited for the expert’s opinion, he would never have waited who knows how many more years to post his papers on arXiv.

    Penny made a mistake in her research. We all make mistakes in our researches, mistakes that sometimes were also pointed out by others. If she decides to admit the mistake, which she did, and move on, I believe that all everybody else needs to do is to show respect.

  9. Pingback: Links to People I Know « Abstract Nonsense

  10. Deane says:

    I agree that Penny made an honest mistake and should be given a break here.

    But I’d also appreciate a little more appreciation of how difficult it is to prove these big theorems. And, again, difficult in the sense of grueling labor over a long period of time and not in the sense of being brilliant.

    For instance, someone has now cited Perelman as someone who doesn’t do what I described. Well, besides the fact that Perelman is a pretty unique singularity that no one should try to emulate, Perelman did in fact do at least some of what I advised. He worked on the Poincare conjecture for approximately seven years. Again, I do not know but am confident that nearly half that time was devoted to checking his work. (why do I think that? Because whenever I’ve proved a new theorem, by the time I’m finished, 90% of my time was spent on checking my work. I’m assuming Perelman is much better at it than I am). Finally, he *did* send emails to other experts consulting them and asking them questions. I concede that he didn’t actually show them his proof, but he did try to verify critical points in his proof with experts. Hamilton did not respond to Perelman, but others did.

    No one should have the illusion that Perelman wrote those short sketchy papers without spending years (yes, years!) verifying the details that underly those papers. I am willing to bet that thousands of pages of calculations and proofs were written down by Perelman before he wrote those papers. And that’s not counting all the calculations and proofs that didn’t work out.

    I think we mathematicians have been all too successful in hiding the fact that doing mathematics is 99.9% grueling labor and approximately 0.1% brilliant ideas (I hope I got the math right).

  11. penny smith says:

    Guess what people.
    Now that my head has cleared.
    The JMAA paper on PERRON is CORRECT!
    The counterexamples were for nondiagonal systems, and that paper was for diagonal systems.

    I was worried about I theorem of Ladyzenskya that I used, but it wasn’t even necessary.

    Several experts wrote saying they couldn’t find an error in that paper.



    The error was in the use of unpublished theorem 4 of the Einstein Paper. That theorem had a very subtle error in the infinite time
    comparison. I have extended it to show that the time of comparison depends ( for the experts) on the C^{1} norms of L(
    sub super solution).

  12. I wrote:

    “Regretably the journal referee did not catch the error or Smith would have corrected it before publication, either restating the result or rewriting the theorem. If the restatement wasn’t strong enough to obtain the immortal smooth solution to the Navier Stokes, she never would have made a claim to have found one. It is possible it was a very subtle error that could only be picked up due to the combined efforts of all the mathematicians who looked at Smith’s work.”

    I do not see this as blaming the referee.

    On another note: the writer of the poem above doesn’t even know undergraduate ordinary differential equations if he doesn’t understand how one takes a second order equation and turns it into a first order system.

  13. penny smith says:

    Deane, I also spend thousands of hours doing calculations, just like you and just like everyone who does math research. That “month” alluded to by Nature was based on a misquote by me. I said:”
    I have spent years ( in fact about ten years) on these hyperbolic
    papers and about a month on NS” –( most of which was checking NS, by the way!).
    I posted on A PREPRINT FILE to get possible priority. I didn’t claim a
    theorem that was unpublished.

    What exactly is your beef with me?

  14. penny smith says:

    And since when, DEANE, did I claim to be brillant?
    What is your need to beat me up?

  15. woit says:

    Please, everyone. Stick to discussing math here, not who to blame about what. I can’t even figure out how to moderate this if people start doing things like writing verse that may or may not contain personal criticisms. Enough already.

  16. Deane says:

    My only point is that it is highly unlikely that anyone can come up with a solution for the Navier-Stokes problem, write it up, and check it properly in less than a year, never mind a month. My guess is that I couldn’t do it if the proof were dictated to me.

    I’m willing to forgive you for your haste, since, as you say, you wanted priority and probably got a little overexcited. But I am dismayed by how many people without seeing your paper appeared to be so willing to believe that this was a reasonable scenario. One of your friends or colleagues should have grabbed you and said, “Penny, wait! Let’s go slow and careful on this.”

    I was not trying to beat you up. I was trying to beat everybody else up. I’m sorry that my aim was so poor.

  17. Daniel Grumiller says:

    Penny: Honestly, I don’t think Deane is beating you up. We all love you for your attempt (there you go – consider how many people would have hated you if you had succeeded…), but the issues Deane mentioned ring true.

  18. TruthSeeker says:


    As an impartial observer reading his comments here, I, too, did not get the impression that Deane was beating you up. His (constructive) criticism was apparently based on the fact that he mistakenly thought you had worked on the problem for only one month. If there’s anyone to blame, it would have to be the journalist (again?) who misunderstood or misquoted you for whatever reason. I hope you will focus your energy on math again and not give up on your passions. The journey of discovery could be just as exciting as the discovery itself. It all depends on your perspective.

  19. penny smith says:

    Dear Truethseeker,
    I love doing math. I just hope people are still willing to read, referee and publish my papers.Thanks.
    As the example of Yau shows beware of journalists.
    Even well intentioned ones like mine. She just missed a point in what I said. All humans make errors–Journalists included–and the MOST important
    thing in life is not intellect but kindness to others.

  20. Brad Stone says:


    Rock on with your math self! I have been fascinated to witness this exchange, and depressed to watch the nature of the media attention around you. I hope that you have the oppurtunity to get away from all of this nonsense and get back to peacefully attacking your PDE research.

    As for Deanne, I simply have to say that his argument about the amount of time you spent on NS research is without force. It saddens me to have to remind her/him of the productivity of one summer in 1905. 3-months and 3-earth shattering papers. Even though you were misquoted by the reporter, Penny, I find some observers’ close-mindedness troubling.

    Good luck with the corrections!

  21. Benni says:

    Dear Penny
    I think the problem is not the nature article, or that it was published so fast. The problem was, that you told the reporter the following words:

    “I’m pretty confident that my result is right, or I would never have submitted it anywhere,” says Smith. She hopes to serve as a role model for women in mathematics. “On the other hand, I certainly want the prize,”

    Dear Penny, why don’t you just have told nature:
    “Sorry. This is only a preprint, which might be wrong. I posted it on Arxiv because it maybe something but I don’t have an Navier stokes expert in my department who I can ask. So it has to be emphasized that my paper might well be wrong”

    Perelman is a good example. He never wrote in his papers something like: “Here I solve a millenium problem”….
    Instead, in the perelman papers, everywhere is caution!

    No mathematician has the right to say, “he is certainly right” when he is attempting a millenium problem, otherwise he is likely to make a fool of himself because in 99% the “proof” will be wrong!

  22. TruthSeeker says:


    There is no reason why people would not want to read or referee your papers, since you’ve already established a good reputation before all this. Mathematicians are generally fair-minded individuals who like to see proof and evidence in everything, rather than going by rumors or gossip. And frankly, there hasn’t even been much of the latter going around anyway. You mentioned that you were depressed upon discovering the mistake in your NS paper. I hope you feel better now.

  23. geometer says:

    According to mathscinet, during 1982–2006 Penelope Smith has published 18 papers which were cited 11 times by 8 authors. Of course mathscinet does not pick up all citations, but surely it picks up more than half. This is a decent record, but I wish Prof. Smith were a bit more modest while talking about her accomplishments.

    BTW, Perelman so far published 17 papers cited 198 times by 111 authors (and this of course does not include his yet unpublished work on Ricci flow).

  24. Genghis Cohen says:

    Mathscinet picks up only reviews since 1998 or so. So geometer’s statistics are pretty irrelevant.

    I don’t know why you’re all being harsh on her. So many mathematicians announce big results that are wrong. Think Poincare Conjecture or Jacobian Conjecture or so many others.. It’s true they spend more time and exercise more caution than she did.. but many of the people behind the false proofs don’t admit making mistakes easily and cause much more of a nuisance for mathematicians than this did. This whole episode lasted what, 3 weeks, and the paper (including its predecessors) was short. So it’s not like the whole field was ground to a halt by this.

  25. Richard says:

    “Penny: Honestly, I don’t think Deane is beating you up.”

    I disagree. There’s a lot of snide condescension in the tone of many of these posts by some who for whatever reason do not have the confidence and backbone to identify themselves, their own backgrounds, and their own accomplishments and failures. Frankly, this discussion has become awfully disgusting, despite the attempt of some to discuss math, and I’m sorry that Penny, who I do not know at all, has been dragged through this.

  26. woit says:

    This endless discussion has become both unpleasant and tedious. No one seems to have anything else to say about mathematics, rather they just want to personally criticize someone or another for something or other. As a result I’m shutting off comments here. If you’ve got something new and substantive to say about this, e-mail me and I’ll consider turning them back on.


    Many people here are posting anonymously and you are right to point this out. Deane is not such a person, he has often identified himself here, he’s Deane Yang, a rather distinguished geometer, one who has worked with Penny in the past.