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. Peter Woit says:


    For better or worse, I’ve had a policy of allowing string theorists to post whatever comments they want here, without deleting them. Some people are convinced that I’m deleting all the intelligent comments with substantive responses to my criticisms of string theory, and just leaving the juvenile, worthless ones. Not so at all.

  2. surlygrad says:

    why isn’t anyone asking the important question: how are we at columbia going to sucker penny smith into coming to present her work?

    while the N-S solution will be fantastic if true, these results on a comparison principle for hyperbolic equations are interesting in their own right and i’d love to see them.

  3. David Purvance says:

    Sounds incredibly coincidental, but a “Fer Product” solution to the 3-space periodic Navier-Stokes was posted on arXiv on 2Oct2006. Has anyone looked at or commented on this paper?

  4. Nets Katz says:

    Perhaps this is not the right place to have a philosophical discussion on the theory of PDE. I confess that I have not attempted to verify the proof line by line, but having looked at the paper, I will nevertheless go out on a limb and say that I have doubts.

    In order to solve a hard problem, generally you have to overcome the difficulty. The difficulty in Navier Stokes has little to do with the linear terms – the parabolicity of the equation – but has more to do with the nonlinear terms – supercriticality. For fixed energy, if we scale down a pulse, the nonlinear term dominates the linear ones.

    What I find very curious about this claimed solution is that supercriticality is never addressed. We replace the equation by a 1st
    order system which can always be done by introducing derivatives of
    the original unknowns as unknowns. We add a tiny bit of differentiation
    in t to make the equation hyperbolic. We claim that hyperbolicity
    is all that matters, and we even get to take a limit making the terms
    that caused hyperbolicity to disappear.

    But there are genuinely hyperbolic problems where supercriticality is
    the issue. Why not try to do the same thing for, say, the 3d quasilinear
    wave equation with 7th power defocusing nonlinear term.
    \Square u – |u|^6 u= 0.
    It is an open problem whether this has “eternal solutions” even in the
    radial case.
    For that matter, why not study the focusing version of this equation
    \Square u + |u|^6 u=0
    (I may have switched the signs.)
    Is there any reason the same method can’t take care of both? That would be odd because the focusing one is known to have short time
    blow up.

    Can anyone explain why my doubts are unfounded?


  5. mathjunkie says:

    I have some doubt about the result that an immortal smooth solution can exist. But, first, I have to admit that I am no expert in turbulence. Navier Stokes equation describes turbulence which is usually chaotic, then it is unlikely that an immortal smooth solution exists.

  6. Navier says:

    I think the paper needs a lot of work to be more readable. And one can check the details. Independently of the validity of the previuos results of the author.
    For instance, based on the definition of the prolongation of the Heywood force, the pressure is given as part of the force F. But then in Theorem 2 an assumption of the regularity of F ( continuous and bounded) makes the pressure continuous (by assumption). However it is well known that if the pressure is continuous then the solution
    (the Leray solution which known to be “immortal”) is regular.
    May be it is a typo and the pressure is not in F, but if it is we are faced with a circular argument.

  7. Euler says:

    It seems that the same proof works for the Euler equations and the Burgers equations!!!!!! If it doesn’t, it would be nice to get an insight on the role of the dissipation in this proof.

  8. NS Fan says:

    1. Does the proof of existence of the solution also mean that the analytical solution is also availble or available soon? If Penny’s solution turns out to be a valid one, she should be awarded $10Million.
    2. Can we construct solutions to some special cases using her method to compare with the known solutions in order to gain more confidence, excitement?

    Penny, no matter what, great job!

  9. Brooks Moses says:

    Nets: This particular paper is essentially one of the “we reduce the problem to a previously-solved problem” sorts of papers — where the previous problem is the question of smooth solutions for hyperbolic equations with this sort of term. You’ll have to read Penny’s other papers (given in the references) to find the things you’re looking for. Specifically, this is handled in the “proof” section of Theorem 2, I believe, which references her previous papers Sm2 and Sm4.

    MathJunkie: Turbulence, so far has been observed, does not contain any cusps or other non-smoothness (and certainly there’s no reason why it would need to!); thus, it’s not at all incompatible with an “immortal smooth solution”.

    Navier: A valid point! However, upon inspecting the equations, it appears that the inclusion of p in F is indeed erroneous; there should not be an undifferentiated p term anywhere in the Navier-Stokes equations, and certainly not in the equation that puts it in.

    Euler: It can’t work in the Euler or Burger’s equations, because both of these have well-known counterexamples involving shock waves. I suspect you’ll find the role of viscosity to show up in the referenced things that I pointed Nets at.

    NS Fan: 1. No, it regrettably does not mean that. 2. The only parts of Penny’s method that construct a solution are the limit-of-hyperbolic-approximations part of the proof. That portion of the method is closely related to existing computational methods which are already quite well-tested. Unfortunately, the rest of the proof doesn’t appear to offer any opportunities for such testing.

  10. Mr. T says:

    >Navier Says:
    >October 7th, 2006 at 8:51 pm
    >I think the paper needs a lot of work to be more readable.

    I agree with Navier. It is not a good idea to use boldface to emphasize what is important and to emphasize. This is mathematics, not literature.

    It seems that she is not familiar with TeX. She should just calm down and do a bit of work to make her paper look fairly neat.

    In addition, I would not advise her to argue mathematically technically on the internet. I do not want to tell why.

    With regards,

    Mr. T

  11. Euler says:

    Brooks Moses: I am sorry to report that the Burgers equation does satisfy the conditions of the hyperbolic systems considered by Prof. Smith. In fact the presence of the dissipation complicates things for NSE and she had to augment them to make NSE fit with her previous theory. My guess is after a day or two, this result will be proven to be incomplete, since Burgers equation to known to be blow up in finite time.

  12. Benni says:

    Penny has withdrawn her paper. It seems that euler is correct.

    the dreams are gone…

    (sometimes it is good, to talk to more people and hear opinions before submitting anything to arxiv or journals. It can damage reputation to be “another one, who tries to get a clay prize”…..)

    Hopefully Penny can correct her paper and make it complete…..

    Good luck, Penny!

  13. anon says:

    The paper has been withdrawn, it seems.

  14. Lianhong Gu says:

    Don’t give up, keep working, Penny!

  15. D R Lunsford says:

    A friend with long experience in turbulence comments as follows:

    “I recently had an interesting discussion with someone who described his own interesting but unsuccessful work on this problem. (He tried to bound the “enstrophy” production – basically dissipation / viscosity).

    “He told me that the solutions absolutely must exist, because if there really were some singularities, they would signal the breakdown of the NSE into something Boltzmann-like, anyway some kinetic level description.
    But the fact is, there are no known breakdowns in any fluid phenomenon known to be governed by the NSE. In other words: non-existence of solns some sort of singularity breakdown of NSE, which we would
    know about.

    “Of course, it’s good to look for real proofs, but this “argument” more or less settled the issue for me.

    “Anyway, it’s curious that according to your first link, the paper was withdrawn by the author due to fatal flaws. Also, it was interesting that she also looked at GR. I heard a talk about the GR equations as a hyperbolic system, with a view to existence of solutions, by Arthur Fischer from UC something or other, years and years ago. This seems more interesting to me — existence of
    solutions to some system the validity of which we really have no independent knowledge of. An existence proof for NSE would very likely have zero impact on turbulence. But who knows, maybe not. Maybe it would reveal some new property.”


  16. Euler says:

    Every serious mathematician who worked on the Navier-Stokes equation spent at least one night thinking they have the proof of the regularity problem, but they usually sober up after a day or two. Smith’s idea to look at NSE as a singular perturbation of a slightly compressible fluid is a well known technique numerically and analytically. However, the slightly compressible NSE is as difficult as the NSE, and may be more. Another problem is the convergence of the solutions to the slightly compressible to the incompressible is not trivial. I hope Smith will take the responsability of a good researcher and clears the status of her preprint with Nature and all the news media who jumped on this.

  17. woit says:

    Polite comments explaining what the problem or problems with Penny Smith’s work are are welcome here, rude ones containing no substance aren’t.

  18. Chris W. says:

    From an interested bystander:

    Leaving aside its apparent failure (pending possible revisions) does this attempt explore significant new territory? Does it suggest ways to approach the problem that haven’t been considered before?

  19. penny smith says:

    Dear Euler,
    I certainly did.
    I loathe the way they wrote articles so quickly.
    Arxiv is supposed to be a preprint file—not a journal or a newspaper.
    This has hurt me a lot.
    I might quit math.

  20. penny smith says:

    Well, no. I wont quit math.
    But, I do feel rather depressed.
    Anyway, life is about ups and downs.

  21. Michael says:

    Hey Penny,

    please keep your head up. See Lubos Motls comment, who is in the business of being blunt and honest, rather than polite:

    “Nevertheless, I think that Prof. Smith has nothing to be ashamed of: serious thinking sometimes requires a trash can.”

    I think Lubos speaks for many colleagues, certainly including myself.

    Surely you were very brave to make this public before a thorough peer review. Oh well — live and learn…

  22. mathjunkie says:

    (sometimes it is good, to talk to more people and hear opinions before submitting anything to arxiv or journals. It can damage reputation to be “another one, who tries to get a clay prize”…..)

    Submitting an article to arxiv is fine. The experts in the field can soon point out the problems with the article. The author soon knows what goes wrong with the article or otherwise it is actually a complete proof.

  23. Benni says:

    mathjunkie wrote: Submitting an article to arxiv is fine.

    But not such thinks. If you are doing this twice or more times, nobody will take you seriously anymore. Such work is best send to a journal with long referee reports before publishing it at

  24. Marty Tysanner says:

    I agree with the comments of “Michael” above. I think it was brave of you to put your results out to a wider audience to get feeback, even though it exposed you to potential public embarassment. Fortunately you discovered your error before things got even more public. In any case, you could do a lot worse than to make errors on such a difficult problem!

    (To “Michael”: It was very refreshing to see a reasonable comment from you. I don’t mean that in a condescending way — I am sincere. If you would take such a reasonable tone when you want to debate Peter, leaving behind the taunting and mocking behavior and ad hominem attacks and sticking with well formed arguments, I and probably others would be more interested in what you have to say.)

  25. Michael says:


    Penny clearly distinguished herself from the likes of Peter Woit by making an honest attempt at solving a very hard problem. Peter only mocks other people’s work and caters to the prejudices of the half-educated. I just do not believe that he deserves the same kind of courtesy — next best thing is to try and expose him for what he really is.

    Is it really so difficult to understand this basic difference? What good are well-formed arguments if the addressee is disingenuous? They are about as effective as stickers at the cockpit door “Please do not hijack this aircraft” against Islamic terrorists. Incidentally, for well-formed arguments you might try the arxiv instead of this blog.

    Take as an example my recent inquiry about Peter’s alleged research activities. Want to make a bet that no research paper of his is forthcoming within a year from now?

  26. To Penny Smith:

    We are made to persist. That’s how we find out who we are.

    — Tobias Wolff

    Best wishes,

  27. Navier says:

    To Penny Smith:

    I understand very well you frustration.
    The mathematicians who thought they proved the regularity of NSE were depressed. In any case there published papers using previous papers which are known to be wrong. This is science, and you need to continue your work.

    I would really like to know what is the error that led you to retract your previous papers?

  28. MathPhys says:

    Isn’t true that Boltzmann killed himself because he couldn’t solve the Boltzmann equation? Well, that was silly.

    I think posting a paper on the archive is just as good, or as bad, as talking to a colleague next door about your work. It’s no big deal.

  29. Arun says:

    Penny Smith,
    Nothing ventured, nothing gained. Don’t give up, please!

  30. surlygrad says:

    so is the earlier work on the comparison principles for hyperbolic equations also void or is it just the NS proof?

  31. Richard says:

    Has anyone else wondered if medals and large monetary prizes in mathematics may have a dark side? I suppose that that they do place mathematics more prominently in the public eye, which is probably a good thing, but recently lawyers were dragged into the picture in another infamous case involving egos and lack of egos, and now Penny has been a victim of premature media hype and heated internet propagation, which probably would not have happened without this “famous” problem having been given such a high monetary profile.

    And Penny — the heartbreak of mistakes is part of the territory. Get right back up on that horse!

  32. Daniel Grumiller says:

    Penny, I don’t know if this will cheer you up, but it is a well-known story about Werner Heisenberg, who graduated on turbulence and later became one of the co-founders of quantum theory and a famous physicist:

    He said he wanted to ask God two questions:
    1. Why Relativity?
    2. Why turbulence?
    He was optimistic to get an answer on 1.

    Don’t give up. And maybe find a simpler and less depressing problem in the meantime.

  33. I have read (in P. A. Davidsons Turbulence) a version of the questions Daniel Grumiller attributed to Heisenberg attributed to Horace Lamb. In that version there were two things he wanted God to explain:

    1. Quantum Electrodynamics
    2. Turbulence

    Since this was dated 1932, you have to say he had a nice insight into at least the order in which God would provide explanations!

  34. Timothy Clemans says:

    One thing to note about the Nature article about Smith’s work is that is clearly addressed the issue of how review would be done. Look problems have come up in many manuscripts that looked like they had promise. The important thing is that the serious flaw was found now and not two years from now. The news media is very important in these kinds of issues. They are real developments that need to be covered.

    “Has anyone else wondered if medals and large monetary prizes in mathematics may have a dark side?” The Wolfskehl prize for Fermat’s Last Theorem generated a lot of manuscripts which overwelmed universities.

  35. D R Lunsford says:

    Dear Penny,

    Thank you for taking on this problem. Hang in there.


  36. D R Lunsford says:

    PS let us know what you find out

  37. Marty Tysanner says:


    Out of respect for everyone else, I don’t want to belabor this point, but I think it is important. You said about Peter,

    I just do not believe that he deserves the same kind of courtesy — next best thing is to try and expose him for what he really is.

    I see two different issues here. The first is whether or not you should show courtesy to Peter. The second is that you want to try to expose him for what you think he is. Key questions are, who are you trying convince in your exposure, and what is the best way to go about it?

    The main problem I see with your past tactics is that you come across as someone who wants to make a lot of noise by heckling Peter rather than someone who has something useful to say. If you were at a talk listening to a well known speaker and somebody in the audience stood up and started shouting and making a lot of noise would that make you sympathetic to that heckler? I would be very surprised if you welcomed it and changed your opinion of the speaker’s point of view because of the outburst. Why should it be any different here? If you want to show anyone how wrong Peter is, you need to use reasoned arguments and counterexamples rather than just trying to “put him in his place.” For one thing, whether or not Peter is right about the role string theory should play in particle physics is quite orthogonal in most people’s minds to whether or not he has an alternative program of his own. (In fact, if he did have his own program, others would probably accuse him of trashing string theory to make his own ideas look better — the “hidden agenda.”) For another, evaluating the merit of a scientific program or idea is meaningful in its own right — grant administrators need to do it, and so does anyone who referees a paper. That’s the way I see it, anyway, and that’s a key part of why I don’t think your tactics of heckling, ad hominem attacks, and ridicule of his scientific activities will resonate much with most impartial observers.

    The other question is, who you are trying to sway with your tactics? If you are trying to use rude comments and interruptions as a way of gaining the praise of other people who already agree with you, then perhaps your tactics are appropriate. But catering to people who already agree with you seems like a really pointless thing to do. Displaying bad manners and disrespect while identifying yourself with string theorists could easily look to everyone else like a reflection of the attitude of some string theorists. In the eyes of the less educated public and other decision makers who ultimately control funding, “some string theorists” could easily become “all string theorists,” and their general perception of string theorists could spill over into attitudes about the viability of string theory in general. Again, that’s the way I see it. Of course, the fraction of the public and decision makers who frequent this blog is probably not substantial, but you are already aware that this blog is a magnet for some journalists who can interpret what is going on here, and thereby disseminate their preceptions to a much wider audience. What kind of perception of string theorists do you personally want to help create in the minds of others?

    So it seems to me that the real audience you should be thinking about is the people who aren’t likely to ever be string theorists: scientists other than physicists, physicists who aren’t string theorists, opinion makers, and the general public who frequent this and other blogs out of interest in science or just a chance to watch a little controversy. There are a lot more of those people than there are theorists in fundamental physics, and they ultimately are a lot more important to the future of string theory funding. And those people aren’t going to be the least bit impressed by tactics like the ones you have favored in the past.

  38. TheGraduate says:


    The information concerning the nature of the error can be found on her website. Perhaps it would be best to look there …

  39. mathjunkie says:

    A side issue.

    Penny said she submitted the article to Journal of Mathematical Analysis and Applications (JMAA).

    Is JMAA’s rejection rate high? Is it a prestigious journal in pde? I just wanted to submit an article on pde.

  40. Sacha says:

    I think possibly every maths person has had moments where they think they’ve solved some hard problem, and then later found an error in some assumption or something with the effect that their proof instantly collapsed. Painful, yes, but helpful, as it means that that path was not the right one to take! And in trying, you learn a huge amount.

    Keep up the research Penny.

  41. a says:

    By the way, why arXiv does not have a trackback to this post, that contains very informative comments?

  42. penny smith says:

    As to earlier work on hyperbolic comparson principles:
    With a necessary ( based on counter examples) additional condition,
    I can still prove these results for short time depending on the initial data.
    I am going to submit a paper on that: fixing my JMAA paper to a journal where it will get a decent referee job. In math that can take a while.
    I have written it in manuscript form already.

    This means the Einstein Result will also be true for short time
    depending on the initial conditions.

  43. penny smith says:

    The necessary condition is on the sub and supersolutions and only such sub and supersolutions will have a comparison principle.
    That’s true for systems.

    For the semilinear wave equation, no extra condition is needed, but there my comparison principle was proved for short time only.

  44. penny smith says:

    This means that my Perron method for hyperbolic systems also works for short time depending on the initial data. And, that is the paper that I am rewriting for submission to a journal.
    Step one: Write and send that paper, with the corrected comparison
    principle included.

  45. penny smith says:

    As to JMAA. It is a great and prestigous journal. It has a high rejection rate.
    Nevertheless, they ( and I ) screwed up on that paper. So did the reviewer for Math Reviews.
    This sort of thing happens. It is my bad luck. If the error had been
    noticed by a referee, I would have fixed it in a few days and resubmitted and that would have been that.
    Now, I look terrible and feel worse.

  46. penny smith says:

    The support here is helping me heal though!

  47. woit says:


    The arXiv policy, set by its moderator Jacques Distler, is to censor any and all trackbacks to this blog, whether they’re about math or physics. I tried for a while to do something about this, finally gave up since it wasn’t worth the time or energy.

  48. D R Lunsford says:


    How did you become interested in the NSE in particular? If you want to discuss offline I am at antimatter33 at yahoo dot com.


  49. Michael says:


    thanks for your effort of explaining. I disagree with your fundamental assumptions, and therefore reach very different conclusions.

    >> would that make you sympathetic to that heckler?

    Are you able to understand the difference between academic discourse and deception for personal gain?

    >> If you want to show anyone how wrong Peter is

    I don’t! Some very accomplished people think that string theory is a waste of time and effort. Plurality of opinion is generally good. What I despise is the way Peter makes no scientific contribution of his own while exploiting his populist criticism for attention and personal gain.

    >> What kind of perception of string theorists do you personally want to help create in the minds of others?

    Do I have to care? Is string theory a public relations affair? The moment this determines more than the level of background noise I have to endure is the moment I quit.

    >> The other question is, who you are trying to sway with your tactics

    Nobody, of course. You are free to believe in a democratic approach to science, I don’t. I have seen that some people dislike my comments here and stick up for Peter as a result. (You are probably in that group.) Maybe it makes you a good buddy, but there would be less positive things to say about it.

  50. Andreas says:

    Oh boy, this blog documents is a real tragedy.

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