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. TheGraduate says:

    very fascinating.

  2. astro says:

    With Cao and now Smith, is Lehigh becoming a new powerhouse in math?

  3. Jud says:

    “With Cao and now Smith, is Lehigh becoming a new powerhouse in math?”

    Becoming? 🙂

    (I grew up in Bethlehem, PA, where Lehigh is located, and live nearby.)

  4. fh says:

    I heard Prof Ehlers talk about her work on GR late last year, it sounds as if her methods are capable of solving some long standing problems in mathematical physics.
    Very impressive.

  5. This certainly looks promising. I wish I was better versed in the relevant details so I could attempt to judge it for myself.

  6. nc says:

    Thanks for reporting on this 9 page paper, ‘Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System’.

    Wikipedia’s page says:

    ‘A $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute to whoever makes substantial progress toward a mathematical theory which will help in the understanding of this phenomenon.’

    Wiki on another page lists the exact problems. So it is solved unless there is an error in

    ‘We prove the existence of a smooth solution for all time – under physicially reasonable hypothesis on the initial data – for the Navier-Stokes System in three dimensions.’

    How long will that take to be properly peer-reviewed before publication? (I trust it will be checked far more carefully than the Bogdanov’s physics papers on string theory…)

  7. Pingback: otro blog ms » Otro problema del milln resuelto?

  8. Pingback: Gaussianos :: Otro Problema del Milenio podría haber sido resuelto :: October :: 2006

  9. fh says:

    nc, this is building on a long string of well established, peer reviewed papers Smith has written over the last few years.

    This should be “relatively” straightforward to check.

    BTW, just for fun, her Mathematical genealogy traces back through Weierstraß to Gauß, and in a different branch via Hilbert and Klein to Poisson and Fourier and on to Lagrange, Euler, Bernoulli and Leibniz.

  10. John Baez says:

    It would be really great if Penny Smith – or anyone! – made some serious progress on the Navier-Stokes equation.

    Does anyone know whether this is the same “Penny” who used to post to sci.math and sci.physics?

  11. jamzik says:

    PSmith9626 seems to fit the bill, Lehigh University, an interest in the Navier-Stokes equation – but it appears she stopped posting in early 2005. She must have found something that occupied her time.

  12. Pingback: Ars Mathematica » Blog Archive » Navier-Stokes Problem Solved?

  13. Pingback: Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System « Sachi’s hyperbolic space

  14. MathPhys says:


    Yes, it’s the same Penny.

  15. erno says:

    Katherine Hayles summarizes Luce Irigaray:

    “The privileging of solid over fluid mechanics, and indeed the inability of science to deal with turbulent flow at all, she attributes to the association of fluidity with femininity. Whereas men have sex organs that protrude and become rigid, women have openings that leak menstrual blood and vaginal fluids… From this perspective it is no wonder that science has not been able to arrive at a successful model for turbulence. The problem of turbulent flow cannot be solved because the conceptions of fluids (and of women) have been formulated so as necessarily to leave unarticulated remainders.”

    And indeed, it takes a woman to solve it! (Maybe) Physics? HA! Postmodern 5th Wave Feminism has the answers.

  16. Yatima says:

    May I suggest some intense sessions with a good Freudian analyst (not that I think that psychanalysis is anything other than ‘not even wrong’)

  17. csrster says:

    Oh gawd, that was something that also lurched unwanted from my suppressed subsconscious when I first read this item. (At the time I first heard it I was a PhD student in Cambridge Applied Maths, a powerhouse of mostly-male fluid dynamicists.)

  18. Dick Thompson says:

    I have always cited the state of fluid mechanics, where the equations have been known forever, but no real analytical progress ever seems to occur, as my comeback to dreams of a final theory.

    Do I have to change my tune now?

  19. Gina says:


    (Of course, we have to remember that the NS problem is infamous for
    “fighting back” as the Hungarian mathematician Paul Erdos used to say about famous problems.) I wonder if the approximation of NS by certain hyperbolic equations used by Smith have physics relevance/intuition on their own, or are just technical tools in the mathematical proof.

  20. TheGraduate says:

    I wish somebody knew enough to place odds. This is intriguing.

  21. Michael says:

    Hey Peter,

    this Navier-Stokes paper is quite interesting. I don’t dare to judge its contents, though. I hope some math colleagues will enlighten me with their expert opinions soon.

    Mind if I ask how your own research is coming? You ignored my question last time I asked.

    You announced in the spring that a paper of yours on some BRST-subtleties would be fourthcoming this year. Can we hope to read it soon?

  22. Walt says:

    Erno: I forgot all about that Irigaray quote. If Smith has really solved it, that would be an incredibly funny development.

  23. Peter Woit says:


    I ignored your question because, from past experience, I don’t think you’re interested in the answer. The answer though is that my work on BRST is not moving as fast as I would like since the response to the book has been kind of overwhelming and it’s taking all of my free time to deal with it. I hope this will die down soon and I’ll get back to making progress on that project.

  24. John Baez says:

    Various unnamed sources say the proof doesn’t seem plausible. Probably best to cool it and wait until the paper is refereed.

  25. penn smith says:

    John, it’s penny. Can you email me and tell what these “unnamed sources” think is wrong?

    Here is one ( not important) error found in “Perron’s Method for Hyperbolic Systems Part I”: In Theorem 21 One needs POSiTIVE
    initial data and positive g. IT then takes a little more work to prove the
    comparison principle for sub and supersol with the SAME initial data.
    Similarly, in Theorem 22. Otherwise counterexamples occur.

    However, in both the Einstein and NS papers, I provide a different comparison principle (used in three space dimensions only) with a different proof.

  26. penn smith says:

    The point is that in the proof of theorem 21 of that paper, to apply the
    Protter maximum principle to the parabolic approximating equation, one needs postive ( not non-nonegative) initial data. In my handwritten version, I had that, and the inequality got miscopied in typeing.
    Again, in the two later papers, a comparison principle with the SAME initial data, is proved and used only in three space dimensions by
    a different method.

  27. penn smith says:

    And what is this ” Unnamed sources tell me that something is implausible” nonsense.
    If they have a real mathematical issue, they should ( out of courtesy and out of mathematical professionalism) email it to me, so I can look it over.

    That is what someone did with the issue that I just posted. I thanked them too!!

    THAT”S HOW MATH WORKS. We are grateful for error correction.
    We are not a cult.

    And I personally care only about truth. I don’t care about ego nonsense.

  28. mathjunkie says:

    Does Penny’s solution mean that it is the solution describing turbulent flow?

  29. Brooks Moses says:

    Well, the various sources that I’ve seen claiming that this “doesn’t look plausible” are one that are responding to claims that this is a constructive proof (with implications that it will be directly useful in simulations of fluid flow). That claim is, in my opinion, quite implausible — but I don’t have the math background to back it up. However, I’m pretty sure that’s not the claim Penny is making; at the least, she’s certainly not claiming that implication, and that’s the really implausible part.

    Meanwhile, to answer mathjunkie’s question: Given that it’s possible to define a non-turbulent initial condition that will lead to turbulent flow in an infinite domain (namely, a sufficiently strong shear flow, at least in a localized region), the solution that Penny’s paper describes does, indeed, have the potential to contain turbulence. Since it’s not a constructive proof, though, it’s not too clear to me what relevance that will end up having to the study of turbulence — though the actual smoothness-of-solution result may well be useful.

    Now, my curiousity is how difficult it will be to apply this to flows with boundaries — and whether the method of doing that will involve the analytical equivalent of the computational “immersed boundary method”, in which a flow with a boundary is simulated by taking a flow in an infinite domain and applying forcing functions to it that cause the velocity to match a prescribed velocity along a certain manifold. Though that may be a bit tricky, given that the forces are usually Dirac delta functions, and thus the velocity isn’t smooth on the manifold. (I think there are ways around that, but I’m a computationalist, not a mathematician.)

  30. penny smith says:

    More detail, for those reading that paper:
    In theorem 22 we require $w_{1}(x) =w_{2}(x)$ and
    This arises because to get nonstrict inequalities Protter ( for the parabolic max principle) adds an exponential NOT IN MY HYPOTHESIZED FUNCTION SPACES.

    IN THEOREM 21 we require $w_{1}= w_{2}$.

    That kills the counterexamples.

    To get Theorem 21 from Theorem 22, on page 365 line 7, we apply the
    proof ( not the statement) of Theorem 22 with a sequence of positive inhomogenous terms and initial data decreasing to zero.

    AGAIN, NONE OF THIS USED OR NEEDED in either the Einstein Cauchy or the NS papers!

  31. penny smith says:

    Thanks Brooks,
    It is NOT a constructive proof. It is an existence theorem.
    The Perron method gives a solution as a lower envelope of a
    transfinite set of supersolutions and are then doing a triply countable
    sequence of those approximations to get the NS solution.

    It is NOT applied computational math.

  32. Brooks Moses says:

    Gina: the approximation by these particular hyperbolic equations does indeed have physical relevance on its own. The incompressible Navier-Stokes equations have some variables (specifically, pressure) that have effectively zero relaxation time. In an incompressible fluid, a pressure wave travels at infinite velocity. One can add a bit of “compressibility” to the equations by giving the pressure a finite relaxation time, and thereby make the problem hyperbolic — and that’s what Penny’s formlation of the hyperbolic equations does. This was a relatively common computational solution method some decades ago, and is referred to as an “artificial compressibility” method. There are also more modern methods that are based off of that, but do a convergence to a truly incompressible solution.

    Since Penny’s forumulation also breaks things up into a first-order set, there’s a similar issue (without as much physical significance to the problem) in the stress tensor, and so she adds a relaxation constant there; this, I haven’t seen before in exactly that form, probably because it doesn’t map to a computational problem the way the zero relaxation time on the pressure does.

  33. Richard says:

    “Various unnamed sources say the proof doesn’t seem plausible.”

    I can’t believe that John actually posted this comment. “Various unnamed sources” has the ring of a Bush administration leak, and “doesn’t seem plausible” is dismissive while having no other content or value of its own.

    This is more typical of a Motl drive-by shooting. I’m demoralized.

  34. John Baez says:

    I should not have mentioned vague second-hand doubts about Penny’s proof on this blog. My only sensible point was that people should calm down and wait until some experts have had a chance to go through this proof.

  35. John Baez says:

    I should add that I would like nothing better for Penny’s proof to be right!

  36. nontrad says:

    To John’s last comment: Hear, hear!

    Beyond the fact that NS is a intricately interesting system of equations (I encourage all to muck around with these equations if they never have; since everything from GR to QCD to EM and, infact, huge swaths of PDE can pop up in suprising ways depending on one’s background) — it would be really truly *great* if Dr Smith’s work really did knock over this historical giant…

    Looking forward to learning more, and hoping for the best on this concern!

  37. Pingback: Brooks Moses: Notes on Divergent Simulations » Penny Smith’s Proof on the Navier-Stokes Equations

  38. penny smith says:

    Brooke is very smart. He understands.
    By the way, I have posted the stuff I wrote here about the corrections to the earlier paper at arxiv. It will appear late on Monday EST. You people got first dibs.
    Thanks for the interest in my paper.

  39. Johan Richter says:

    What would the physical relevance, if any, be if this paper held up?

    Also how long do you think it will take before the paper is checked? It is short so we should not have to wait as long as we did with Poincare should we?

  40. MathPhys says:

    Perelman’s papers are not that long. That was the trouble 🙂

  41. Pingback: Nonoscience / Analytical Solution to the Navier Stokes Equation Reported

  42. surlygrad says:

    pardon my ignorance, but is the analogous result known for the IBVP, on some reasonable domain?

  43. jeremy says:


    Can you write an “expository paper” for your work on NS equation, meaning to have all the details in one paper? It makes a difficult read having to check all the references; some of them are yet to appear. Hopefully this would not require too much of work.

  44. Michael says:


    no need to doubt that I wanted to know what I was asking. Yes, I dislike your populist anti-science crusade. But that makes me all the more interested in what you have might have to say on a more technical level.

    Let me freely admit that I don’t believe you are doing any research at all. I believe you are pretending to make yourself look more serious and competent than you are. I would happily stand corrected, however, if you could demonstrate the ability to write a decent research paper — no matter if it’s math or physics. I’ll be patient and check back with you every few months to see if you are making any progress. Please keep us posted. Thanks.

  45. penny smith says:

    This doesn’t affect either my proof of NS or Einstein Cauchy.

    I posted some fixups here of theorems 21 and 22 of “Perron’s Method for
    Quasilinear Hyperbolic Systems PartI” and have now found another fixup needed
    so I have decided to ditch those theorems and replace them by Theorem 4
    of my Einstein Cauchy Arxiv paper math.DG/ 0605352.

    This gives the results of the Part I Perron paper above in all space dimensions that are odd and bigger than or equal to three. Lately, I seem to care only about three
    space dimensions ( smile).

    The correction will appear at arxiv late on Monday evening.

  46. Peter Woit says:


    Thanks for your interest, although I think you have me confused with someone else (Lenny Susskind?), since I’ve never been on a “populist anti-science crusade”.

  47. penny smith says:

    Dear Jeremy,
    The Corrigendum to the old paper is at both JMAA (electronically through science direct), and at Christina’s Somani’s webpage for me.

    When the dust settles and I get more than two hours a night sleep, I may indeed write something expository, but odds are that someone else will do it.

  48. jeremy says:


    Thanks for the information. I hope you will get some more sleep and get refreshed.

    All the best.

  49. Michael says:


    believe me: I wouldn’t confuse you with Susskind in a million years. You see, he is a maverick with incredible skill and talent. I don’t agree with many of his views, but I sure admire his many important and beautiful papers.

  50. Marty Tysanner says:

    re: “Michael”

    There is an interesting discussion of a potentially very important mathematical result going on here, and what does this “Michael” contribute? Pure noise and jarring distraction that has absolutely nothing to do with the topic at hand; his demonstrating thatcomments here are unsubstantive by any measure. As usual, “Michael” is acting like someone who is clueless about what is going on around him, someone who has an adolescent fixation on heckling Peter because he disagrees with Peter’s broader viewpoints about the direction of research in string theory and particle theory in general.

    But this is what we have come to expect from “Michael” — nothing of substance, just air-headed commentary that is almost invariably on the same topic: So, Peter, how’s your research coming? I think you are a charlatan. Blah, blah, blah… I don’t know what he thinks he is accomplishing by this immature mode of discourse, but to rudely interrupt a meaningful discussion with mindless, irrelevant blather like his above comments is bound to convince others that he is nothing more than an insensitive, anti-intellectual boor.

    I wish Peter would be more merciless about deleting such worthless comments.

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