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Another Millenium Problem May Have Been Solved 134

S3D writes "After recent verification of the proof of the Poincaré conjecture, another of the Clay Institute's Millenium Problems may have been solved. This new solution is for Navier-Stokes equations under physically reasonable conditions. Navier-Stocks equations describe the motion of fluid substances such as liquids and gases. Penny Smith has posted an Arxiv paper entitled 'Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System' which may prove the existence of such solutions."
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Another Millenium Problem May Have Been Solved

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  • If this turns out to be true, then it's a pretty big deal. I remember studying this kinda stuff a few years ago... suffice to say that it really makes my head hurt, even now. Having had a quick look at the article, it does promise to be a very interesting read, at the very least.
  • Hm. (Score:5, Funny)

    by ZombieRoboNinja ( 905329 ) on Saturday October 07, 2006 @03:37AM (#16345905)
    I have no idea what any of that means, but rest assured that by the time this thread ends I will have developed ironclad opinions on the subject.

    LOUD ones.
  • pr0n (Score:2, Funny)

    This new solution is for Navier-Stokes equations under physically reasonable conditions. Navier-Stocks equations describe the motion of fluid substances such as liquids

    who needs a description of the motion of fluid substances? I want video, perferably in slow-motion and from multiple angles.
  • I bet if I put on a pimp hat and read it while drinking a glass of Courvoisier, I could make it "The Even Smoother Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System".

    Don't player hate, player appreciate baby.
  • Neat indeed (Score:3, Interesting)

    by Zx-man ( 759966 ) on Saturday October 07, 2006 @04:08AM (#16345993)
    As a math major I may say the this is impressive: after understanding the significance and complexity of the problem seeing a solution has been found is really exciting. Although I'm looking forward to see something done about the most significant of the Millennium Problems (IMO and from the pure maths POV) -- the Riemann hypothesis [].

    Note: Not considering P vs. NP as it is quite possibly unprovable.
    • How could it be unprovable?

      It certainly couldn't ever be proven unprovable, like some things can be, since proving it unprovable would also prove there was no way to implement a conversion P = NP, and, therefore, P != NP.

      Just because we can't prove it doesn't mean it's unprovable.
      • Re: (Score:3, Informative)

        by Famatra ( 669740 )
        How could it be unprovable?

        Just because we can't prove it doesn't mean it's unprovable.

        Godel's incompleteness theorems []
        • Re: (Score:3, Interesting)

          by ZorbaTHut ( 126196 )
          That is true. However, note that unlike Godel's incompleteness theorem, P = NP has direct and obvious connections to the real world. We're not choosing between competing logical theories that exist in a vacuum. P = NP allows us to do certain interesting things on computers. If it turns out we can prove we'll never be able to do those, that is the same thing as saying it is impossible.
          • Re:Neat indeed (Score:5, Insightful)

            by Garse Janacek ( 554329 ) on Saturday October 07, 2006 @11:22AM (#16348287)

            Not necessarily -- it is conceivable that there exists a poly-time algorithm for an NP-complete problem, but there is no proof (within ZFC, say) that it is correct. The physical truth is certain -- but what we can know about the physical truth is limited.

            Now, I'm with you in believing that that's extraordinarily improbable, but math doesn't always respect what we consider to be likely.

            In my opinion (as a complexity theory grad student), the "maybe P=NP is independent" speculation is bunk. There are genuine, interesting results talking about the limits of how we can resolve P vs. NP, but none of them come anywhere near logical independence, and giving up on a field-defining problem after 30-odd years is just very odd considering how long the really major open problems often take to solve. I believe the solution exists, and I hope it is found soon, but I will be unsurprised if it takes another 100 years or so while we get a better handle on what computation really means.

            • by AxelBoldt ( 1490 )

              it is conceivable that there exists a poly-time algorithm for an NP-complete problem, but there is no proof (within ZFC, say) that it is correct.

              Yes, that's conceivable but seems unlikely. A more likely scenario (and in fact my money is on it) is that we can eventually prove that ZFC can neither prove nor disprove P=NP, and in that case we don't know whether your scenario above is correct, or if on the contrary no such algorithm exists but ZFC is simply too weak to establish that.

            • by gkhan1 ( 886823 )
              It should probably pointed out for all people that hasn't studied advanced mathematics that ZFC is Zermelo-Frankel set theory [], the most common way to define the foundation of mathematics. See article for some axiomatic set theory fun!
              • by gkhan1 ( 886823 )
                That should be Zermelo-Fraenkel set theory. Apologies to Fraenkel.
                • Well, actually, ZFC is Zermelo-Fraenkel with the Axiom of Choice. Although AC is almost certainly "true", there's a lot of exceedingly interesting mathematics you can do without it, or even with various strong forms of its negation.
                  • by gkhan1 ( 886823 )

                    Yes, the axiom of choice would be the Z. Who was it that said that great quote, "The axiom of choice is obviously true, the well-ordering principle is obviously false, and who the hell knows about Zorn's lemma!"?

                    (the joke being, all three of those things are mathematically equivalent, if you accept the axiom of choice, you accept the other two).

          • by gkhan1 ( 886823 )

            You are fundamentally misunderstanding the incompleteness theorem. The incompletness theorem states that in a powerful axiomatic system, there are statements that are true, yet they cannot be proven using the system. That is, it is possible for there to be a polynomial algorithm for an NP-complete problem, but we cannot prove that it is polynomial-time. Conversely, it could be true that there is infact no such algorithm (ie NP != P), but we cannot prove this fact, even though it is still true. Pick up G

      • It can be independent of the current accepted axioms.
        • That's true. However, P = NP is interesting because it has practical uses. So, while there is obviously a mathematical meaning to it, the part that I am personally interested in is "is it doable on modern computer hardware".

          So even if it does turn out to be independent of current accepted axioms, which I will admit I'm skeptical about, I feel that knowing that allows us to immediately add an axiom to make that view of math approach "physical computer hardware". Or, alternatively, to define a subset of all N
          • by AxelBoldt ( 1490 )
            "is it doable on modern computer hardware"
            The question P=NP is thoroughly uninteresting when restricted to existing computers. Every existing computer has a finite and bounded amount of memory and storage, and hence a finite set of internal states, and is therefore a finite state machine. Everything a finite state machine does can be done in linear time.
            • by Retric ( 704075 )
              The are several things that a finite state machine can do faster than linear time and several that will take longer than linear time.

              AKA find index of Y in a sorted set X vs sort set X.

              • by AxelBoldt ( 1490 )
                Yes, some things a finite-state machine does don't require linear time (e.g. checking whether a given input string is empty or not); nothing a finite-machine does ever requires more than linear time. Sorting cannot be done with a finite-state machine.
                • by Retric ( 704075 )
                  You don't understand what a finite state machine is. Read []

                  Then look at this a finite state machine which takes a binary number and counts down.

                  11111 =
                  11110 =
                  11101 =
                  11100 =
                  11011 = ...

                  Starting with a set of N binary digits counting down with this finite state machine takes at worst (N^2)-1 steps.
                  • by AxelBoldt ( 1490 )
                    I think you mean (2^N)-1 steps. And no, there is no such finite-state machine. As Wikipedia correctly explains, a finite-state machine is fed an input string letter by letter from start to finish, and in each step, based on currently read letter and current internal state, the machine changes its internal state (and, optionally, outputs a result letter). Once the string is read in full, the computation is over. Thus the computation always takes linear time.
            • by azipsun ( 180681 )
              In general, computers have some form of external removable storage be it floppy, tape drive, or USB key. So, with just a little human intervention, you can have unbounded storage on an existing computer. I'll admit that it isn't particularly practical and you still have the problem that there are only a finite number of floppies in the universe, but the notion that the P=NP question is uninteresting to existing computers is also ridiculous.
              • by AxelBoldt ( 1490 )
                there are only a finite number of floppies in the universe
                That's precisely the point. If you are willing to imagine that you have an unlimited supply of floppy disks, then P=NP becomes an interesting mathematical question. If you stick to the real world with a finite supply of floppy disks, every algorithm is linear and P=NP is uninteresting.
      • There have been some problems (e.g. the weak pigeonhole principle) where it's been shown that any proof must be intractably large. There is some evidence / ublicationszSzPchina.pdf/raz02np.pdfthat [] this is true of P!=NP.

        If you think about this, there's a certain amount of poetic justice. NP-hard problems are solvable in principle, just not in practice. And the conjecture that P!=NP may be true in principle

      • You could certainly prove that the equation is wrong or incomplete.
    • by jonadab ( 583620 )
      Indeed. When it became evident that the Poincaire conjecture is not going to be considered an open problem much longer, my thoughts too were of the Reimann Hypothesis. Throughout most of the twentieth century those were widely considered to be the two biggest open problems in pure math, and it'd be really cool to see them both solved in our generation.

      This Navier-Stokes thing seems to be more of an applied-math problem, and although I'm sure it's important, it's just not as exciting to me as the more abst
      • by rogerdr ( 745180 )
        I'd say that this can be considered a 4D generalization of complex analysis, if true. In that case, it could go far beyond applied math into Relativity.
      • by be-fan ( 61476 )
        See, from my POV (I'm an engineer), an analytic solution to Navier-Stokes would be far more important. It would mea a huge advance for our understanding of aerodynamics (among other fluid-flow problems).
        • by jd ( 1658 )
          Navier-Stokes turns chaotic is you sneeze the wrong way, which makes me think that no analytic solutions exist for the problems likely to be of interest. There is a flip-side to that. If an analytic, computable solution does indeed exist to the completely generalized Navier-Stokes, then it goes far beyond any "applied" solution (such as the solution of compressible or incompressible flows, say in aerodynamics, climate modelling or even canal maintenance!) - this would smash into the heart of chaos theory an
        • by jonadab ( 583620 )
          > See, from my POV (I'm an engineer), an analytic solution to Navier-Stokes
          > would be far more important.

          I don't deny that applied math can be _important_. I only said that I find pure math more _interesting_. Plenty of things in this world are more interesting than important, or vice versa.
      • Re: (Score:3, Informative)

        by AxelBoldt ( 1490 )

        This Navier-Stokes thing seems to be more of an applied-math problem

        Not really. Actually solving Navier-Stokes for concretely given boundary conditions is very much an applied math problem, maybe the most important one of them all, and it is done with computers and algorithms from numerical analysis. But the paper we're discussing here is pure math: she proves that for a certain class of boundary conditions a solution must exist, without saying what it looks like or how to get it. It's of pure intellectua

    • by noigmn ( 929935 )
      I don't think the Riemann Hypothesis will have that much value. It's just a big challenge. We know it holds to at least squillions, and could guess that it is most likely right without proving it.

      P vs. NP on the other hand would be more than quite significant. It would turn the world upside down, inside out, and potentially make our mathematical abilities unstoppable. The day someone does an RSA challenge number by hand or in their head will be a grand day indeed.
  • Quite impressive (Score:5, Informative)

    by adityamalik ( 997063 ) on Saturday October 07, 2006 @04:18AM (#16346037)
    As a mechanical engineer, I have some idea of what this means.. Fluid dynamics is a fairly pervasive subject which goes into the design of airplanes, irrigation canals, industrial machinery, turbines and a lot of other places. The solution of the navier stokes' equation in three dimensions is quite fabulous, since without such a mathematical tool it's not possible to estimate how a fluid will flow in three dimensions.. Till now, we typically use either special conditions (ex. along a turbine blade, constant pressure) or fractional element methods (think of fluid as lots of tiny balls) or physical modelling for such problems. To put some perspective, it's about as cool as being able to determine the movement of n planets simultaneously attracting each other gravitationally.. quite tough!
    • Re:Quite impressive (Score:5, Informative)

      by S3D ( 745318 ) on Saturday October 07, 2006 @04:51AM (#16346187)
      That is not "the solution" of the Navier-Stocks system - they could be solved only numerically (fractional element methods or other discretization), but this is the next best thing - proof of the existance of such solution. From the practical point of view that mean, if you have correct physical starting conditions and working numerical method you will get correct result after calculation. Until now, you couldn't have been sure if you will get physyically reasonable result of numerical calculations, even if starting conditions would be correct.
      • Re:Quite impressive (Score:5, Informative)

        by vogon jeltz ( 257131 ) on Saturday October 07, 2006 @06:40AM (#16346563)
        it's about the existence of a solution for certain boundary / initial conditions of the NSEs. This is still a very big deal because you can now expect correct results when doing numerical calculations. By the way you probably meant FEM (Finite Element Method), not "fractional element methods". FEM is rarely, if not at all used for solving the NSEs, you'd rather use Finite Volume Methods (applicable for structured and unstructured grids, as are FEM).
        • Why do you think that FEM is rarely used to solve the Navier-Stokes equations? A quick Google search will indicate otherwise.

          The choice of method for solving the equations does seem to vary quite a bit between disciplines. Engineers tend to love FEM, while, say, atmospheric modelers seem to prefer finite-volume or finite-difference approaches.
      • Re: (Score:1, Offtopic)

        IANA Expert in Fluid Dynamics; however, though an exact solution exists, this doesn't mean that it'll be easy to find even if we have a method or formula to solve it exactly.

        Here's an example. Two board, one 3m and one 2m are laying crisscross in an alley, with one end in each corner of the alley, and laying the other end on the opposite wall.

        Their intersection is exactly 1m from the ground, how wide is the alley?

        This problem is very easy to find a numeric solution, but suprisingly difficult to fi
        • Your analogy is completely misleading.

          You take a quartic equation [] and choose to call "exact" what is called "a solution by radicals".

          Yes, a solution by radicals can be hard to find even when it turns out to exist. (Indeed quartics weren't solved by radicals until Ferrari in 1540.)

          But the question whether a solution by radicals exists has nothing to do with whether a solution (period) exists. Indeed polynomial of higher degree have the latter (Gauss' fundamental theorem of algebra) but not always the f

    • by pipingguy ( 566974 ) * on Saturday October 07, 2006 @06:13AM (#16346471)
      I agree. Fluid dynamics is very fascinating. Since I'm not so smart I've devoted my limited abilities to trying to understand the things we put conventional fluids into so that we can transmit them.
      • Since I'm not so smart I've devoted my limited abilities to trying to understand the things we put conventional fluids into so that we can transmit them.

        You mean the series of tubes that make up the Internet?

    • Re: (Score:3, Funny)

      by legrimpeur ( 594896 )
      Actually the whole thing IS NOT about FINDING SOLUTION of the Navier-Stokes equations,
      but rather the PROOF of THE EXISTENCE OF A FORMAL SOLUTION. You still have to find it,
        either analytically or (most probably) numerically.

      Bottom line: about this a mathematician gets horny, an engineer says SO WHAT!!!

      • It's kinda like trying to find an exact analytical solution for something as innocuous as the current distribution on a current carrying conductor for more accurate impedance calculations. Major PITA.
      • by pyite ( 140350 )
        Actually, engineers care about this a lot. Inexperienced people take a formula and assume it works under all conditions. However, when you take a [good] numerical analysis course, you'll do more than just learn how to use a formula. You'll spend time doing a lot of real analysis that you don't necessarily enjoy doing (I didn't) but the point is clear. A lot of times we really, really, care about existence of solutions. At times, we even care about the uniqueness of such solutions. Or, how about convergence
  • Whuh? (Score:5, Funny)

    by LiquidEdge ( 774076 ) on Saturday October 07, 2006 @04:24AM (#16346063) Homepage
    Man, I haven't had a date in like 4 years, and even *I'm* not nerdy enough to know why this matters...
  • by Simulacrus ( 1003107 ) on Saturday October 07, 2006 @04:35AM (#16346111)
    While I know they perform many, many computer simulations, I think aerodynamics is still regarded as one of the "black arts" in the field. Wind tunnels are still used extensively (it's often about who can build the better wind tunnel, never mind car). Maybe complete solutions of fluid movement will mean some odd-looking cars in 2007!
    • by Overzeetop ( 214511 ) on Saturday October 07, 2006 @07:44AM (#16346839) Journal
      Not really. This proof of the existance of the solution won't substatially affect the real-world application of fluid dynamics (including aerodynamics) for quite a ling time (maybe within my lifetime, probably not). Numerical and real simulation will still guide the principal advances at the full assembly level. Nonetheless, this is a pretty cool event. I remember studying N-S in undergrad. Still makes the hair on the backof my neck stand up is apprehension. (tensor math and pdes both make me ill).
    • Re: (Score:2, Insightful)

      by quanminoan ( 812306 )
      Using the Finite Element Method (FEM) will give you very good results. I've worked with Comsol and Floworks simulations designing a variety of things - but mostly cooling loops. This is where the problem lies - these simulations are very computer intensive and even a simple simulation such as a cooling loop through copper (one bend) can take over a day to converge to a solution (and i would make all sorts of assumptions to cut the time down, like perfectly smooth walls). A desktop computer wouldn't even be
  • by fatphil ( 181876 ) on Saturday October 07, 2006 @04:40AM (#16346137) Homepage
    Well, at least contributors to arXiv between them seem to. (The 'GM' section in mathematics has been dubbed by some serious mathematicians "garbage machine", for example.)

    Wait for the peer review to begin. I've not seen anyone familiar with the field say anything about the paper yet, only then does it gain credibility.

  • the article could have very well been in french or latin
  • I thought it was "Millennium"? Certainly that's how the linked-to website spells it.
  • by john-da-luthrun ( 876866 ) on Saturday October 07, 2006 @06:14AM (#16346477) getting people to spell it "millennium". Cracking that one would be a million dollars of anybody's money...
  • by 140Mandak262Jamuna ( 970587 ) on Saturday October 07, 2006 @07:28AM (#16346747) Journal
    Abstract of this post

    It is a big deal for the mathematicians. That is all

    The N-S Eqn has been "solved" in 2D using Velocity Potential, Stream Function approach. But in 3D stream function does not exist and the method does not extend. But in practice the only problem that is really "solved" even in 2D was was this driven cavity problem, a box with a moving wall.

    Take the much more simple to solve for a hundred years, the Heat Equation. Analytical solutions exist for simple domains like a semi infinite plate or a box with Dirichlet boundaries. But in practice ANSYS sells numerical solutions to Heat Equations and the industry has been buying millions dollars worth every year. Similarly FLUENT (Recently acquired by ANSYS) does not have to worry its market has fallen out of the bottom. For real life geometries we will be using numerical solutions of NS Eqn for the foreseeable future.

    Further though I could not see any geometry restrictions in the paper, it appears as though they have just proved solutions exist, and not actually solved it. Depending on the assumptions made and terms neglected, engineers may be able to build better turbulence ing out of this.

    Caveat: Though I started out in CFD I have not read CFD papers for some 12 years. and frankly I dont understand much of the math in this paper.

    • Re: (Score:1, Insightful)

      by Anonymous Coward
      Caveat: Though I started out in CFD I have not read CFD papers for some 12 years. and frankly I dont understand much of the math in this paper.

      That's OK - this is slashdot.

      Most commenters won't have even read the article, let get as far as failing to understand it!

    • It is a big deal for the mathematicians. That is all

      I wouldn't go so far as to say it is only interesting for mathematicians. Fluid dynamics and Navier-Stokes especially, is what, for example, many 3D engines use to simulate water by now. Granted, they use simplified equations, usually only taking the surface into consideration, but any breakthrough in the theory their models are based on might have implications for those models as well. I'd say let's wait until a) those new findings have been properly p
  • An important step (Score:5, Informative)

    by Orp ( 6583 ) on Saturday October 07, 2006 @07:57AM (#16346919) Homepage
    As a previous commenter stated, this is a mathematical proof that such a solution exists. You cannot explicitly solve the Navier Stokes equations as written. If you could, my job would be much easier (I model thunderstorms at very high resolution on massively parallel supercomputers). The Navier Stokes equations, along with some other conservation laws, and some physical parameterizations, can be "closed" such that you can approximate a solution using numerical tehcniques, given an initial state and boundary conditions. It is not easy. From a practical standpoint, dealing with massively parallel computers is not much fun. I've spent the past couple of months debugging my own stupid coding errors, competing with hundreds of other scientists running their models, and finding ways to manage the terabytes of data these models produce when they do run succesfully.

    Back to the paper... While I am not a mathematician, the paper appears kind of rough to me - lots of punctuation errors, commas in the wrong place, unclosed parehtneses... I suspect this paper has not been fully through the peer review process. I don't know how the mathematicians do it, but I would say this paper is a draft (not discrediting the work - I am not quallfied to judge it - but it looks rough).

    • Re: (Score:2, Informative)

      by Anonymous Coward
      I suspect this paper has not been fully through the peer review process. I don't know how the mathematicians do it, but I would say this paper is a draft (not discrediting the work - I am not quallfied to judge it - but it looks rough).

      Not that I think you are making an attack on mathematicians here, but I just want to comment on this for anyone that might construe it as such.

      Mathematicians do subject papers to full peer review before being published in any reputable journal, but the arXiv is not a journal
      • by jschrod ( 172610 )
        [On arXiv,] there are dozens of papers there that claim to have solved the Goldbach conjecture, or the Riemann hypothesis, or proven that the real numbers are countable, etc.
        The difference is that the authors of these papers have no track record of getting articles accepted in the best math journals, also have no track record of previous ground breaking new work in math, and have not caused a stir in the community that is as positive as it is this time. Penny Smith has.
    • by Ibag ( 101144 )
      The arXiv (pronounced "archive") is a preprint server where people post their papers. Sometimes the papers are awaiting publication, sometimes they aren't going to be published, and sometimes they are just rough things like lecture notes that people just figure others might appreciate. As such, it is fairly unsurprising that the paper would be a rough draft. That doesn't mean that the ideas aren't all there, but it most likely hasn't been peer reviewed in any broad sense yet. Putting it on the arXiv is
    • by egork ( 449605 )
      I wonder, if what they wanted to say was "this is more then proved"?
      Proof: This is less than proved in the brilliant paper of [H]. QED.
    • my job would be much easier (I model thunderstorms at very high resolution on massively parallel supercomputers)
      Oh, you're a weatherman?
  • it's about time someone named an institute after Clay Aiken []
  • A generic solution of Navier-Stokes under any kind of realistic conditions is huge. I'm sure it will still be necessary to discretize for most aircraft or boats. I'm also certain the solution is "strange" (as in "highly sensitive to boundary conditions") in many cases. Still, this is a major breakthrough if it verifies.
  • I'm holding out for a solution to the Navel-Strokes system.
  • not that WOW, but the surprise! If there do exist solutions to the navier-stokes equations be prepared for something huge. These equations play a dominating role in anything where something is moving in the presence of many other moving particles (think water, gas, landslides etc) The significance of this finding cannot be emphasized enough, if it proves to be true. Current solutions revolve around using computers to actually simulate the ENTIRE section of the system you want, then you test for condition
    • Even if the result ends up holding, it's not time to close the book on simulation methods yet. The result shows (or claims to show) existence and uniqueness. It does this through an indirect method that is not constructive. (i.e. It doesn't tell you what the solution is, only that it's out there somewhere.)

      If this holds up, the methods used are doubtless going to lead to better approximations and possibly - after a lot more research - to constructive methods. It's going to be exciting to see what ha

  • Hmmm... the arxiv, of course, has a bit of a 'reputation'. They'll take anything, and more power to them for being willing to do so. However, it does tend to mean that if one's a non-specialist, the cranks can look awfully convincing. Without, obviously, wishing to ascribe that appellation to the good Associate Professor, I would note that this paper carries some of the hallmarks: an extremely dodgy abstract, poor punctuation (as described above in comments), ropey spelling, dubious use of English (whass
    • Re: (Score:2, Informative)

      Check the last link in the summary. The author is a highly-respected mathematician in the field and this follows previous work that has been peer-reviewed. That doesn't mean it is *right*, but that does make it newsworthy.
      • Hokay, missed that. All the same, I'd wait for final verification before breaking out any bubbly (and then proceedng to analyse its egress from bottle to glass).
  • Can I just say "As a mathematician/engineer/chef," to get modded up now?
  • Brooks Moses has some additional explanation which I found helpful: f-on-the-navier-stokes-equations/ []
  • I read her very entertaining posts [] for many years, until she suddenly quit killing time on Usenet.
  • Withdrawn (Score:4, Informative)

    by mathcam ( 937122 ) on Sunday October 08, 2006 @12:30PM (#16355523)
    Well, I guess peer review has already taken its toll. The paper has been withdrawn from the arXiv due to "serious flaws."
  • I was trying to find a copy of the paper on the preprint server and it now says that it has been withdrawn for serious errors. It's too bad.
  • As a Ph.D. student in partial differential equations, I was very excited to hear about the possibility of NS being cracked finally. I was even more shocked to see that the techniques used in this paper to prove the existence were 'oldschool'. Quite literally, the core of the technique (perron's method) has been around since the early-mid 1900s. The regularity extension pulls on difference quotient methods, another classical technique from decades past. It is widely thought that to prove existence of NS with

One can't proceed from the informal to the formal by formal means.