It is always dangerous to venture an opinion as to why a problem is hard (cf. Clarke’s first law), but I’m going to stick my neck out on this one, because (a) it seems that there has been a lot of effort expended on this problem recently, sometimes perhaps without full awareness of the main difficulties, and (b) I would love to be proved wrong on this opinion :-) .

The global regularity problem for Navier-Stokes is of course a Clay Millennium Prize problem and it would be redundant to describe it again here. I will note, however, that it asks for existence of global smooth solutions to a Cauchy problem for a nonlinear PDE. There are countless other global regularity results of this type for many (but certainly not all) other nonlinear PDE; for instance, global regularity is known for Navier-Stokes in two spatial dimensions rather than three (this result essentially dates all the way back to Leray’s thesis in 1933!). Why is the three-dimensional Navier-Stokes global regularity problem considered so hard, when global regularity for so many other equations is easy, or at least achievable?

(For this post, I am only considering the global regularity problem for Navier-Stokes, from a purely mathematical viewpoint, and in the precise formulation given by the Clay Institute; I will not discuss at all the question as to what implications a rigorous solution (either positive or negative) to this problem would have for physics, computational fluid dynamics, or other disciplines, as these are beyond my area of expertise. But if anyone qualified in these fields wants to make a comment along these lines, by all means do so.)

The standard response to this question is *turbulence* – the behaviour of three-dimensional Navier-Stokes equations at fine scales is much more nonlinear (and hence unstable) than at coarse scales. I would phrase the obstruction slightly differently, as *supercriticality*. Or more precisely, all of the globally controlled quantities for Navier-Stokes evolution which we are aware of (and we are not aware of very many) are either *supercritical* with respect to scaling, which means that they are much weaker at controlling fine-scale behaviour than controlling coarse-scale behaviour, or they are *non-coercive*, which means that they do not really control the solution at all, either at coarse scales or at fine. (I’ll define these terms more precisely later.) At present, all known methods for obtaining global smooth solutions to a (deterministic) nonlinear PDE Cauchy problem require either

- Exact and explicit solutions (or at least an exact, explicit transformation to a significantly simpler PDE or ODE);
- Perturbative hypotheses (e.g. small data, data close to a special solution, or more generally a hypothesis which involves an somewhere); or
- One or more globally controlled quantities (such as the total energy) which are both coercive and either critical or subcritical.

(Note that the presence of (1), (2), or (3) are currently *necessary *conditions for a global regularity result, but far from *sufficient*; otherwise, papers on the global regularity problem for various nonlinear PDEs would be substantially shorter :-) . In particular, there have been many good, deep, and highly non-trivial papers recently on global regularity for Navier-Stokes, but they all assume either (1), (2) or (3) via additional hypotheses on the data or solution. For instance, in recent years we have seen good results on global regularity assuming (2), as well as good results on global regularity assuming (3); a complete bibilography of recent results is unfortunately too lengthy to be given here.)

The Navier-Stokes global regularity problem for arbitrary large smooth data lacks all of these three ingredients. Reinstating (2) is impossible without changing the statement of the problem, or adding some additional hypotheses; also, in perturbative situations the Navier-Stokes equation evolves almost linearly, while in the non-perturbative setting it behaves very nonlinearly, so there is basically no chance of a reduction of the non-perturbative case to the perturbative one unless one comes up with a highly nonlinear transform to achieve this (e.g. a naive scaling argument cannot possibly work). Thus, one is left with only three possible strategies if one wants to solve the full problem:

- Solve the Navier-Stokes equation exactly and explicitly (or at least transform this equation exactly and explicitly to a simpler equation);
- Discover a new globally controlled quantity which is both coercive and either critical or subcritical; or
- Discover a new method which yields global smooth solutions even in the absence of the ingredients (1), (2), and (3) above.

For the rest of this post I refer to these strategies as “Strategy 1”, “Strategy 2”, and “Strategy 3”.

Much effort has been expended here, especially on Strategy 3, but the supercriticality of the equation presents a truly significant obstacle which already defeats all known methods. Strategy 1 is probably hopeless; the last century of experience has shown that (with the very notable exception of completely integrable systems, of which the Navier-Stokes equations is *not* an example) most nonlinear PDE, even those arising from physics, do not enjoy explicit formulae for solutions from *arbitrary* data (although it may well be the case that there are interesting exact solutions from special (e.g. symmetric) data). Strategy 2 may have a little more hope; after all, the Poincaré conjecture became solvable (though still very far from trivial) after Perelman introduced a new globally controlled quantity for Ricci flow (the *Perelman entropy*) which turned out to be both coercive and critical. (See also my exposition of this topic.) But we are still not very good at discovering new globally controlled quantities; to quote Klainerman, “the discovery of any new bound, stronger than that provided by the energy, for general solutions of *any* of our basic physical equations would have the significance of a major event” (emphasis mine).

I will return to Strategy 2 later, but let us now discuss Strategy 3. The first basic observation is that the Navier-Stokes equation, like many other of our basic model equations, obeys a *scale invariance*: specifically, given any scaling parameter , and any smooth velocity field solving the Navier-Stokes equations for some time T, one can form a new velocity field to the Navier-Stokes equation up to time , by the formula

(Strictly speaking, this scaling invariance is only present as stated in the absence of an external force, and with the non-periodic domain rather than the periodic domain . One can adapt the arguments here to these other settings with some minor effort, the key point being that an approximate scale invariance can play the role of a perfect scale invariance in the considerations below. The pressure field gets rescaled too, to , but we will not need to study the pressure here. The viscosity remains unchanged.)

We shall think of the rescaling parameter as being large (e.g. ). One should then think of the transformation from u to as a kind of “magnifying glass”, taking fine-scale behaviour of u and matching it with an identical (but rescaled, and slowed down) coarse-scale behaviour of . The point of this magnifying glass is that it allows us to treat both fine-scale and coarse-scale behaviour on an equal footing, by identifying both types of behaviour with something that goes on at a fixed scale (e.g. the unit scale). Observe that the scaling suggests that fine-scale behaviour should play out on much smaller time scales than coarse-scale behaviour (T versus ). Thus, for instance, if a unit-scale solution does something funny at time 1, then the rescaled fine-scale solution will exhibit something similarly funny at spatial scales and at time . Blowup can occur when the solution shifts its energy into increasingly finer and finer scales, thus evolving more and more rapidly and eventually reaching a singularity in which the scale in both space and time on which the bulk of the evolution is occuring has shrunk to zero. In order to prevent blowup, therefore, we must arrest this motion of energy from coarse scales (or low frequencies) to fine scales (or high frequencies). (There are many ways in which to make these statements rigorous, for instance using Littlewood-Paley theory, which we will not discuss here, preferring instead to leave terms such as “coarse-scale” and “fine-scale” undefined.)

Now, let us take an arbitrary large-data smooth solution to Navier-Stokes, and let it evolve over a very long period of time [0,T), assuming that it stays smooth except possibly at time T. At very late times of the evolution, such as those near to the final time T, there is no reason to expect the solution to resemble the initial data any more (except in perturbative regimes, but these are not available in the arbitrary large-data case). Indeed, the only control we are likely to have on the late-time stages of the solution are those provided by globally controlled quantities of the evolution. Barring a breakthrough in Strategy 2, we only have two really useful globally controlled (i.e. bounded even for very large T) quantities:

- The
*maximum kinetic energy*; and - The
*cumulative energy dissipation*.

Indeed, the energy conservation law implies that these quantities are both bounded by the initial kinetic energy E, which could be large (we are assuming our data could be large) but is at least finite by hypothesis.

The above two quantities are *coercive*, in the sense that control of these quantities imply that the solution, even at very late times, stays in a bounded region of some function space. However, this is basically the only thing we know about the solution at late times (other than that it is smooth until time T, but this is a qualitative assumption and gives no bounds). So, unless there is a breakthrough in Strategy 2, we cannot rule out the worst-case scenario that the solution near time T is essentially an *arbitrary* smooth divergence-free vector field which is bounded both in kinetic energy and in cumulative energy dissipation by E. In particular, near time T the solution could be concentrating the bulk of its energy into fine-scale behaviour, say at some spatial scale . (Of course, cumulative energy dissipation is not a function of a single time, but is an integral over all time; let me suppress this fact for the sake of the current discussion.)

Now, let us take our magnifying glass and blow up this fine-scale behaviour by to create a coarse-scale solution to Navier-Stokes. Given that the fine-scale solution could (in the worst-case scenario) be as bad as an arbitrary smooth vector field with kinetic energy and cumulative energy dissipation at most E, the rescaled unit-scale solution can be as bad as an arbitrary smooth vector field with kinetic energy and cumulative energy dissipation at most , as a simple change-of-variables shows. Note that the control given by our two key quantities has worsened by a factor of ; because of this worsening, we say that these quantities are *supercritical* – they become increasingly useless for controlling the solution as one moves to finer and finer scales. This should be contrasted with *critical* quantities (such as the energy for *two-dimensional* Navier-Stokes), which are invariant under scaling and thus control all scales equally well (or equally poorly), and *subcritical* quantities, control of which becomes increasingly powerful at fine scales (and increasingly useless at very coarse scales).

Now, suppose we know of examples of unit-scale solutions whose kinetic energy and cumulative energy dissipation are as large as , but which can shift their energy to the next finer scale, e.g. a half-unit scale, in a bounded amount O(1) of time. Given the previous discussion, we cannot rule out the possibility that our rescaled solution behaves like this example. Undoing the scaling, this means that we cannot rule out the possibility that the original solution will shift its energy from spatial scale to spatial scale in time . If this bad scenario repeates over and over again, then convergence of geometric series shows that the solution may in fact blow up in finite time. Note that the bad scenarios do not have to happen immediately after each other (the *self-similar* blowup scenario); the solution could shift from scale to , wait for a little bit (in rescaled time) to “mix up” the system and return to an “arbitrary” (and thus potentially “worst-case”) state, and then shift to , and so forth. While the cumulative energy dissipation bound can provide a little bit of a bound on how long the system can “wait” in such a “holding pattern”, it is far too weak to prevent blowup in finite time. To put it another way, we have no rigorous, deterministic way of preventing Maxwell’s demon from plaguing the solution at increasingly frequent (in absolute time) intervals, invoking various rescalings of the above scenario to nudge the energy of the solution into increasingly finer scales, until blowup is attained.

Thus, in order for Strategy 3 to be successful, we basically need to rule out the scenario in which unit-scale solutions with *arbitrarily large *kinetic energy and cumulative energy dissipation shift their energy to the next highest scale. But every single analytic technique we are aware of (except for those involving *exact* solutions, i.e. Strategy 1) requires at least one bound on the size of solution in order to have any chance at all. Basically, one needs at least one bound in order to control all nonlinear errors – and any strategy we know of which does not proceed via exact solutions will have at least one nonlinear error that needs to be controlled. The only thing we have here is a bound on the *scale* of the solution, which is not a bound in the sense that a norm of the solution is bounded; and so we are stuck.

To summarise, any argument which claims to yield global regularity for Navier-Stokes via Strategy 3 must inevitably (via the scale invariance) provide a radically new method for providing non-trivial control of nonlinear unit-scale solutions of arbitrary large size for unit time, which looks impossible without new breakthroughs on Strategy 1 or Strategy 2. (There are a couple of loopholes that one might try to exploit: one can instead try to refine the control on the “waiting time” or “amount of mixing” between each shift to the next finer scale, or try to exploit the fact that each such shift requires a certain amount of energy dissipation, but one can use similar scaling arguments to the preceding to show that these types of loopholes cannot be exploited without a new bound along the lines of Strategy 2, or some sort of argument which works for arbitrarily large data at unit scales.)

To rephrase in an even more jargon-heavy manner: the “energy surface” on which the dynamics is known to live in, can be quotiented by the scale invariance. After this quotienting, the solution can stray arbitrarily far from the origin even at unit scales, and so we lose all control of the solution unless we have exact control (Strategy 1) or can significantly shrink the energy surface (Strategy 2).

The above was a general critique of Strategy 3. Now I’ll turn to some known specific attempts to implement Strategy 3, and discuss where the difficulty lies with these:

*Using weaker or approximate notions of solution*(e.g. viscosity solutions, penalised solutions, super- or sub- solutions, etc.). This type of approach dates all the way back to Leray. It has long been known that by weakening the nonlinear portion of Navier-Stokes (e.g. taming the nonlinearity), or strengthening the linear portion (e.g. introducing hyperdissipation), or by performing a discretisation or regularisation of spatial scales, or by relaxing the notion of a “solution”, one can get global solutions to approximate Navier-Stokes equations. The hope is then to take limits and recover a smooth solution, as opposed to a mere global*weak*solution, which was already constructed by Leray for Navier-Stokes all the way back in 1933. But in order to ensure the limit is smooth, we need convergence in a strong topology. In fact, the same type of scaling arguments used before basically require that we obtain convergence in either a critical or subcritical topology. Absent a breakthrough in Strategy 2, the only type of convergences we have are in very rough – in particular, in supercritical – topologies. Attempting to upgrade such convergence to critical or subcritical topologies is the qualitative analogue of the quantitative problems discussed earlier, and ultimately faces the same problem (albeit in very different language) of trying to control unit-scale solutions of arbitrarily large size. Working in a purely qualitative setting (using limits, etc.) instead of a quantitative one (using estimates, etc.) can disguise these problems (and, unfortunately, can lead to errors if limits are manipulated carelessly), but the qualitative formalism does not magically make these problems disappear. Note that weak solutions are already known to be badly behaved for the closely related Euler equation. More generally, by recasting the problem in a sufficiently abstract formalism (e.g. formal limits of near-solutions), there are a number of ways to create an abstract object which could be considered as a kind of generalised solution, but the moment one tries to establish actual control on the regularity of this generalised solution one will encounter all the supercriticality difficulties mentioned earlier.*Iterative methods*(e.g. contraction mapping principle, Nash-Moser iteration, power series, etc.)*in a function space*. These methods are perturbative, and require*something*to be small: either the data has to be small, the nonlinearity has to be small, or the time of existence desired has to be small. These methods are excellent for constructing*local*solutions for large data, or global solutions for*small*data, but cannot handle global solutions for large data (running into the same problems as any other Strategy 3 approach). These approaches are also typically rather insensitive to the specific structure of the equation, which is already a major warning sign since one can easily construct (rather artificial) systems similar to Navier-Stokes for which blowup is known to occur. The optimal perturbative result is probably very close to that established by Koch-Tataru, for reasons discussed in that paper.*Exploiting blowup criteria*. Perturbative theory can yield some highly non-trivial blowup criteria – that certain norms of the solution must diverge if the solution is to blow up. For instance, a celebrated result of Beale-Kato-Majda shows that the maximal vorticity must have a divergent time integral at the blowup point. However, all such blowup criteria are subcritical or critical in nature, and thus, barring a breakthrough in Strategy 2, the known globally controlled quantities cannot be used to reach a contradiction. Scaling arguments similar to those given above show that perturbative methods cannot achieve a supercritical blowup criterion.*Asymptotic analysis of the blowup point(s)*. Another proposal is to rescale the solution near a blowup point and take some sort of limit, and then continue the analysis until a contradiction ensues. This type of approach is useful in many other contexts (for instance, in understanding Ricci flow). However, in order to actually extract a useful limit (in particular, one which still solves Navier-Stokes in a strong sense, and does collapse to the trivial solution), one needs to uniformly control all rescalings of the solution – or in other words, one needs a breakthrough in Strategy 2. Another major difficulty with this approach is that blowup can occur not just at one point, but can conceivably blow up on a one-dimensional set; this is another manifestation of supercriticality.*Analysis of a minimal blowup solution*. This is a strategy, initiated by Bourgain, which has recently been very successful in establishing large data global regularity for a variety of equations with a critical conserved quantity, namely to assume for contradiction that a blowup solution exists, and then extract a*minimal*blowup solution which minimises the conserved quantity. This strategy (which basically pushes the perturbative theory to its natural limit) seems set to become the standard method for dealing with large data critical equations. It has the appealing feature that there is enough compactness (or almost periodicity) in the minimal blowup solution (once one quotients out by the scaling symmetry) that one can begin to use subcritical and supercritical conservation laws and monotonicity formulae as well (see my survey on this topic). Unfortunately, as the strategy is currently understood, it does not seem to be directly applicable to a supercritical situation (unless one simply assumes that some critical norm is globally bounded) because it is impossible, in view of the scale invariance, to minimise a non-scale-invariant quantity.*Abstract approaches*(avoiding the use of properties specific to the Navier-Stokes equation). At its best, abstraction can efficiently organise and capture the key difficulties of a problem, placing the problem in a framework which allows for a direct and natural resolution of these difficulties without being distracted by irrelevant concrete details. (Kato’s semigroup method is a good example of this in nonlinear PDE; regrettably for this discussion, it is limited to subcritical situations.) At its worst, abstraction conceals the difficulty within some subtle notation or concept (e.g. in various types of convergence to a limit), thus incurring the risk that the difficulty is “magically” avoided by an inconspicuous error in the abstract manipulations. An abstract approach which manages to breezily ignore the supercritical nature of the problem thus looks very suspicious. More substantively, there are many equations which enjoy a coercive conservation law yet still can exhibit finite time blowup (e.g. the mass-critical focusing NLS equation); an abstract approach thus would have to exploit some subtle feature of Navier-Stokes which is not present in all the examples in which blowup is known to be possible. Such a feature is unlikely to be discovered abstractly before it is first discovered concretely; the field of PDE has proven to be the type of mathematics where progress generally starts in the concrete and then flows to the abstract, rather than vice versa.

If we abandon Strategy 1 and Strategy 3, we are thus left with Strategy 2 – discovering new bounds, stronger than those provided by the (supercritical) energy. This is not *a priori* impossible, but there is a huge gap between simply wishing for a new bound and actually discovering and then rigorously establishing one. Simply sticking in the existing energy bounds into the Navier-Stokes equation and seeing what comes out will provide a few more bounds, but they will all be supercritical, as a scaling argument quickly reveals. The only other way we know of to create global non-perturbative deterministic bounds is to discover a new conserved or monotone quantity. In the past, when such quantities have been discovered, they have always been connected either to geometry (symplectic, Riemmanian, complex, etc.), to physics, or to some consistently favourable (defocusing) sign in the nonlinearity (or in various “curvatures” in the system). There appears to be very little usable geometry in the equation; on the one hand, the Euclidean structure enters the equation via the diffusive term and by the divergence-free nature of the vector field, but the nonlinearity is instead describing transport by the velocity vector field, which is basically just an arbitrary volume-preserving infinitesimal diffeomorphism (and in particular does not respect the Euclidean structure at all). One can try to quotient out by this diffeomorphism (i.e. work in material coordinates) but there are very few geometric invariants left to play with when one does so. (In the case of the Euler equations, the vorticity vector field is preserved modulo this diffeomorphism, as observed for instance by Li, but this invariant is very far from coercive, being almost purely topological in nature.) The Navier-Stokes equation, being a system rather than a scalar equation, also appears to have almost no favourable sign properties, in particular ruling out the type of bounds which the maximum principle or similar comparison principles can give. This leaves physics, but apart from the energy, it is not clear if there are any physical quantities of fluids which are *deterministically* monotone. (Things look better on the stochastic level, in which the laws of thermodynamics might play a role, but the Navier-Stokes problem, as defined by the Clay institute, is deterministic, and so we have Maxwell’s demon to contend with.) It would of course be fantastic to obtain a fourth source of non-perturbative controlled quantities, not arising from geometry, physics, or favourable signs, but this looks somewhat of a long shot at present. Indeed given the turbulent, unstable, and chaotic nature of Navier-Stokes, it is quite conceivable that in fact no reasonable globally controlled quantities exist beyond that which arise from the energy.

Of course, given how hard it is to show global regularity, one might try instead to establish finite time blowup instead (this also is acceptable for the Millennium prize). Unfortunately, even though the Navier-Stokes equation is known to be very unstable, it is not clear at all how to pass from this to a rigorous demonstration of a blowup solution. All the rigorous finite time blowup results (as opposed to mere instability results) that I am aware of rely on one or more of the following ingredients:

- Exact blowup solutions (or at least an exact transformation to a significantly simpler PDE or ODE, for which blowup can be established);
- An ansatz for a blowup solution (or approximate solution), combined with some nonlinear stability theory for that ansatz;
- A comparison principle argument, dominating the solution by another object which blows up in finite time, taking the solution with it; or
- An indirect argument, constructing a functional of the solution which must attain an impossible value in finite time (e.g. a quantity which is manifestly non-negative for smooth solutions, but must become negative in finite time).

It may well be that there is some exotic symmetry reduction which gives (1), but no-one has located any good exactly solvable special case of Navier-Stokes (in fact, those which have been found, are known to have global smooth solutions). (2) is problematic for two reasons: firstly, we do not have a good ansatz for a blowup solution, but perhaps more importantly it seems hopeless to establish a stability theory for any such ansatz thus created, as this problem is essentially a more difficult version of the global regularity problem, and in particular subject to the main difficulty, namely controlling the highly nonlinear behaviour at fine scales. (One of the ironies in pursuing method (2) is that in order to establish rigorous *blowup* in some sense, one must first establish rigorous *stability* in some other (renormalised) sense.) Method (3) would require a comparison principle, which as noted before appears to be absent for the non-scalar Navier-Stokes equations. Method (4) suffers from the same problem, ultimately coming back to the “Strategy 2” problem that we have virtually no globally monotone quantities in this system to play with (other than energy monotonicity, which clearly looks insufficient by itself). Obtaining a new type of mechanism to force blowup other than (1)-(4) above would be quite revolutionary, not just for Navier-Stokes; but I am unaware of even any proposals in these directions, though perhaps topological methods might have some effectiveness.

So, after all this negativity, do I have any positive suggestions for how to solve this problem? My opinion is that Strategy 1 is impossible, and Strategy 2 would require either some exceptionally good intuition from physics, or else an incredible stroke of luck. Which leaves Strategy 3 (and indeed, I think one of the main reasons why the Navier-Stokes problem is interesting is that it *forces* us to create a Strategy 3 technique). Given how difficult this strategy seems to be, as discussed above, I only have some extremely tentative and speculative thoughts in this direction, all of which I would classify as “blue-sky” long shots:

*Work with ensembles of data, rather than a single initial datum*. All of our current theory for deterministic evolution equations deals only with a single solution from a single initial datum. It may be more effective to work with parameterised familes of data and solutions, or perhaps probability measures (e.g. Gibbs measures or other invariant measures). One obvious partial result to shoot for is to try to establish global regularity for*generic*large data rather than*all*large data; in other words, acknowledge that Maxwell’s demon might exist, but show that the probability of it actually intervening is very small. The problem is that we have virtually no tools for dealing with generic (average-case) data other than by treating all (worst-case) data; the enemy is that the Navier-Stokes flow itself might have some perverse entropy-reducing property which somehow makes the average case drift towards (or at least recur near) the worst case over long periods of time. This is incredibly unlikely to be the truth, but we have no tools to prevent it from happening at present.*Work with a much simpler (but still supercritical) toy model*. The Navier-Stokes model is parabolic, which is nice, but is complicated in many other ways, being relatively high-dimensional and also non-scalar in nature. It may make sense to work with other, simplified models which still contain the key difficulty that the only globally controlled quantities are supercritical. Examples include the Katz-Pavlovic dyadic model for the Euler equations (for which blowup can be demonstrated by a monotonicity argument; see this survey for more details), or the spherically symmetric defocusing supercritical nonlinear wave equation.*Develop non-perturbative tools to control deterministic non-integrable dynamical systems*. Throughout this post we have been discussing PDEs, but actually there are similar issues arising in the nominally simpler context of finite-dimensional dynamical systems (ODEs). Except in perturbative contexts (such as the neighbourhood of a fixed point or invariant torus), the long-time evolution of a dynamical system for deterministic data is still largely only controllable by the classical tools of exact solutions, conservation laws and monotonicity formulae; a discovery of a new and effective tool for this purpose would be a major breakthrough. One natural place to start is to better understand the long-time, non-perturbative dynamics of the classical three-body problem, for which there are still fundamental unsolved questions.*Establish really good bounds for critical or nearly-critical problems*. Recently, I showed that having a very good bound for a critical equation essentially implies that one also has a global regularity result for a slightly supercritical equation. The idea is to use a monotonicity formula which does weaken very slightly as one passes to finer and finer scales, but such that each such passage to a finer scale costs a significant amount of monotonicity; since there is only a bounded amount of monotonicity to go around, it turns out that the latter effect just barely manages to overcome the former in my equation to recover global regularity (though by doing so, the bounds worsen from polynomial in the critical case to double exponential in my logarithmically supercritical case). I severely doubt that my method can push to non-logarithmically supercritical equations, but it does illustrate that having very strong bounds at the critical level may lead to some modest progress on the problem.*Try a topological method*. This is a special case of (1). It may well be that a primarily topological argument may be used either to construct solutions, or to establish blowup; there are some precedents for this type of construction in elliptic theory. Such methods are very global by nature, and thus not restricted to perturbative or nearly-linear regimes. However, there is no obvious topology here (except possibly for that generated by the vortex filaments) and as far as I know, there is not even a “proof-of-concept” version of this idea for any evolution equation. So this is really more of a wish than any sort of concrete strategy.*Understand pseudorandomness*. This is an incredibly vague statement; but part of the difficulty with this problem, which also exists in one form or another in many other famous problems (e.g. Riemann hypothesis, , , twin prime and Goldbach conjectures, normality of digits of , Collatz conjecture, etc.) is that we expect any sufficiently complex (but deterministic) dynamical system to behave “chaotically” or “pseudorandomly”, but we still have very few tools for actually making this intuition precise, especially if one is considering deterministic initial data rather than generic data. Understanding pseudorandomness in other contexts, even dramatically different ones, may indirectly shed some insight on the turbulent behaviour of Navier-Stokes.

In conclusion, while it is good to occasionally have a crack at impossible problems, just to try one’s luck, I would personally spend much more of my time on other, more tractable PDE problems than the Clay prize problem, though one should certainly keep that problem in mind if, in the course on working on other problems, one indeed does stumble upon something that smells like a breakthrough in Strategy 1, 2, or 3 above. (In particular, there are many other serious and interesting questions in fluid equations that are not anywhere near as difficult as global regularity for Navier-Stokes, but still highly worthwhile to resolve.)

## 534 comments

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25 February, 2019 at 1:18 am

R. Kh.As this is noticed by M Struve: O. A LADYZHENSKAYA would not have agreed with this formulation of the problem -corresponding regularity question – for the Sixht Problem of the Millenium – for 0. A – the MAIN PROBLEM IS EXISTENCE AND UNIQUENESS – OBVIOUSLY

R. KH; Z

25 February, 2019 at 5:37 am

victorivriiI am really puzzled by Mr Zeytounian comments (as long as I can decrypt his writings).

1) Indeed, Ch. Fefferman made an omission in the official description https://www.claymath.org/sites/default/files/navierstokes.pdf , not mentioning that in the periodic framework not only the gradient of the pressure but the pressure itself must be periodic.

However, this hardly warrants “very badli” statement, Furthermore, this omission was corrected on the 6th page of the official description [well, it was a patch; the original description originally contained 5 pages, with the 5th page having only few lines and then the 6th page was sloppily added 61/2 years later: it follows from it appearance that it was simply inserted into pdf file and looking at the shape of symbols one can see that it’s author was a German mathematician)

2) This problem was not resolved as far as I know.

3) For Olga Ladyzhenskaya definitely the main question was the existence and uniqueness; however both of them are known, but in different classes. If the conjecture of the global regularity holds then the existence and uniqueness would be established in the class of smooth functions.

However, if the conjecture of the global regularity fails, the next problem indeed would be to find the class in which both existence and uniqueness hold.

25 February, 2019 at 6:36 am

victorivriiA bit lame analogy: for uniqueness holds in the class of continuous functions, and the global existence holds in the class of bounded functions. However in the former class there is no global existence and in the latter there is no unicity.

It was established 50+ y.a. (if I am not mistaken) that in the class of bounded functions there are both global existence and uniqueness, if to this equation dissipativity condition is added.

8 May, 2019 at 1:23 pm

Bob McCannPhysics perspective: Navier-Stokes is a model that applies to bulk fluids where the scale size is large enough that individual particle motion is not relevant. Nearing that scale it becomes necessary to switch models to something including distributions and collisions, e.g., the Vlasov or Boltzmann equations. As an exercise in pure math, the Millennium problem is an excellent challenge for extending the reach of our analysis tool set. Here is an idea I have used in Plasma Physics. If we can bound the problem between two simpler problems that are better behaved, it may be possible to constrain the more difficult behavior. For instance, if we use a Fourier expansion with index “n” and the behavior of the first few terms is well behaved but high order terms are too difficult, then expanding in (1/n) and moving to the limit of n->infinity sometimes results in a well behaved equation that bounds the small scale behavior. Using analytic matching, analytic continuation, or the Wiener-Hopf technique (depending on the character of the small and large scale equations) can then provide tighter bounds on the more difficult intermediate scale behavior.

16 May, 2019 at 5:09 am

ferhatmohamed Ferhatafter 10 years of effort i think this puzzle requires to develope a new variational methods

17 July, 2020 at 3:52 am

Colin McLartyTerrence: just a question about terminology. for use in a talk to historians and philosophers of math. I think that when you say “exact and explicit” you mean something like a closed form solution (where “closed form” is also not a precisely defined expression). Are you happy with using “exact solution” for a solution which is in no sense closed form?

17 July, 2020 at 5:19 am

AnonymousI’m not Terence but I’d expect it to be possible to make a formal definition in terms of computable reals. E.g. an infinite series solution that converges to within epsilon after terms where f is computable using an explicitly stated algorithm, seems explicit to me.

17 July, 2020 at 6:25 pm

Terence TaoYes, in this text I mean solutions that are both exact (not approximate) and explicit (describable in terms of operations simpler than “solve the Navier-Stokes equations”), with the point being that it should be an easier task to determine the regularity or singularity of such explicitly described solutions than arbitrary (exact) solutions to the Navier-Stokes equations. On the other hand, explicit approximate solutions are typically not, by themselves, enough to settle these sorts of questions: just because an explicit approximate solution exhibits an explicit singularity at a point, it is often not feasible to obtain a sufficiently strong perturbative theory to establish that there is also an exact solution close to the approximate solution that also exhibits a singularity. (Perturbation theory tends to only work well when the functions involved are relatively small, but singular solutions by their nature tend to be quite large in various norms. One can sometimes get around this by working in multiple norms simultaneously, but it can be quite delicate.)

20 August, 2020 at 1:32 am

ChristianDear Prof. Tao,

I wonder if it’s possible to explain your words “the behaviour of three-dimensional Navier-Stokes equations at fine scales is much more nonlinear (and hence unstable) than at coarse scales” with estimates using $u^(\lambda)$ and $u.$

Many thanks for your effort.

4 December, 2020 at 6:12 pm

DanielA proposed solution to the NS problem is at https://vixra.org/pdf/1911.0343v8.pdf

27 December, 2020 at 1:17 am

AnonymousI recently came across this article, describing a fluid mechanics experiment (i.e. with actual water tanks and stuff) that seems to have uncovered some previously unknown conservation laws in near-turbulence that might extend into the turbulent regime. It probably doesn’t reach the NS regularity question, but I wonder if it’s been studied mathematically and is considered mathematically significant:

https://www.quantamagazine.org/an-unexpected-twist-lights-up-the-secrets-of-turbulence-20200903/

That’s a popular-level article but it links to the research article that was in Science Magazine a while earlier.

17 February, 2021 at 2:09 am

RobertIf the Navier-Stokes problem cannot be solved by arguably the greatest mathematical mind on Earth, could that mean that the premise, i.e., the formulation of the Navier-Stokes equation is in itself incorrect?

Can anyone tell me whether proving that the premise is incorrect would be considered sufficient as a counter example?

In Professor Tao’s article he speaks of:

“Discover a new globally controlled quantity which is both coercive and either critical or subcritical”

If this was discovered, then this would render the premise incorrect since this new globally controlled quantity is not in the equation. Am I right or wrong here? I am just a layman in the field of mathematics and physics but I like to solve problems.

17 February, 2021 at 10:52 am

Anonymous1. It is a valid interesting problem regardless of whether Terry (or any other expert) can solve it.

2. A hypothetical new globally controlled quantity would already be “in the equation,” just not yet identified by mathematicians.

28 December, 2021 at 11:45 pm

Dwight WalshAs “just a layman”, you seem to have much insight into what’s happening in regard to this millennium problem. To answer your question, I believe the problem is correctly formulated in the Official Problem Description which came out in the year 2000. Traditionally, however, actual attempts at proving existence and smoothness of solutions to the NS equation have generally started with the work of the French mathematician Jean Leray in 1934 in which he showed that a certain class of “weak” solutions to the NS equation always exist. Unfortunately, it seems that this tradition has carried over to the point to where the mathematical establishment will not view any proposed solution to the NS millennium problem as credible if it is not based on Leray’s work.

Although Prof. Tao may have a great mathematical mind and genius level IQ, I believe he has followed with this tradition and made some bad mistakes in the process. The problem with his arguments is that he fails to constrain div u = 0. In each of his online forums and presentations on the NS equation, he makes an assertion at some point that we can neglect the scalar pressure since it only keeps div u = 0. Unfortunately, however, “neglecting” p and grad p is the same as setting them equal to zero, and from that point on, he no longer mentions p, grad p, or div u. Instead, he only makes “re-scaling” arguments with the nonlinear transport term to claim that energy could be redistributed into successively smaller scale sizes, thereby resulting in a blowup in energy density. In general, his analysis is vague, intuitive, and doesn’t prove anything.

By failing to constrain div u = 0, Dr. Tao’s arguments support only an under-determined version of the problem where there are more unknowns than equations. This, of course, gives him (and the entire mathematics community) great freedom to find “counter-examples” to existence and smoothness that are not actually solutions to the NS equation. In this case, it would be ABSOLUTELY IMPOSSIBLE for the mathematics community to recognize a valid proof of existence and smoothness of solutions to the NS equation.

Therefore, while I believe the actual formulation of the NS millennium problem is correct, there have been serious errors in pursuing solutions. And, unless the mathematics community approaches this problem with open minds, it won’t be solved for a long, long time.

29 December, 2021 at 11:31 am

Dwight WalshI forgot to tell you in my above posting that I started my own chain of comments which are a little way down the page. In particular, I posted my own solution to the NS millennium problem to the website at

http://www.navierstokessmoothsolutions.com

As you may imagine, however, peer reviewed journals don’t seem to want to touch it with ten foot pole, probably because my approach of leaving the NS equation intact breaks with tradition. Thus far, it has been rejected by about a dozen such journals. A few readers have suggested that I post the paper to a moderated archive, but to do that, I still need to be sponsored by a reputable mathematician, and after 30 years of no professional contacts, no such individual would even know me.

And so, the mathematics community moves forward with their attempted “solutions” to this millennium problem unaware that their most impressive discoveries (including the fact that they can “prove” existence and smoothness in two dimensions but not three) are merely artifacts of their failure to properly account for p and grad p.

25 April, 2021 at 7:51 am

PolihronovDear Prof. Tao,

Thank you for a very informative blog, I have read it many times!

I wanted to ask a question about the viscosity and NSE scale-invariance. In the blog you discuss one of the strategies and then mention that pressure gets rescaled too; then you have said that the viscosity remains unchanged (under rescaling).

Would you be able to share your thoughts on the meaning of this. In the Millennium problem, how is the viscosity set to behave under rescaling.

My attempt to understand it:

In the dimensionalized NSE, the viscosity is a fixed constant; however, since it has dimension/units; then, must be rescaled when the NSE is rescaled because the units are rescaled.

Coming to the nondimensionalized NSE: here the viscosity is replaced by the Reynolds number . Since in the Millennium problem, the viscosity is a fixed parameter, it would then follow that the Reynolds number is a fixed number as well.

About the scalability of :

It appears that one can look upon the nondimensionalized NSE in its own merit, and written in the nondimensionalized variables , it can be thought of having scale-invariance in these variables. Leray’s self-similar solutions seem to suggest this as well. They are written for the nondimensionalized NSE; their name, “self-similar” refers to rescaling. Their rescaling always yields the proper rescaling of the velocity and pressure . In other words, must be scalable in the nondimensionalized NSE. Is then the Reynolds number scalable just like the viscosity is scalable?

Would you be able to comment on this. Would there be a mathemathical reason to set or as non-scalable. I’m wondering, since there is no unified viscosity theory, and the molecular origins of viscosity are unclear.

26 April, 2021 at 9:23 am

Terence TaoMathematically, there is no distinction between a “rescaleable” or “non-rescalable” quantity (or a “dimensional” or “dimensionless” quantity), as long as we are describing quantities such as position coordinates , time , and viscosity in terms of real numbers. (If one wants to describe such quantities using –torsors instead, then there are distinctions of this type, but this is not usually how the mathematical equations are formulated.)

In particular, one is free to apply any (smooth, invertible) change of variables one wants to the fields to transform the Navier-Stokes equation to another equation. Sometimes when one does so, the transformed equation is the same as the original equation (with no change in the viscosity parameter); in other cases, it transforms to a Navier-Stokes equation with a different viscosity parameter; and in yet other cases it transforms to a completely different equation which is not of Navier-Stokes type. All three such types of transformations are mathematically useful, though in any specific context one such transformation may be more useful than another.

More specifically: for any , the change of variables defined by

transforms solutions of the Navier-Stokes equation at a given viscosity to solutions of the Navier-Stokes equation with the same viscosity . This gives a scale-invariance to the set of solutions to such an equation (with fixed viscosity ) which is very useful for understanding the structure of the space of such solutions, as is discussed in this post.

More generally, though, one can consider dimensional scalings for any defined by

which transform a solution to Navier-Stokes at a given viscosity to a solution to Navier-Stokes at viscosity . The previous scaling is the special case in which . This more general scaling is useful when one wants to analyse a specific time scale and a specific length scale , or if one wishes to investigate limits when the viscosity tends to zero or to infinity. It is not used in this post, but this does not mean that it is not useful in other contexts.

As an example of the third type of transformation, the transformation , where

is the vorticity field, transforms the usual formulation of Navier-Stokes into the vorticity-stream formulation (consisting of the vorticity equation and the Biot-Savart law), in which the pressure is absent. This is a useful formulation for many purposes (particularly in two dimensions, when vorticity becomes a scalar transported by the velocity field).

27 April, 2021 at 7:22 am

PolihronovDear Prof. Tao,

Thank you for your reply! Yes, then the solutions

$\latex u^{(\lambda)}(t,x):=\frac{1}{\lambda}u(\frac{t}{\lambda^2},\frac{x}{\lambda})$;

$\latex p^{(\lambda)}(t,x):=\frac{1}{\lambda^2}p(\frac{t}{\lambda^2},\frac{x}{\lambda})$

would fall into the scenario of Energy-supercritical NSE as you had said above.

These solutions are scalable; they are well-known as Leray’s self-similar solutions. One arrives at them when the viscosity $\latex \nu$ is not changed by rescaling, as you mentioned. Their form is

$\latex u(t,x):=\frac{1}{\sqrt{t}}u(\frac{x}{\sqrt{t}})$;

$\latex p(t,x):=\frac{1}{t}p(\frac{x}{\sqrt{t}})$

and are known not to be strong solutions (Cannone and Planchon /1996). It would not be feasible then to use Beale-Kato-Majda to study them, since they are not strong.

Also, the Millennium problem requires smoothness of solutions at $\latex t=0$. Leray’s self-similar solutions are not defined at $\latex t=0$ and are divergent at the origin (or, along the coordinate axes).

To sum up, if viscosity $\latex \nu$ is not changed by rescaling, we arrive at an energy-supercritical NSE with scalable solutions that are not strong solutions; and are divergent at the initial moment and along the coord. axes. Such approach to show NSE regularity would be bound not to succeed.

If the viscosity $\latex \nu$ is allowed to scale, all of the above issues are resolved. Namely, smoothness at $\latex t=0$, along the axes and criticality are no longer a problem. Beale-Kato-Majda show strong solutions at $\latex T=\infty$. Would this be an acceptable approach to the Millennium problem?

3 May, 2021 at 7:06 am

PolihronovDear Prof. Tao,

If I could add this detail –

The solutions ; and form a group of scaling transformations. It is a Lie group, and the solutions have important invariant properties. From these properties, one can derive their functional form

and

where and are arbitrary functions. Here and are derived under the assumption that the viscosity is unchanged under scaling. One can see that are not defined at and are not smooth functions.

It is true that supercriticality is an obstruction to proving NSE regularity; although supercriticality comes to view at a deeper level of analysis. It seems that one has more urgent issues coming from the NSE scaling transformation itself. What would then be the justification of analyzing NSE regularity through , ? How could they be useful for understanding the structure of the space of NSE solutions.

4 May, 2021 at 2:16 pm

Terence TaoThe rescaled solutions are not required to equal the original solutions ; typically, they are a different solution to the Navier-Stokes equations, with a different set of initial data (a rescaling of the original set). So they do not need to take the scale-invariant form you describe.

5 May, 2021 at 8:51 am

PolihronovThe Lie scaling transformation preserves the NSE; If is to remain unchanged, the transformation is

where denotes the transformed entity .

Then

.

and transform as expected. , are rescaled versions of the original .

10 May, 2021 at 12:42 pm

PolihronovI would add this as well –

The questions arise due to the complexity of Lie’s invariant theory. When speaking of invariants, the assumption is

,

while it is not the case; the correct invariance is

.

Understanding differential invariants in detail is crucial in them being highly relevant to the NSE.

This is true specifically on the functional form of the various differential invariants.

Charles L. Bouton, a Harvard Math professor has written an excellent article on this subject:

Invariants of the General Linear Differential Equation and Their Relation to the Theory of Continuous Groups

American Journal of Mathematics, Vol. 21, No. 1 (Jan., 1899), pp. 25-84

http://www.jstor.org/stable/2369876

This article settles the matter; and of the importance of the viscosity in NSE regularity. Which importance goes even further in posing the question on how behaves at various scales in fluids.

8 July, 2021 at 5:01 am

J. YongThere is a paper by A. Tsionskiy and M. Tsionskiy with the title “Existence, uniquess and smoothness of a solution for 3D Navier-Stokes equations with any smooth initial velocity. A priori estimate of this solution” published in Adv. Theor. Math. Phys.19 (2015), 701-743. Is there any comment on that? Does it mean that the problem has been solved? Or the paper has some problems?

8 July, 2021 at 8:44 am

AnonymousIt’s clear at a glance that this paper has basically no chance of being correct. Surely no expert has wasted their time checking it.

30 November, 2021 at 1:27 am

Grigori RozenblumThis reference to Tsionskiy is fake. In the Journal, on these pages, different papers are published. https://www.intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0019/0003/index.php

6 September, 2021 at 4:46 am

Dwight WalshIn January 2020, I came up with a proof of existence, smoothness, and uniqueness of solutions to the incompressible, zero driving-force Navier-Stokes equation, consistent with statement (A) of the Official Problem Description for the Navier-Stokes Millennium Problem. The interesting part is that I was able to show this result using only vector analysis, Poisson’s integral, and some elementary and partial DEs. Thus far, however, my proof has been rejected by eight journals for reasons not related to mathematical correctness. In fact, one reviewer stated “the proof is so easy even an undergraduate reader could follow”. Now, my Ph.D. is in physics and not mathematics, but I have worked some with the Navier-Stokes equation in one of my research project while I was still employed. Also, it sometimes takes a newcomer to shed new light on a problem that has had the experts stumped for a long time.

In my paper, I was able to show that any solution of the NS equation that is initially smooth must remain smooth for as long as it exists. Therefore, if a blowup occurs in finite time, it must be a “smooth blowup” where the solution remains spatially smooth at all times, even when approaching blowup. Also, this blowup point must be a global maximum, and I was able to show that the fluid velocity at such a point cannot reach infinite velocities even in infinite time. Thus, this smooth blowup cannot happen and the solution remains smooth for all time.

If you would like to review my paper, you can download it at

http://www.navierstokessmoothsolutions.com

since you may not see this paper in a journal for good long time.

8 September, 2021 at 10:49 am

AnonymousThere are some problems with the assumptions you make about t_b, x_b, and K_b; however the biggest flaw with this paper is that you assume integral |grad p| dt is finite up to the blowup time. This is only necessarily true for every t *less* than the blowup time as it is a supercritical quantity (see Tao’s article above).

8 September, 2021 at 10:50 am

Anonymous*subcritical, sorry

11 September, 2021 at 2:34 am

Dwight WalshI do not assume that the time integral of |grad p| is bounded up to the blowup time. I PROVE IT. Please re-read the section titled “Existence of Pressure Gradient Integral over Time”, and let me know if you find any errors in the proof.

9 January, 2022 at 10:54 pm

Dwight WalshI should have recognized this months ago, but your very use of the term “blowup time” implies that you are assuming there is a blowup in finite time. Isn’t it rather difficult to prove a smooth solution when a blowup is one of the “givens”?

11 September, 2021 at 1:21 pm

Dwight WalshFrom the comments I have received from various sources, I sense there is one crucial point I need to emphasize. That is, this paper shows that the time integral of |grad p| is actually BOUNDED on the ENTIRE open interval (0,T). The first step in proving this is to show that the spatial integral of the function q over R^3 (or any subset thereof) is bounded above. This is done in the discussion preceding inequality (108). In this discussion, the only assumption made is that the u_i remain smooth according to inequality (79). No mention is made about how large the u_i values become. Therefore, inequality (108) remains valid even as our smooth solution approaches blowup. Then, since all spatial integrals of q are bounded (regardless of the size of the u_i), it follows that the time integral of |grad p| is also bounded over all points in the open interval (0,T). This, as we show later, becomes the final argument proving that a “smooth blowup” is not possible.

12 September, 2021 at 3:19 am

Dwight WalshWould any of you “thumbs down” folks care to leave a constructive comment on what you find erroneous about my paper, or do you not understand the problem that well yourselves?

12 September, 2021 at 3:44 am

victorivrii1. In your “proof” that |\nabla p| is bounded, after (117) you wrote “Since the function q(x, t) is bounded and continuous” . Why this function, defined by (112) is bounded and continuous?

2. There is a simple criteria to distinguish proofs which may be correct, from those which are wrong for sure: the latter “work” in any dimension. “Elementary methods” like yours, which do not use even imbedding theorems, definitely fail this criteria.

13 September, 2021 at 12:37 am

Dwight WalshI did not claim that |grad p| is bounded. I stated and proved that the time integral of |grad p| is bounded over the entire open time interval (0,T_b) for which the fluid velocity remains smooth. This, of course, implies that a “smooth blowup” is not possible.

In regard to your second comment, I quite frankly don’t know what you are talking about. The fact is that despite all of the journal rejections and less than positive feedback my paper has received on this site, at least one reviewer seemed to believe I had a valid proof. You, however, have not attacked my paper with a single factual statement. Also, I have seen lots of theorems proven but have never heard of “imbedding theorems” before.

13 September, 2021 at 6:59 am

Gandhi ViswanathanI am a theoretical physicist, not a professional mathematician. So my knowledge of real analysis is admittedly limited. But even I know the importance of the Sobolev embedding theorems. If you have not heard of Sobolev spaces, then it will be very difficult for you to be taken seriously by researchers in the field. Try to see first whether your “proof” works for simpler model systems, such as the 2D viscous quasi-geostrophic equations. Or even simpler models, such as the 3D or even 1D Burger’s eq. with a fractional laplacian. And if others don’t understand or agree with your point of view, try not to blame them. Instead, try to see their point of view. After all, professional mathematicians have devoted many years of their lives to learn their trade.

29 November, 2021 at 3:29 pm

Daniel HayesWhy would a proposed proof of global regularity for NS in any dimension be wrong for sure?

12 September, 2021 at 12:48 pm

DanielEquation (6) is wrong

13 September, 2021 at 12:43 am

Dwight WalshWhat do you mean “equation (6) is wrong”. It’s part of the problem description. If you have issues with that, you will need to take them up with Charles Fefferman or the CMI.

13 September, 2021 at 6:43 am

AnonymousAsking renowned mathematicians on their website to review a result is simply bad form – rather you should submit your result to a journal

13 September, 2021 at 7:07 am

victorivriiActually, no: the partial derivatives in (6) should be with respect to $x,y,z$ respectively.

13 September, 2021 at 7:30 pm

Dwight WalshIn response to various naysayers about my work with the NS Millennium Problem, I have posted a new version of my paper to my website at

https://www.navierstokessmoothsolutions.com

In this new version, I went through the section titled “Existence of Pressure Gradient Integral over Time” with a fine-toothed comb and believe I have left little doubt that the time integral of |grad p| is not only finite, but also is bounded uniformly over the entire open time interval (0,T) by a single upper bound. Every statement I make in this section about the time dependence of q and |grad p| is backed by previously proven equations, and the result is that that the time integral of |grad p| is bounded regardless of how large the fluid velocity becomes. I simply don’t know how I can possibly make the proof any clearer.

Anyway, those who are genuinely interested but were unable to follow this section in my previous paper are welcome to download this new version and see if they get a better understanding.

Also, I apologize to victorivrii and Ghandi Viswanathan for coming up with a proof that does not use Sobolev embedding theorems and for which the result falls out nicely in three dimensions.

Finally, in regard to the comment from Anonymous, I must say that just because I am not renown does not mean I am not capable. My career, while it lasted, was primarily with the US defense industry, and therefore I had little opportunity to publish, especially in peer reviewed journals. One of my research project did involve the NS equation, but it only resulted in a few conference papers. When the “Cold War” ended in the early 1990s, so did my research career, and I have been professionally ignored ever since that time. I submitted my paper to eight journals so far, and have received nothing but flat rejections or possibly rejections having nothing to do with mathematical correctness, which seems quite odd for a problem of such “importance”.

Now, as I see it, for someone with my degree and capabilities to be so ignored for the past thirty years is also “bad form”. So what do you suggest I do!?

14 September, 2021 at 1:52 am

victorivriiIn you new version the second equality in (117) is obviously wrong. Please find the error by yourself.

14 September, 2021 at 7:41 pm

Dwight WalshThanks so much!

19 September, 2021 at 1:25 am

Dwight WalshFor those still interested in my work and possible contributions to solving the NS millennium problem, I just made a major breakthrough in regard to the time integral of |grad p|. As I stated in previous comments, much of the negative feedback I received concerning my proposed solution to the Navier-Stokes Millennium Problem has been claims that I have not adequately proven that this time integral is finite and in fact uniformly bounded over the entire open time interval immediately preceding the blowup. That is, there may be agreement that the time integral of |grad p| is finite at all times prior to the blowup, but may not be bounded as t approaches the blowup time. Well, in my break-through, I was able to show that the time integral of |grad p| is in fact bounded above by E_0 / (2\pi\nu) on the entire time interval 0 < t < T_b, where E_0 is the initial energy of fluid motion, \nu is the normalized viscosity coefficient of the fluid, and T_b is the blowup time. Now, I still believe my previous versions were generally correct, though perhaps not as well pulled together. In view of how nicely my new writing of the proof came together with arguments that are easy to follow and verify along with a clear, simple final result; I would dare claim that the "naysayers" would have one monster of a challenge in debunking it.

Again, for those interested in my work, and even those who would like to shoot it down (LOL), you can download my latest version of the paper (Version 26) from my website at

https://www.navierstokessmoothsolutions.com

Now, I acknowledge that Dr. Tao far surpasses me in mathematics education and intellect as do probably most others making comments in this blog. Nevertheless, a proof is a proof, regardless of whether elementary or highly advanced concepts are used, or what educational background the author has. Also, arguing that the time integral of |grad p| cannot be uniformly bounded (with a single upper bound) because it is a "supercritical" quantity is invalid since this term applies only to "weak" solutions of the "averaged" NS equation instead of actual solutions to the original one. In general, I believe that the whole idea of "weak solutions" has been far overused, and too many "important" results are actually artifacts of adjustments made to the equations in order to arrive at "solutions" of some sort. Discussion and sometimes an entire lingo then centers around these results rather than those of the original equations, and individuals trying to bring new insight to the problem are not well received by the established "experts". This, I sense, is what I am up against and has very much curtailed any actual progress in solving the NS millennium problem.

20 September, 2021 at 1:54 am

AnonymousIn order to reduce the (understandable) doubts of Math journals editors toward an “outsider” mathematical credibility, you may try to register to a university as a mathematics student, and then (as a mathematics student) try to gradually publish your results as a series of well-written small papers (with “less ambitious” titles).

20 September, 2021 at 12:45 pm

AnonymousThis inequality you claim is dimensionally inconsistent…normalizing nu=1, integral grad p dt has dimensions of length^-1, while the energy has units of length. This issue is the essence of Tao’s above article about supercriticality.

29 September, 2021 at 1:05 pm

Dwight WalshIn my most recent effort to established an upper bound on the time integral of |grad p|, I came up with some interesting results. While I found it impossible to actually pin down a value for this upper bound (which I would expect depends on initial conditions), I believe I was able to prove its existence and that it must be finite for all t and as t approaches infinity. This would in turn prove existence and smoothness of solutions to the NS equation for all t > 0.

Anyway, those interested are welcome to download my most recent paper from the website at

http://www.navierstokessmoothsolutions.com

and check on the arguments leading to equation (113).

30 September, 2021 at 2:22 pm

Dwight WalshIn view of the fact that my postings seems to have drawn some highly unprofessional responses, I won’t be posting anything further unless specifically requested by another participant of this thread. My website at

http://www.navierstokessmoothsolutions.com

is up and running again after being down for a few days, and those who are genuinely interested in my ideas about the NS Millennium Problem are welcome to visit the site, download my most recent version of the paper, and send me their comments with the posted email address.

12 September, 2021 at 5:37 am

PolihronovSorry to interject, I did not receive constructive feedback either…

Perhaps Prof. Tao could elaborate what, as an example, would be any other forms of .

For the folks that left a “thumbs up”, I appreciate this; if you would comment on what part of the argument appealed to you as correct; it would be a great part of this discussion.

20 September, 2021 at 11:37 am

rbcoulterI can’t follow the reasoning starting at (26). It seems as all the effort in the proof is in showing that there exist a system of curves (meeting the Clay criteria) that bound the solution (and its derivatives) of the Poisson equation. However, I can’t see how this proves the time evolution of a flow scenario that is initially bounded by the Clay criteria, and stays bounded for all t. It is though some sort of circular logic is going on which involves assuming the derivatives in the NS equations exist for all time. Most of the effort in the paper seems dedicated to constructing the bounding curves. This is trivial task if one first assumes the derivatives don’t blow up.

21 September, 2021 at 2:57 pm

victorivriiOut of 7 Clay Millenium Problems one was solved and 6, including NSE, remain unsolved. There were several proofs published by well-established mathematicians and containing subtle (or not so subtle) crucial errors, several pranks (https://vixra.org/author/jorma_jormakka, for NSE the author exploited the lack of peridicity condition with respect to spatial variables for a pressure in the original Clay text) and many “solutions” and “proofs” written by cranks and freeks.

It is not dissimalar to Fermat’s Last Theorem: even after Wiles’ proof there is a flow of simple and elementary “proofs”.

One did not need a crystall ball to predict this side effect of the Millenium Problems project. 😀

21 September, 2021 at 10:14 pm

Dwight WalshI acknowledge I made a goof in equation (115) in Version 26 of the paper, and it will probably be a few days before I have a corrected version posted. However, I still believe I had this part correct in previous versions, but with the repeated rejections, I keep trying to find ways to make my arguments shorter and simpler without compromising validity. But, as we can see, I make mistakes like everyone else. Anyway, I am still working on clarifying the argument that the time integral of |grad p| is uniformly bounded on the entire interval 0 <= t < T_b.

22 September, 2021 at 1:35 am

Dwight WalshI would like to address the concerns that some individuals are having in regard to possible circular logic in which we assume existence and smoothness of solutions in order to prove existence and smoothness of solutions.

In this problem, we are given through the CMI criterion that a smooth solution exists initially, but we don’t know how long it stays that way. So, in equations/inequalities (17)-(79), we show that the CMI criterion implies that a solution u must be smooth IF IT EXISTS (ie. remains finite). This would preclude functions u from evolving such as sin[1/(x – X)] which may be bounded but still show wild behavior at one or more points. Such functions would violate the ranges of the higher spatial derivatives imposed by the CMI smoothness criterion. A smooth solution, however, can still approach infinite values in finite time unless proven otherwise. Stated more precisely, we can get the fluid velocity to be arbitrarily high by choosing t sufficiently close to T_b.

Therefore, we are NOT assuming that the derivatives in the NS equation or anything else exists for all time. In equations/inequalities (80)-(125), the idea is to see whether a smooth solution could could evolve into a blowup at some point which we call x_b and (finite) blowup time which we call T_b. If such a blowup does occur, the fluid velocity function u must still stay smooth at all points and all times on the interval 0 <= t < T_b since u is still finite at those times and therefore smooth according to equations (75)-(79). This is what is often called a "smooth blowup" and it is the only kind of blowup we can have in this problem.

Anticipating the possibility of such a smooth blowup, the section titled "Existence and Smoothness of Solution over Time" explains that in a smooth blowup, a global maximum must first form at the blowup point (which we call x_b) at a time t_b where t_b < T_b. The equations and discussion then shows that for the blowup to occur, the time integral of |grad p| at this point must become infinite at T_b. Therefore, the "crux" of my proof is in showing that this time integral of |grad p| remains finite and in fact uniformly bounded on the entire time interval 0 <= t < T_b, and there is certainly some controversy at this point over whether I have adequately shown this.

Anyway, the important point to realize concerning "circular logic" in my arguments is that smoothness (over space) does not imply binding of the solution over time. They must be established separately.

22 September, 2021 at 8:19 am

AnonymousIf you actually prove a bound on the time integral of grad p, concluding global regularity would be routine. However it is extremely unlikely such a bound exists for the reasons listed above in Tao’s article…

13 January, 2022 at 12:10 am

Dwight WalshWhy don’t you just read my paper and see for yourself if I have shown a bounded time integral of grad p. If you have any questions, just send me an email.

23 September, 2021 at 3:38 am

Dwight WalshI just posted Version 27 of my paper to the website at

https://www.navierstokessmoothsolutions.com

which I believe fixes the obvious error in equation (115) of Version 26 and does some minor tweaking of arguments and explanation to hopefully make them more understandable. It is the section titled “Existence of Pressure Gradient Integral over Time” leading up to inequality (117) that contains my argument for claiming that this integral is finite for all times the fluid velocity remains smooth, regardless of how high the fluid velocity gets. You might want to go to this website and check it out.

25 September, 2021 at 2:32 am

Dwight WalshFor those interested in my work in establishing a uniform bound on the integral of |grad p|, please check out Version 28 of my paper at

https://www.navierstokessmoothsolutions.com

In this version, I completely overhauled the section titled “Existence of Pressure Gradient Integral over Time”, giving much stronger, clearer arguments showing uniform binding of the time integral of |grad p|, which in turn is part of the proof that “smooth blowups” in this problem are not possible.

26 September, 2021 at 12:24 pm

Dwight WalshTo the one giving me the “thumbs down”:

Would you care to give a material response to my comment, stating what error(s) you have found in my new arguments for the time integral of |grad p| being bounded? As I see it, this is a blog where individuals interested in the NS problem can share ideas on this topic, and not a political rally.

26 September, 2021 at 12:52 pm

Gandhi ViswanathanThe reason somebody (not me) probably gave you thumbs down is that you seem not to be willing to accept that your paper was rejected by the journals and by the expert mathematicians for a good reason. You are insisting on an argument that is unconvincing. Let me ask you: Assuming that your proof is correct, you should be able easily to tell us what is the controlling norm. What is it? Does this norm depend on time? If yes, how so? Is it a conserved quantity, or does it satisfy an exponential or double exponential bound etc.? Why doesn’t it blow up? Does it scale correctly under scale transformations? And what is the new idea that you had that allowed you (but nobody else before you) to find this controlling norm? Without convincing answers to such questions, many people will assume that your “proof” is not correct (and hence not worth going through). Extraordinary claims require extraordinary evidence, as they say.

26 September, 2021 at 4:09 pm

AnonymousThis is well said, Gandhi

22 December, 2021 at 11:29 pm

Dwight WalshActually, most of your questions and claims are easily addressed. First, no “controlling norm” has even been defined in this proof. We simply don’t use that particular concept — whatever it is. Expecting me to come up with such a “norm” in this paper could be compared with expecting me to use calculus and differential equations to balance a checkbook. BTW, you are the only who has thrown the “controlling norm” term at me thus far.

Also, if you read the paper, you will see why the solution (or “controlling norm”) doesn’t blow up. Assuming I am communicating with professional mathematicians, they should be able to readily follow my more “elementary” arguments and point out actual flaws in the proof without posting anonymous “thumbs down”. These and personally insulting comments only convince me more than ever that my proof is correct.

In regard to “scale transformations”, this becomes an issue only if we neglect the scalar pressure (and its gradient) which Dr. Tao does in this this article. (Please see his quote “…, but we will not need to study the pressure here.”). Therefore, it is no mystery to me “Why global regularity for Navier-Stokes is hard”. In my paper, where I correctly take into account the pressure gradient as well as the nonlinear transport term, I show that a (smooth) blowup is not possible.

Finally, your statement that extraordinary claims require extraordinary evidence is applicable only if my evidence is read in an unbiased manner. Otherwise, my efforts constitute a waste of time.

24 December, 2021 at 7:05 am

AnonymousIt seems that the only way to convince the experts that your proof is correct, is by showing its correctness via an (objective!) automatic formal proof verification (using a standard software for such formal proof verification). Such formal verification is known to be much more reliable than the usual (human!) peer review verification.

26 September, 2021 at 2:18 pm

roland5999Hello Dwight Walsh !!

i follow this thread for 10+ years and occasionally put some comments here as well (sometimes less and sometimes more politely; now washed away by the kind winds of time).

I learnt quite a couple things from this (and elsewhere) in the meantime.

It is worthwhile to consider the kind remarks of Gandhi Viswathan carefully. Basically, this thread here is the WRONG one for you (and myself).

You are (with an obvious passion) pointing to something relevant to the physics-realm.

Such arguments are not fit to be uttered in this forum,

which is dedicated to the Feynmannian “dry water” version of the NS formulation. Pressure as such does not even exist there-in (i.e. can be eliminated by a mathematical trick).

i am willing to interact with your pointed website and talk elsewhere.

Dear other writers/readers and dear administrator,

please ignore these offtopical words.

with kind regards: nobody.

30 September, 2021 at 4:37 am

victorivriiDo we need a flamewar? Yes, I already explained why there is no chance that the simple and elementary solution of NSE is correct. Decision to ban any user belongs to the master of the blog, not to us. And it would be fair to issue the warning first.

Unfortunately (but not unexpectedly) the master of the blog does not seem to care about this part of the blog anymore. I am not giving up following this topic because I hope that rather soon the real mathematician will publish solution (positive or negative) somewhere (probably in arXiv first) and post a link here.

12 December, 2021 at 1:00 am

Dwight WalshI realize I stated on Sept. 30 that I would not be posting anything further to this thread, but since no one else has posted anything since that date either, I thought that some readers might be interested in my latest developments — maybe even those who don’t particularly like me.

In my latest paper (posted on navierstokessmoothsolutions.com), I show from three different angles that the time integral of |grad p| must be finite at all points in R^3. For example, if there is some point x_b where the scalar pressure p blows up at time T, then grad p(x_b,t) approaches zero as t approaches T. This is due to the fact that in a smooth blowup, the spatial max/min values of p can only occur at points where grad p=0, and since blowups can evolve only from global maximums, it follows that grad p smoothly approaches zero as the blowup happens. Therefore, |grad p| at x_b remains finite over the entire finite closed time interval [0,T].

Also, I should point out that in re-examining previous versions of the paper, I believe my arguments were correct in version 24 and even earlier, although there may have been some room for improvement on clarity. In fact, one reviewer of version 5 seemed to believe I did have a valid proof of existence and smoothness, although the paper was rejected over grammatical and formatting issues. It seems, ironically, that my difficulty lately has not been with the mathematics but with the “experts” who simply don’t believe my work is worth their time. This is clearly demonstrated in their comments to me in which they have jumped to conclusions and made claims about my arguments that are way off base. Unfortunately, this has resulted in me making a dozen more revisions of the paper, hoping to address the concerns they wouldn’t have if they had only read the paper in the first place. And since my arguments are “elementary”, they should have no trouble following them.

12 December, 2021 at 11:00 am

AnonymousHello Dwight, The key error is in equation (118). This is finite for any epsilon individually but you should expect the integral to be unbounded as epsilon -> 0. You mention the Dirac delta as the worst-case behavior at 0, but really you should imagine something like r^{-100}, say.

12 December, 2021 at 2:14 pm

Dwight WalshNo, this is not an error. In this section we are still within the realm of a smooth, existing solution and do not introduce the concept of (smooth) blowups until the next section. Therefore, the function q must still be smooth, and r^{-100} is not smooth at r=0. In the arguments following equation (118), however, I show that even if this smoothness requirement for q is relaxed to allow delta function singularities (which are the strongest singularities allowed by inequality (109)), the time integral of |grad p| is still finite at the singularities.

I decided to include these arguments in the paper not because they are actually needed in the proof, but to ease the minds of several readers who seemed to believe I was using “circular logic” in assuming a smooth solution to prove a smooth solution.

12 December, 2021 at 2:58 pm

AnonymousI only meant to make the philosophical point about why the delta function is the wrong heuristic. If you want a literal example, suppose q is a smooth function which is =delta^(-3) when |x|2delta. (Note importantly that this is consistent with your (109).) When epsilon << delta, the integral in (117) is approximately delta^(-2).

The fact that we can satisfy (109) but find counterexamples to (117) and (118) is the exact point made by Tao in this article. (109) is a supercritical bound while (117) and (118) are subcritical.

12 December, 2021 at 3:00 pm

AnonymousSorry I meant: q is a smooth function such that q=delta^(-3) when |x| 2delta.

12 December, 2021 at 3:01 pm

AnonymousIt won’t render it for some reason. Again: is a smooth function such that when and when .

12 December, 2021 at 9:11 pm

Dwight WalshI’m afraid I don’t quite understand what you are talking about, but equations (109), (117), and (118) all arise as necessary conditions for the fluid velocity and scalar pressure functions to satisfy the non-compressible Navier-Stokes equation. Therefore, counter-examples to (117) and (118) could not be actual solutions to the Navier-Stokes Millennium Problem, but only “weak” solutions.

14 December, 2021 at 12:02 pm

AnonymousThe example I give above is not a counterexample of the Navier-Stokes problem, it is a counterexample of the deduction of (118) from (117), with the added feature that it is still consistent with (109). Since the choice of is arbitrary, you can take it to be, say where $T_*$ is the hypothetical first blowup time.

15 December, 2021 at 9:08 am

Dwight WalshWrong. In obtaining (118) from (117), all we did was take the lim as epsilon -> 0 indicated in (117), which was a rather trivial step since smoothness of u (and therefore q) prior to blowup was already proven in the discussion leading to inequality (79). Your “counterexample” however, assumes a (highly unlikely) time dependence of the solution that has nothing to do with the Navier-Stokes equation.

10 January, 2022 at 6:46 pm

AnonymousI’m not going to argue with you Dwight. You like to claim that no mathematician has looked at your work but it seems the real problem is you refuse to listen when they point out an error…

11 January, 2022 at 8:01 pm

Dwight Walsh> I’m not going to argue with you Dwight.

Good! You are making absolutely zero sense. And if you are so confident in your claims, why do you not put your name on them?

12 December, 2021 at 12:48 pm

AnonymousIf the proof is sufficiently simple, is it possible to formalize it and then try to give it a formal verification (as done e.g. for the extremely complicated proof of the Kepler conjecture)?

14 December, 2021 at 1:46 am

Dwight WalshI’m not sure what you mean by “formalize” the proof and then try to give it a “formal verification”. Are you referring to getting it posted in an archive of some sort where other mathematicians can review it for free? If so, then the answer is a plain simple “no”. With my career fizzle 30 years ago, no mathematician nowadays would sponsor me. In fact, look at the insults and thumbs down I have had on this forum alone.

14 December, 2021 at 10:05 am

AnonymousI meant that if the proof is sufficiently simple, there are several software programs that can give it (after translating the proof into a symbolic input syntax which is needed by the automatic verification program) automated(!) rigorous(!) computerized verification.

Several complicated proofs (too complicated to manually(!) verify all their details) were already given such automated(!) computerized verification – see e.g. the Wikipedia article on the Kepler conjecture and the automatic symbolic verification of its extremely complicated proof.

15 December, 2021 at 12:12 am

Dwight WalshI’m afraid you would know much more about computerized verification than I do. In fact, the only time I heard anything about computer-aided proofs was with the proof of the four-color map conjecture in 1976, and in that case there was some controversy over the validity of such a proof. Setting those issues aside, however, I don’t sense that the NS Millennium Problem would lend itself as well to such an analysis since it involves only a few logical arguments and not imassive numbers of logical statements to be evaluated.

18 December, 2021 at 9:16 pm

Dwight WalshDear Dr. Tao,

I read you comments concerning the maximum kinetic energy and cumulative energy dissipation, and I understand that these two quantities cannot by themselves establish existence and smoothness of solutions to the NS equation. From my study of this problem, however, I believe that the global maximum of the kinetic energy density K = u^2(x,t)/2 provides the “missing link” that allows proof of existence and smoothness. First note that a “smooth blowup” is occurs iff this value becomes infinite in finite time. But then, since we are working with a viscous fluid, it is possible to show that no maximum point can attain infinite velocity in finite time. If you are interested, please load my paper from the website at

http://www.navierstokessmoothsolutions.com

19 December, 2021 at 5:09 am

AnonymousIs energy subcritical, supercritical or critical in 3D ? What about in 2D ? Professor Tao’s post is partially about this issue. Once one understands the concept of how quantities scale, it becomes clear why kinetic energy cannot be a controlling norm.

19 December, 2021 at 11:53 am

Dwight WalshI realize that total kinetic energy cannot be a controlling norm by itself. What I am saying is that knowledge of the maximum kinetic energy density (ie. joules/meter^3), which is a global but time-dependent quantity, along with the two supercritical quantities indicated by Prof. Tao (total kinetic energy and cumulative dissipation energy) is sufficient to establish existence and smoothness of solutions to the incompressible NS equation.

I did not consider any 2-dimensional version of the problem since the NS equation is formulated in 3 dimensions, not 2, and the Green’s function used in many of my arguments applies only to 3 dimensions.

Also, how quantities scale is completely independent of existence or smoothness of solutions to any equation. The rule is that one chooses a scale size (or equivalently, a system of units) that is most convenient for the problem at hand, and uses it consistently.

19 December, 2021 at 11:57 pm

Dwight WalshDear Prof. Tao

Could you please explain why “we will not need to study the pressure here”? The grad p term cannot be negligible compared to the nonlinear transport term since the sum of these two terms must have a zero divergence, and in the analysis that follows, you discuss the possibility of a blowup occurring at some time T as a result of energy being shifted to successively smaller space scales.

In the paper I posted at http://www.navierstokessmoothsolutions.com, however, I show that when grad p is correctly taken into account, then no such blowup is possible.

29 December, 2021 at 8:09 am

Antoine DeleforgeI am just a random researcher in acoustics & signal processing so most of the theory surrounding Navier-Stoke goes way above my head, but even I can tell that you are completely misguided when you say that Prof. Tao is “neglecting the grad p term”. He literally wrote “we will not need to study the pressure here”, how is that equivalent to “neglecting the grad p term” according to you? Are you a native English speaker (genuine question, I am not) ? The two expressions have basically nothing to do with each other. There is a huge difference between choosing *not to study* a particular quantity and *neglecting a term* in an equation. You see this, right?

29 December, 2021 at 8:57 pm

Dwight WalshI’m afraid that in this problem, not studying the pressure is the same as setting the pressure to zero. Even if the scalar pressure p is totally unknown, the grad p term must still be carried in the NS equation simply for mathematical correctness. Dr. Tao, however, never once mentions this term after making the assertion that “we will not need to study the pressure here”. Instead, he discusses the problem only in terms of “rescaling”, “supercritical and subcritical quantities”, and “coercive and noncoercive quantities”. The presence or absence of the grad p term does not affect his arguments one iota. Therefore, by not including this term in his analysis, Dr. Tao is by default setting it equal to zero. Hence, his methods do not in general apply to solutions of the incompressible NS equation

29 December, 2021 at 10:33 pm

Dwight WalshI should also point out that in the comment by Robert (17 Feb. 2021), he quotes Prof. Tao’s proposed strategy

“Discover a new globally controlled quantity which is both coercive and either critical or subcritical”

and correctly points out that discovery of such a quantity would render the formulation of the problem invalid since this new quantity was not part of the original Official Problem Description. Fortunately, Dr. Tao at least recognizes that his arguments are not complete, but the problem is not some missing coercive quantity. It is failure to maintain the incompressibility constraint (div u = 0).

In my paper at http://www.navierstokessmoothsolutions.com, however, I correctly included the grad p term in the NS equation, and there no issues related to an incomplete problem description.

30 December, 2021 at 12:54 am

victorivrii> I’m afraid that in this problem, not studying the pressure is the same as setting the pressure to zero.

Absolutely wrong. One of the completely valid strategies is to

excludepressure. How to do it: if the system has a form, $\latex \nabla \cdot u =0$ where $\latex p$ belongs to a prescribed class (f.e. in the

periodicsettings it must be periodic with respect to spatial variables). Here is the nonlinear operator.Consider projector $\latex P$ in the space of $\latex L^2(\mathbb{R}^3;\mathbb{R}^3)$ on the subspace orthogonal to the space formed by $\latex \nabla p$ for all such $\latex p$. Then since $\latex \nabla \cdot u =0$ we see that $Pu=u$ and then the problem becomes

with $\latex u:\ \nabla \cdot u =0$. That’s all. This problem is

equivalentto the original problem.On the other hand, simply putting would force us to drop condition , so the problem would be which is not equivalent to the original problem. But nobody suggests to do this.

Dwight Walshstop fighting with the straw man.RemarkIf we exclude non-linear part from $\latex A$, taking $\latex A=\nu \Delta u$ then $\latex PA[u] =A[Pu]$ and then indeed we would have equation with a compatible to it condition $\latex \nabla \cdot u =0$ but it is not the case.> “Discover a new globally controlled quantity which is both coercive and either critical or subcritical” and correctly points out that discovery of such a quantity would render the formulation of the problem invalid since this new quantity was not part of the original Official Problem Description.

Again wrong. This quantitity would be a tool. F.e. considering heat equation one can observe that for solution $\latex \iiint u dx $ is constant and $\latex \iiint u^2 dx$ decays but those quantities are not the poart of the original problem description.

I hope this LaTeX works

30 December, 2021 at 11:00 am

Dwight WalshVery good! Now would you like to explain exactly how Prof. Tao implements all of this to maintain a zero divergence fluid velocity and actual solutions to the NS equation as opposed to “weak” solutions? Keep in mind, of course, that he does not use equations of inequalities in this article — just vague intuitive arguments that work for compressible as well as incompressible fluids.

30 December, 2021 at 2:41 pm

AnonymousHi Dwight, if you want to learn about the basic theory of the Navier-Stokes equations, I would recommend you start with Tao’s lecture notes on the subject. Start here: https://terrytao.wordpress.com/2018/09/16/254a-notes-1-local-well-posedness-of-the-navier-stokes-equations/

In particular, sections 2 and 4 will dispel your misunderstanding about the pressure. Hopefully this text will be accessible to you.

30 December, 2021 at 11:50 pm

Dwight WalshNo, I believe I am well acquainted with the NS equation including the role of the scalar pressure and how it is obtained. Have you read my paper at http://www.navierstokessmoothsolutions.com where I prove existence, smoothness, and uniqueness of solutions to this equation? I may not use the same notation as Prof. Tao, but the concepts are still there.

My concern with this current article by Dr. Tao is what seems to be arguments based solely on “re-scaling” of the solution to claim the possibility of energy being transferred to successively smaller scale sizes, eventually resulting in a blowup. His arguments are vague and intuitive, but he does assert at one point that we do not need to study the pressure here. This leads me to believe that we will not be considering the scalar pressure in whatever it is we are doing, and in fact, Dr. Tao doesn’t mention pressure again in this article. In this case, however, how do we maintain incompressibility of the fluid? In general, WHAT IS HE DOING AND HOW DO WE KNOW IT IS APPLICABLE TO THE NS EQUATION?

Finally, I should note once again that in the paper I posted at

http://www.navierstokessmoothsolutions.com

grad p and the nonlinear transport term are correctly taken into account, and regularity is established without such issues arising.

31 December, 2021 at 1:02 am

victorivrii> Now would you like to explain exactly how Prof. Tao implements all of this to maintain a zero divergence fluid velocity and

This is a common knowledge; procedure was well understood even before T. Tao was born

> actual solutions to the NS equation as opposed to “weak” solution

It works for both smooth solutions and week solutions as well. Simply smooth/wee solutions of the reduced problem correspond to smooth/week solutions of the original problem. Please read materials suggested to you.

31 December, 2021 at 9:12 am

Dwight Walsh>This is a common knowledge

This is a common cop-out for people who don’t know themselves.

31 December, 2021 at 9:44 am

victorivriiExplaing:

Consider reduced equation . Applying (remember, is an orthogonal projector) we get

. Thus if $latex(I-P)u=0$ as , it will be so for any as well. And $latex(I-P)u=0$ is equivalent to (Calculus II).

Happy

1 January, 2022 at 12:13 pm

Dwight WalshYes, and I believe this is equivalent to how I propagate the NS solution in my paper, although I do not use the Leray projector notation. Unfortunately, however, Prof. Tao’s vague writing style (often without equations) makes it extremely difficult for a newcomer to the NS equation to follow, and is wide open for misunderstanding. Anyway, after a few more reviews of his arguments about the fluid energy being transferred to successively smaller scale sizes, I believe I have a somewhat better understanding what he is saying, and will not press issues on his arguments not being consistent with div u = 0 or the NS equation in general.

I do, however, have a much more serious concern about his belief that a blowup solution is always theoretically possible unless a new quantity is discovered that is coercive and critical or subcritical. This, of course, would change the problem statement for the NS global regularity. In my paper, however, I show that as any “smooth” blowup ensues, the work done by the (retarding) viscous forces acting on a volume element at the blowup point exceeds the work done by grad p forces on this volume element. Hence, no such blowup would be possible. This, I believe, is an EXTREMELY important result that Dr. Tao misses in his equation-free arguments.

31 December, 2021 at 1:09 am

victorivrii> No, I believe I am well acquainted with the NS equation including the role of the scalar pressure and how it is obtained

You are presumably well with the acquainted how NSE is derived but your knowledge of mathematical aspects is marginal at best. And this ignorance is aggravated by your arrogance. Sorry for blunt assessment but

When, O Catiline, do you mean to cease abusing our patience?31 December, 2021 at 9:20 am

Dwight WalshI will continue to do whatever it takes to get the word out about my solution to the Navier-Stokes Millennium Problem unless I am shown to be wrong mathematically. And if you don’t like it, that is your problem.

31 December, 2021 at 9:47 am

victorivrii🤣🤣🤣🤣🤣🤣

31 December, 2021 at 9:55 am

Anonymous“NS Don Quixote”

4 January, 2022 at 12:25 am

Dwight WalshI should indicate at this point that several statements I made over the past week or two concerning Dr. Tao’s are probably not accurate due to my misunderstandings of what he is trying to show. This includes my claims of his possible inconsistency with the NS equation and incompressibility. Unfortunately, however, his mostly verbal and intuitive descriptions were extremely difficult for me to follow. Therefore, I no longer have the same issues about his arguments that I recently posted in my comments.

My current concern about his article in not so much in what he has but what he is missing. I agree that the nonlinear terms appearing in the transport and grad p terms could channel fluid energy into smaller scale sizes and could lead to a blowup if unchecked, but does this actually happen? Dr. Tao seems to recognize that the currently known coercive, critical, and super-critical quantities along with scale-size arguments is insufficient to establish global regularity. However, his suggestion of finding a new coercive/critical parameter would change the statement for the NS global regularity problem (ie.NS Millennium Problem), and claims it to be an “impossible problem” in his final paragraph.

What I believe Dr. Tao is missing in his arguments is the fact even a blowup solution must be smooth until the actual blowup time. Therefore, the blowup point must originate from a global maximum, and since the fluid kinetic energy density is still a smooth function, this maximum point must also be a level point. Additionally, the Laplacian of this function at the maximum point must be negative or zero. In my paper at

http://www.navierstokessmoothsolutions.com

these properties of a smooth blowup are used to show that no such blowup is possible for solutions of the NS equation.

7 January, 2022 at 7:23 am

AnonymusShush

14 January, 2022 at 6:46 pm

Dwight WalshWOW! What gems of wisdom!

10 January, 2022 at 6:36 pm

AnonymousMathematicians have known that solutions stay smooth until the blow up time since Leray’s thesis in 1934.

11 January, 2022 at 10:09 pm

Dwight WalshYes, I realize this and in fact proved it my paper, assuming of course there is a blowup. Unfortunately, however, many reviewers seem to have DEFINED a blowup time T, and claimed that I have only proven a smooth solution on the semi-open interval [0,T) while there still could be a blowup at time T. My question to them is exactly what happens at this time T? How did they choose it? Could it be that they think of it as an actual time at which the fluid velocity becomes infinite at some point, as they seem to be calling it? If so, then it’s no wonder the NS regularity problem has not been solved. Proving existence and smoothness when blowup is a given would be a monumental task!

12 January, 2022 at 12:51 pm

AnonymousThe blowup time is defined as the supremum over all finite nonnegative times t for which there is a smooth solution on [0, t). you need to show rigorously(!) that this supremum is

13 January, 2022 at 7:43 pm

Dwight WalshGood! Now can you look at equation (128) and find a particular value T of t such that K_b(t) -> infinity as t -> T? If not, then there is no blowup time and we have a smooth solution. Right? Remember, of course, that in the discussion leading to equations (119) and (120), we showed in two ways that the time integral of |grad(p)| has a finite upper bound that applies for all t > 0.

13 January, 2022 at 10:27 pm

Dwight WalshAlso, I should add that all of my arguments and equations/inequalities hold on the semi-open time interval [0,T) where the solution exists and is smooth, and make no claims about what happens at t = T. To prove existence and smoothness of a solution, I show that there is no value of T such that u(x_b,t) becomes arbitrarily large for t sufficiently close to T, but still less than T. Therefore, THERE IS NO CIRCULAR LOGIC WHATSOEVER in my arguments, and I strongly believe that one would be very hard pressed to come up with a presentation that is clearer and more rigorous.

10 January, 2022 at 4:17 pm

PolihronovHello Prof. Tao and Everyone,

Thank you for your post of May 4, 2021. I have spent time reading your replies as well as reading sections of the main blog post. I do appreciate this web site, the comments and ideas presented and the concept of criticality through which the NSE solutions are analyzed; and have followed the blog posts made through the years. The information I posted in 2021 above was given in parts; some references to articles were made; for this reason, if you would allow me, I would like to summarize it in a more contained form; as well as to request feedback.

In the article “T. Tao, Proceedings: Current Developments in Mathematics International Press, 255 (2006)” we find the definitions of -critical, -subcritical and -supercritical PDEs. All of them require a scaling law for a solution ; and the criticality of the value then depends on the scaling behavior of .

As I understand it (please correct me if I’m wrong), NSE criticality is estimated through

because it is part of its definition.

According to Lloyd, “S. P. Lloyd, Acta Mech. 38, 85 (1981)”, this transformation is a Lie group of scaling transformations. Charles L. Bouton, an American mathematician and a Harvard mathematics professor was a student of Sophus Lie; Bouton wrote a number of articles on Lie transformations and their invariants; however Bouton’s doctoral thesis is especially interesting. It is published in the literature as “C. L. Bouton, Am. J. Math. 21, 25 (1899)”.

In his article, Bouton examines a very similar scaling transformation. He determines the functional form of the invariants of this transformation. Repeating Bouton’s analysis for the case of the NSE shows that under the above Lie transformation, the NSE solutions must have the form

This conclusion follows from Bouton’s first theorem and is a result of his analysis. In his article, he lists a number of characteristics of the Lie invariants and they are well defined mathematically.

All this seemed interesting to me since it links together NSE criticality, the invariance of and the possibility of scaling-induced blow-up.

I am very interested to hear other’s opinions on the invariance of .

10 January, 2022 at 5:21 pm

victorivriiThere may be an invariant solution (which is non-smooth as ) but it does not help to prove ordisprove that any solution smooth as is smooth for all .

10 January, 2022 at 7:14 pm

PolihronovDear Prof. Ivrii,

Thank you for replying! (U of T is awesome :)

It would be very interesting to see if the Bouton-form of the solutions

is the only form and can have when we use the scaling transformation shown above. According to Bouton’s first theorem, this must be the case since they should both transform identically to the jet-space variables and of the Lie scaling transformation. Once this is established, one can find solutions that would not be subject to blow-up; I have a link with more information on how to accomplish this; but would first need Prof. Tao’s permission to post such a link here in his blog.

I was respectfully seeking feedback on the above expressions for and ; the fact that they must have this form would have strong implications on how one estimates NSE criticality.

Many thanks.