Conserved quantities for the surface quasi-geostrophic equation

As in the previous post, all computations here are at the formal level only.

In the previous blog post, the Euler equations for inviscid incompressible fluid flow were interpreted in a Lagrangian fashion, and then Noether’s theorem invoked to derive the known conservation laws for these equations. In a bit more detail: starting with Lagrangian space and Eulerian space , we let be the space of volume-preserving, orientation-preserving maps from Lagrangian space to Eulerian space. Given a curve , we can define the Lagrangian velocity field as the time derivative of , and the Eulerian velocity field . The volume-preserving nature of ensures that is a divergence-free vector field:

If we formally define the functional

then one can show that the critical points of this functional (with appropriate boundary conditions) obey the Euler equations

for some pressure field . As discussed in the previous post, the time translation symmetry of this functional yields conservation of the Hamiltonian

the rigid motion symmetries of Eulerian space give conservation of the total momentum

and total angular momentum

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

for any closed loop in , or equivalently pointwise conservation of the Lagrangian vorticity , where is the -form associated with the vector field using the Euclidean metric on , with denoting pullback by .

It turns out that one can generalise the above calculations. Given any self-adjoint operator on divergence-free vector fields , we can define the functional

as we shall see below the fold, critical points of this functional (with appropriate boundary conditions) obey the generalised Euler equations

for some pressure field , where in coordinates is with the usual summation conventions. (When , , and this term can be absorbed into the pressure , and we recover the usual Euler equations.) Time translation symmetry then gives conservation of the Hamiltonian

If the operator commutes with rigid motions on , then we have conservation of total momentum

and total angular momentum

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

or pointwise conservation of the Lagrangian vorticity . These applications of Noether’s theorem proceed exactly as the previous post; we leave the details to the interested reader.

One particular special case of interest arises in two dimensions , when is the inverse derivative . The vorticity is a -form, which in the two-dimensional setting may be identified with a scalar. In coordinates, if we write , then

Since is also divergence-free, we may therefore write

where the stream function is given by the formula

If we take the curl of the generalised Euler equation (2), we obtain (after some computation) the surface quasi-geostrophic equation

This equation has strong analogies with the three-dimensional incompressible Euler equations, and can be viewed as a simplified model for that system; see this paper of Constantin, Majda, and Tabak for details.

Now we can specialise the general conservation laws derived previously to this setting. The conserved Hamiltonian is

(a law previously observed for this equation in the abovementioned paper of Constantin, Majda, and Tabak). As commutes with rigid motions, we also have (formally, at least) conservation of momentum

(which up to trivial transformations is also expressible in impulse form as , after integration by parts), and conservation of angular momentum

(which up to trivial transformations is ). Finally, diffeomorphism invariance gives pointwise conservation of Lagrangian vorticity , thus is transported by the flow (which is also evident from (3). In particular, all integrals of the form for a fixed function are conserved by the flow.

Noether’s theorem, and the conservation laws for the Euler equations

Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. (Rigorous justification of the conservation laws and other identities arising from the formal manipulations below can usually be established in an a posteriori fashion once the identities are in hand, without the need to rigorously justify the manipulations used to come up with these identities).

It is a remarkable fact in the theory of differential equations that many of the ordinary and partial differential equations that are of interest (particularly in geometric PDE, or PDE arising from mathematical physics) admit a variational formulation; thus, a collection of one or more fields on a domain taking values in a space will solve the differential equation of interest if and only if is a critical point to the functional

involving the fields and their first derivatives , where the Lagrangian is a function on the vector bundle over consisting of triples with , , and a linear transformation; we also usually keep the boundary data of fixed in case has a non-trivial boundary, although we will ignore these issues here. (We also ignore the possibility of having additional constraints imposed on and , which require the machinery of Lagrange multipliers to deal with, but which will only serve as a distraction for the current discussion.) It is common to use local coordinates to parameterise as and as , in which case can be viewed locally as a function on .

Example 1 (Geodesic flow) Take and to be a Riemannian manifold, which we will write locally in coordinates as with metric for . A geodesic is then a critical point (keeping fixed) of the energy functional

or in coordinates (ignoring coordinate patch issues, and using the usual summation conventions)

As discussed in this previous post, both the Euler equations for rigid body motion, and the Euler equations for incompressible inviscid flow, can be interpreted as geodesic flow (though in the latter case, one has to work really formally, as the manifold is now infinite dimensional).

More generally, if is itself a Riemannian manifold, which we write locally in coordinates as with metric for , then a harmonic map is a critical point of the energy functional

or in coordinates (again ignoring coordinate patch issues)

If we replace the Riemannian manifold by a Lorentzian manifold, such as Minkowski space , then the notion of a harmonic map is replaced by that of a wave map, which generalises the scalar wave equation (which corresponds to the case ).

Example 2 (-particle interactions) Take and ; then a function can be interpreted as a collection of trajectories in space, which we give a physical interpretation as the trajectories of particles. If we assign each particle a positive mass , and also introduce a potential energy function , then it turns out that Newton’s laws of motion in this context (with the force on the particle being given by the conservative force ) are equivalent to the trajectories being a critical point of the action functional

Formally, if is a critical point of a functional , this means that

whenever is a (smooth) deformation with (and with respecting whatever boundary conditions are appropriate). Interchanging the derivative and integral, we (formally, at least) arrive at

Write for the infinitesimal deformation of . By the chain rule, can be expressed in terms of . In coordinates, we have

where we parameterise by , and we use subscripts on to denote partial derivatives in the various coefficients. (One can of course work in a coordinate-free manner here if one really wants to, but the notation becomes a little cumbersome due to the need to carefully split up the tangent space of , and we will not do so here.) Thus we can view (2) as an integral identity that asserts the vanishing of a certain integral, whose integrand involves , where vanishes at the boundary but is otherwise unconstrained.

A general rule of thumb in PDE and calculus of variations is that whenever one has an integral identity of the form for some class of functions that vanishes on the boundary, then there must be an associated differential identity that justifies this integral identity through Stokes’ theorem. This rule of thumb helps explain why integration by parts is used so frequently in PDE to justify integral identities. The rule of thumb can fail when one is dealing with “global” or “cohomologically non-trivial” integral identities of a topological nature, such as the Gauss-Bonnet or Kazhdan-Warner identities, but is quite reliable for “local” or “cohomologically trivial” identities, such as those arising from calculus of variations.

In any case, if we apply this rule to (2), we expect that the integrand should be expressible as a spatial divergence. This is indeed the case:

Proposition 1 (Formal) Let be a critical point of the functional defined in (1). Then for any deformation with , we have

where is the vector field that is expressible in coordinates as

Proof: Comparing (4) with (3), we see that the claim is equivalent to the Euler-Lagrange equation

The same computation, together with an integration by parts, shows that (2) may be rewritten as

Since is unconstrained on the interior of , the claim (6) follows (at a formal level, at least).

Many variational problems also enjoy one-parameter continuous symmetries: given any field (not necessarily a critical point), one can place that field in a one-parameter family with , such that

for all ; in particular,

which can be written as (2) as before. Applying the previous rule of thumb, we thus expect another divergence identity

whenever arises from a continuous one-parameter symmetry. This expectation is indeed the case in many examples. For instance, if the spatial domain is the Euclidean space , and the Lagrangian (when expressed in coordinates) has no direct dependence on the spatial variable , thus

then we obtain translation symmetries

for , where is the standard basis for . For a fixed , the left-hand side of (7) then becomes

where . Another common type of symmetry is a pointwise symmetry, in which

for all , in which case (7) clearly holds with .

If we subtract (4) from (7), we obtain the celebrated theorem of Noether linking symmetries with conservation laws:

Theorem 2 (Noether’s theorem) Suppose that is a critical point of the functional (1), and let be a one-parameter continuous symmetry with . Let be the vector field in (5), and let be the vector field in (7). Then we have the pointwise conservation law

In particular, for one-dimensional variational problems, in which , we have the conservation law for all (assuming of course that is connected and contains ).

Noether’s theorem gives a systematic way to locate conservation laws for solutions to variational problems. For instance, if and the Lagrangian has no explicit time dependence, thus

then by using the time translation symmetry , we have

as discussed previously, whereas we have , and hence by (5)

and so Noether’s theorem gives conservation of the Hamiltonian

For instance, for geodesic flow, the Hamiltonian works out to be

so we see that the speed of the geodesic is conserved over time.

For pointwise symmetries (9), vanishes, and so Noether’s theorem simplifies to ; in the one-dimensional case , we thus see from (5) that the quantity

is conserved in time. For instance, for the -particle system in Example 2, if we have the translation invariance

for all , then we have the pointwise translation symmetry

for all , and some , in which case , and the conserved quantity (11) becomes

as was arbitrary, this establishes conservation of the total momentum

Similarly, if we have the rotation invariance

for any and , then we have the pointwise rotation symmetry

for any skew-symmetric real matrix , in which case , and the conserved quantity (11) becomes

since is an arbitrary skew-symmetric matrix, this establishes conservation of the total angular momentum

Below the fold, I will describe how Noether’s theorem can be used to locate all of the conserved quantities for the Euler equations of inviscid fluid flow, discussed in this previous post, by interpreting that flow as geodesic flow in an infinite dimensional manifold.

Conserved quantities for the Euler equations

The Euler equations for incompressible inviscid fluids may be written as

where is the velocity field, and is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that is bounded. We will take the dimension to be at least two, with the three-dimensional case being of course especially interesting.

The Euler equations are the inviscid limit of the Navier-Stokes equations; as discussed in my previous post, one potential route to establishing finite time blowup for the latter equations when is to be able to construct “computers” solving the Euler equations, which generate smaller replicas of themselves in a noise-tolerant manner (as the viscosity term in the Navier-Stokes equation is to be viewed as perturbative noise).

Perhaps the most prominent obstacles to this route are the conservation laws for the Euler equations, which limit the types of final states that a putative computer could reach from a given initial state. Most famously, we have the conservation of energy

(assuming sufficient decay of the velocity field at infinity); thus for instance it would not be possible for a computer to generate a replica of itself which had greater total energy than the initial computer. This by itself is not a fatal obstruction (in this paper of mine, I constructed such a “computer” for an averaged Euler equation that still obeyed energy conservation). However, there are other conservation laws also, for instance in three dimensions one also has conservation of helicity

and (formally, at least) one has conservation of momentum

and angular momentum

(although, as we shall discuss below, due to the slow decay of at infinity, these integrals have to either be interpreted in a principal value sense, or else replaced with their vorticity-based formulations, namely impulse and moment of impulse). Total vorticity

is also conserved, although it turns out in three dimensions that this quantity vanishes when one assumes sufficient decay at infinity. Then there are the pointwise conservation laws: the vorticity and the volume form are both transported by the fluid flow, while the velocity field (when viewed as a covector) is transported up to a gradient; among other things, this gives the transport of vortex lines as well as Kelvin’s circulation theorem, and can also be used to deduce the helicity conservation law mentioned above. In my opinion, none of these laws actually prohibits a self-replicating computer from existing within the laws of ideal fluid flow, but they do significantly complicate the task of actually designing such a computer, or of the basic “gates” that such a computer would consist of.

Below the fold I would like to record and derive all the conservation laws mentioned above, which to my knowledge essentially form the complete set of known conserved quantities for the Euler equations. The material here (although not the notation) is drawn from this text of Majda and Bertozzi.

Polymath8b, IX: Large quadratic programs

This is the ninth thread for the Polymath8b project to obtain new bounds for the quantity

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page.

The focus is now on bounding unconditionally (in particular, without resorting to the Elliott-Halberstam conjecture or its generalisations). We can bound whenever one can find a symmetric square-integrable function supported on the simplex such that

Our strategy for establishing this has been to restrict to be a linear combination of symmetrised monomials (restricted of course to ), where the degree is small; actually, it seems convenient to work with the slightly different basis where the are restricted to be even. The criterion (1) then becomes a large quadratic program with explicit but complicated rational coefficients. This approach has lowered down to , which led to the bound .

Actually, we know that the more general criterion

will suffice, whenever and is supported now on and obeys the vanishing marginal condition whenever . The latter is in particular obeyed when is supported on . A modification of the preceding strategy has lowered slightly to , giving the bound which is currently our best record.

However, the quadratic programs here have become extremely large and slow to run, and more efficient algorithms (or possibly more computer power) may be required to advance further.

Polymath8b, VIII: Time to start writing up the results?

This is the eighth thread for the Polymath8b project to obtain new bounds for the quantity

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page.

The big news since the last thread is that we have managed to obtain the (sieve-theoretically) optimal bound of assuming the generalised Elliott-Halberstam conjecture (GEH), which pretty much closes off that part of the story. Unconditionally, our bound on is still . This bound was obtained using the “vanilla” Maynard sieve, in which the cutoff was supported in the original simplex , and only Bombieri-Vinogradov was used. In principle, we can enlarge the sieve support a little bit further now; for instance, we can enlarge to , but then have to shrink the J integrals to , provided that the marginals vanish for . However, we do not yet know how to numerically work with these expanded problems.

Given the substantial progress made so far, it looks like we are close to the point where we should declare victory and write up the results (though we should take one last look to see if there is any room to improve the bounds). There is actually a fair bit to write up:

• Improvements to the Maynard sieve (pushing beyond the simplex, the epsilon trick, and pushing beyond the cube);
• Asymptotic bounds for and hence ;
• Explicit bounds for (using the Polymath8a results)
• ;
• on GEH (and parity obstructions to any further improvement).

I will try to create a skeleton outline of such a paper in the Polymath8 Dropbox folder soon. It shouldn’t be nearly as big as the Polymath8a paper, but it will still be quite sizeable.

“New equidistribution estimates of Zhang type, and bounded gaps between primes” – and a retrospective

There are multiple purposes to this blog post.

The first purpose is to announce the uploading of the paper “New equidistribution estimates of Zhang type, and bounded gaps between primes” by D.H.J. Polymath, which is the main output of the Polymath8a project on bounded gaps between primes, to the arXiv, and to describe the main results of this paper below the fold.

The second purpose is to roll over the previous thread on all remaining Polymath8a-related matters (e.g. updates on the submission status of the paper) to a fresh thread. (Discussion of the ongoing Polymath8b project is however being kept on a separate thread, to try to reduce confusion.)

The final purpose of this post is to coordinate the writing of a retrospective article on the Polymath8 experience, which has been solicited for the Newsletter of the European Mathematical Society. I suppose that this could encompass both the Polymath8a and Polymath8b projects, even though the second one is still ongoing (but I think we will soon be entering the endgame there). I think there would be two main purposes of such a retrospective article. The first one would be to tell a story about the process of conducting mathematical research, rather than just describe the outcome of such research; this is an important aspect of the subject which is given almost no attention in most mathematical writing, and it would be good to be able to capture some sense of this process while memories are still relatively fresh. The other would be to draw some tentative conclusions with regards to what the strengths and weaknesses of a Polymath project are, and how appropriate such a format would be for other mathematical problems than bounded gaps between primes. In my opinion, the bounded gaps problem had some fairly unique features that made it particularly amenable to a Polymath project, such as (a) a high level of interest amongst the mathematical community in the problem; (b) a very focused objective (“improve !”), which naturally provided an obvious metric to measure progress; (c) the modular nature of the project, which allowed for people to focus on one aspect of the problem only, and still make contributions to the final goal; and (d) a very reasonable level of ambition (for instance, we did not attempt to prove the twin prime conjecture, which in my opinion would make a terrible Polymath project at our current level of mathematical technology). This is not an exhaustive list of helpful features of the problem; I would welcome other diagnoses of the project by other participants.

With these two objectives in mind, I propose a format for the retrospective article consisting of a brief introduction to the polymath concept in general and the polymath8 project in particular, followed by a collection of essentially independent contributions by different participants on their own experiences and thoughts. Finally we could have a conclusion section in which we make some general remarks on the polymath project (such as the remarks above). I’ve started a dropbox subfolder for this article (currently in a very skeletal outline form only), and will begin writing a section on my own experiences; other participants are of course encouraged to add their own sections (it is probably best to create separate files for these, and then input them into the main file retrospective.tex, to reduce edit conflicts. If there are participants who wish to contribute but do not currently have access to the Dropbox folder, please email me and I will try to have you added (or else you can supply your thoughts by email, or in the comments to this post; we may have a section for shorter miscellaneous comments from more casual participants, for people who don’t wish to write a lengthy essay on the subject).

As for deadlines, the EMS Newsletter would like a submitted article by mid-April in order to make the June issue, but in the worst case, it will just be held over until the issue after that.

Finite time blowup for an averaged three-dimensional Navier-Stokes equation

I’ve just uploaded to the arXiv the paper “Finite time blowup for an averaged three-dimensional Navier-Stokes equation“, submitted to J. Amer. Math. Soc.. The main purpose of this paper is to formalise the “supercriticality barrier” for the global regularity problem for the Navier-Stokes equation, which roughly speaking asserts that it is not possible to establish global regularity by any “abstract” approach which only uses upper bound function space estimates on the nonlinear part of the equation, combined with the energy identity. This is done by constructing a modification of the Navier-Stokes equations with a nonlinearity that obeys essentially all of the function space estimates that the true Navier-Stokes nonlinearity does, and which also obeys the energy identity, but for which one can construct solutions that blow up in finite time. Results of this type had been previously established by Montgomery-Smith, Gallagher-Paicu, and Li-Sinai for variants of the Navier-Stokes equation without the energy identity, and by Katz-Pavlovic and by Cheskidov for dyadic analogues of the Navier-Stokes equations in five and higher dimensions that obeyed the energy identity (see also the work of Plechac and Sverak and of Hou and Lei that also suggest blowup for other Navier-Stokes type models obeying the energy identity in five and higher dimensions), but to my knowledge this is the first blowup result for a Navier-Stokes type equation in three dimensions that also obeys the energy identity. Intriguingly, the method of proof in fact hints at a possible route to establishing blowup for the true Navier-Stokes equations, which I am now increasingly inclined to believe is the case (albeit for a very small set of initial data).

To state the results more precisely, recall that the Navier-Stokes equations can be written in the form

for a divergence-free velocity field and a pressure field , where is the viscosity, which we will normalise to be one. We will work in the non-periodic setting, so the spatial domain is , and for sake of exposition I will not discuss matters of regularity or decay of the solution (but we will always be working with strong notions of solution here rather than weak ones). Applying the Leray projection to divergence-free vector fields to this equation, we can eliminate the pressure, and obtain an evolution equation

purely for the velocity field, where is a certain bilinear operator on divergence-free vector fields (specifically, . The global regularity problem for Navier-Stokes is then equivalent to the global regularity problem for the evolution equation (1).

An important feature of the bilinear operator appearing in (1) is the cancellation law

(using the inner product on divergence-free vector fields), which leads in particular to the fundamental energy identity

This identity (and its consequences) provide essentially the only known a priori bound on solutions to the Navier-Stokes equations from large data and arbitrary times. Unfortunately, as discussed in this previous post, the quantities controlled by the energy identity are supercritical with respect to scaling, which is the fundamental obstacle that has defeated all attempts to solve the global regularity problem for Navier-Stokes without any additional assumptions on the data or solution (e.g. perturbative hypotheses, or a priori control on a critical norm such as the norm).

Our main result is then (slightly informally stated) as follows

Theorem 1 There exists an averaged version of the bilinear operator , of the form

for some probability space , some spatial rotation operators for , and some Fourier multipliers of order , for which one still has the cancellation law

and for which the averaged Navier-Stokes equation

admits solutions that blow up in finite time.

(There are some integrability conditions on the Fourier multipliers required in the above theorem in order for the conclusion to be non-trivial, but I am omitting them here for sake of exposition.)

Because spatial rotations and Fourier multipliers of order are bounded on most function spaces, automatically obeys almost all of the upper bound estimates that does. Thus, this theorem blocks any attempt to prove global regularity for the true Navier-Stokes equations which relies purely on the energy identity and on upper bound estimates for the nonlinearity; one must use some additional structure of the nonlinear operator which is not shared by an averaged version . Such additional structure certainly exists – for instance, the Navier-Stokes equation has a vorticity formulation involving only differential operators rather than pseudodifferential ones, whereas a general equation of the form (2) does not. However, “abstract” approaches to global regularity generally do not exploit such structure, and thus cannot be used to affirmatively answer the Navier-Stokes problem.

It turns out that the particular averaged bilinear operator that we will use will be a finite linear combination of local cascade operators, which take the form

where is a small parameter, are Schwartz vector fields whose Fourier transform is supported on an annulus, and is an -rescaled version of (basically a “wavelet” of wavelength about centred at the origin). Such operators were essentially introduced by Katz and Pavlovic as dyadic models for ; they have the essentially the same scaling property as (except that one can only scale along powers of , rather than over all positive reals), and in fact they can be expressed as an average of in the sense of the above theorem, as can be shown after a somewhat tedious amount of Fourier-analytic symbol manipulations.

If we consider nonlinearities which are a finite linear combination of local cascade operators, then the equation (2) more or less collapses to a system of ODE in certain “wavelet coefficients” of . The precise ODE that shows up depends on what precise combination of local cascade operators one is using. Katz and Pavlovic essentially considered a single cascade operator together with its “adjoint” (needed to preserve the energy identity), and arrived (more or less) at the system of ODE

where are scalar fields for each integer . (Actually, Katz-Pavlovic worked with a technical variant of this particular equation, but the differences are not so important for this current discussion.) Note that the quadratic terms on the RHS carry a higher exponent of than the dissipation term; this reflects the supercritical nature of this evolution (the energy is monotone decreasing in this flow, so the natural size of given the control on the energy is ). There is a slight technical issue with the dissipation if one wishes to embed (3) into an equation of the form (2), but it is minor and I will not discuss it further here.

In principle, if the mode has size comparable to at some time , then energy should flow from to at a rate comparable to , so that by time or so, most of the energy of should have drained into the mode (with hardly any energy dissipated). Since the series is summable, this suggests finite time blowup for this ODE as the energy races ever more quickly to higher and higher modes. Such a scenario was indeed established by Katz and Pavlovic (and refined by Cheskidov) if the dissipation strength was weakened somewhat (the exponent has to be lowered to be less than ). As mentioned above, this is enough to give a version of Theorem 1 in five and higher dimensions.

On the other hand, it was shown a few years ago by Barbato, Morandin, and Romito that (3) in fact admits global smooth solutions (at least in the dyadic case , and assuming non-negative initial data). Roughly speaking, the problem is that as energy is being transferred from to , energy is also simultaneously being transferred from to , and as such the solution races off to higher modes a bit too prematurely, without absorbing all of the energy from lower modes. This weakens the strength of the blowup to the point where the moderately strong dissipation in (3) is enough to kill the high frequency cascade before a true singularity occurs. Because of this, the original Katz-Pavlovic model cannot quite be used to establish Theorem 1 in three dimensions. (Actually, the original Katz-Pavlovic model had some additional dispersive features which allowed for another proof of global smooth solutions, which is an unpublished result of Nazarov.)

To get around this, I had to “engineer” an ODE system with similar features to (3) (namely, a quadratic nonlinearity, a monotone total energy, and the indicated exponents of for both the dissipation term and the quadratic terms), but for which the cascade of energy from scale to scale was not interrupted by the cascade of energy from scale to scale . To do this, I needed to insert a delay in the cascade process (so that after energy was dumped into scale , it would take some time before the energy would start to transfer to scale ), but the process also needed to be abrupt (once the process of energy transfer started, it needed to conclude very quickly, before the delayed transfer for the next scale kicked in). It turned out that one could build a “quadratic circuit” out of some basic “quadratic gates” (analogous to how an electrical circuit could be built out of basic gates such as amplifiers or resistors) that achieved this task, leading to an ODE system essentially of the form

where is a suitable large parameter and is a suitable small parameter (much smaller than ). To visualise the dynamics of such a system, I found it useful to describe this system graphically by a “circuit diagram” that is analogous (but not identical) to the circuit diagrams arising in electrical engineering:

The coupling constants here range widely from being very large to very small; in practice, this makes the and modes absorb very little energy, but exert a sizeable influence on the remaining modes. If a lot of energy is suddenly dumped into , what happens next is roughly as follows: for a moderate period of time, nothing much happens other than a trickle of energy into , which in turn causes a rapid exponential growth of (from a very low base). After this delay, suddenly crosses a certain threshold, at which point it causes and to exchange energy back and forth with extreme speed. The energy from then rapidly drains into , and the process begins again (with a slight loss in energy due to the dissipation). If one plots the total energy as a function of time, it looks schematically like this:

As in the previous heuristic discussion, the time between cascades from one frequency scale to the next decay exponentially, leading to blowup at some finite time . (One could describe the dynamics here as being similar to the famous “lighting the beacons” scene in the Lord of the Rings movies, except that (a) as each beacon gets ignited, the previous one is extinguished, as per the energy identity; (b) the time between beacon lightings decrease exponentially; and (c) there is no soundtrack.)

There is a real (but remote) possibility that this sort of construction can be adapted to the true Navier-Stokes equations. The basic blowup mechanism in the averaged equation is that of a von Neumann machine, or more precisely a construct (built within the laws of the inviscid evolution ) that, after some time delay, manages to suddenly create a replica of itself at a finer scale (and to largely erase its original instantiation in the process). In principle, such a von Neumann machine could also be built out of the laws of the inviscid form of the Navier-Stokes equations (i.e. the Euler equations). In physical terms, one would have to build the machine purely out of an ideal fluid (i.e. an inviscid incompressible fluid). If one could somehow create enough “logic gates” out of ideal fluid, one could presumably build a sort of “fluid computer”, at which point the task of building a von Neumann machine appears to reduce to a software engineering exercise rather than a PDE problem (providing that the gates are suitably stable with respect to perturbations, but (as with actual computers) this can presumably be done by converting the analog signals of fluid mechanics into a more error-resistant digital form). The key thing missing in this program (in both senses of the word) to establish blowup for Navier-Stokes is to construct the logic gates within the laws of ideal fluids. (Compare with the situation for cellular automata such as Conway’s “Game of Life“, in which Turing complete computers, universal constructors, and replicators have all been built within the laws of that game.)

Polymath8b, VII: Using the generalised Elliott-Halberstam hypothesis to enlarge the sieve support yet further

This is the seventh thread for the Polymath8b project to obtain new bounds for the quantity

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page.

The current focus is on improving the upper bound on under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from to . Very recently, we have been able to exploit GEH more fully, leading to a promising new expansion of the sieve support region. The problem now reduces to the following:

Problem 1 Does there exist a (not necessarily convex) polytope with quantities , and a non-trivial square-integrable function supported on such that

• 
• when ;
• when ;
• when ;

and such that we have the inequality

An affirmative answer to this question will imply on GEH. We are “within two percent” of this claim; we cannot quite reach yet, but have got as far as . However, we have not yet fully optimised in the above problem. In particular, the simplex

is now available, and should lead to some noticeable improvement in the numerology.

There is also a very slim chance that the twin prime conjecture is now provable on GEH. It would require an affirmative solution to the following problem:

Problem 2 Does there exist a (not necessarily convex) polytope with quantities , and a non-trivial square-integrable function supported on such that

• 

  

• when ;
• when ;

and such that we have the inequality

We suspect that the answer to this question is negative, but have not formally ruled it out yet.

For the rest of this post, I will justify why positive answers to these sorts of variational problems are sufficient to get bounds on (or more generally ).

Polymath8b, VI: A low dimensional variational problem

This is the sixth thread for the Polymath8b project to obtain new bounds for the quantity

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page (which has recently returned to full functionality, after a partial outage).

The current focus is on improving the upper bound on under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from to , which looks to be the limit of the method (see this previous comment for a semi-rigorous reason as to why is not possible with this method). With the most general Selberg sieve available, the problem reduces to the following three-dimensional variational one:

Problem 1 Does there exist a (not necessarily convex) polytope with quantities , and a non-trivial square-integrable function supported on such that

• 
• when ;
• when ;
• when ;

and such that we have the inequality

(Initially it was assumed that was convex, but we have now realised that this is not necessary.)

An affirmative answer to this question will imply on GEH. We are “within almost two percent” of this claim; we cannot quite reach yet, but have got as far as . However, we have not yet fully optimised in the above problem.

The most promising route so far is to take the symmetric polytope

with symmetric as well, and (we suspect that the optimal will be roughly ). (However, it is certainly worth also taking a look at easier model problems, such as the polytope , which has no vanishing marginal conditions to contend with; more recently we have been looking at the non-convex polytope .) Some further details of this particular case are given below the fold.

There should still be some progress to be made in the other regimes of interest – the unconditional bound on (currently at ), and on any further progress in asymptotic bounds for for larger – but the current focus is certainly on the bound on on GEH, as we seem to be tantalisingly close to an optimal result here.

Polymath8b, V: Stretching the sieve support further

This is the fifth thread for the Polymath8b project to obtain new bounds for the quantity

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page (which has recently returned to full functionality, after a partial outage). In particular, the upper bound for has been shaved a little from to , and we have very recently achieved the bound on the generalised Elliott-Halberstam conjecture GEH, formulated as Conjecture 1 of this paper of Bombieri, Friedlander, and Iwaniec. We also have explicit bounds for for , both with and without the assumption of the Elliott-Halberstam conjecture, as well as slightly sharper asymptotics for the upper bound for as .

The basic strategy for bounding still follows the general paradigm first laid out by Goldston, Pintz, Yildirim: given an admissible -tuple , one needs to locate a non-negative sieve weight , supported on an interval for a large , such that the ratio

is asymptotically larger than as ; this will show that . Thus one wants to locate a sieve weight for which one has good lower bounds on the numerator and good upper bounds on the denominator.

One can modify this paradigm slightly, for instance by adding the additional term to the numerator, or by subtracting the term from the numerator (which allows one to reduce the bound to ); however, the numerical impact of these tweaks have proven to be negligible thus far.

Despite a number of experiments with other sieves, we are still relying primarily on the Selberg sieve

where is the divisor sum

with , is the level of distribution ( if relying on Bombieri-Vinogradov, if assuming Elliott-Halberstam, and (in principle) if using Polymath8a technology), and is a smooth, compactly supported function. Most of the progress has come by enlarging the class of cutoff functions one is permitted to use.

The baseline bounds for the numerator and denominator in (1) (as established for instance in this previous post) are as follows. If is supported on the simplex

and we define the mixed partial derivative by

then the denominator in (1) is

where

and

Similarly, the numerator of (1) is

where

Thus, if we let be the supremum of the ratio

whenever is supported on and is non-vanishing, then one can prove whenever

We can improve this baseline in a number of ways. Firstly, with regards to the denominator in (1), if one upgrades the Elliott-Halberstam hypothesis to the generalised Elliott-Halberstam hypothesis (currently known for , thanks to Motohashi, but conjectured for ), the asymptotic (2) holds under the more general hypothesis that is supported in a polytope , as long as obeys the inclusion

examples of polytopes obeying this constraint include the modified simplex

the prism

the dilated simplex

and the truncated simplex

See this previous post for a proof of these claims.

With regards to the numerator, the asymptotic (3) is valid whenever, for each , the marginals vanish outside of . This is automatic if is supported on , or on the slightly larger region , but is an additional constraint when is supported on one of the other polytopes mentioned above.

More recently, we have obtained a more flexible version of the above asymptotic: if the marginals vanish outside of for some , then the numerator of (1) has a lower bound of

where

A proof is given here. Putting all this together, we can conclude

Theorem 1 Suppose we can find and a function supported on a polytope obeying (4), not identically zero and with all marginals vanishing outside of , and with

Then implies .

In principle, this very flexible criterion for upper bounding should lead to better bounds than before, and in particular we have now established on GEH.

Another promising direction is to try to improve the analysis at medium (more specifically, in the regime ), which is where we are currently at without EH or GEH through numerical quadratic programming. Right now we are only using and using the baseline analysis, basically for two reasons:

• We do not have good numerical formulae for integrating polynomials on any region more complicated than the simplex in medium dimension.
• The estimates produced by Polymath8a involve a parameter, which introduces additional restrictions on the support of (conservatively, it restricts to where and ; it should be possible to be looser than this (as was done in Polymath8a) but this has not been fully explored yet). This then triggers the previous obstacle of having to integrate on something other than a simplex.

However, these look like solvable problems, and so I would expect that further unconditional improvement for should be possible.