255B, Notes 2: Onsager’s conjecture

We consider the incompressible Euler equations on the (Eulerian) torus , which we write in divergence form as

 

 

where is the (inverse) Euclidean metric. Here we use the summation conventions for indices such as (reserving the symbol for other purposes), and are retaining the convention from Notes 1 of denoting vector fields using superscripted indices rather than subscripted indices, as we will eventually need to change variables to Lagrangian coordinates at some point. In principle, much of the discussion in this set of notes (particularly regarding the positive direction of Onsager’s conjecture) could also be modified to also treat non-periodic solutions that decay at infinity if desired, but some non-trivial technical issues do arise non-periodic settings for the negative direction.

As noted previously, the kinetic energy

is formally conserved by the flow, where is the Euclidean metric. Indeed, if one assumes that are continuously differentiable in both space and time on , then one can multiply the equation (1) by and contract against to obtain

which rearranges using (2) and the product rule to

and then if one integrates this identity on and uses Stokes’ theorem, one obtains the required energy conservation law

 

It is then natural to ask whether the energy conservation law continues to hold for lower regularity solutions, in particular weak solutions that only obey (1), (2) in a distributional sense. The above argument no longer works as stated, because is not a test function and so one cannot immediately integrate (1) against . And indeed, as we shall soon see, it is now known that once the regularity of is low enough, energy can “escape to frequency infinity”, leading to failure of the energy conservation law, a phenomenon known in physics as anomalous energy dissipation.

But what is the precise level of regularity needed in order to for this anomalous energy dissipation to occur? To make this question precise, we need a quantitative notion of regularity. One such measure is given by the Hölder space for , defined as the space of continuous functions whose norm

is finite. The space lies between the space of continuous functions and the space of continuously differentiable functions, and informally describes a space of functions that is “ times differentiable” in some sense. The above derivation of the energy conservation law involved the integral

that roughly speaking measures the fluctuation in energy. Informally, if we could take the derivative in this integrand and somehow “integrate by parts” to split the derivative “equally” amongst the three factors, one would morally arrive at an expression that resembles

which suggests that the integral can be made sense of for once . More precisely, one can make

Conjecture 1 (Onsager’s conjecture) Let and , and let .

• (i) If , then any weak solution to the Euler equations (in the Leray form ) obeys the energy conservation law (3).
• (ii) If , then there exist weak solutions to the Euler equations (in Leray form) which do not obey energy conservation.

This conjecture was originally arrived at by Onsager by a somewhat different heuristic derivation; see Remark 7. The numerology is also compatible with that arising from the Kolmogorov theory of turbulence (discussed in this previous post), but we will not discuss this interesting connection further here.

The positive part (i) of Onsager conjecture was established by Constantin, E, and Titi, building upon earlier partial results by Eyink; the proof is a relatively straightforward application of Littlewood-Paley theory, and they were also able to work in larger function spaces than (using -based Besov spaces instead of Hölder spaces, see Exercise 3 below). The negative part (ii) is harder. Discontinuous weak solutions to the Euler equations that did not conserve energy were first constructed by Sheffer, with an alternate construction later given by Shnirelman. De Lellis and Szekelyhidi noticed the resemblance of this problem to that of the Nash-Kuiper theorem in the isometric embedding problem, and began adapting the convex integration technique used in that theorem to construct weak solutions of the Euler equations. This began a long series of papers in which increasingly regular weak solutions that failed to conserve energy were constructed, culminating in a recent paper of Isett establishing part (ii) of the Onsager conjecture in the non-endpoint case in three and higher dimensions ; the endpoint remains open. (In two dimensions it may be the case that the positive results extend to a larger range than Onsager’s conjecture predicts; see this paper of Cheskidov, Lopes Filho, Nussenzveig Lopes, and Shvydkoy for more discussion.) Further work continues into several variations of the Onsager conjecture, in which one looks at other differential equations, other function spaces, or other criteria for bad behavior than breakdown of energy conservation. See this recent survey of de Lellis and Szekelyhidi for more discussion.

In these notes we will first establish (i), then discuss the convex integration method in the original context of the Nash-Kuiper embedding theorem. Before tackling the Onsager conjecture (ii) directly, we discuss a related construction of high-dimensional weak solutions in the Sobolev space for close to , which is slightly easier to establish, though still rather intricate. Finally, we discuss the modifications of that construction needed to establish (ii), though we shall stop short of a full proof of that part of the conjecture.

We thank Phil Isett for some comments and corrections.

Jean Bourgain

I have just learned that Jean Bourgain passed away last week in Belgium, aged 64, after a prolonged battle with cancer.  He and Eli Stein were the two mathematicians who most influenced my early career; it is something of a shock to find out that they are now both gone, having died within a few days of each other.

Like Eli, Jean remained highly active mathematically, even after his cancer diagnosis.  Here is a video profile of him by National Geographic, on the occasion of his 2017 Breakthrough Prize in Mathematics, doing a surprisingly good job of describing in lay terms the sort of mathematical work he did:

 

When I was a graduate student in Princeton, Tom Wolff came and gave a course on recent progress on the restriction and Kakeya conjectures, starting from the breakthrough work of Jean Bourgain in a now famous 1991 paper in Geom. Func. Anal..  I struggled with that paper for many months; it was by far the most difficult paper I had to read as a graduate student, as Jean would focus on the most essential components of an argument, treating more secondary details (such as rigorously formalising the uncertainty principle) in very brief sentences.  This image of my own annotated photocopy of this article may help convey some of the frustration I had when first going through it:

Eventually, though, and with the help of Eli Stein and Tom Wolff, I managed to decode the steps which had mystified me – and my impression of the paper reversed completely.  I began to realise that Jean had a certain collection of tools, heuristics, and principles that he regarded as “basic”, such as dyadic decomposition and the uncertainty principle, and by working “modulo” these tools (that is, by regarding any step consisting solely of application of these tools as trivial), one could proceed much more rapidly and efficiently.  By reading through Jean’s papers, I was able to add these tools to my own “basic” toolkit, which then became a fundamental starting point for much of my own research.  Indeed, a large fraction of my early work could be summarised as “take one of Jean’s papers, understand the techniques used there, and try to improve upon the final results a bit”.  In time, I started looking forward to reading the latest paper of Jean.  I remember being particularly impressed by his 1999 JAMS paper on global solutions of the energy-critical nonlinear Schrodinger equation for spherically symmetric data.  It’s hard to describe (especially in lay terms) the experience of reading through (and finally absorbing) the sections of this paper one by one; the best analogy I can come up with would be watching an expert video game player nimbly navigate his or her way through increasingly difficult levels of some video game, with the end of each level (or section) culminating in a fight with a huge “boss” that was eventually dispatched using an array of special weapons that the player happened to have at hand.  (I would eventually end up spending two years with four other coauthors trying to remove that spherical symmetry assumption; we did finally succeed, but it was and still is one of the most difficult projects I have been involved in.)

While I was a graduate student at Princeton, Jean worked at the Institute for Advanced Study which was just a mile away.  But I never actually had the courage to set up an appointment with him (which, back then, would be more likely done in person or by phone rather than by email). I remember once actually walking to the Institute and standing outside his office door, wondering if I dared knock on it to introduce myself.  (In the end I lost my nerve and walked back to the University.)

I think eventually Tom Wolff introduced the two of us to each other during one of Jean’s visits to Tom at Caltech (though I had previously seen Jean give a number of lectures at various places).  I had heard that in his younger years Jean had quite the competitive streak; however, when I met him, he was extremely generous with his ideas, and he had a way of condensing even the most difficult arguments to a few extremely information-dense sentences that captured the essence of the matter, which I invariably found to be particularly insightful (once I had finally managed to understand it).  He still retained a certain amount of cocky self-confidence though.  I remember posing to him (some time in early 2002, I think) a problem Tom Wolff had once shared with me about trying to prove what is now known as a sum-product estimate for subsets of a finite field of prime order, and telling him that Nets Katz and I would be able to use this estimate for several applications to Kakeya-type problems.  His initial reaction was to say that this estimate should easily follow from a Fourier analytic method, and promised me a proof the following morning.  The next day he came up to me and admitted that the problem was more interesting than he had initially expected, and that he would continue to think about it.  That was all I heard from him for several months; but one day I received a two-page fax from Jean with a beautiful hand-written proof of the sum-product estimate, which eventually became our joint paper with Nets on the subject (and the only paper I ended up writing with Jean).  Sadly, the actual fax itself has been lost despite several attempts from various parties to retrieve a copy, but a LaTeX version of the fax, typed up by Jean’s tireless assistant Elly Gustafsson, can be seen here.

About three years ago, Jean was diagnosed with cancer and began a fairly aggressive treatment.  Nevertheless he remained extraordinarily productive mathematically, authoring over thirty papers in the last three years, including such breakthrough results as his solution of the Vinogradov conjecture with Guth and Demeter, or his short note on the Schrodinger maximal function and his paper with Mirek, Stein, and Wróbel on dimension-free estimates for the Hardy-Littlewood maximal function, both of which made progress on problems that had been stuck for over a decade.  In May of 2016 I helped organise, and then attended, a conference at the IAS celebrating Jean’s work and impact; by then Jean was not able to easily travel to attend, but he gave a superb special lecture, not announced on the original schedule, via videoconference that was certainly one of the highlights of the meeting.  (UPDATE: a video of his talk is available here.  Thanks to Brad Rodgers for the link.)

I last met Jean in person in November of 2016, at the award ceremony for his Breakthrough Prize, though we had some email and phone conversations after that date.  Here he is with me and Richard Taylor at that event (demonstrating, among other things, that he wears a tuxedo much better than I do):

Jean was a truly remarkable person and mathematician.  Certainly the world of analysis is poorer with his passing.

[UPDATE, Dec 31: Here is the initial IAS obituary notice for Jean.]

[UPDATE, Jan 3: See also this MathOverflow question “Jean Bourgain’s Relatively Lesser Known Significant Contributions”.]

Polymath15, eleventh thread: Writing up the results, and exploring negative t

This is the eleventh research thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

There are currently two strands of activity.  One is writing up the paper describing the combination of theoretical and numerical results needed to obtain the new bound .  The latest version of the writeup may be found here, in this directory.  The theoretical side of things have mostly been written up; the main remaining tasks to do right now are

1. giving a more detailed description and illustration of the two major numerical verifications, namely the barrier verification that establishes a zero-free region for for , and the Dirichlet series bound that establishes a zero-free region for ; and
2. giving more detail on the conditional results assuming more numerical verification of RH.

Meanwhile, several of us have been exploring the behaviour of the zeroes of for negative ; this does not directly lead to any new progress on bounding (though there is a good chance that it may simplify the proof of ), but there have been some interesting numerical phenomena uncovered, as summarised in this set of slides.  One phenomenon is that for large negative , many of the complex zeroes begin to organise themselves near the curves

(An example of the agreement between the zeroes and these curves may be found here.)  We now have a (heuristic) theoretical explanation for this; we should have an approximation

in this region (where are defined in equations (11), (15), (17) of the writeup, and the above curves arise from (an approximation of) those locations where two adjacent terms , in this series have equal magnitude (with the other terms being of lower order).

However, we only have a partial explanation at present of the interesting behaviour of the real zeroes at negative t, for instance the surviving zeroes at extremely negative values of appear to lie on the curve where the quantity is close to a half-integer, where

The remaining zeroes exhibit a pattern in coordinates that is approximately 1-periodic in , where

A plot of the zeroes in these coordinates (somewhat truncated due to the numerical range) may be found here.

We do not yet have a total explanation of the phenomena seen in this picture.  It appears that we have an approximation

where is the non-zero multiplier

and

The derivation of this formula may be found in this wiki page.  However our initial attempts to simplify the above approximation further have proven to be somewhat inaccurate numerically (in particular giving an incorrect prediction for the location of zeroes, as seen in this picture).  We are in the process of using numerics to try to resolve the discrepancies (see this page for some code and discussion).

 

Elias Stein

I was deeply saddened to learn that Elias Stein died yesterday, aged 87.

I have talked about some of Eli’s older mathematical work in these blog posts.  He continued to be quite active mathematically in recent years, for instance finishing six papers (with various co-authors including Jean Bourgain, Mariusz Mirek, Błażej Wróbel, and Pavel Zorin-Kranich) in just this year alone.  I last met him at Wrocław, Poland last September for a conference in his honour; he was in good health (and good spirits) then.   Here is a picture of Eli together with several of his students (including myself) who were at that meeting (taken from the conference web site):

 Eli was an amazingly effective advisor; throughout my graduate studies I think he never had fewer than five graduate students, and there was often a line outside his door when he was meeting with students such as myself.   (The Mathematics Geneaology Project lists 52 students of Eli, but if anything this is an under-estimate.)  My weekly meetings with Eli would tend to go something like this: I would report on all the many different things I had tried over the past week, without much success, to solve my current research problem; Eli would listen patiently to everything I said, concentrate for a moment, and then go over to his filing cabinet and fish out a preprint to hand to me, saying “I think the authors in this paper encountered similar problems and resolved it using Method X”.  I would then go back to my office and read the preprint, and indeed they had faced something similar and I could often adapt the techniques there to resolve my immediate obstacles (only to encounter further ones for the next week, but that’s the way research tends to go, especially as a graduate student).  Amongst other things, these meetings impressed upon me the value of mathematical experience, by being able to make more key progress on a problem in a handful of minutes than I was able to accomplish in a whole week.  (There is a well known story about the famous engineer Charles Steinmetz fixing a broken piece of machinery by making a chalk mark; my meetings with Eli often had a similar feel to them.)

Eli’s lectures were always masterpieces of clarity.  In one hour, he would set up a theorem, motivate it, explain the strategy, and execute it flawlessly; even after twenty years of teaching my own classes, I have yet to figure out his secret of somehow always being able to arrive at the natural finale of a mathematical presentation at the end of each hour without having to improvise at least a little bit halfway during the lecture.  The clear and self-contained nature of his lectures (and his many books) were a large reason why I decided to specialise as a graduate student in harmonic analysis (though I would eventually return to other interests, such as analytic number theory, many years after my graduate studies).

Looking back at my time with Eli, I now realise that he was extraordinarily patient and understanding with the brash and naive teenager he had to meet with every week.  A key turning point in my own career came after my oral qualifying exams, in which I very nearly failed due to my overconfidence and lack of preparation, particularly in my chosen specialty of harmonic analysis.  After the exam, he sat down with me and told me, as gently and diplomatically as possible, that my performance was a disappointment, and that I seriously needed to solidify my mathematical knowledge.  This turned out to be exactly what I needed to hear; I got motivated to actually work properly so as not to disappoint my advisor again.

So many of us in the field of harmonic analysis were connected to Eli in one way or another; the field always felt to me like a large extended family, with Eli as one of the patriarchs.  He will be greatly missed.

[UPDATE: Here is Princeton’s obituary for Elias Stein.]

255B, Notes 1: The Lagrangian formulation of the Euler equations

These lecture notes are a continuation of the 254A lecture notes from the previous quarter.

We consider the Euler equations for incompressible fluid flow on a Euclidean space ; we will label as the “Eulerian space” (or “Euclidean space”, or “physical space”) to distinguish it from the “Lagrangian space” (or “labels space”) that we will introduce shortly (but the reader is free to also ignore the or subscripts if he or she wishes). Elements of Eulerian space will be referred to by symbols such as , we use to denote Lebesgue measure on and we will use for the coordinates of , and use indices such as to index these coordinates (with the usual summation conventions), for instance denotes partial differentiation along the coordinate. (We use superscripts for coordinates instead of subscripts to be compatible with some differential geometry notation that we will use shortly; in particular, when using the summation notation, we will now be matching subscripts with superscripts for the pair of indices being summed.)

In Eulerian coordinates, the Euler equations read

 

where is the velocity field and is the pressure field. These are functions of time and on the spatial location variable . We will refer to the coordinates as Eulerian coordinates. However, if one reviews the physical derivation of the Euler equations from 254A Notes 0, before one takes the continuum limit, the fundamental unknowns were not the velocity field or the pressure field , but rather the trajectories , which can be thought of as a single function from the coordinates (where is a time and is an element of the label set ) to . The relationship between the trajectories and the velocity field was given by the informal relationship

 

We will refer to the coordinates as (discrete) Lagrangian coordinates for describing the fluid.

In view of this, it is natural to ask whether there is an alternate way to formulate the continuum limit of incompressible inviscid fluids, by using a continuous version of the Lagrangian coordinates, rather than Eulerian coordinates. This is indeed the case. Suppose for instance one has a smooth solution to the Euler equations on a spacetime slab in Eulerian coordinates; assume furthermore that the velocity field is uniformly bounded. We introduce another copy of , which we call Lagrangian space or labels space; we use symbols such as to refer to elements of this space, to denote Lebesgue measure on , and to refer to the coordinates of . We use indices such as to index these coordinates, thus for instance denotes partial differentiation along the coordinate. We will use summation conventions for both the Eulerian coordinates and the Lagrangian coordinates , with an index being summed if it appears as both a subscript and a superscript in the same term. While and are of course isomorphic, we will try to refrain from identifying them, except perhaps at the initial time in order to fix the initialisation of Lagrangian coordinates.

Given a smooth and bounded velocity field , define a trajectory map for this velocity to be any smooth map that obeys the ODE

 

in view of (2), this describes the trajectory (in ) of a particle labeled by an element of . From the Picard existence theorem and the hypothesis that is smooth and bounded, such a map exists and is unique as long as one specifies the initial location assigned to each label . Traditionally, one chooses the initial condition

 

for , so that we label each particle by its initial location at time ; we are also free to specify other initial conditions for the trajectory map if we please. Indeed, we have the freedom to “permute” the labels by an arbitrary diffeomorphism: if is a trajectory map, and is any diffeomorphism (a smooth map whose inverse exists and is also smooth), then the map is also a trajectory map, albeit one with different initial conditions .

Despite the popularity of the initial condition (4), we will try to keep conceptually separate the Eulerian space from the Lagrangian space , as they play different physical roles in the interpretation of the fluid; for instance, while the Euclidean metric is an important feature of Eulerian space , it is not a geometrically natural structure to use in Lagrangian space . We have the following more general version of Exercise 8 from 254A Notes 2:

Exercise 1 Let be smooth and bounded.

• If is a smooth map, show that there exists a unique smooth trajectory map with initial condition for all .
• Show that if is a diffeomorphism and , then the map is also a diffeomorphism.

Remark 2 The first of the Euler equations (1) can now be written in the form

 

which can be viewed as a continuous limit of Newton’s first law .

Call a diffeomorphism (oriented) volume preserving if one has the equation

 

for all , where the total differential is the matrix with entries for and , where are the components of . (If one wishes, one can also view as a linear transformation from the tangent space of Lagrangian space at to the tangent space of Eulerian space at .) Equivalently, is orientation preserving and one has a Jacobian-free change of variables formula

for all , which is in turn equivalent to having the same Lebesgue measure as for any measurable set .

The divergence-free condition then can be nicely expressed in terms of volume-preserving properties of the trajectory maps , in a manner which confirms the interpretation of this condition as an incompressibility condition on the fluid:

Lemma 3 Let be smooth and bounded, let be a volume-preserving diffeomorphism, and let be the trajectory map. Then the following are equivalent:

• on .
• is volume-preserving for all .

Proof: Since is orientation-preserving, we see from continuity that is also orientation-preserving. Suppose that is also volume-preserving, then for any we have the conservation law

for all . Differentiating in time using the chain rule and (3) we conclude that

for all , and hence by change of variables

which by integration by parts gives

for all and , so is divergence-free.

To prove the converse implication, it is convenient to introduce the labels map , defined by setting to be the inverse of the diffeomorphism , thus

for all . By the implicit function theorem, is smooth, and by differentiating the above equation in time using (3) we see that

where is the usual material derivative

 

acting on functions on . If is divergence-free, we have from integration by parts that

for any test function . In particular, for any , we can calculate

and hence

for any . Since is volume-preserving, so is , thus

Thus is volume-preserving, and hence is also.

Exercise 4 Let be a continuously differentiable map from the time interval to the general linear group of invertible matrices. Establish Jacobi’s formula

and use this and (6) to give an alternate proof of Lemma 3 that does not involve any integration.

Remark 5 One can view the use of Lagrangian coordinates as an extension of the method of characteristics. Indeed, from the chain rule we see that for any smooth function of Eulerian spacetime, one has

and hence any transport equation that in Eulerian coordinates takes the form

for smooth functions of Eulerian spacetime is equivalent to the ODE

where are the smooth functions of Lagrangian spacetime defined by

In this set of notes we recall some basic differential geometry notation, particularly with regards to pullbacks and Lie derivatives of differential forms and other tensor fields on manifolds such as and , and explore how the Euler equations look in this notation. Our discussion will be entirely formal in nature; we will assume that all functions have enough smoothness and decay at infinity to justify the relevant calculations. (It is possible to work rigorously in Lagrangian coordinates – see for instance the work of Ebin and Marsden – but we will not do so here.) As a general rule, Lagrangian coordinates tend to be somewhat less convenient to use than Eulerian coordinates for establishing the basic analytic properties of the Euler equations, such as local existence, uniqueness, and continuous dependence on the data; however, they are quite good at clarifying the more algebraic properties of these equations, such as conservation laws and the variational nature of the equations. It may well be that in the future we will be able to use the Lagrangian formalism more effectively on the analytic side of the subject also.

Remark 6 One can also write the Navier-Stokes equations in Lagrangian coordinates, but the equations are not expressed in a favourable form in these coordinates, as the Laplacian appearing in the viscosity term becomes replaced with a time-varying Laplace-Beltrami operator. As such, we will not discuss the Lagrangian coordinate formulation of Navier-Stokes here.

254A, Supplemental: Weak solutions from the perspective of nonstandard analysis (optional)

Note: this post is not required reading for this course, or for the sequel course in the winter quarter.

In a Notes 2, we reviewed the classical construction of Leray of global weak solutions to the Navier-Stokes equations. We did not quite follow Leray’s original proof, in that the notes relied more heavily on the machinery of Littlewood-Paley projections, which have become increasingly common tools in modern PDE. On the other hand, we did use the same “exploiting compactness to pass to weakly convergent subsequence” strategy that is the standard one in the PDE literature used to construct weak solutions.

As I discussed in a previous post, the manipulation of sequences and their limits is analogous to a “cheap” version of nonstandard analysis in which one uses the Fréchet filter rather than an ultrafilter to construct the nonstandard universe. (The manipulation of generalised functions of Columbeau-type can also be comfortably interpreted within this sort of cheap nonstandard analysis.) Augmenting the manipulation of sequences with the right to pass to subsequences whenever convenient is then analogous to a sort of “lazy” nonstandard analysis, in which the implied ultrafilter is never actually constructed as a “completed object“, but is instead lazily evaluated, in the sense that whenever membership of a given subsequence of the natural numbers in the ultrafilter needs to be determined, one either passes to that subsequence (thus placing it in the ultrafilter) or the complement of the sequence (placing it out of the ultrafilter). This process can be viewed as the initial portion of the transfinite induction that one usually uses to construct ultrafilters (as discussed using a voting metaphor in this post), except that there is generally no need in any given application to perform the induction for any uncountable ordinal (or indeed for most of the countable ordinals also).

On the other hand, it is also possible to work directly in the orthodox framework of nonstandard analysis when constructing weak solutions. This leads to an approach to the subject which is largely equivalent to the usual subsequence-based approach, though there are some minor technical differences (for instance, the subsequence approach occasionally requires one to work with separable function spaces, whereas in the ultrafilter approach the reliance on separability is largely eliminated, particularly if one imposes a strong notion of saturation on the nonstandard universe). The subject acquires a more “algebraic” flavour, as the quintessential analysis operation of taking a limit is replaced with the “standard part” operation, which is an algebra homomorphism. The notion of a sequence is replaced by the distinction between standard and nonstandard objects, and the need to pass to subsequences disappears entirely. Also, the distinction between “bounded sequences” and “convergent sequences” is largely eradicated, particularly when the space that the sequences ranged in enjoys some compactness properties on bounded sets. Also, in this framework, the notorious non-uniqueness features of weak solutions can be “blamed” on the non-uniqueness of the nonstandard extension of the standard universe (as well as on the multiple possible ways to construct nonstandard mollifications of the original standard PDE). However, many of these changes are largely cosmetic; switching from a subsequence-based theory to a nonstandard analysis-based theory does not seem to bring one significantly closer for instance to the global regularity problem for Navier-Stokes, but it could have been an alternate path for the historical development and presentation of the subject.

In any case, I would like to present below the fold this nonstandard analysis perspective, quickly translating the relevant components of real analysis, functional analysis, and distributional theory that we need to this perspective, and then use it to re-prove Leray’s theorem on existence of global weak solutions to Navier-Stokes.

Fourier uniformity of bounded multiplicative functions in short intervals on average

Kaisa Matomäki, Maksym Radziwill, and I just uploaded to the arXiv our paper “Fourier uniformity of bounded multiplicative functions in short intervals on average“. This paper is the outcome of our attempts during the MSRI program in analytic number theory last year to attack the local Fourier uniformity conjecture for the Liouville function . This conjecture generalises a landmark result of Matomäki and Radziwill, who show (among other things) that one has the asymptotic

whenever and goes to infinity as . Informally, this says that the Liouville function has small mean for almost all short intervals . The remarkable thing about this theorem is that there is no lower bound on how goes to infinity with ; one can take for instance . This lack of lower bound was crucial when I applied this result (or more precisely, a generalisation of this result to arbitrary non-pretentious bounded multiplicative functions) a few years ago to solve the Erdös discrepancy problem, as well as a logarithmically averaged two-point Chowla conjecture, for instance it implies that

The local Fourier uniformity conjecture asserts the stronger asymptotic

under the same hypotheses on and . As I worked out in a previous paper, this conjecture would imply a logarithmically averaged three-point Chowla conjecture, implying for instance that

This particular bound also follows from some slightly different arguments of Joni Teräväinen and myself, but the implication would also work for other non-pretentious bounded multiplicative functions, whereas the arguments of Joni and myself rely more heavily on the specific properties of the Liouville function (in particular that for all primes ).

There is also a higher order version of the local Fourier uniformity conjecture in which the linear phase is replaced with a polynomial phase such as , or more generally a nilsequence ; as shown in my previous paper, this conjecture implies (and is in fact equivalent to, after logarithmic averaging) a logarithmically averaged version of the full Chowla conjecture (not just the two-point or three-point versions), as well as a logarithmically averaged version of the Sarnak conjecture.

The main result of the current paper is to obtain some cases of the local Fourier uniformity conjecture:

Theorem 1 The asymptotic (2) is true when for a fixed .

Previously this was known for by the work of Zhan (who in fact proved the stronger pointwise assertion for in this case). In a previous paper with Kaisa and Maksym, we also proved a weak version

of (2) for any growing arbitrarily slowly with ; this is stronger than (1) (and is in fact proven by a variant of the method) but significantly weaker than (2), because in the latter the worst-case is permitted to depend on the parameter, whereas in (3) must remain independent of .

Unfortunately, the restriction is not strong enough to give applications to Chowla-type conjectures (one would need something more like for this). However, it can still be used to control some sums that had not previously been manageable. For instance, a quick application of the circle method lets one use the above theorem to derive the asymptotic

whenever for a fixed , where is the von Mangoldt function. Amusingly, the seemingly simpler question of establishing the expected asymptotic for

is only known in the range (from the work of Zaccagnini). Thus we have a rare example of a number theory sum that becomes easier to control when one inserts a Liouville function!

We now give an informal description of the strategy of proof of the theorem (though for numerous technical reasons, the actual proof deviates in some respects from the description given here). If (2) failed, then for many values of we would have the lower bound

for some frequency . We informally describe this correlation between and by writing

for (informally, one should view this as asserting that “behaves like” a constant multiple of ). For sake of discussion, suppose we have this relationship for all , not just many.

As mentioned before, the main difficulty here is to understand how varies with . As it turns out, the multiplicativity properties of the Liouville function place a significant constraint on this dependence. Indeed, if we let be a fairly small prime (e.g. of size for some ), and use the identity for the Liouville function to conclude (at least heuristically) from (4) that

for . (In practice, we will have this sort of claim for many primes rather than all primes , after using tools such as the Turán-Kubilius inequality, but we ignore this distinction for this informal argument.)

Now let and be primes comparable to some fixed range such that

Then we have both

and

on essentially the same range of (two nearby intervals of length ). This suggests that the frequencies and should be close to each other modulo , in particular one should expect the relationship

Comparing this with (5) one is led to the expectation that should depend inversely on in some sense (for instance one can check that

would solve (6) if ; by Taylor expansion, this would correspond to a global approximation of the form ). One now has a problem of an additive combinatorial flavour (or of a “local to global” flavour), namely to leverage the relation (6) to obtain global control on that resembles (7).

A key obstacle in solving (6) efficiently is the fact that one only knows that and are close modulo , rather than close on the real line. One can start resolving this problem by the Chinese remainder theorem, using the fact that we have the freedom to shift (say) by an arbitrary integer. After doing so, one can arrange matters so that one in fact has the relationship

whenever and obey (5). (This may force to become extremely large, on the order of , but this will not concern us.)

Now suppose that we have and primes such that

For every prime , we can find an such that is within of both and . Applying (8) twice we obtain

and

and thus by the triangle inequality we have

for all ; hence by the Chinese remainder theorem

In practice, in the regime that we are considering, the modulus is so huge we can effectively ignore it (in the spirit of the Lefschetz principle); so let us pretend that we in fact have

whenever and obey (9).

Now let be an integer to be chosen later, and suppose we have primes such that the difference

is small but non-zero. If is chosen so that

(where one is somewhat loose about what means) then one can then find real numbers such that

for , with the convention that . We then have

which telescopes to

and thus

and hence

In particular, for each , we expect to be able to write

for some . This quantity can vary with ; but from (10) and a short calculation we see that

whenever obey (9) for some .

Now imagine a “graph” in which the vertices are elements of , and two elements are joined by an edge if (9) holds for some . Because of exponential sum estimates on , this graph turns out to essentially be an “expander” in the sense that any two vertices can be connected (in multiple ways) by fairly short paths in this graph (if one allows one to modify one of or by ). As a consequence, we can assume that this quantity is essentially constant in (cf. the application of the ergodic theorem in this previous blog post), thus we now have

for most and some . By Taylor expansion, this implies that

on for most , thus

But this can be shown to contradict the Matomäki-Radziwill theorem (because the multiplicative function is known to be non-pretentious).

Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion

I’ve just uploaded to the arXiv my paper “Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion“, submitted to Revista Matematica Iberoamericana. This paper concerns the extent to which one can accurately embed the metric structure of the Heisenberg group

into Euclidean space, which we can write as with the notation

Here we give the right-invariant Carnot-Carathéodory metric coming from the right-invariant vector fields

but not from the commutator vector field

This gives the geometry of a Carnot group. As observed by Semmes, it follows from the Carnot group differentiation theory of Pansu that there is no bilipschitz map from to any Euclidean space or even to , since such a map must be differentiable almost everywhere in the sense of Carnot groups, which in particular shows that the derivative map annihilate almost everywhere, which is incompatible with being bilipschitz.

On the other hand, if one snowflakes the Heisenberg group by replacing the metric with for some , then it follows from the general theory of Assouad on embedding snowflaked metrics of doubling spaces that may be embedded in a bilipschitz fashion into , or even to for some depending on .

Of course, the distortion of this bilipschitz embedding must degenerate in the limit . From the work of Austin-Naor-Tessera and Naor-Neiman it follows that may be embedded into with a distortion of , but no better. The Naor-Neiman paper also embeds into a finite-dimensional space with independent of , but at the cost of worsening the distortion to . They then posed the question of whether this worsening of the distortion is necessary.

The main result of this paper answers this question in the negative:

Theorem 1 There exists an absolute constant such that may be embedded into in a bilipschitz fashion with distortion for any .

To motivate the proof of this theorem, let us first present a bilipschitz map from the snowflaked line (with being the usual metric on ) into complex Hilbert space . The map is given explicitly as a Weierstrass type function

where for each , is the function

and are an orthonormal basis for . The subtracting of the constant is purely in order to make the sum convergent as . If are such that for some integer , one can easily check the bounds

with the lower bound

at which point one finds that

as desired.

The key here was that each function oscillated at a different spatial scale , and the functions were all orthogonal to each other (so that the upper bound involved a factor of rather than ). One can replicate this example for the Heisenberg group without much difficulty. Indeed, if we let be the discrete Heisenberg group, then the nilmanifold is a three-dimensional smooth compact manifold; thus, by the Whitney embedding theorem, it smoothly embeds into . This gives a smooth immersion which is -automorphic in the sense that for all and . If one then defines to be the function

where is the scaling map

then one can repeat the previous arguments to obtain the required bilipschitz bounds

for the function

To adapt this construction to bounded dimension, the main obstruction was the requirement that the took values in orthogonal subspaces. But if one works things out carefully, it is enough to require the weaker orthogonality requirement

for all , where is the bilinear form

One can then try to construct the for bounded dimension by an iterative argument. After some standard reductions, the problem becomes this (roughly speaking): given a smooth, slowly varying function whose derivatives obey certain quantitative upper and lower bounds, construct a smooth oscillating function , whose derivatives also obey certain quantitative upper and lower bounds, which obey the equation

 

We view this as an underdetermined system of differential equations for (two equations in unknowns; after some reductions, our can be taken to be the explicit value ). The trivial solution to this equation will be inadmissible for our purposes due to the lower bounds we will require on (in order to obtain the quantitative immersion property mentioned previously, as well as for a stronger “freeness” property that is needed to close the iteration). Because this construction will need to be iterated, it will be essential that the regularity control on is the same as that on ; one cannot afford to “lose derivatives” when passing from to .

This problem has some formal similarities with the isometric embedding problem (discussed for instance in this previous post), which can be viewed as the problem of solving an equation of the form , where is a Riemannian manifold and is the bilinear form

The isometric embedding problem also has the key obstacle that naive attempts to solve the equation iteratively can lead to an undesirable “loss of derivatives” that prevents one from iterating indefinitely. This obstacle was famously resolved by the Nash-Moser iteration scheme in which one alternates between perturbatively adjusting an approximate solution to improve the residual error term, and mollifying the resulting perturbation to counteract the loss of derivatives. The current equation (1) differs in some key respects from the isometric embedding equation , in particular being linear in the unknown field rather than quadratic; nevertheless the key obstacle is the same, namely that naive attempts to solve either equation lose derivatives. Our approach to solving (1) was inspired by the Nash-Moser scheme; in retrospect, I also found similarities with Uchiyama’s constructive proof of the Fefferman-Stein decomposition theorem, discussed in this previous post (and in this recent one).

To motivate this iteration, we first express using the product rule in a form that does not place derivatives directly on the unknown :

 

This reveals that one can construct solutions to (1) by solving the system of equations

 

for . Because this system is zeroth order in , this can easily be done by linear algebra (even in the presence of a forcing term ) if one imposes a “freeness” condition (analogous to the notion of a free embedding in the isometric embedding problem) that are linearly independent at each point , which (together with some other technical conditions of a similar nature) one then adds to the list of upper and lower bounds required on (with a related bound then imposed on , in order to close the iteration). However, as mentioned previously, there is a “loss of derivatives” problem with this construction: due to the presence of the differential operators in (3), a solution constructed by this method can only be expected to have two degrees less regularity than at best, which makes this construction unsuitable for iteration.

To get around this obstacle (which also prominently appears when solving (linearisations of) the isometric embedding equation ), we instead first construct a smooth, low-frequency solution to a low-frequency equation

 

where is a mollification of (of Littlewood-Paley type) applied at a small spatial scale for some , and then gradually relax the frequency cutoff to deform this low frequency solution to a solution of the actual equation (1).

We will construct the low-frequency solution rather explicitly, using the Whitney embedding theorem to construct an initial oscillating map into a very low dimensional space , composing it with a Veronese type embedding into a slightly larger dimensional space to obtain a required “freeness” property, and then composing further with a slowly varying isometry depending on and constructed by a quantitative topological lemma (relying ultimately on the vanishing of the first few homotopy groups of high-dimensional spheres), in order to obtain the required orthogonality (4). (This sort of “quantitative null-homotopy” was first proposed by Gromov, with some recent progress on optimal bounds by Chambers-Manin-Weinberger and by Chambers-Dotterer-Manin-Weinberger, but we will not need these more advanced results here, as one can rely on the classical qualitative vanishing for together with a compactness argument to obtain (ineffective) quantitative bounds, which suffice for this application).

To perform the deformation of into , we must solve what is essentially the linearised equation

 

of (1) when , (viewed as low frequency functions) are both being deformed at some rates (which should be viewed as high frequency functions). To avoid losing derivatives, the magnitude of the deformation in should not be significantly greater than the magnitude of the deformation in , when measured in the same function space norms.

As before, if one directly solves the difference equation (5) using a naive application of (2) with treated as a forcing term, one will lose at least one derivative of regularity when passing from to . However, observe that (2) (and the symmetry ) can be used to obtain the identity

 

and then one can solve (5) by solving the system of equations

for . The key point here is that this system is zeroth order in both and , so one can solve this system without losing any derivatives when passing from to ; compare this situation with that of the superficially similar system

that one would obtain from naively linearising (3) without exploiting the symmetry of . There is still however one residual “loss of derivatives” problem arising from the presence of a differential operator on the term, which prevents one from directly evolving this iteration scheme in time without losing regularity in . It is here that we borrow the final key idea of the Nash-Moser scheme, which is to replace by a mollified version of itself (where the projection depends on the time parameter). This creates an error term in (5), but it turns out that this error term is quite small and smooth (being a “high-high paraproduct” of and , it ends up being far more regular than either or , even with the presence of the derivatives) and can be iterated away provided that the initial frequency cutoff is large and the function has a fairly high (but finite) amount of regularity (we will eventually use the Hölder space on the Heisenberg group to measure this).

The nonlinearity of the Fefferman-Stein decomposition

The celebrated decomposition theorem of Fefferman and Stein shows that every function of bounded mean oscillation can be decomposed in the form

 

modulo constants, for some , where are the Riesz transforms. A technical note here a function in BMO is defined only up to constants (as well as up to the usual almost everywhere equivalence); related to this, if is an function, then the Riesz transform is well defined as an element of , but is also only defined up to constants and almost everywhere equivalence.

The original proof of Fefferman and Stein was indirect (relying for instance on the Hahn-Banach theorem). A constructive proof was later given by Uchiyama, and was in fact the topic of the second post on this blog. A notable feature of Uchiyama’s argument is that the construction is quite nonlinear; the vector-valued function is defined to take values on a sphere, and the iterative construction to build these functions from involves repeatedly projecting a potential approximant to this function to the sphere (also, the high-frequency components of this approximant are constructed in a manner that depends nonlinearly on the low-frequency components, which is a type of technique that has become increasingly common in analysis and PDE in recent years).

It is natural to ask whether the Fefferman-Stein decomposition (1) can be made linear in , in the sense that each of the depend linearly on . Strictly speaking this is easily accomplished using the axiom of choice: take a Hamel basis of , choose a decomposition (1) for each element of this basis, and then extend linearly to all finite linear combinations of these basis functions, which then cover by definition of Hamel basis. But these linear operations have no reason to be continuous as a map from to . So the correct question is whether the decomposition can be made continuously linear (or equivalently, boundedly linear) in , that is to say whether there exist continuous linear transformations such that

 

modulo constants for all . Note from the open mapping theorem that one can choose the functions to depend in a bounded fashion on (thus for some constant , however the open mapping theorem does not guarantee linearity. Using a result of Bartle and Graves one can also make the depend continuously on , but again the dependence is not guaranteed to be linear.

It is generally accepted folklore that continuous linear dependence is known to be impossible, but I had difficulty recently tracking down an explicit proof of this assertion in the literature (if anyone knows of a reference, I would be glad to know of it). The closest I found was a proof of a similar statement in this paper of Bourgain and Brezis, which I was able to adapt to establish the current claim. The basic idea is to average over the symmetries of the decomposition, which in the case of (1) are translation invariance, rotation invariance, and dilation invariance. This effectively makes the operators invariant under all these symmetries, which forces them to themselves be linear combinations of the identity and Riesz transform operators; however, no such non-trivial linear combination maps to , and the claim follows. Formal details of this argument (which we phrase in a dual form in order to avoid some technicalities) appear below the fold.

254A, Notes 3: Local well-posedness for the Euler equations

We now turn to the local existence theory for the initial value problem for the incompressible Euler equations

For sake of discussion we will just work in the non-periodic domain , , although the arguments here can be adapted without much difficulty to the periodic setting. We will only work with solutions in which the pressure is normalised in the usual fashion:

Formally, the Euler equations (with normalised pressure) arise as the vanishing viscosity limit of the Navier-Stokes equations

that was studied in previous notes. However, because most of the bounds established in previous notes, either on the lifespan of the solution or on the size of the solution itself, depended on , it is not immediate how to justify passing to the limit and obtain either a strong well-posedness theory or a weak solution theory for the limiting equation (1). (For instance, weak solutions to the Navier-Stokes equations (or the approximate solutions used to create such weak solutions) have lying in for , but the bound on the norm is and so one could lose this regularity in the limit , at which point it is not clear how to ensure that the nonlinear term still converges in the sense of distributions to what one expects.)

Nevertheless, by carefully using the energy method (which we will do loosely following an approach of Bertozzi and Majda), it is still possible to obtain local-in-time estimates on (high-regularity) solutions to (3) that are uniform in the limit . Such a priori estimates can then be combined with a number of variants of these estimates obtain a satisfactory local well-posedness theory for the Euler equations. Among other things, we will be able to establish the Beale-Kato-Majda criterion – smooth solutions to the Euler (or Navier-Stokes) equations can be continued indefinitely unless the integral

becomes infinite at the final time , where is the vorticity field. The vorticity has the important property that it is transported by the Euler flow, and in two spatial dimensions it can be used to establish global regularity for both the Euler and Navier-Stokes equations in these settings. (Unfortunately, in three and higher dimensions the phenomenon of vortex stretching has frustrated all attempts to date to use the vorticity transport property to establish global regularity of either equation in this setting.)

There is a rather different approach to establishing local well-posedness for the Euler equations, which relies on the vorticity-stream formulation of these equations. This will be discused in a later set of notes.