- The NSF-CBMS regional research conferences are now requesting proposals for the 2020 conference series. (I was the principal lecturer for one of these conferences back in 2005; it was a very intensive experience, but quite enjoyable, and I am quite pleased with the book that resulted from it.)
- The awardees for the Sloan Fellowships for 2019 have now been announced. (I was on the committee for the mathematics awards. For the usual reasons involving the confidentiality of letters of reference and other sensitive information, I will be unfortunately be unable to answer any specific questions about our committee deliberations.)

In coordinates, the Euler equations read

where is the pressure field and is the velocity field, and denotes the Levi-Civita connection with the usual Penrose abstract index notation conventions; we restrict attention here to the case where are smooth and is compact, smooth, orientable, connected, and without boundary. Let’s call an *Euler flow* on (for the time interval ) if it solves the above system of equations for some pressure , and an *incompressible flow* if it just obeys the divergence-free relation . Thus every Euler flow is an incompressible flow, but the converse is certainly not true; for instance the various conservation laws of the Euler equation, such as conservation of energy, will already block most incompressible flows from being an Euler flow, or even being approximated in a reasonably strong topology by such Euler flows.

However, one can ask if an incompressible flow can be *extended* to an Euler flow by adding some additional dimensions to . In my paper, I formalise this by considering warped products of which (as a smooth manifold) are products of with a torus, with a metric given by

for , where are the coordinates of the torus , and are smooth positive coefficients for ; in order to preserve the incompressibility condition, we also require the volume preservation property

though in practice we can quickly dispose of this condition by adding one further “dummy” dimension to the torus . We say that an incompressible flow is *extendible to an Euler flow* if there exists a warped product extending , and an Euler flow on of the form

for some “swirl” fields . The situation here is motivated by the familiar situation of studying axisymmetric Euler flows on , which in cylindrical coordinates take the form

The base component

of this flow is then a flow on the two-dimensional plane which is not quite incompressible (due to the failure of the volume preservation condition (2) in this case) but still satisfies a system of equations (coupled with a passive scalar field that is basically the square of the swirl ) that is reminiscent of the Boussinesq equations.

On a fixed -dimensional manifold , let denote the space of incompressible flows , equipped with the smooth topology (in spacetime), and let denote the space of such flows that are extendible to Euler flows. Our main theorem is

Theorem 1

- (i) (Generic inextendibility) Assume . Then is of the first category in (the countable union of nowhere dense sets in ).
- (ii) (Non-rigidity) Assume (with an arbitrary metric ). Then is somewhere dense in (that is, the closure of has non-empty interior).

More informally, starting with an incompressible flow , one usually cannot extend it to an Euler flow just by extending the manifold, warping the metric, and adding swirl coefficients, even if one is allowed to select the dimension of the extension, as well as the metric and coefficients, arbitrarily. However, many such flows can be *perturbed* to be extendible in such a manner (though different perturbations will require different extensions, in particular the dimension of the extension will not be fixed). Among other things, this means that conservation laws such as energy (or momentum, helicity, or circulation) no longer present an obstruction when one is allowed to perform an extension (basically this is because the swirl components of the extension can exchange energy (or momentum, etc.) with the base components in a basically arbitrary fashion.

These results fall short of my hopes to use the ability to extend the manifold to create universal behaviour in Euler flows, because of the fact that each flow requires a different extension in order to achieve the desired dynamics. Still it does seem to provide a little bit of support to the idea that high-dimensional Euler flows are quite “flexible” in their behaviour, though not completely so due to the generic inextendibility phenomenon. This flexibility reminds me a little bit of the flexibility of weak solutions to equations such as the Euler equations provided by the “-principle” of Gromov and its variants (as discussed in these recent notes), although in this case the flexibility comes from adding additional dimensions, rather than by repeatedly adding high-frequency corrections to the solution.

The proof of part (i) of the theorem basically proceeds by a dimension counting argument (similar to that in the proof of Proposition 9 of these recent lecture notes of mine). Heuristically, the point is that an arbitrary incompressible flow is essentially determined by independent functions of space and time, whereas the warping factors are functions of space only, the pressure field is one function of space and time, and the swirl fields are technically functions of both space and time, but have the same number of degrees of freedom as a function just of space, because they solve an evolution equation. When , this means that there are fewer unknown functions of space and time than prescribed functions of space and time, which is the source of the generic inextendibility. This simple argument breaks down when , but we do not know whether the claim is actually false in this case.

The proof of part (ii) proceeds by direct calculation of the effect of the warping factors and swirl velocities, which effectively create a forcing term (of Boussinesq type) in the first equation of (1) that is a combination of functions of the Eulerian spatial coordinates (coming from the warping factors) and the Lagrangian spatial coordinates (which arise from the swirl velocities, which are passively transported by the flow). In a non-empty open subset of , the combination of these coordinates becomes a non-degenerate set of coordinates for spacetime, and one can then use the Stone-Weierstrass theorem to conclude. The requirement that be topologically a torus is a technical hypothesis in order to avoid topological obstructions such as the hairy ball theorem, but it may be that the hypothesis can be dropped (and it may in fact be true, in the case at least, that is dense in all of , not just in a non-empty open subset).

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- The 2019 National Math Festival will be held in Washington D.C. on May 4 (together with some satellite events at other US cities). This festival will have numerous games, events, films, and other activities, which are all free and open to the public. (I am on the board of trustees of MSRI, which is one of the sponsors of the festival.)
- The Institute for Pure and Applied Mathematics (IPAM) is now accepting applications for its second Industrial Short Course for May 16-17 2019, with the topic of “Deep Learning and the Latest AI Algorithms“. (I serve on the Scientific Advisory Board of this institute.) This is an intensive course (in particular requiring active participation) aimed at industrial mathematicians involving both the theory and practice of deep learning and neural networks, taught by Xavier Bresson. (Note: space is very limited, and there is also a registration fee of $2,000 for this course, which is expected to be in high demand.)

The International Congress of Mathematicians (ICM) is widely considered to be the premier conference for mathematicians. It is held every four years; for instance, the 2018 ICM was held in Rio de Janeiro, Brazil, and the 2022 ICM is to be held in Saint Petersburg, Russia. The most high-profile event at the ICM is the awarding of the 10 or so prizes of the International Mathematical Union (IMU) such as the Fields Medal, and the lectures by the prize laureates; but there are also approximately twenty plenary lectures from leading experts across all mathematical disciplines, several public lectures of a less technical nature, about 180 more specialised invited lectures divided into about twenty section panels, each corresponding to a mathematical field (or range of fields), as well as various outreach and social activities, exhibits and satellite programs, and meetings of the IMU General Assembly; see for instance the program for the 2018 ICM for a sample schedule. In addition to these official events, the ICM also provides more informal networking opportunities, in particular allowing mathematicians at all stages of career, and from all backgrounds and nationalities, to interact with each other.

For each Congress, a Program Committee (together with subcommittees for each section) is entrusted with the task of selecting who will give the lectures of the ICM (excluding the lectures by prize laureates, which are selected by separate prize committees); they also have decided how to appropriately subdivide the entire field of mathematics into sections. Given the prestigious nature of invitations from the ICM to present a lecture, this has been an important and challenging task, but one for which past Program Committees have managed to fulfill in a largely satisfactory fashion.

Nevertheless, in the last few years there has been substantial discussion regarding ways in which the process for structuring the ICM and inviting lecturers could be further improved, for instance to reflect the fact that the distribution of mathematics across various fields has evolved over time. At the 2018 ICM General Assembly meeting in Rio de Janeiro, a resolution was adopted to create a new Structure Committee to take on some of the responsibilities previously delegated to the Program Committee, focusing specifically on the structure of the scientific program. On the other hand, the Structure Committee is *not* involved with the format for prize lectures, the selection of prize laureates, or the selection of plenary and sectional lecturers; these tasks are instead the responsibilities of other committees (the local Organizing Committee, the prize committees, and the Program Committee respectively).

The first Structure Committee was constituted on 1 Jan 2019, with the following members:

- Terence Tao [Chair from 15 Feb, 2019]
- Carlos Kenig [IMU President (from 1 Jan 2019),
*ex officio*] - Nalini Anantharaman
- Alexei Borodin
- Annalisa Buffa
- Irene Fonseca
- János Kollár
- Laci Lovász [Chair until 15 Feb, 2019]
- Terry Lyons
- Stephane Mallat
- Hiraku Nakajima
- Éva Tardos
- Peter Teichner
- Akshay Venkatesh
- Anna Wienhard

As one of our first actions, we on the committee are using this blog post to solicit input from the mathematical community regarding the topics within our remit. Among the specific questions (in no particular order) for which we seek comments are the following:

- Are there suggestions to change the format of the ICM that would increase its value to the mathematical community?
- Are there suggestions to change the format of the ICM that would encourage greater participation and interest in attending, particularly with regards to junior researchers and mathematicians from developing countries?
- What is the correct balance between research and exposition in the lectures? For instance, how strongly should one emphasize the importance of good exposition when selecting plenary and sectional speakers? Should there be “Bourbaki style” expository talks presenting work not necessarily authored by the speaker?
- Is the balance between plenary talks, sectional talks, and public talks at an optimal level? There is only a finite amount of space in the calendar, so any increase in the number or length of one of these types of talks will come at the expense of another.
- The ICM is generally perceived to be more important to pure mathematics than to applied mathematics. In what ways can the ICM be made more relevant and attractive to applied mathematicians, or should one not try to do so?
- Are there structural barriers that cause certain areas or styles of mathematics (such as applied or interdisciplinary mathematics) or certain groups of mathematicians to be under-represented at the ICM? What, if anything, can be done to mitigate these barriers?

Of course, we do not expect these complex and difficult questions to be resolved within this blog post, and debating these and other issues would likely be a major component of our internal committee discussions. Nevertheless, we would value constructive comments towards the above questions (or on other topics within the scope of our committee) to help inform these subsequent discussions. We therefore welcome and invite such commentary, either as responses to this blog post, or sent privately to one of the members of our committee. We would also be interested in having readers share their personal experiences at past congresses, and how it compares with other major conferences of this type. (But in order to keep the discussion focused and constructive, we request that comments here refrain from discussing topics that are out of the scope of this committee, such as suggesting specific potential speakers for the next congress, which is a task instead for the 2022 ICM Program Committee.)

]]>The problem involves word maps on a matrix group, which for sake of discussion we will take to be the special orthogonal group of real matrices (one of the smallest matrix groups that contains a copy of the free group, which incidentally is the key observation powering the Banach-Tarski paradox). Given any abstract word of two generators and their inverses (i.e., an element of the free group ), one can define the word map simply by substituting a pair of matrices in into these generators. For instance, if one has the word , then the corresponding word map is given by

for . Because contains a copy of the free group, we see the word map is non-trivial (not equal to the identity) if and only if the word itself is nontrivial.

Anyway, here is the problem:

Problem.Does there exist a sequence of non-trivial word maps that converge uniformly to the identity map?

To put it another way, given any , does there exist a non-trivial word such that for all , where denotes (say) the operator norm, and denotes the identity matrix in ?

As I said, I don’t want to spoil the fun of working out this problem, so I will leave it as a challenge. Readers are welcome to share their thoughts, partial solutions, or full solutions in the comments below.

]]>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.

** — 1. Energy conservation for sufficiently regular weak solutions — **

We now prove the positive part (i) of Onsager’s conjecture, which turns out to be a straightforward application of Littlewood-Paley theory. We need the following relation between Hölder spaces and Littlewood-Paley projections:

Let be a weak solution to the Euler equations for some , thus

To show (3), it will suffice from dominated convergence to show that

as . Applying to (4), we have

From Bernstein’s inequality we conclude that , and thus . Thus solves the PDE

in the classical sense. We can then apply the fundamental theorem of calculus to write

and so it will suffice to show that

We can integrate by parts to place the Leray projection onto the divergence-free factor , at which point it may be removed. Moving the derivative over there as well, we now reduce to showing that

On the other hand, the expression is a total derivative (as is divergence-free), and thus has vanishing integral. Thus it will remain to show that

From Bernstein’s inequality, Exercise 2, and the triangle inequality one has for any time that

as is finite, it thus suffices to establish the pointwise bound

We split the left-hand side into the sum of

To treat (7), we use Exercise 2 to conclude that

and so the quantity (7) is , which is acceptable since . Now we turn to (5). This is a commutator of the form

where . Observe that this commutator would vanish if were replaced by , thus we may write this commutator as

where . If we write

for a suitable Schwartz function of total mass one, we have

writing , we thus have the bound

But from Bernstein’s inequality and Exercise 2 we have

and

and so we see that (5) is also of size , which is acceptable since . A similar argument gives (6), and the claim follows.

As shown by Constantin, E, and Titi, the Hölder regularity in the above result can be relaxed to Besov regularity, at least in non-endpoint cases:

Exercise 3Let . Define the Besov space to be the space of functions such that the Besov space normShow that if is a weak solution to the Euler equations, the energy is conserved in time.

The endpoint case of the above exercise is still open; however energy conservation in the slightly smaller space is known thanks to the work of Cheskidov, Constantin, Friedlander, and Shvydkoy (see also this paper of Isett for further discussion, and this paper of Isett and Oh for an alternate argument that also works on Riemannian manifolds).

As observed by Isett (see also the recent paper of Colombo and de Rosa), the above arguments also give some partial information about the energy in the low regularity regime:

Exercise 4Let be a weak solution to the Euler equations for .

- (i) If , show that the energy is a function of time. (
Hint:express the energy as the uniform limit of .)- (ii) If , show that the energy is a function of time.

Exercise 5Let be a weak solution to the Navier-Stokes equations for some with initial data . Establish the energy identity

Remark 6An alternate heuristic derivation of the threshold for the Onsager conjecture is as follows. If , then from Exercise 2 we see that the portion of that fluctuates at frequency has amplitude at most ; in particular, the amount of energy at frequencies is at most . On the other hand, by the heuristics in Remark 11 of 254A Notes 3, the time needed for the portion of the solution at frequency to evolve to a higher frequency scale such as is of order . Thus the rate ofenergy fluxat frequency should be . For , the energy flux goes to zero as , and so energy cannot escape to frequency infinity in finite time.

Remark 7Yet another alternate heuristic derivation of the threshold arises by considering the dynamics of individual Fourier coefficients. Using a Fourier expansionthe Euler equations may be written as

In particular, the energy at a single Fourier mode evolves according to the equation

If , then we have for any , hence by Plancherel’s theorem

which suggests that (up to logarithmic factors) one would expect to be of magnitude about . Onsager posited that for typically “turbulent” or “chaotic” flows, the main contributions to (8) come when have magnitude roughly comparable to that of , and that the various summands should not be correlated strongly to each other. For , one might expect about significant terms in the sum, which according to standard “square root cancellation heuristics” (cf. the central limit theorem) suggests that the sum is about as large as times the main term. Thus the total flux of energy in or out of a single mode would be expected to be of size , and so the total flux in or out of the frequency range (which consists of modes ) should be about . As such, for the energy flux should decay in and so there is no escape of energy to frequency infinity, whereas for such an escape should be possible. Related heuristics can also support Kolmogorov’s 1941 model of the distribution of energy in the vanishing viscosity limit; see this blog post for more discussion. On the other hand, while Onsager did discuss the dynamics of individual Fourier coefficients in his paper, it appears that he arrived at the threshold by a more physical space based approach, a rigorous version of which was eventually established by Duchon and Robert; see this survey of Eyink and Sreenivasan for more discussion.

** — 2. The (local) isometric embedding problem — **

Before we develop the convex integration method for fluid equations, we first discuss the simpler (and historically earlier) instance of this technique for the isometric embedding problem for Riemannian manifolds. To avoid some topological technicalities that are not the focus of the present set of notes, we only consider the *local problem* of embedding a small neighbourhood of the origin in into Euclidean space .

Let be an open neighbourhood of in . A (smooth) Riemannian metric on , when expressed in coordinates, is a family of smooth maps for , such that for each point , the matrix is symmetric and positive definite. Any such metric gives the structure of an (incomplete) Riemannian manifold . An *isometric embedding* of this manifold into a Euclidean space is a map which is continuously differentiable, injective, and obeys the equation

pointwise on , where is the usual inner product (or dot product) on . In the differential geometry language from Notes 1, we are looking for an injective map such that the Euclidean metric on pulls back to via : .

The *isometric embedding problem* asks, given a Riemannian manifold such as , whether there is an isometric embedding from this manifold to a Euclidean space ; for simplicity we only discuss the simpler *local isometric embedding problem* of constructing an isometric immersion of into for some sufficiently small neighbourhood of the origin. In particular for the local problem we do not need to worry about injectivity since (9) ensures that the derivative map is injective at the origin, and hence is injective near the origin by the inverse function theorem (indeed it is an immersion near the origin).

It is a celebrated theorem of Nash (discussed in this previous blog post) that the isometric embedding problem is possible in the smooth category if the dimension is large enough. For sake of discussion we just present the local version:

Theorem 8 (Nash embedding theorem)Suppose that is sufficiently large depending on . Then for any smooth metric on a neighbourhood of the origin, there is a smooth local isometric embedding on some smaller neighbourhood of the origin.

The optimal value of depending on is not completely known, but it grows roughly quadratically in . Indeed, in this paper of Günther it is shown that one can take

In the other direction, one cannot take below :

Proposition 9Suppose that . Then there exists a smooth Riemannian metric on an open neighbourhood of the origin in such that there is no smooth embedding from any smaller neighbourhood of the origin to .

*Proof:* Informally, the reason for this is that the given field has degrees of freedom (which is the number of independent fields after accounting for the symmetry ), but there are only degrees of freedom for the unknown . To make this rigorous, we perform a Taylor expansion of both and around the origin up to some large order , valid for a sufficiently small neighbourhood :

Equating coefficients, we see that the coefficients

are a polynomial function of the coefficients

this polynomial can be written down explicitly if desired, but its precise form will not be relevant for the argument. Observe that the space of possible coefficients contains an open ball, as can be seen by considering arbitrary perturbations of the Euclidean metric on (here it is important to restrict to in order to avoid the symmetry constraint ; also, the positive definiteness of the metric will be automatic as long as one restricts to sufficiently small perturbations). Comparing dimensions, we conclude that if every smooth metric had a smooth embedding , one must have the inequality

Dividing by and sending , we conclude that . Taking contrapositives, the claim follows.

Remark 10If one replaces “smooth” with “analytic”, one can reverse the arguments here using the Cauchy-Kowaleski theorem and show that any analytic metric on can be locally analytically embedded into ; this is a classical result of Cartan and Janet.

Apart from the slight gap in dimensions, this would seem to settle the question of when a -dimensional metric may be locally isometrically embedded in . However, all of the above arguments required the immersion map to be smooth (i.e., ), whereas the definition of an isometric embedding only required the regularity of .

It is a remarkable (and somewhat counter-intuitive) result of Nash and Kuiper that if one only requires the embedding to be in , then one can embed into a much lower dimensional space:

Theorem 11 (Nash-Kuiper embedding theorem)Let . Then for any smooth metric on a neighbourhood of the origin, there is a local isometric embedding on some smaller neighbourhood of the origin.

Nash originally proved this theorem with the slightly weaker condition ; Kuiper then obtained the optimal condition . The case fails due to curvature obstructions; for instance, if the Riemannian metric has positive scalar curvature, then small Riemannian balls of radius will have (Riemannian) volume slightly less than their Euclidean counterparts, whereas any embedding into will preserve both Riemannian length and volume, preventing such an isometric embedding from existing.

Remark 12One striking illustration of the distinction between the and smooth categories comes when considering isometric embeddings of the round sphere (with the usual metric) into Euclidean space . It is a classical result (see e.g., Spivak’s book) that the only isometric embeddings of in are the obvious ones coming from composing the inclusion map with an isometry of the Euclidean space; however, the Nash-Kuiper construction allows one to create an embedding of into an arbitrarily small ball! Thus the embedding problem lacks the “rigidity” of the embedding problem. This is an instance of a more general principle that nonlinear differential equations such as (10) can become much less rigid when one weakens the regularity hypotheses demanded on the solution.

To prove this theorem we work with a relaxation of the isometric embedding problem. We say that is a *short isometric embedding* on if , solve the equation

on with the matrix symmetric and positive definite for all . With the additional unknown field it is much easier to solve the short isometric problem than the true problem. For instance:

Proposition 13Let , and let be a smooth Riemannian metric on a neighbourhood of the origin in . There is at least one short isometric embedding .

*Proof:* Set and for a sufficiently small , where is the standard embedding, and the Euclidean metric on ; this will be a short isometric embedding on some neighbourhood of the origin.

To create a true isometric embedding , we will first construct a sequence of short embeddings with converging to zero in a suitable sense, and then pass to a limit. The key observation is then that by using the fact that the positive matrices lie in the convex hull of the rank one matrices, one can add a high frequency perturbation to the first component of a short embedding to largely erase the error term , replacing it instead with a much higher frequency error.

We now prove Theorem 11. The key iterative step is

Theorem 14 (Iterative step)Let , let be a closed ball in , let be a smooth Riemannian metric on , and let be a short isometric embedding on . Then for any , one can find a short isometric embedding to (11) on with

Suppose for the moment that we had Theorem 14. Starting with the short isometric embedding on a ball provided by Proposition 13, we can iteratively apply the above theorem to obtain a sequence of short isometric embeddings on with

for . From this we see that converges uniformly to zero, while converges in norm to a limit , which then solves (10) on , giving Theorem 11. (Indeed, this shows that the space of isometric embeddings is dense in the space of short maps in the topology.)

We prove Theorem 14 through a sequence of reductions. Firstly, we can rearrange it slightly:

Theorem 15 (Iterative step, again)Let , let be a closed ball in , let be a smooth immersion, and let be a smooth Riemannian metric on . Then there exists a sequence of smooth immersions for obeying the boundsuniformly on for , where denotes a quantity that goes to zero as (for fixed choices of ).

Let us see how Theorem 15 implies Theorem 14. Let the notation and hypotheses be as in Theorem 14. We may assume to be small. Applying Theorem 15 with replaced by (which will be positive definite for small enough), we obtain a sequence of smooth immersions obeying the estimates

If we set

then is smooth, symmetric, and (from (13)) will be positive definite for large enough. By construction, we thus have solving (11), and Theorem 14 follows.

To prove Theorem 15, it is convenient to break up the metric into more “primitive” pieces that are rank one matrices:

Lemma 16 (Rank one decomposition)Let be a closed ball in , and let be a smooth Riemannian metric on . Then there exists a finite collection of unit vectors in , and smooth functions , such thatfor all . Furthermore, for each , at most of the are non-zero.

Remark 17Strictly speaking, the unit vectors should belong to the dual space of rather than itself, in order to have the index appear as subscripts instead of superscripts. A similar consideration occurs for the frequency vectors from Remark 7. However, we will not bother to distinguish between and here (since they are identified using the Euclidean metric).

*Proof:* Fix a point in . Then the matrix is symmetric and positive definite; one can thus write , where is an orthonormal basis of (column) eigenvectors of and are the eigenvalues (we suppress for now the dependence of these objects on ). Using the parallelogram identity

for some positive real numbers , where are the unit vectors of the form for , enumerated in an arbitrary order. From the further parallelogram identity

we see that every sufficiently small symmetric perturbation of also has a representation of the form (14) with slightly different coefficients that depend smoothly on the perturbation. As is smooth, we thus see that for sufficiently close to we have the decomposition

for some positive quantities varying smoothly with . This gives the lemma in a small ball centred at ; the claim then follows by covering by a finite number balls of the form (say), covering these balls by balls of a fixed radius smaller than all the in the finite cover, in such a way that any point lies in at most of the balls , constructing a smooth partition of unity adapted to the , multiplying each of the decompositions of previously obtained on (which each lie in one of the ) by , and summing to obtain the required decomposition on .

Remark 18Informally, Lemma 16 lets one synthesize a metric as a “convex integral” of rank one pieces, so that if the problem at hand has the freedom to “move” in the direction of each of these rank one pieces, then it also has the freedom to move in the direction , at least if one is working in low enough regularities that one can afford to rapidly change direction from one rank one perturbation to another. This convex integration technique was formalised by Gromov in his interpretation of the Nash-Kuiper method as part of his “-principle“, which we will not discuss further here.

One can now deduce Theorem 15 from

Theorem 19 (Iterative step, rank one version)Let , let be a closed ball in , let be a smooth immersion, let be a unit vector, and let be smooth. Then there exists a sequence of smooth immersions for obeying the boundsuniformly on for . Furthermore, the support of is contained in the support of .

Indeed, suppose that Theorem 19 holds, and we are in the situation of Theorem 15. We apply Lemma 16 to obtain the decomposition

with the stated properties. On taking traces we see that

for all . Applying Theorem 19 times (and diagonalising the sequences as necessary), we obtain sequences of smooth immersions for such that and one has

an such that the support of is contained in that of . The claim then follows from the triangle inequality, noting that the implied constant in (12) will not depend on because of the bounded overlap in the supports of the .

It remains to prove Theorem 19. We note that the requirement that be an immersion will be automatic from (15) for large enough since was already an immersion, making the matrix positive definite uniformly for , and this being unaffected by the addition of the perturbation and the positive semi-definite rank one matrix .

Writing , it will suffice to find a sequence of smooth maps supported in the support of and obeying the approximate difference equation

To locate these functions , we use the *method of slow and fast variables*. First we observe by applying a rotation that we may assume without loss of generality that is the unit vector , thus . We then use the ansatz

where is a smooth function independent of to be chosen later; thus is a function both of the “slow” variable and the “fast” variable taking values in the“fast torus” . (We adopt the convention here of using boldface symbols to denote functions of both the fast and slow variables. The fast torus is isomorphic to the Eulerian torus from the introduction, but we denote them by slightly different notation as they play different roles.) Thus is a low amplitude but highly oscillating perturbation to . The fast variable oscillation means that will not be bounded in regularity norms higher than (and so this ansatz is not available for use in the smooth embedding problem), but because we only wish to control and type quantities, we will still be able to get adequate bounds for the purposes of embedding. Now that we have twice as many variables, the problem becomes more “underdetermined” and we can arrive at a simpler PDE by decoupling the role of the various variables (in particular, we will often work with PDE where the derivatives of the main terms are in the fast variables, but the coefficients only depend on the slow variables, and are thus effectively constant coefficient with respect to the fast variables).

Remark 20Informally, one should think of functions that are independent of the fast variable as being of “low frequency”, and conversely functions that have mean zero in the fast variable (thus for all ) as being of “high frequency”. Thus for instance any smooth function on can be uniquely decomposed into a “low frequency” component and a “high frequency” component, with the two components orthogonal to each other. In later sections we will start inverting “fast derivatives” on “high frequency” functions, which will effectively gain important factors of in the analysis. See also the table below for the dictionary between ordinary physical coordinates and fast-slow coordinates.

Position | Slow variable |

Fast variable | Fast variable |

Function | Function |

Low-frequency function | Function independent of |

High-frequency function | Function mean zero in |

N/A | Slow derivative |

N/A | Fast derivative |

If we expand out using the chain rule, using and to denote partial differentiation in the coordinates of the slow and fast variables respectively, and noting that all terms with at least one power of can be absorbed into the error, we see that we will be done as long as we can construct to obey the bounds

where are viewed as functions of the slow variable only. The original approach of Nash to solve this equation was to use a function that was orthogonal to the entire gradient of , thus

for . Taking derivatives in one would conclude that

and similarly

and one now just had to solve the equation

For this, Nash used a “spiral” construction

where were unit vectors varying smoothly with respect to the slow variable; this obeys (22) and (19), and would also obey (21) if the vectors and were both always orthogonal to the entire gradient of . This is not possible in (as cannot then support linearly independent vectors), but there is no obstruction for :

Lemma 21 (Constructing an orthogonal frame)Let be an immersion. If , then there exist smooth vector fields such that at every point , are unit vectors orthogonal to each other and to for .

*Proof:* Applying the Gram-Schmidt process to the linearly independent vectors for , we can find an orthonormal system of vectors , depending smoothly on , whose span is the same as the span of the . Our task is now to find smooth functions solving the system of equations

For this is possible at the origin from the Gram-Schmidt process. Now we extend in the direction to the line segment . To do this we evolve the fields by the parallel transport ODE

on this line segment. From the Picard existence and uniqueness theorem we can uniquely extend smoothly to this segment with the specified initial data at , and a simple calculation using Gronwall’s inequality shows that the system of equations (23), (24), (25) is preserved by this evolution. Then, one can extend to the disk by using the previous extension to the segment as initial data and solving the parallel transport ODE

Iterating this procedure we obtain the claim.

This concludes Nash’s proof of Theorem 11 when . Now suppose that . In this case we cannot locate two unit vector fields orthogonal to each other and to the entire gradient of ; however, we may still obtain one such vector field by repeating the above arguments. By Gram-Schmidt, we can then locate a smooth unit vector field which is orthogonal to and to for , but for which the quantity is positive. If we use the “Kuiper corrugation” ansatz

for some smooth functions , one is reduced to locating such functions that obey the bounds

and the ODE

This can be done by an explicit construction:

Exercise 22 (One-dimensional corrugation)For any positive and any , show that there exist smooth functions solving the ODEand which vary smoothly with (even at the endpoint ), and obey the bounds

(

Hint:one can renormalise . The problem is basically to locate a periodic function mapping to the circle of mean zero and Lipschitz norm that varies smoothly with . Choose for some smooth and small that is even and compactly supported in with mean zero on each interval, and then choose to be odd.)

This exercise supplies the required functions , completing Kuiper’s proof of Theorem 11 when .

Remark 23For sake of discussion let us restrict attention to the surface case . For the local isometric embedding problem, we have seen that we have rigidity at regularities at or above , but lack of regularity at . The precise threshold at which rigidity occurs is not completely known at present: a result of Borisov (also reproven here) gives rigidity at the level for , while a result of de Lellis, Inauen, and Szekelyhidi (building upon a series of previous results) establishes non-rigidity when . For recent results in higher dimensions, see this paper of Cao and Szekelyhidi.

** — 3. Low regularity weak solutions to Navier-Stokes in high dimensions — **

We now turn to constructing solutions (or near-solutions) to the Euler and Navier-Stokes equations. For minor technical reasons it is convenient to work with solutions that are periodic in both space and time, and normalised to have zero mean at every time (although the latter restriction is not essential for our arguments, since one can always reduce to this case after a Galilean transformation as in 254A Notes 1). Accordingly, let denote the periodic spacetime

and let denote the space of smooth periodic functions that have mean zero and are divergence-free at every time , thus

and

We use as an abbreviation for for various vector spaces (the choice of which will be clear from context).

Let (for now, our discussion will apply both to the Navier-Stokes equations and the Euler equations ). Smooth solutions to Navier-Stokes equations then take the form

for some and smooth . Here of course denotes the spatial Laplacian.

Much as we replaced the equation (10) in the previous section with (11), we will consider the relaxed version

of the Navier-Stokes equations, where we have now introduced an additional field , known as the *Reynolds stress* (cf. the Cauchy stress tensor from 254A Notes 0). If , , are smooth solutions to (26), (27), (28), with having mean zero at every time, then we call a *Navier-Stokes-Reynolds flow* (or *Euler-Reynolds flow*, if ). Note that if then we recover a solution to the true Navier-Stokes equations. Thus, heuristically, the smaller is, the closer and should become to a solution to the true Navier-Stokes equations. (The Reynolds stress tensor here is a rank tensor, as opposed to the rank tensor used in the previous section to measure the failure of isometric embedding, but this will not be a particularly significant distinction.)

Note that if is a Navier-Stokes-Reynolds flow, and , , are smooth functions, then will also be a Navier-Stokes-Reynolds flow if and only if has mean zero at every time, and obeys the difference equation

When this occurs, we say that is a *difference Navier-Stokes-Reynolds flow* at .

It will be thus of interest to find, for a given , difference Navier-Stokes-Reynolds flows at with small, as one could hopefully iterate this procedure and take a limit to construct weak solutions to the true Euler equations. The main strategy here will be to choose a highly oscillating (and divergence-free) correction velocity field such that approximates up to an error which is also highly oscillating (and somewhat divergence-free). The effect of this error can then eventually be absorbed efficiently into the new Reynolds stress tensor . Of course, one also has to manage the other terms , , , appearing in (29). In high dimensions it turns out that these terms can be made very small in norm, and can thus be easily disposed of. In three dimensions the situation is considerably more delicate, particularly with regards to the and terms; in particular, the transport term term is best handled by using a local version of Lagrangian coordinates. We will discuss these subtleties in later sections.

To execute above strategy, it will be convenient to have an even more flexible notion of solution, in which is no longer required to be perfectly divergence-free and mean zero, and is also allowed to be slightly inaccurate in solving (29). We say that is an *approximate difference Navier-Stokes-Reynolds flow* at if , , are smooth functions obeying the system

If the error terms , as well as the mean of , are all small, one can correct an approximate difference Navier-Stokes-Reynolds flow to a true difference Navier-Stokes-Reynolds flow with only small adjustments:

Exercise 24 (Removing the error terms)Let be a Navier-Stokes-Reynolds flow, and let be an approximate difference Navier-Stokes-Reynolds flow at . Show that is an approximate difference Navier-Stokes-Reynolds flow at , whereand is the mean of , thus

(

Hint:one will need at some point to show that has mean zero in space at every time; this can be achieved by integrating (32) in space.)

Because of this exercise we will be able to tolerate the error terms if they (and the mean ) are sufficiently small.

As a simple corollary of Exercise 24, we have the following analogue of Proposition 13:

Proposition 25Let . Then there exist smooth fields , such that is a Navier-Stokes-Reynolds flow. Furthermore, if is supported in for some compact time interval , then can be chosen to also be supported in this region.

*Proof:* Clearly is an approximate difference Navier-Stokes-Reynolds flow at , where

Applying Exercise 24, we can construct an difference Navier-Stokes-Reynolds flow at , which then verifies the claimed properties.

Now, we show that, in sufficiently high dimension, a Navier-Stokes-Reynolds flow can be approximated (in an sense) as the limit of Navier-Stokes-Reynolds flows , with the Reynolds stress going to zero.

Proposition 26 (Weak improvement of Navier-Stokes-Reynolds flows)Let , and let be sufficiently large depending on . Let be a Navier-Stokes-Reynolds flow. Then for sufficiently large , there exists a Navier-Stokes-Reynolds flow obeying the estimatesFurthermore, if is supported in for some interval , then one can arrange for to be supported on for any interval containing in its interior (at the cost of allowing the implied constants in the above to depend also on ).

This proposition can be viewed as an analogue of Theorem 14. For an application at the end of this section it is important that the implied constant in (36) is uniform in the choice of initial flow . The estimate (35) can be viewed as asserting that the new velocity field is oscillating at frequencies , at least in an sense. In the next section, we obtain a stronger version of this proposition with more quantitative estimates that can be iterated to obtain higher regularity weak solutions.

To simplify the notation we adopt the following conventions. Given an -dimensional vector of differential operators, we use to denote the -tuple of differential operators with . We use to denote the -tuple formed by concatenating for . Thus for instance the estimate (35) can be abbreviated as

for all . Informally, one should read the above estimate as asserting that is bounded in with norm , and oscillates with frequency in time and in space (or equivalently, with a temporal wavelength of and a spatial wavelength of ).

*Proof:* We can assume that is non-zero, since if we can just take . We may assume that is supported in for some interval (which may be all of ), and let be an interval containing in its interior. To abbreviate notation, we allow all implied constants to depend on .

Assume is sufficiently large. Using the ansatz

and the triangle inequality, it suffices to construct a difference Navier-Stokes-Reynolds flow at supported on and obeying the bounds

for all .

It will in fact suffice to construct an *approximate* difference Navier-Stokes-Reynolds flow at supported on and obeying the bounds

for , since an application of Exercise 24 and some simple estimation will then give a difference Navier-Stokes-Reynolds flow obeying the desired estimates (using in particular the fact that is bounded on and , as can be seen from Littlewood-Paley theory; also note that (39) can be used to ensure that the mean of is very small).

To construct this approximate solution, we again use the method of fast and slow variables. Set , and introduce the fast-slow spacetime , which we coordinatise as ; we use to denote partial differentiation in the coordinates of the slow variable , and to denote partial differentiation in the coordinates of the fast variable . We also use as shorthand for . Define an *approximate fast-slow solution* to the difference Navier-Stokes-Reynolds equation at (at the frequency scale ) to be a tuple of smooth functions , , that obey the system of equations

Here we think of as a “low-frequency” function (in the sense of Remark 20) that only depends on and the slow variable , but not on the fast variable .

Let denote the tuple . Suppose that for any sufficiently large , we can construct an approximate fast-slow solution to the difference equations at supported on supported on that obeys the bounds

for all . (Informally, the presence of the derivatives means that the fields involved are allowed to oscillate in time at wavelength , in the slow variable at wavelength , and in the fast variable at wavelength .) From (46) and the choice of we then have

for all , and similarly for (48), (49). (Note here that there was considerable room in the estimates with regards to regularity in the variable; this room will be exploited more in the next section.) For any shift , we see from the chain rule that is an approximate difference Navier-Stokes-Reynolds flow at supported on , where

Also from (46) and Fubini’s theorem we have

for all . By Markov’s inequality and (52), we see that for each , we have

for all outside of an exceptional set of measure (say) . Similarly for the other equations above. Applying the union bound, we can then find a such that obeys all the required bounds (37)-\eqref[bd-5} simultaneously for all . (This is an example of the probabilistic method, originally developed in combinatorics; one can think of probabilistically as a shift drawn uniformly at random from the torus , in order to relate the fast-slow Lebesgue norms to the original Lebesgue norms .)

It remains to construct an approximate fast-slow solution supported on with the required bounds (46)–(51). Actually, in this high-dimensional setting we can afford to simplify the situation here by removing some of the terms (and in particular eliminating the role of the reference velocity field ). Define a *simplified fast-slow solution* at to be a tuple of smooth functions on obeying the simplified equations

If we can find a simplified fast-slow solution of smooth functions on supported on obeying the bounds

for all , then the will be an approximate fast-slow solution supported on obeying the required bounds (46)–(51), where

Now we need to construct a simplified fast-slow solution supported on obeying the bounds (56)–(60). We do this in stages, first finding a solution that cancels off the highest order terms and , and also such that has mean zero in the fast variable (so that it is “high frequency” in the sense of Remark 20). This still leads to fairly large values of and , but we will then apply a “divergence corrector” to almost completely eliminate , followed by a “stress corrector” that almost completely eliminates , at which point we will be done.

We turn to the details. Our preliminary construction of the velocity field will be a “Mikado flow”, consisting of flows along narrow tubes. (Earlier literature used other flows, such as Beltrami flows; however, Mikado flows have the advantage of being localisable to small subsets of spacetime, which is particularly useful in high dimensions.) We need the following modification of Lemma 16:

Exercise 27Let be a compact subset of the space of positive definite matrices . Show that there exist non-zero lattice vectors and smooth functions for some such thatfor all . (This decomposition is essentially due to de Lellis and Szekelyhidi. The subscripting and superscripting here is reversed from that in Lemma 16; this is because we are now trying to decompose a rank tensor rather than a rank tensor.)

We would like to apply this exercise to the matrix with entries . We thus need to select the pressure so that this matrix is positive definite. There are many choices available for this pressure; we will take

where is the Frobenius norm of . Then is smooth and is positive definite on all of the compact spacetime (recall that we can assume to not be identically zero), and in particular ranges in a compact subset of positive definite matrices. Applying the previous exercise and composing with the function , we conclude that there exist non-zero lattice vectors and smooth functions for some such that

for all . As depend only on , and is a component of , all norms of these quantities are bounded by ; they are independent of . Furthermore, on taking traces and integrating on , we obtain the estimate

(note here that the implied constant is uniform in , ). By applying a smooth cutoff in time that equals on and vanishes outside of , we may assume that the are supported in .

Now for each , the closed subgroup is a one-dimensional subset of , so the -neighbourhood of this subgroup has measure ; crucially, this will be a large negative power of when is very large. let be a translate of this -neighbourhood such that all the are disjoint; this is easily accomplished by the probabilistic method for large enough, translating each of the by an independent random shift and noting that the probability of a collision goes to zero as (here we need the fact that we are in at least three dimensions).

Let be a large integer (depending on ) to be chosen later. For any , let be a scalar test function supported on that is constant in the direction, thus

and is not identically zero, which implies that the iterated Laplacian of is also not identically zero (thanks to the unique continuation property of harmonic functions). We can normalise so that

and we can also arrange to have the bounds

for all (basically by first constructing a version of on a standard cylinder and the applying an affine transformation to map onto ).

Let denote the function

intuitively this represents the velocity field of a fluid traveling along the tube , with the presence of the Laplacian ensuring that this function is extremely well balanced (for instance, it will have mean zero, and thus “high frequency” in the sense of Remark 20). Clearly is divergence free, and one also has the steady-state Euler equation

for all and . If we then set

then one easily checks that is a simplified fast-slow solution supported in . Direct calculation using the Leibniz rule then gives the bounds

for all , while from (64) one has

(note here that the implied constant is uniform in ).

This looks worse than (56)–(60). However, observe that is supported on the set , which has measure , which for large enough can be taken to be (say) . Thus by Cauchy-Schwarz one has

for all . Also, from construction (particularly (66)) we see that is of mean zero in the variable (thus it is “high frequency” in the sense of Remark 20).

We are now a bit closer to (56)–(60), but our bounds on are not yet strong enough. We now apply a “divergence corrector” to make much smaller. Observe from construction that where

and is supported on and obeys the estimates

We abbreviate the differential operator as . Iterating the above identity times, we obtain

where

and

In particular, is supported in . Observe that is a simplified fast-slow solution supported in , where

From (72) we have

so in particular for large enough

for any . Meanwhile, another appeal to (72) yields

for any , and hence by (67) and the triangle inequality

Similarly one has

Since continues to be supported on the thin set , we can apply Hölder as before to conclude that

for any . Also, from (73) and Hölder we have

We have now achieved the bound (59); the remaining estimate that needs to be corrected for is (60). This we can do by a modification of the previous argument, where we now work to reduce the size of rather than . Observe that as is “high frequency” (mean zero in the variable), one can write

where is the linear operator on smooth vector-valued functions on of mean zero defined by the formula

Note that also has mean zero. We can thus iterate and obtain

where

and is a smooth function whose exact form is explicit but irrelevant for our argument. We then see that is a simplified fast-slow solution supported in . Since is bounded in , we see from (69) that

and

if is large enough. Thus obeys the required bounds (56)–(60), concluding the proof.

As an application of this proposition we construct a low-regularity weak solution to high-dimensional Navier-Stokes that does not obey energy conservation. More precisely, for any , let be the Banach space of periodic functions which are divergence free, and of mean zero at every time. For , define a *time-periodic weak solution* of the Navier-Stokes (or Euler, if ) equations to be a function that solves the equation

in the sense of distributions. (Note that one may easily define on functions in a distributional sense, basically because the adjoint operator maps test functions to bounded functions.)

Corollary 28 (Low regularity non-trivial weak solutions)Assume that the dimension is sufficiently large. Then for any , there exists a periodic weak solution to Navier-Stokes which equals zero at time , but is not identically zero. In particular, periodic weak solutions are not uniquely determined by their initial data, and do not necessarily obey the energy inequality

*Proof:* Let be an element of that is supported on and is not identically zero (it is easy to show that such an element exists). By Proposition 25, we may then find a Navier-Stokes-Reynolds flow also supported on . Let be sufficiently large. By applying Proposition 26 repeatedly (with say ) and with a sufficiently rapidly increasing sequence , we can find a sequence of Navier-Stokes-Reynolds flows supported on (say) obeying the bounds

(say) for . For sufficiently rapidly growing, this implies that converges strongly in to zero, while converges strongly in to some limit supported in . From the triangle inequality we have

(if is sufficiently rapidly growing) and hence is not identically zero if is chosen large enough. Applying Leray projections to the Navier-Stokes-Reynolds equation we have

in the sense of distributions (where is the vector field with components for ); taking distributional limits as , we conclude that is a periodic weak solution to the Navier-Stokes equations, as required.

** — 4. High regularity weak solutions to Navier-Stokes in high dimensions — **

Now we refine the above arguments to give a higher regularity version of Corollary 28, in which we can give the weak solutions almost half a derivative of regularity in the Sobolev scale:

Theorem 29 (Non-trivial weak solutions)Let , and assume that the dimension is sufficiently large depending on . Then for any , there exists a periodic weak solution which equals zero at time , but is not identically zero. In particular, periodic weak solutions are not uniquely determined by their initial data, and do not necessarily obey the energy inequality (74).

This result is inspired by a three-dimensional result of Buckmaster and Vicol (with a small value of ) and a higher dimensional result of Luo (taking , and restricting attention to time-independent solutions). In high dimensions one can create fairly regular functions which are large in type norms but tiny in type norms; when using the Sobolev scale to control the solution (and type norms to measure an associated stress tensor), this has the effect of allowing one to treat as negligible the linear terms in (variants of) the Navier-Stokes equation, as well as interaction terms between low and high frequencies. As such, the analysis here is simpler than that required to establish the Onsager conjecture. The construction used to prove this theorem shows in fact that periodic weak solutions are in some sense “dense” in , but we will not attempt to quantify this fact here.

In the proof of Corollary 28, we took the frequency scales to be extremely rapidly growing in . This will no longer be good enough for proving Theorem 29, and in fact we need to take a fairly dense set of frequency scales in which for a small . In order to do so, we have to replace Proposition 26 with a more quantitative version in which the dependence of bounds on the size of the original Navier-Stokes-Reynolds flow is made much more explicit.

We turn to the details. We select the following parameters:

- A regularity ;
- A quantity , assumed to be sufficiently small depending on ;
- An integer , assumed to be sufficiently large depending on ; and
- A dimension , assumed to be sufficiently large depending on .

Then we let . To simplify the notation we allow all implied constants to depend on unless otherwise specified. We recall from the previous section the notion of a Navier-Stokes-Reynolds flow . The basic strategy is to start with a Navier-Stokes-Reynolds flow and repeatedly adjust by increasingly high frequency corrections in order to significantly reduce the size of the stress (at the cost of making both of these expressions higher in frequency).

As before, we abbreviate as . We write for the spatial gradient to distinguish it from the time derivative .

The main iterative statement (analogous to Theorem 14) starts with a Navier-Stokes-Reynolds flow oscillating at spatial scales up to some small wavelength , and modifies it to a Navier-Stokes-Reynolds flow oscillating at a slightly smaller wavelength , with a smaller Reynolds stress. It can be viewed as a more quantitative version of Proposition 26.

Theorem 30 (Iterative step)Let be sufficiently large depending on the parameters . Set . Suppose that one has a Navier-Stokes-Reynolds flow obeying the estimatesfor some . Set . Then there exists a Navier-Stokes-Reynolds flow obeying the estimates

Furthermore, if is supported on for some interval , then one can ensure that is supported in , where is the -neighbourhood of .

Let us assume Theorem 30 for the moment and establish Theorem 29. Let be chosen to be supported on (say) and not be identically zero. By Proposition 25, we can then find a Navier-Stokes-Reynolds flow supported on . Let be a sufficiently large parameter, and set , then the hypotheses (75), (76) will be obeyed for large enough. Set for all . By iteratively applying Theorem 30, we may find a sequence of Navier-Stokes-Reynolds flows, all supported on (say) , obeying the bounds

for . In particular, the converge weakly to zero on , and we have the bound

from Plancherel’s theorem, and hence by Sobolev embedding in time

Thus converges strongly in (and in particular also in for some ) to some limit ; as the are all divergence-free, is also. From applying Leray projections to (26) one has

Taking weak limits we conclude that is a weak solution to Navier-Stokes. Also, from construction one has

(say), and so for large enough is not identically zero. This proves Theorem 29.

It remains to establish Theorem 30. It will be convenient to introduce the intermediate frequency scales

where

is slightly larger than , and

is slightly smaller than (and constrained to be integer).

Before we begin the rigorous argument, we first give a heuristic explanation of the numerology. The initial solution has about degrees of regularity controlled at . For technical reasons we will upgrade this to an infinite amount of regularity, at the cost of worsening the frequency bound slightly from to . Next, to cancel the Reynolds stress up to a smaller error , we will perturb by some high frequency correction , basically oscillating at spatial frequency (and temporal frequency ), so that is approximately equal to (minus a pressure term) after averaging at spatial scales . Given the size bound (76), one expects to achieve this with of norm about . By exploiting the small gap between and , we can make concentrate on a fairly small measure set (of density something like ), which in high dimension allows us to make linear terms such as and (as well as the correlation terms and ) negligible in size (as measured using type norms) when compared against quadratic terms such as (cf. the proof of Proposition 26). The defect will then oscillate at frequency , but can be selected to be of size about in norm, because can choose to cancel off all the high-frequency (by which we mean or greater) contributions to this term, leaving only low frequency contributions (at frequencies or below). Using the ellipticity of the Laplacian, we can then express this defect as where the norm of is of order

When , this is slightly less than , allowing one to close the argument.

We now turn to the rigorous details. In a later part of the argument we will encounter a *loss of derivatives*, in that the new Navier-Stokes-Reynolds flow has lower amounts of controlled regularity (in both space and time) than the Navier-Stokes-Reynolds flow used to construct it. To counteract this loss of derivatives we need to perform an initial *mollification step*, which improves the amount of regularity from derivatives in space and one in time to an unlimited number of derivatives in space and two derivatives in time, at the cost of worsening the estimates on slightly (basically by replacing with ).

Proposition 31 (Mollification)Let the notation and hypotheses be as above. Then we can find a Navier-Stokes-Reynolds flow obeying the estimatesFurthermore, if is supported on for some interval , then one can ensure that is supported in , where is the -neighbourhood of .

We remark that this sort of mollification step is now a standard technique in any iteration scheme that involves loss of derivatives, including the Nash-Moser iteration scheme that was first used to prove Theorem 8.

*Proof:* Let be a bump function (depending only on ) supported on the region of total mass , and define the averaging operator on smooth functions by the formula

From the fundamental theorem of calculus we have

where is the identity operator and

The operators and will behave like low and high frequency Littlewood-Paley projections. (We cannot directly use these projections here because their convolution kernels are not localised in time.)

Observe that are convolution operators and thus commute with each other and with the partial derivatives . If we apply the operator to (26), (27), (28), we see that is Navier-Stokes-Reynolds flow, where

Since , is a linear combination of the operators . In particular, we see that is supported on .

We abbreviate . For any , we have

and therefore deduce the bounds

for any , thanks to Young’s inequality. A similar application of Young’s inequality gives

From (81) and decomposing as linear combinations of , we have

for any , and hence (77) follows from (75). In a similar spirit, from (82), (75) one has

if is large enough, and this gives (79), (80).

Finally we prove (78). By the triangle inequality it suffices to show that

for any . The claim (83) follows from (81), (76), after writing as a linear combination of and noting that . For (84), if we again write as a linear combination of and uses (81) and the Leibniz rule, one can bound the left-hand side of (84) by

and hence by (75) (bounding by ) this is bounded by

This gives (84) when . For , we rewrite the expression

as

The contribution of the first term to (84) can be bounded using (82), (81) (splitting ) by

which by the Leibniz rule, bounding by , and (75) is bounded by

which is again an acceptable contribution to (84) since is large. The other terms are treated similarly.

We return to the proof of Theorem 30. We abbreviate . Let be the Navier-Stokes-Reynolds flow constructed by Proposition 31. By using the ansatz

and the triangle inequality, it will suffice to locate a difference Navier-Stokes-Reynolds flow at supported on , obeying the estimates

From (77) we have

so by the triangle inequality we can replace (85) by

and then (85), (87) may then be replaced by the single estimate

(say). By using Exercise 24 as in the previous section, it then suffices to construct an *approximate* difference Navier-Stokes-Reynolds flow to the difference equation at supported on supported in obeying the bounds (86), (89),

Now, we pass to fast and slow variables. Let denote the tuple

informally, the use of is consistent with oscillations in time of wavelength , in the slow variable of wavelength , and in the fast variable of wavelength .

Exercise 32By using the method of fast and slow variables as in the previous section, show that to construct the approximate Navier-Stokes-Reynolds flow at obeying the bounds (86), (89), (90), (91), it suffices to locate an approximate fast-slow solution to the difference Navier-Stokes-Reynolds equation at (at frequency scale rather than ) and supported in that obey the bounds

As in the previous section, we can then pass to simplified fast-slow soutions:

Exercise 33Show that to construct the approximate fast-slow solution to the difference equation at obeying the estimates of the previous exercise, it will in fact suffice to locate a simplified fast-slow solution at (again at frequency scale ) supported on , obeying the bounds (92), (93), (95), (96) and(

Hint:one will need the estimatefrom Proposition 31.)

Now we need to construct a simplified fast-slow solution at supported on obeying the bounds (92), (93), (95), (96), (97). As in the previous section, we do this in stages, first finding a solution that cancels off the top order terms and , and also such that is “ high frequency” (mean zero in ). Then we apply a divergence corrector to completely eliminate , followed by a stress corrector that almost completely eliminates .

As before, we need to select so that is positive definite. In the previous section we essentially took to be a large multiple of , but now we will need good control on the derivatives of , which requires a little more care. Namely, we will need the following technical lemma:

Lemma 34 (Smooth polar-type decomposition)There exists a factorisation , where , are smooth, supported on , and obey the estimates

*Proof:* We may assume that is not identically zero, since otherwise the claim is trivial. For brevity we write and . From (78) we have

Let denote the spacetime cylinder , and let denote the maximal function

From the fundamental theorem of calculus (or Sobolev embedding) one has the pointwise estimate

thus by Fubini’s theorem and (101)

We do not have good control on the derivatives of , so we apply a smoothing operator. Let denote the function

where , then by Fubini’s theorem (or Young’s inequality)

Also, is smooth and strictly positive everywhere, and from differentation under the integral sign and integration by parts we have

for any . Also, from construction one has

Write . From many applications of the chain rule (or the Faá di Bruno formula), we see that for any , is a linear combination of terms of the form

where sum up to (more precisely, each component of is a linear combination of expressions of the above form in which one works with individual components of each factor rather than the full tuple ). From (103) we thus have the pointwise estimate

for any , and (98) now follows from (102). A similar argument gives

for any , hence if we set , then by the product rule

and (99) now follows from (104).

Strictly speaking we are not quite done because is not supported in , but if one applies a smooth cutoff function in time that equals on (where is supported in time) and vanishes outside of , we obtain the required support property without significantly affecting the estimates.

Let be the factorisation given by the above lemma. If we set for a sufficiently large constant depending only on , then

For large enough, we see from (99) that the matrix with entries takes values in a compact subset of positive definite matrices) that depends only on . Applying Exercise 27, we conclude that there exist non-zero lattice vectors and smooth functions for some such that

for all , and furthermore (from (99) the chain rule) we have the derivative estimates

for . Setting , we thus have

and from the Leibniz rule and (98) we have

Let be the disjoint tubes in from the previous section, with width rather than . Construct the functions as in the previous section, and again set

Then as before, each is divergence free, and obeys the identities (65), (66) and the counds

for all and . As in the preceding section, we then set

and one easily checks that is a simplified fast-slow solution supported in . Direct calculation using the Leibniz rule and (105), (107) then gives the bounds

As before, is “high frequency” (mean zero in the variable). Also, is supported on the set , and for large enough the latter set has measure (say) . Thus by Cauchy-Schwarz (in just the variable) one has

The divergence corrector can be applied without difficulty:

Exercise 35Show that there is a simplified fast-slow solution supported in obeying the estimates

The crucial thing here is the tiny gain in the third estimate, with the first factor coming from a “slow” derivative and the second factor coming from essentially inverting a “fast” derivative .

Finally, we apply a stress corrector:

Exercise 36Show that there is a simplified fast-slow solution supported in obeying the estimates

Again, we have a crucial gain of coming from applying a slow derivative and inverting a fast one.

Since

(with implied constant in the exponent uniform in ) and , we see (for small enough) that

and the desired estimates (92), (93), (95), (96), (97) now follow.

** — 5. Constructing low regularity weak solutions to Euler — **

Throughout this section, we specialise to the Euler equations in the three-dimensional case (although all of the arguments here also apply without much modification to as well). In this section we establish an analogue of Corollary 28:

Proposition 37 (Low regularity non-trivial weak solutions)There exists a periodic weak solution to Euler which equals zero at time , but is not identically zero.

This result was first established by de Lellis and Szekelyhidi. Our approach will deviate from the one in that paper in a number of technical respects (for instance, we use Mikado flows in place of Beltrami flows, and we place more emphasis on the method of fast and slow variables). A key new feature, which was not present in the high-dimensional Sobolev-scale setting, is that the material derivative term in the difference Euler-Reynolds equations is no longer negligible, and needs to be treated by working with an ansatz in Lagrangian coordinates (or equivalently, an ansatz transported by the flow). (This use of Lagrangian coordinates is implicit in the thesis of Isett, this paper of de Lellis and Szekelyhidi, and in the later work of Isett.)

Just as Corollary 28 was derived from Proposition 26, the above proposition may be derived from

Proposition 38 (Weak improvement of Euler-Reynolds flows)Let be an Euler-Reynolds flow supported on a strict compact subinterval . Let be another interval in containing in its interior. Then for sufficiently large , there exists a Euler-Reynolds flow supported in obeying the estimates

The point of the decomposition (115) is that it (together with the smallness bounds (116)) asserts that the velocity correction is mostly “high frequency” in nature, in that its low frequency components are small. Together with (112), the bounds roughly speaking assert that it is only the frequency components of that can be large in norm. Unlike the previous estimates, it will be important for our arguments that is supported in a strict subinterval of , because we will not be able to extend Lagrangian coordinates periodically around the circle. Actually the long-time instability of Lagrangian coordinates causes significant technical difficulties to overcome when one wants to construct solutions in higher regularity Hölder spaces , and in particular for close to ; we discuss this in the next section.

Exercise 39Deduce Proposition 37 from Proposition 38. (The decomposition (116) is needed to keep close to in a very weak topology – basically the topology – but one which is still sufficent to ensure that the limiting solution constructed is not identically zero.)

We now begin the proof of Proposition 38, repeating many of the steps used to prove Proposition 26. As before we may assume that is non-zero, and that is supported in . We can assume that is also a strict subinterval of .

Assume is sufficiently large; by rounding we may assume that is a natural number. Using the ansatz

and the triangle inequality, it suffices to construct a difference Euler-Reynolds flow at supported on and obeying the bounds

for all , and for which we have a decomposition obeying (116).

As before, we permit ourselves some error:

Exercise 40Show that it suffices to construct anapproximatedifference Euler-Reynolds flow at supported on and obeying the boundsfor , and for which we have a decomposition obeying (116).

It still remains to construct the approximate difference Euler-Reynolds flow obeying the claimed estimates. By definition, has to obey the system of equations

As is divergence-free, the first equation (122) may be rewritten as

where is the material Lie derivative of , thus

The lower order terms in (126) will turn out to be rather harmless; the main new difficulty is dealing with the material Lie derivative term . We will therefore invoke Lagrangian coordinates in order to convert the material Lie derivative into the more tractable time derivative (at the cost of mildly complicating all the other terms in the system).

We introduce a “Lagrangian torus” that is an isomorphic copy of the Eulerian torus ; as in the previous section, we paramterise this torus by , and adopt the usual summation conventions for the indices . Let be a trajectory map for , that is to say a smooth map such that for every time , the map is a diffeomorphism and one obeys the ODE

for all . The existence of such a trajectory map is guaranteed by the Picard existence theorem (it is important here that is not all of the torus ); see also Exercise 1 from Notes 1. From (the periodic version of) Lemma 3 of Notes 1, we can ensure that the map is volume-preserving, thus

Recall from Notes 1 that

- (i) Any Eulerian scalar field on can be pulled back to a Lagrangian scalar field on by the formula
- (ii) Any Eulerian vector field on can be pulled back to a Lagrangian vector field on by the formula
where is the inverse of the matrix , defined by

and

- (iii) Any Eulerian rank tensor on , can be pulled back to a Lagrangian rank tensor on by the formula

(One can pull back other tensors also, but these are the only ones we will need here.) Each of these pullback operations may be inverted by the corresponding pullback operation for the labels map (also known as *pushforward* by ). One can compute how these pullbacks interact with divergences:

Exercise 41 (Pullback and divergence)

- (i) If is a smooth Eulerian vector field, show that the pullback of the divergence of equals the divergence of the pullback of :
(

Hint:you will need to use the fact that is volume-preserving. Similarly to Lemma 3 and Exercise 4 of Notes 1, one can establish this either using the laws of integration or the laws of differentiation.).- (ii) Show that there exist smooth functions for with the following property: for any smooth Eulerian rank tensor on , with divergence , one has
(In fact, can be given explicitly as , where is the Kronecker delta and is the Christoffel symbol associated with the pullback of the Euclidean metric – but we will not need this precise formula. The right-hand side may also be written (in Penrose abstract index notation) as , where is the covariant derivative associated to .)

As remarked upon in the exercise, these calculations can be streamlined using the theory of the covariant derivative in Riemannian geometry; we will not develop this theory further here, but see for instance these two blog posts.

If one now applies the pullback operation to the system (126), (123), (124), (125) (and uses Exercise 16 from Notes 1 to convert the material Lie derivative into the ordinary time derivative) one obtain the equivalent system

where denotes the rank tensor

Thus, if one introduces the Lagrangian fields

and also

then (from many applications of the chain rule) we see that our task has now transformed to that of obtaining a supported on obeying the equations

for , where denotes the supremum on the Lagrangian spacetime . (In (137) we have to move back to Eulerian coordinates because the coefficients in the pushforward depend on , and we want an estimate here uniform in .)

This problem looks complicated, but the net effect of moving to the Lagrangian formulation is to arrive at a problem that is nearly identical to the Eulerian one, but in which the material Lie derivative has been replaced by the ordinary time derivative , and several lower order terms with smooth variable coefficients have been added to the system.

Now that the dangerous transport term in the material Lie derivative has been eliminated, it is now safe to use the method of fast-slow variables, but now on the Lagrangian torus rather than the Eulerian torus . We now parameterise the fast torus by (thus we think of now as a “Lagrangian fast torus” rather than a “Eulerian fast torus”) and use the ansatz

so that the equations of motion now become

where we think of as “low frequency” functions of time and the slow Lagrangian variable only (thus they are independent of the fast Lagrangian variable ). Set

It will now suffice to find a smooth solution to the above system supported on obeying the estimates

and obeying the pointwise estimate

If is chosen to be “high frequency” (mean zero in the fast variable ), then we can automatically obtain the estimate (148), as one may obtain the decomposition (142) with

and

at which point the estimates (148) follow from (144). Thus we may drop (148) and (142) from our requirements as long as we instead require to be high frequency.

As in previous sections, we can relax the conditions on and :

Exercise 42

- (i) (Stress corrector) Show that we may replace the condition (147) with the condition that
for all , and that is of high frequency (mean zero in ). (

Hint:add a corrector of size to , so that the main term now cancels off , and the other terms created by the correction are of size and of mean zero in . Then iterate.)- (ii) (Divergence corrector) After performing the modifications in (i), show that we may replace the condition (146) with the condition that
for all , and that has mean zero in . (Note that correcting for will modify by , but this will make a negligible contribution to (149) for large enough.)

We can now repeat the Mikado flow construction from previous sections:

Exercise 43Set . Construct a smooth vector field supported on , with of mean zero in , obeying the equationsand such that

is of mean zero in , obeying the bounds (144) for all . Also show that for a sufficiently large absolute constant not depending on , one can ensure that the matrix with entries

If we now set

one can verify that the equations (139), (140), (141) hold, and that have mean zero in . Furthermore, by pushing forward (150) by , we conclude that the matrix with entries

is positive semi-definite for all ; taking traces one concludes (149). Thus we have obtained all the properties required in Exercise 43, concluding the proof of Proposition 38.

** — 6. Constructing high regularity weak solutions to Euler — **

We now informally discuss how to modify the arguments above to establish the negative direction (ii) of Onsager’s conjecture. The full details are rather complicated (and arranged slightly differently from the presentation here), and we refer to Isett’s original paper for details. See also a subsequent paper of Buckmaster, De Lellis, Szekelyhidi, and Vicol for a simplified argument establishing this statement (as well as some additional strengthenings of it).

Let , let be a small quantity, and let be a large integer. The main iterative step, analogous to Theorem 30, roughly speaking (ignoring some technical logarithmic factors) takes an Euler-Reynolds flow obeying the estimates that look like

for some sufficiently large , and obtains a new Euler-Reynolds flow close to that obeys the estimates

where ; see Lemma 2.1 of Isett’s original paper for a more precise statement (in a slightly different notation), which also includes various estimates on the difference that we omit here. In contrast to previous arguments, it is useful for technical reasons to not impose time regularity in the estimates. Once this claim is formalised and proved, conclusions such as Onsager’s conjecture follow from the usual iteration arguments.

To achieve this iteration step, the first step is a mollification step analogous to Proposition 31 in which one perturbs the initial flow to obtain additional spatial regularity on the solution. Roughly speaking, this mollification allows one to replace the purely spatial differential operator appearing in (151), (153) with for much larger than (in practice there are some slight additional losses, which we will ignore here).

Now one has to solve the difference equation. We focus on the equation (126) and omit the small errors involving . Suppose for the time being that we could magically replace the material Lie derivative here by an ordinary time derivative, thus we would be trying to construct solving an equation such as

As before, we can use the method of fast and slow variables to construct a of amplitude roughly , oscillating at frequency , such that is high frequency (mean zero in the fast variable) and has amplitude about . We can also arrange matters (using something like (65)) so that the fast derivative component of vanishes, leaving only a slower derivative of size about . This makes the expression of magnitude about and oscillating at frequency about , which can be cancelled by a stress corrector in of magnitude about . Such a term would be acceptable (smaller than ) for as large as , in the spirit of Theorem 29.

However, one also has the terms and , which (in contrast to the high-dimensional Sobolev scale setting) cannot be ignored in this low-dimensional Hölder scale problem. The natural time scale of oscillation here is , coming from the usual heuristics concerning the Euler equation (see Remark 11 from 254A Notes 3). With this heuristic, and should both behave like , these expressions would be expected to have amplitude . They still oscillate at the high frequency , though, and lead to a stress corrector of magnitude about . This however remains acceptable for up to , which in principle resolves Onsager’s conjecture.

Now we have to remove the “cheat” of replacing the material Lie derivative by the ordinary time derivative. As we saw in the previous section, the natural way to fix this is to work in Lagrangian coordinates. However, we encounter a new problem: if one initialises the trajectory flow map to be the identity at some given time , then it turns out that one only gets good control on the flow map and its derivatives for times within the natural time scale of that initial time ; beyond this, the Gronwall-type arguments used to obtain bounds start to deteriorate exponentially. Because of this, one cannot rely on a “global” Lagrangian coordinate system as in the previous section. To get around this, one needs to partition the time domain into intervals of length about , and construct a separate trajectory map adapted to each such interval. One can then use these “local Lagrangian coordinates” to construct local components of the velocity perturbation that obey the required properties on each such interval. This construction is essentially the content of the “convex integration lemma” in Lemma 3.3 of Isett’s paper.

However, a new problem arises when trying to “glue” these local corrections together: two consecutive time intervals will overlap, and their corresponding local corrections will also overlap. This leads to some highly undesirable interaction terms between and (such as the fast derivative of ) which are very difficult to make small (for instance, one cannot simply ensure that have disjoint spatial supports as they are constructed using different local Lagrangian coordinate systems). On the other hand, if the original Reynolds stress had a special structure, namely that it was only supported on every other interval (i.e., on the for all even, or the for all odd), then these interactions no longer occur and the iteration step can proceed.

One could try to then resolve the problem by correcting the odd and even interval components of the stress in separate stages (cf. how Theorem 19 can be iterated to establish Theorem 15), but this is inefficient with regards to the parameter, in particular this makes the argument stop well short of the optimal threshold. To attain this threshold one needs the final ingredient of Isett’s argument, namely a “gluing approximation” (see Lemma 3.2 of Isett’s paper), in which one tales the (mollified) initial Euler-Reynolds flow and replaces it with a nearby flow in which the new Reynolds stress is only supported on every other interval . Combining this gluing approximation lemma with the mollification lemma and convex integration lemma gives the required iteration step. (One technical point is that this gluing has to create some useful additional regularity along the material derivative, in the spirit of Remark 38 of Notes 1, as such regularity will be needed in order to justify the convex integration step.)

To obtain this gluing approximation, one takes an -separated sequence of times , and for each such time , one solves the *true* Euler equations with initial data at to obtain smooth Euler solutions on a time interval centred at of lifespan , that agree with at time . (It is non-trivial to check that these solutions even exist on such an interval, let alone obey good estimates, but this can be done if the initial data was suitably mollified, as is consistent with the heuristics in Remark 11 from 254A Notes 3.) One can then glue these solutions together around the reference solution by defining

for a suitable partition of unity in time. This gives fields that solve the Euler equation near each time . To finish the proof of the gluing approximation lemma, one needs to then find a matching Reynolds stress for the intervals at which the Euler equation is not solved exactly. Isett’s original construction of this stress was rather intricate; see Sections 7-10 of Isett’s paper for the technical details. However, with improved estimates, a simpler construction was used in a subsequent paper of Buckmaster, De Lellis, Szekelyhidi, and Vicol, leading to a simplified proof of (the non-endpoint version of) this direction of Onsager’s conjecture.

]]>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”.]

]]>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

- 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
- 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).

]]>

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.]

]]>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 1Let 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 2The first of the Euler equations (1) can now be written in the formwhich 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 3Let 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 4Let be a continuously differentiable map from the time interval to the general linear group of invertible matrices. Establish Jacobi’s formulaand use this and (6) to give an alternate proof of Lemma 3 that does not involve any integration in space.

Remark 5One 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 hasand 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 6One 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.

** — 1. Pullbacks and Lie derivatives — **

In order to efficiently change coordinates, it is convenient to use the language of differential geometry, which is designed to be almost entirely independent of the choice of coordinates. We therefore spend some time recalling the basic concepts of differential geometry that we will need. Our presentation will be based on explicitly working in coordinates; there are of course more coordinate-free approaches to the subject (for instance setting up the machinery of vector bundles, or of derivations), but we will not adopt these approaches here.

Throughout this section, we fix a diffeomorphism from Lagrangian space to Eulerian space ; one can for instance take where is a diffeomorphic trajectory map and is some time. Then all the differential geometry structures on Eulerian space can be pulled back via to Lagrangian space . For instance, a physical point can be pulled back to a label , and similarly a subset of physical space can be pulled back to a subset of label space. A scalar field can be pulled back to a scalar field , defined by pre-composition:

These operations are all compatible with each other in various ways; for instance, if , , and , and then

- if and only if .
- if and only if .
- The map is an isomorphism of -algebras.
- The map is an algebra isomorphism.

**Differential forms.** The next family of structures we will pull back are that of differential forms, which we will define using coordinates. (See also my previous notes on this topic for more discussion on differential forms.) For any , a *-form* on will be defined as a family of functions for which is totally antisymmetric with respect to permutations of the indices , thus if one interchanges and for any , then flips to . Thus for instance

- A -form is just a scalar field ;
- A -form, when viewed in coordinates, is a collection of scalar functions;
- A -form, when viewed in coordinates, is a collection of scalar functions with (so in particular );
- A -form, when viewed in coordinates, is a collection of scalar functions with , , and .

The antisymmetry makes the component of a -form vanish whenever two of the indices agree. In particular, if , then the only -form that exists is the zero -form . A -form is also known as a volume form; amongst all such forms we isolate the *standard volume form* , defined by setting for any permutation (with being the sign of the permutation), and setting all other components of equal to zero. For instance, in three dimensions one has equal to when , when , and otherwise. We use to denote the space of -forms on .

If is a scalar field and , we can define the product by pointwise multiplication of components:

More generally, given two forms , , we define the wedge product to be the -form given by the formula

where is the symmetric group of permutations on . For instance, for a scalar field (so ), . Similarly, if and , we have the pointwise identities

Exercise 7Show that the wedge product is a bilinear map from to that obeys the supercommutative propertyfor and , and the associative property

for , , . (In other words, the space of formal linear combinations of forms, graded by the parity of the order of the forms, is a supercommutative algebra. Very roughly speaking, the prefix “super” means that “odd order objects anticommute with each other rather than commute”.)

If is continuously differentiable, we define the exterior derivative in coordinates as

It is easy to verify that this is indeed a -form. Thus for instance:

- If is a continously differentiable scalar field, then .
- If is a continuously differentiable -form, then .
- If is a continuously differentiable -form, then .

Exercise 8If and are continuously differentiable, establish the antiderivation (or super-Leibniz) lawand if is twice continuously differentiable, establish the chain complex law

Each of the coordinates , can be viewed as scalar fields . In particular, the exterior derivatives , are -forms. It is easy to verify the identity

for any with the usual summation conventions (which, in this differential geometry formalism, assert that we sum indices whenever they appear as a subscript-superscript pair). In particular the volume form can be written as

One can of course define differential forms on Lagrangian space as well, changing the indices from Roman to Greek. For instance, if is continuously differentiable, then is given in coordinates as

If , we define the pullback form by the formula

with the usual summation conventions. Thus for instance

- If is a scalar field, then the pullback is given by the same formula as before.
- If is a -form, then the pullback is given by the formula .
- If is a -form, then the pullback is given by the formula

It is easy to see that pullback is a linear map from to . It also preserves the exterior algebra and exterior derivative:

Exercise 9Let . Show thatand if is continuously differentiable, show that

One can integrate -forms on oriented -manifolds. Suppose for instance that an oriented -manifold has a parameterisation , where is an open subset of and is an injective immersion. Then any continuous compactly supported -form can be integrated on by the formula

with the usual summation conventions. It can be shown that this definition is independent of the choice of parameterisation. For a more general manifold , one can use a partition of unity to decompose the integral into parameterised manifolds, and define the total integral to be the sum of the components; again, one can show (after some tedious calculation) that this is independent of the choice of parameterisation. If is all of (with the standard orientation), and , then we have the identity

linking integration on differential forms with the Lebesgue (or Riemann) integral. We also record Stokes’ theorem

whenever is a smooth orientable -manifold with smooth boundary , and is a continuous, compactly supported -form. The regularity conditions on here can often be relaxed by the usual limiting arguments; for the purposes of this set of notes, we shall proceed formally and assume that identities such as (14) hold for all manifolds and forms under consideration.

From the change of variables formula we see that pullback also respects integration on manifolds, in that

whenever is a smooth orientable -manifold, and a continuous compactly supported -form.

Exercise 10Establish the identityConclude in particular that is volume-preserving if and only if

**Vector fields.** Having pulled back differential forms, we now pull back vector fields. A vector field on , when viewed in coordinates, is a collection , of scalar functions; superficially, this resembles a -form , except that we use superscripts instead of subscripts to denote the components. On the other hand, we will transform vector fields under pullback in a different manner from -forms. For each , a basic example of a vector field is the coordinate vector field , defined by setting to equal when and otherwise. Then every vector field may be written as

where we multiply scalar functions against vector fields in the obvious fashion; compare this with the expansion of a -form into its components . The space of all vector fields will be denoted . One can of course define vector fields on similarly.

The pullback of is defined to be the unique vector field such that

for all (so that is the pushforward of ). Equivalently, if is the inverse matrix to the total differential (which we recall in coordinates is ), so that

with denoting the Kronecker delta, then

From the inverse function theorem one can also write

thus is also the pullback of by .

If is a -form and are vector fields, one can form the scalar field by the formula

Thus for instance if is a -form and are vector fields, then

It is clear that is a totally antisymmetric form in the . If is a -form for some and is a vector field, we define the contraction (or *interior product*) in coordinates by the formula

or equivalently that

for . Thus for instance if is a -form, and is a vector field, then is the -form

If is a vector field and is a continuously differentiable scalar field, then is just the directional derivative of along the vector field :

The contraction is also denoted in the literature. If one contracts a vector field against the standard volume form , one obtains a -form which we will call (by slight abuse of notation) the Hodge dual of :

This can easily be seen to be a bijection between vector fields and -forms. The inverse of this operation will also be denoted by the Hodge star :

In a similar spirit, the Hodge dual of a scalar field will be defined as the volume form

and conversely the Hodge dual of a volume form is a scalar field:

More generally one can form a Hodge duality relationship between -vector fields and -forms for any , but we will not do so here as we will not have much use for the notion of a -vector field for any .

These operations behave well under pullback (if one assumes volume preservation in the case of the Hodge star):

Exercise 11

- (i) If and , show that
- (ii) If for some and , show that
- (iii) If is volume-preserving, show that
whenever is a scalar field, vector field, -form, or -form on .

**Riemannian metrics.** A Riemannian metric on , when expressed in coordinates is a collection of scalar functions such that for each point , the matrix is symmetric and strictly positive definite. In particular it has an inverse metric , which is a collection of scalar functions such that

where denotes the Kronecker delta; here we have abused notation (and followed the conventions of general relativity) by allowing the inverse on the metric to be omitted when expressed in coordinates (relying instead on the superscripting of the indices, as opposed to subscripting, to indicate the metric inversion). The Euclidean metric is an example of a metric tensor, with equal to when and zero otherwise; the coefficients of the inverse Euclidean metric is similarly equal to when and otherwise. Given two vector fields and a Riemannian metric , we can form the scalar field by

this is a symmetric bilinear form in .

We can define the pullback metric by the formula

this is easily seen to be a Riemannian metric on , and one has the compatibility property

for all . It is then not difficult to check that if we pull back the inverse metric by the formula

then we have the expected relationship

Exercise 12If is a diffeomorphism, show thatfor any , and similarly

for any , and

for any Riemannian metric .

Exercise 13Show that is an isometry (with respect to the Euclidean metric on both and ) if and only if .

Every Riemannian metric induces a musical isomorphism between vector fields on with -forms: if is a vector field, the associated -form (also denoted or simply ) is defined in coordinates as

and similarly if , the associated vector field (also denoted or ) is defined in coordinates as

These operations clearly invert each other: and . Note that can still be defined if is not positive definite, though it might not be an isomorphism in this case. Observe the identities

The musical isomorphism interacts well with pullback, provided that one also pulls back the metric :

Exercise 14If is a Riemannian metric, show thatfor all , and

for all .

We can now interpret some classical operations on vector fields in this differential geometry notation. For instance, if are vector fields, the dot product can be written as

and also

and for , the cross product can be written in differential geometry notation as

Exercise 15Formulate a definition for the pullback of a rank tensor field (which in coordinates would be given by for ) that generalises the pullback of differential forms, vector fields, and Riemannian metrics. Argue why your definition is the natural one.

**Lie derivatives.** Let is a continuously differentiable vector field, and is a continuously differentiable -form, we will define the Lie derivative of along by the *Cartan formula*

with the convention that vanishes if is a -form. Thus for instance:

- If is a continuously differentiable scalar field, then is just the directional derivative of along : .
- If is a continuously differentiable -form, then is the -form
- If is a continuously differentiable -form, then is the -form

One can interpret the Lie derivative as the infinitesimal version of pullback:

Exercise 16Let be smooth and bounded (so that can be viewed as a smooth vector field on for each ), and let be a trajectory map. If is a smooth -form, show thatMore generally, if is a smooth -form that varies smoothly in , show that

where denotes the

material Lie derivative

Note that the material Lie derivative specialises to the material derivative when applied to scalar fields. The above exercise shows that the trajectory map intertwines the ordinary time derivative with the material (Lie) derivative.

Remark 17If one lets be the trajectory map associated to a time-independent vector field with initial condition (4) (thus and , then the above exercise shows that for any differential form . This can be used as an alternate definition of the Lie derivative (and has the advantage of readily extending to other tensors than differential forms, for which the Cartan formula is not available).

The Lie derivative behaves very well with respect to exterior product and exterior derivative:

Exercise 18Let be continuously differentiable, and let also be continuously differentiable. Establish the Leibniz ruleIf is twice continuously differentiable, also establish the commutativity

of exterior derivative and Lie derivative.

Exercise 19Let be continuously differentiable. Show thatwhere is the divergence of . Use this and Exercise 16 to give an alternate proof of Lemma 3.

Exercise 20Let be continuously differentiable. For any smooth compactly supported volume form , show thatConclude in particular that if is divergence-free then

for any .

The Lie derivative of a continuously differentiable vector field is defined in coordinates as

and the Lie derivative of a continuously differentiable rank tensor is defined in coordinates as

Thus for instance the Lie derivative of the Euclidean metric is expressible in coordinates as

(compare with the *deformation tensor* used in Notes 0).

We have similar properties to Exercise 18:

Exercise 21Let be continuously differentiable.

- (i) If and are continuously differentiable, establish the Leibniz rule
If , and , establish the variant Leibniz rule

- (ii) If is a continuously differentiable rank tensor and are continuously differentiable, establish the Leibniz rule
similarly, for , show that

- (iii) Establish the analogue of Exercise 16 in which the differential form is replaced by a vector field or a rank -tensor .
- (iv) If is divergence-free, show that
whenever is a continuously differential scalar field, vector field, -form, or -form on .

Exercise 22If is continuously differentiable, establish the identitywhenever is a continuously differentiable differential form, vector field, or metric tensor.

Exercise 23If are smooth, define theLie bracketby the formulaEstablish the anti-symmetry (so in particular ) and the Jacobi identity

and also

whenever are smooth, and is a smooth differentiable form, vector field, or metric tensor.

Exercise 24Formulate a definition for the Lie derivative of a (continuously differentiable) rank tensor field along a vector field that generalises the Lie derivative of differential forms, vector fields, and Riemannian metrics. Argue why your definition is the natural one.

** — 2. The Euler equations in differential geometry notation — **

Now we write the Euler equations (1) in differential geometry language developed in the above section. This will make it relatively painless to change coordinates. As in the rest of this set of notes, we work formally, assuming that all fields are smooth enough to justify the manipulations below.

The Euler equations involve a time-dependent scalar field , which can be viewed as an element of , and a time-dependent velocity field , which can be viewed as an element of . The second of the Euler equations simply asserts that this vector field is divergence-free:

or equivalently (by Exercise 19 and the definition of material Lie derivative )

For the first equation, it is convenient to work instead with the *covelocity field* , formed by applying the Euclidean musical isomorphism to :

In coordinates, we have , thus for . The Euler equations can then be written in coordinates as

The left-hand side is close to the component of the material Lie derivative of . Indeed, from (20) we have

and so the first Euler equation becomes

Since , we can express the right-hand side as a total derivative , where is the *modified pressure*

We thus see that the Euler equations can be transformed to the system

Using the Cartan formula (19), one can also write (22) as

where is another modification of the pressure:

In coordinates, (25) becomes

One advantage of the formulation (22)–(24) is that one can pull back by an arbitrary diffeomorphic change of coordinates (both time-dependent and time-independent), with the only things potentially changing being the material Lie derivative , the metric , and the volume form . (Another, related, advantage is that this formulation readily suggests an extension to more general Riemannian manifolds, by replacing with a general Riemannian metric and with the associated volume form, without the need to explicitly introduce other Riemannian geometry concepts such as covariant derivatives or Christoffel symbols.)

For instance, suppose , and we wish to view the Euler equations in cylindrical coordinates , thus pulling back under the time-independent map defined by

Strictly speaking, this is not a diffeomorphism due to singularities at , but we ignore this issue for now by only working away from the axis . As is well known, the metric pulls back under this change of coordinates as

thus the pullback metric is diagonal in coordinates with entries

The volume form similarly pulls back to the familiar cylindrical coordinate volume form

If (by slight abuse of notation) we write the components of as , and the components of as , then the second equation (23) in our current formulation of the Euler equations now becomes

and the third equation (24) is

which by the product rule and Exercise 19 becomes

or after expanding in coordinates

If one substitutes (27) into (26) in the coordinates to eliminate the variables, we thus see that the cylindrical coordinate form of the Euler equations is

One should compare how readily one can derive these equations using the differential geometry formalism with the more pedestrian aproach using the chain rule:

Exercise 25Starting with a smooth solution to the Euler equations (1) in , and transforming to cylindrical coordinates , establish the chain rule formulaeand use this and the identity

to rederive the system (28)–(31) (away from the axis) without using the language of differential geometry.

Exercise 26Turkington coordinates are a variant of cylindrical coordinates , defined by the formulaethe advantage of these coordinates are that the map from Cartesian coordinates to Turkington coordinates is volume preserving. Show that in these coordinates, the Euler equations become

(These coordinates are particularly useful for studying solutions to Euler that are “axisymmetric with swirl”, in the sense that the fields do not depend on the variable, so that all the terms involving vanish; one can specialise further to the case of solutions that are “axisymmetric without swirl”, in which case also vanishes.)

We can use the differential geometry formalism to formally verify the conservation laws of the Euler equation. We begin with conservation of energy

Formally differentiating this in time (and noting that the form is symmetric in ) we have

Using (22), we can write

From the Cartan formula (19) one has ; from Exercise 23 one has , and hence by the Leibniz rule (Exercise 18(i)) we thus can write as a total derivative:

From Exercise 20 we thus formally obtain the conservation law .

Now suppose that is a time-independent vector field that is a Killing vector field for the Euclidean metric , by which we mean that

Taking traces in (21), this implies in particular that is divergence-free, or equivalently

(Geometrically, this implication arises because the volume form can be constructed from the Euclidean metric (up to a choice of orientation).) Consider the formal quantity

As is the only time-dependent quantity here, we may formally differentiate to obtain

Using (22), the left-hand side is

By Cartan’s formula, is a total derivative , and hence this contribution to the integral formally vanishes as is divergence-free. The quantity can be written using the Leibniz rule as the difference of the total derivative and the quantity . The former quantity also gives no contribution to the integral as is divergence free, thus

By Exercise 23, we have . Since (and hence ) is annihilated by , and the form is symmetric in , we can express as a total derivative

and so this integral also vanishes. Thus we obtain the conservation law . If we set the Killing vector field equal to the constant vector field for some , we obtain conservation of the momentum components

for ; if we instead set the Killing vector field equal to the rotation vector field ) (which one can easily verify to be Killing using (21)) we obtain conservation of the angular momentum components

for . Unfortunately, this essentially exhausts the supply of Killing vector fields:

Exercise 27Let be a smooth Killing vector field of the Euclidean metric . Show that is a linear combination (with real coefficients) of the constant vector fields , and the rotation vector fields , . (Hint: use (21) to show that all the second derivatives of components of vanish.)

The *vorticity -form* is defined as the exterior derivative of the covelocity:

It already made an appearance in Notes 3 from the previous quarter. Taking exterior derivatives of (22) using (10) and Exercise 18 we obtain the appealingly simple *vorticity equation*

In two and three dimensions we may take the Hodge dual of the velocity -form to obtain either a scalar field (in dimension ) or a vector field (in dimension ), and then Exercise 18(iv) implies that

In two dimensions, this gives us a lot of conservation laws, since one can apply the scalar chain rule to then formally conclude that

for any , which upon integration on using Exercise 20 gives the conservation law

for any such function . Thus for instance the norms of are formally conserved for every , and hence also for by a limiting argument, recovering Proposition 24 from Notes 3 of the previous quarter.

In three dimensions there is also an interesting conservation law involving the vorticity. Observe that the wedge product of the covelocity and the vorticity is a -form and can thus be integrated over . The helicity

is a formally conserved quantity of the Euler equations. Indeed, formally differentiating and using Exercise 20 we have

From the Leibniz rule and (32) we have

Applying (22) we can write this expression as . From (10) we have , hence this expression is also a total derivative . From Stokes’ theorem (14) we thus formally obtain the conservation of helicity: ; this was first observed by Moreau.

Exercise 28Formally verify the conservation of momentum, angular momentum, and helicity directly from the original form (1) of the Euler equations.

Exercise 29In even dimensions , show that the integral (formed by taking the exterior product of copies of ) is conserved by the flow, while in odd dimensions , show that the generalised helicity is conserved by the flow. (This observation is due to Denis Serre, as well as unpublished work of Tartar.)

As it turns out, there are no further conservation laws for the Euler equations in Eulerian coordinates that are linear or quadratic integrals of the velocity field and its derivatives, at least in three dimensions; see this paper of Denis Serre. In particular, the Euler equations are not believed to be completely integrable. (But there are a few more conserved integrals of motion in the Lagrangian formalism; see Exercise 40.)

Exercise 30Let be a smooth solution to the Euler equations in three dimensions , let be the vorticity vector field, and let be an arbitrary smooth scalar field. EstablishErtel’s theorem

Exercise 31 (Clebsch variables)Let be a smooth solution to the Euler equations. Suppose that at time zero, the covelocity takes the formfor some smooth scalar fields .

- (i) Show that at all subsequent times , the covelocity takes the form
where are smooth scalar fields obeying the transport equations

- (ii) Suppose that we are in the classical case initially studied by Clebsch in 1859. (The extension to general was observed by Constantin.) Show that the vorticity vector field is given by
and conclude in particular that are annihilated by this vector field:

To put it another way, the vortex lines of lie in the joint level sets of and (and indeed, if are transverse to each other, then the vortex lines are locally the intersection of the two level sets, away from critical points at least).

** — 3. Viewing the Euler equations in Lagrangian coordinates — **

Throughout this section, is a smooth solution to the Euler equations on , and let be a volume-preserving trajectory map.

We pull back the Euler equations (22), (23), (24), to create a Lagrangian velocity field , a Lagrangian covelocity field , a Lagrangian modified pressure field , and a Lagrangian vorticity field by the formulae

By Exercise 16, the Euler equations now take the form

and the vorticity is given by

and obeys the vorticity equation

We thus see that the Lagrangian vorticity is a *pointwise* conserved quantity:

This lets us solve for the Eulerian vorticity in terms of the trajectory map. Indeed, from (12), (35) we have

applying the inverse of the linear transformation , we thus obtain the *Cauchy vorticity formula*

If we normalise the trajectory map by (4), then , and we thus have

Thus for instance, we see that the support of the vorticity is transported by the flow:

Among other things, this shows that the volume and topology of the support of the vorticity remain constant in time. It also suggests that the Euler equations admit a number of “vortex patch” solutions in which the vorticity is compactly supported.

Exercise 32Assume the normalisation (4).

- (i) In the two-dimensional case , show that the Cauchy vorticity formula simplifies to
Thus in this case, vorticity is simply transported by the flow.

- (ii) In the three-dimensional case , show that the Cauchy vorticity formula can be written using the Hodge dual of the vorticity as
Thus we see that the vorticity is transported and also stretched by the flow, with the stretching given by the matrix .

One can also phrase the conservation of vorticity in an integral form. If is a two-dimensional oriented surface in that does not vary in time, then from (37) we see that the integral

is formally conserved in time:

Composing this with the trajectory map using (35), we conclude that

Writing and using Stokes’ theorem (14), we arrive at the Kelvin circulation theorem

The integral of the covelocity along a loop is known as the *circulation* of the fluid along the loop; the Kelvin circulation theorem then asserts that this circulation remains constant over time as long as the loop evolves along the flow.

Exercise 33 (Cauchy invariants)

- (i) Use (3) to establish the identity
expressing the Lagrangian covelocity in terms of the Euclidean metric and the trajectory map .

- (ii) Use (i) and (36) to establish the Lagrangian equation of motion
- (iii) Show that is also the pullback of the unmodified Eulerian pressure , thus
and recover

Newton’s first law(5).- (iv) Use (ii) to conclude the
Cauchy invariantsare pointwise conserved in time.

- (v) Show that the Cauchy invariants are precisely the components of the Lagrangian vorticity, thus the conservation of the Cauchy invariants is equivalent to the Cauchy vorticity formula.
For more discussion of Cauchy’s investigation of the Cauchy invariants and vorticity formula, see this article of Frisch and Villone.

Exercise 34 (Transport of vorticity lines)Suppose we are in three dimensions , so that the Hodge dual of vorticity is a vector field. A smooth curve (either infinite on both ends, or a closed loop) in is said to be a vortex line (or vortex ring, in the case of a closed loop) at time if at every point of the curve , the tangent to at is parallel to the vorticity at that point. Suppose that the trajectory map is normalised using (4). Show that if is a vortex line at time , then is a vortex line at any other time ; thus, vortex lines (or vortex rings) flow along with the fluid.

Exercise 35 (Conservation of helicity in Lagrangian coordinates)

- (i) In any dimension, establish the identity
in Lagrangian spacetime.

- (ii) Conclude that in three dimensions , the quantity
is formally conserved in time. Explain why this conserved quantity is the same as the helicity (34).

- (iii) Continue assuming . Define a
vortex tubeat time to be a region in which, at every point on the boundary , the vorticity vector field is tangent to . Show that if is a vortex tube at time , then is a vortex tube at time , and the helicity on the vortex tube is formally conserved in time.- (iv) Let . If the covelocity can be expressed in Clebsch variables (Exercise 31) with , show that the local helicity formally vanishes on every vortex tube . This provides an obstruction to the existence of Clebsch variables. (On the other hand, it is easy to find Clebsch variables on with for an arbitrary covelocity , simply by setting equal to the coordinate functions .)

Exercise 36In the three-dimensional case , show that the material derivative commutes with operation of differentiation along the (Hodge dual of the) vorticity.

The Cauchy vorticity formula (39) can be used to obtain an integral representation for the velocity in terms of the trajectory map , leading to the *vorticity-stream formulation* of the Euler equations. Recall from 254A Notes 3 that if one takes the divergence of the (Eulerian) vorticity , one obtains the Laplacian of the (Eulerian) covelocity :

where are the partial derivatives raised by the Euclidean metric. For , we can use the fundamental solution of the Laplacian (see Exercise 18 of 254A Notes 1) that (formally, at least)

Integrating by parts (after first removing a small ball around , and observing that the boundary terms from this ball go to zero as one shrinks the radius to zero) one obtains the Biot-Savart law

for the covelocity, or equivalently

for the velocity.

Exercise 37Show that this law is also valid in the two-dimensional case .

Changing to Lagrangian variables, we conclude that

Using the Cauchy vorticity formula (39) (assuming the normalisation (4)), we obtain

Combining this with (3), we obtain an integral-differential equation for the evolution of the trajectory map:

This is known as the *vorticity-stream formulation* of the Euler equations. In two and three dimensions, the formulation can be simplified using the alternate forms of the vorticity formula in Exercise 32. While the equation (42) looks complicated, it is actually well suited for Picard-type iteration arguments (of the type used in 254A Notes 1), due to the relatively small number of derivatives on the right-hand side. Indeed, it turns out that one can iterate this equation with the trajectory map placed in function spaces such as ; see Chapter 4 of Bertozzi-Majda for details.

Remark 38Because of the ability to solve the Euler equations in Lagrangian coordinates by an iteration method, the local well-posedness theory is slightly stronger in some respects in Lagrangian coordinates than it is in Eulerian coordinates. For instance, in this paper of Constantin Kukavica and Vicol it is shown that Lagrangian coordinate Euler equations are well-posed in Gevrey spaces, while Eulerian coordinate Euler equations are not. It also happens that the trajectory maps are real-analytic in even if the initial data is merely smooth; see for instance this paper of Constantin-Vicol-Wu and the references therein. An example of this phenomenon is given in the exercise below.

Exercise 39 (DiPerna-Majda example)Let and be smooth functions.

- (i) Show that the DiPerna-Majda flow defined by
solves the three-dimesional Euler equations (with zero pressure).

- (ii) Show that the trajectory map with initial condition (4) is given by
in particular the trajectory map is analytic in the time variable , even though the Eulerian velocity field need not be.

- (iii) Show that the Lagrangian covelocity field is given by
and the Lagrangian vorticity field is given by

In particular the Lagrangian vorticity is conserved in time (as it ought to).

Exercise 40Show that the integralis formally conserved in time. (

Hint:some of the terms arising from computing the derivative are more easily treated by moving to Eulerian coordinates and performing integration by parts there, rather than in Lagrangian coordinates. One can also proceed by rewriting the terms in this integral using the Eulerian covelocity and the Lagrangian covelocity .) With the normalisation (4), conclude in particular thatThis conservation law is related to a scaling symmetry of the Euler equations in Lagrangian coordinates, and is due to Shankar. It does not have a local expression in purely Eulerian coordinates (mainly because of the appearance of the labels coordinate ).

We summarise the dictionary between Eulerian and Lagrangian coordinates in the following table:

Eulerian spacetime | Lagrangian spacetime |

Time | Time |

Eulerian position | Trajectory map |

Labels map | Lagrangian position |

Eulerian velocity | Lagrangian velocity |

Eulerian covelocity | Lagrangian covelocity |

Eulerian vorticity | Lagrangian vorticity |

Eulerian pressure | Lagrangian pressure |

Euclidean metric | Pullback metric |

Standard volume form | Standard volume form |

Material Lie derivative | Time derivative |

** — 4. Variational characterisation of the Euler equations — **

Our computations in this section will be even more formal than in previous sections.

From Exercise 1, a (smooth, bounded) vector field (together with a choice of initial map ) gives rise to a trajectory map . From Lemma 3, we see that that is volume preserving for all times if and only if is volume preserving and if is divergence-free. Given such a trajectory map, let us formally define the *Lagrangian* by the formula

As observed by Arnold, the Euler equations can be viewed as the Euler-Lagrange equations for this Lagrangian, subject to the constraint that the trajectory map is always volume-preserving:

Proposition 41Let be a smooth bounded divergence-free vector field with a volume-preserving trajectory map . Then the following are formally equivalent:

- (i) There is a pressure field such that solves the Euler equations.
- (ii) The trajectory map is a critical point of the Lagrangian with respect to all compactly supported infinitesimal perturbations of in that preserve the volume-preserving nature of the trajectory map.

*Proof:* First suppose that (i) holds. Consider an infinitesimal deformation of the trajectory map, with compactly supported in , where one can view either as an infinitesimal or as a parameter tending to zero (in this formal analysis we will not bother to make the setup more precise than this). If this deformation is still volume-preserving, then we have

differentiating at using Exercise 4 we see that

Writing , we thus see from the chain rule that the Eulerian vector field is divergence-free:

Now, let us compute the infinitesimal variation of the Lagrangian:

Formally differentiating under the integral sign, this expression becomes

which by symmetry simplifies to

We integrate by parts in time to move the derivative off of the perturbation , to arrive at

Using Newton’s first law (41), this becomes

Writing , we can change to Eulerian variables to obtain

We can now integrate by parts and use (45) and conclude that this variation vanishes. Thus is a formal critical point of the Lagrangian.

Conversely, if is a formal critical point, then the above analysis shows that the expression (46) vanishes whenever obeys (45). Changing variables to Euclidean space, this expression becomes

Hodge theory (cf. Exercise 16 of 254A Notes 1) then implies (formally) that must be a differential , which is equivalent to Newton’s first law (41), which is in turn equivalent to the Euler equations (recalling that is assumed to be divergence-free).

Remark 42The above analysis reveals that the pressure field can be interpreted as a Lagrange multiplier arising from the constraint that the trajectory map be volume-preserving.

Following Arnold, one can use Proposition 41 to formally interpret the Euler equations as a geodesic flow on an infinite dimensional Riemannian manifold. Indeed, for a finite-dimensional Riemannian manifold , it is well known that (constant speed) geodesics are formal critical points of the energy functional

Thus we see that if we formally take to be the infinite-dimensional space of volume-preserving diffeomorphisms , with the formal Riemannian metric at a point in the directions of two infinitesimal perturbations defined by

then Proposition 41 asserts, formally, that solutions to the Euler equations coincide with constant speed geodesic flows on . As it turns out, a number of other physical equations, including several further fluid equations, also have such a geodesic interpretation, such as Burgers’ equation, the Korteweg-de Vries equation, and the Camassa-Holm equations; see for instance this paper of Vizman for a survey. In principle this means that the tools of Riemannian geometry could be deployed to obtain a better understanding of the Euler equations (and of the other equations mentioned above), although to date this has proven to be somewhat elusive (except when discussing conservation laws, as in Remark 43 below) for a number of reasons, not the least of which is that rigorous Riemannian geometry on infinite-dimensional manifolds is technically quite problematic. (Nevertheless, one can at least recover the local existence theory for the Euler equations this way; see the aforementioned work of Ebin and Marsden.)

Remark 43Noether’s theorem tells us that one should expect a one-to-one correspondence between symmetries of a Lagrangian and conservation laws of the corresponding Euler-Lagrange equation. Applying this to Proposition 41, we conclude that the conservation laws of the Euler equations should correspond to symmetries of the Lagrangian (43). There are basically two obvious symmetries of this Lagrangian; one coming from isometries of Eulerian spacetime , and in particular time translation, spatial translation, and spatial rotation; and the other coming from volume-preserving diffeomorphisms of Lagrangian space . One can check that time translation corresponds to energy conservation, spatial translation corresponds to momentum conservation, and spatial rotation corresponds to angular momentum conservation, while Lagrangian diffeomorphism invariance corresponds to conservation of Lagrangian vorticity (or equivalently, the Cauchy vorticity formula). In three dimensions, if one specialises to the specific Lagrangian diffeomorphism created by flow along the vorticity vector field , one also recovers conservation of helicity; see this previous blog post for more discussion.

Remark 44There are also Hamiltonian formul