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

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

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

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

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

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

$\displaystyle \int_X^{2X} |\sum_{x \leq n \leq x+H} \lambda(n)|\ dx = o(HX) \ \ \ \ \ (1)$

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

$\displaystyle \sum_{n \leq X} \frac{\lambda(n) \lambda(n+1)}{n} = o(\log X).$

The local Fourier uniformity conjecture asserts the stronger asymptotic

$\displaystyle \int_X^{2X} \sup_{\alpha \in {\bf R}} |\sum_{x \leq n \leq x+H} \lambda(n) e(-\alpha n)|\ dx = o(HX) \ \ \ \ \ (2)$

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

$\displaystyle \sum_{n \leq X} \frac{\lambda(n) \lambda(n+1) \lambda(n+2)}{n} = o(\log X).$

This particular bound also follows from some slightly different arguments of Joni Teräväinen and myself, but the implication would also work for other non-pretentious bounded multiplicative functions, whereas the arguments of Joni and myself rely more heavily on the specific properties of the Liouville function (in particular that ${\lambda(p)=-1}$ for all primes ${p}$).

There is also a higher order version of the local Fourier uniformity conjecture in which the linear phase ${{}e(-\alpha n)}$ is replaced with a polynomial phase such as ${e(-\alpha_d n^d - \dots - \alpha_1 n - \alpha_0)}$, or more generally a nilsequence ${\overline{F(g(n) \Gamma)}}$; as shown in my previous paper, this conjecture implies (and is in fact equivalent to, after logarithmic averaging) a logarithmically averaged version of the full Chowla conjecture (not just the two-point or three-point versions), as well as a logarithmically averaged version of the Sarnak conjecture.

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

Theorem 1 The asymptotic (2) is true when ${H = X^\theta}$ for a fixed ${\theta > 0}$.

Previously this was known for ${\theta > 5/8}$ by the work of Zhan (who in fact proved the stronger pointwise assertion ${\sup_{\alpha \in {\bf R}} |\sum_{x \leq n \leq x+H} \lambda(n) e(-\alpha n)|= o(H)}$ for ${X \leq x \leq 2X}$ in this case). In a previous paper with Kaisa and Maksym, we also proved a weak version

$\displaystyle \sup_{\alpha \in {\bf R}} \int_X^{2X} |\sum_{x \leq n \leq x+H} \lambda(n) e(-\alpha n)|\ dx = o(HX) \ \ \ \ \ (3)$

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

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

$\displaystyle \sum_{h \leq H} \sum_{n \leq X} \lambda(n) \Lambda(n+h) \Lambda(n+2h) = o( H X )$

whenever ${H = X^\theta}$ for a fixed ${\theta > 0}$, where ${\Lambda}$ is the von Mangoldt function. Amusingly, the seemingly simpler question of establishing the expected asymptotic for

$\displaystyle \sum_{h \leq H} \sum_{n \leq X} \Lambda(n+h) \Lambda(n+2h)$

is only known in the range ${\theta \geq 1/6}$ (from the work of Zaccagnini). Thus we have a rare example of a number theory sum that becomes easier to control when one inserts a Liouville function!

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

$\displaystyle |\sum_{x \leq n \leq x+H} \lambda(n) e(-\alpha_x n)| \gg 1$

for some frequency ${\alpha_x \in{\bf R}}$. We informally describe this correlation between ${\lambda(n)}$ and ${e(\alpha_x n)}$ by writing

$\displaystyle \lambda(n) \approx e(\alpha_x n) \ \ \ \ \ (4)$

for ${n \in [x,x+H]}$ (informally, one should view this as asserting that ${\lambda(n)}$ “behaves like” a constant multiple of ${e(\alpha_x n)}$). For sake of discussion, suppose we have this relationship for all ${x \in [X,2X]}$, not just many.

As mentioned before, the main difficulty here is to understand how ${\alpha_x}$ varies with ${x}$. As it turns out, the multiplicativity properties of the Liouville function place a significant constraint on this dependence. Indeed, if we let ${p}$ be a fairly small prime (e.g. of size ${H^\varepsilon}$ for some ${\varepsilon>0}$), and use the identity ${\lambda(np) = \lambda(n) \lambda(p) = - \lambda(n)}$ for the Liouville function to conclude (at least heuristically) from (4) that

$\displaystyle \lambda(n) \approx e(\alpha_x n p)$

for ${n \in [x/p, x/p + H/p]}$. (In practice, we will have this sort of claim for many primes ${p}$ rather than all primes ${p}$, after using tools such as the Turán-Kubilius inequality, but we ignore this distinction for this informal argument.)

Now let ${x, y \in [X,2X]}$ and ${p,q \sim P}$ be primes comparable to some fixed range ${P = H^\varepsilon}$ such that

$\displaystyle x/p = y/q + O( H/P). \ \ \ \ \ (5)$

Then we have both

$\displaystyle \lambda(n) \approx e(\alpha_x n p)$

and

$\displaystyle \lambda(n) \approx e(\alpha_y n q)$

on essentially the same range of ${n}$ (two nearby intervals of length ${\sim H/P}$). This suggests that the frequencies ${p \alpha_x}$ and ${q \alpha_y}$ should be close to each other modulo ${1}$, in particular one should expect the relationship

$\displaystyle p \alpha_x = q \alpha_y + O( \frac{P}{H} ) \hbox{ mod } 1. \ \ \ \ \ (6)$

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

$\displaystyle \alpha_x = T/x \ \ \ \ \ (7)$

would solve (6) if ${T = O( X / H^2 )}$; by Taylor expansion, this would correspond to a global approximation of the form ${\lambda(n) \approx n^{iT}}$). One now has a problem of an additive combinatorial flavour (or of a “local to global” flavour), namely to leverage the relation (6) to obtain global control on ${\alpha_x}$ that resembles (7).

A key obstacle in solving (6) efficiently is the fact that one only knows that ${p \alpha_x}$ and ${q \alpha_y}$ are close modulo ${1}$, rather than close on the real line. One can start resolving this problem by the Chinese remainder theorem, using the fact that we have the freedom to shift (say) ${\alpha_y}$ by an arbitrary integer. After doing so, one can arrange matters so that one in fact has the relationship

$\displaystyle p \alpha_x = q \alpha_y + O( \frac{P}{H} ) \hbox{ mod } p \ \ \ \ \ (8)$

whenever ${x,y \in [X,2X]}$ and ${p,q \sim P}$ obey (5). (This may force ${\alpha_q}$ to become extremely large, on the order of ${\prod_{p \sim P} p}$, but this will not concern us.)

Now suppose that we have ${y,y' \in [X,2X]}$ and primes ${q,q' \sim P}$ such that

$\displaystyle y/q = y'/q' + O(H/P). \ \ \ \ \ (9)$

For every prime ${p \sim P}$, we can find an ${x}$ such that ${x/p}$ is within ${O(H/P)}$ of both ${y/q}$ and ${y'/q'}$. Applying (8) twice we obtain

$\displaystyle p \alpha_x = q \alpha_y + O( \frac{P}{H} ) \hbox{ mod } p$

and

$\displaystyle p \alpha_x = q' \alpha_{y'} + O( \frac{P}{H} ) \hbox{ mod } p$

and thus by the triangle inequality we have

$\displaystyle q \alpha_y = q' \alpha_{y'} + O( \frac{P}{H} ) \hbox{ mod } p$

for all ${p \sim P}$; hence by the Chinese remainder theorem

$\displaystyle q \alpha_y = q' \alpha_{y'} + O( \frac{P}{H} ) \hbox{ mod } \prod_{p \sim P} p.$

In practice, in the regime ${H = X^\theta}$ that we are considering, the modulus ${\prod_{p \sim P} p}$ is so huge we can effectively ignore it (in the spirit of the Lefschetz principle); so let us pretend that we in fact have

$\displaystyle q \alpha_y = q' \alpha_{y'} + O( \frac{P}{H} ) \ \ \ \ \ (10)$

whenever ${y,y' \in [X,2X]}$ and ${q,q' \sim P}$ obey (9).

Now let ${k}$ be an integer to be chosen later, and suppose we have primes ${p_1,\dots,p_k,q_1,\dots,q_k \sim P}$ such that the difference

$\displaystyle q = |p_1 \dots p_k - q_1 \dots q_k|$

is small but non-zero. If ${k}$ is chosen so that

$\displaystyle P^k \approx \frac{X}{H}$

(where one is somewhat loose about what ${\approx}$ means) then one can then find real numbers ${x_1,\dots,x_k \sim X}$ such that

$\displaystyle \frac{x_j}{p_j} = \frac{x_{j+1}}{q_j} + O( \frac{H}{P} )$

for ${j=1,\dots,k}$, with the convention that ${x_{k+1} = x_1}$. We then have

$\displaystyle p_j \alpha_{x_j} = q_j \alpha_{x_{j+1}} + O( \frac{P}{H} )$

which telescopes to

$\displaystyle p_1 \dots p_k \alpha_{x_1} = q_1 \dots q_k \alpha_{x_1} + O( \frac{P^k}{H} )$

and thus

$\displaystyle q \alpha_{x_1} = O( \frac{P^k}{H} )$

and hence

$\displaystyle \alpha_{x_1} = O( \frac{P^k}{H} ) \approx O( \frac{X}{H^2} ).$

In particular, for each ${x \sim X}$, we expect to be able to write

$\displaystyle \alpha_x = \frac{T_x}{x} + O( \frac{1}{H} )$

for some ${T_x = O( \frac{X^2}{H^2} )}$. This quantity ${T_x}$ can vary with ${x}$; but from (10) and a short calculation we see that

$\displaystyle T_y = T_{y'} + O( \frac{X}{H} )$

whenever ${y, y' \in [X,2X]}$ obey (9) for some ${q,q' \sim P}$.

Now imagine a “graph” in which the vertices are elements ${y}$ of ${[X,2X]}$, and two elements ${y,y'}$ are joined by an edge if (9) holds for some ${q,q' \sim P}$. Because of exponential sum estimates on ${\sum_{q \sim P} q^{it}}$, this graph turns out to essentially be an “expander” in the sense that any two vertices ${y,y' \in [X,2X]}$ can be connected (in multiple ways) by fairly short paths in this graph (if one allows one to modify one of ${y}$ or ${y'}$ by ${O(H)}$). As a consequence, we can assume that this quantity ${T_y}$ is essentially constant in ${y}$ (cf. the application of the ergodic theorem in this previous blog post), thus we now have

$\displaystyle \alpha_x = \frac{T}{x} + O(\frac{1}{H} )$

for most ${x \in [X,2X]}$ and some ${T = O(X^2/H^2)}$. By Taylor expansion, this implies that

$\displaystyle \lambda(n) \approx n^{iT}$

on ${[x,x+H]}$ for most ${x}$, thus

$\displaystyle \int_X^{2X} |\sum_{x \leq n \leq x+H} \lambda(n) n^{-iT}|\ dx \gg HX.$

But this can be shown to contradict the Matomäki-Radziwill theorem (because the multiplicative function ${n \mapsto \lambda(n) n^{-iT}}$ is known to be non-pretentious).