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A proof of concept tool to verify estimates

1 May, 2025 in math.AP, math.CA | Tags: Software | by Terence Tao | 30 comments

This post was inspired by some recent discussions with Bjoern Bringmann.

Symbolic math software packages are highly developed for many mathematical tasks in areas such as algebra, calculus, and numerical analysis. However, to my knowledge we do not have similarly sophisticated tools for verifying asymptotic estimates – inequalities that are supposed to hold for arbitrarily large parameters, with constant losses. Particularly important are functional estimates, where the parameters involve an unknown function or sequence (living in some suitable function space, such as an ${L^p}$ space); but for this discussion I will focus on the simpler situation of asymptotic estimates involving a finite number of positive real numbers, combined using arithmetic operations such as addition, multiplication, division, exponentiation, and minimum and maximum (but no subtraction). A typical inequality here might be the weak arithmetic mean-geometric mean inequality

$\displaystyle (abc)^{1/3} \lesssim a+b+c \ \ \ \ \ (1)$

where ${a,b,c}$ are arbitrary positive real numbers, and the ${\lesssim}$ here indicates that we are willing to lose an unspecified (multiplicative) constant in the estimates.

I have wished in the past (e.g., in this MathOverflow answer) for a tool that could automatically determine whether such an estimate was true or not (and provide a proof if true, or an asymptotic counterexample if false). In principle, simple inequalities of this form could be automatically resolved by brute force case splitting. For instance, with (1), one first observes that ${a+b+c}$ is comparable to ${\max(a,b,c)}$ up to constants, so it suffices to determine if

$\displaystyle (abc)^{1/3} \lesssim \max(a,b,c). \ \ \ \ \ (2)$

Next, to resolve the maximum, one can divide into three cases: ${a \gtrsim b,c}$ ; ${b \gtrsim a,c}$ ; and ${c \gtrsim a,b}$ . Suppose for instance that ${a \gtrsim b,c}$ . Then the estimate to prove simplifies to

$\displaystyle (abc)^{1/3} \lesssim a,$

and this is (after taking logarithms) a positive linear combination of the hypotheses ${a \gtrsim b}$ , ${a \gtrsim c}$ . The task of determining such a linear combination is a standard linear programming task, for which many computer software packages exist.

Any single such inequality is not too difficult to resolve by hand, but there are applications in which one needs to check a large number of such inequalities, or split into a large number of cases. I will take an example at random from an old paper of mine (adapted from the equation after (51), and ignoring some epsilon terms for simplicity): I wanted to establish the estimate

$\displaystyle \frac{\langle N_2 \rangle^{1/2}}{\langle N_1 \rangle^{1/4} L_1^{1/2} L_2^{1/2} } L_{\min}^{1/2} N^{-1} (N_1 N_2 N_3)^{1/2} \lesssim 1 \ \ \ \ \ (3)$

for any ${N_1,N_2,N_3,L_1,L_2,L_3 > 0}$ obeying the constraints

$\displaystyle N_{\max} \sim N_{\mathrm{med}} \sim N; \quad L_{\max} \sim L_{\mathrm{med}} \gtrsim N_1 N_2 N_3$

where ${N_{\max}}$ , ${N_{\mathrm{med}}}$ , and ${N_{\min}}$ are the maximum, median, and minimum of ${N_1, N_2, N_3}$ respectively, and similarly for ${L_{\max}}$ , ${L_{\mathrm{med}}}$ , and ${L_{\min}}$ , and ${\langle N \rangle := (1+N^2)^{1/2}}$ . This particular bound could be dispatched in three or four lines from some simpler inequalities; but it took some time to come up with those inequalities, and I had to do a dozen further inequalities of this type. This is a task that seems extremely ripe for automation, particularly with modern technology.

Recently, I have been doing a lot more coding (in Python, mostly) than in the past, aided by the remarkable facility of large language models to generate initial code samples for many different tasks, or to autocomplete partially written code. For the most part, I have restricted myself to fairly simple coding tasks, such as computing and then plotting some mildly complicated mathematical functions, or doing some rudimentary data analysis on some dataset. But I decided to give myself the more challenging task of coding a verifier that could handle inequalities of the above form. After about four hours of coding, with frequent assistance from an LLM, I was able to produce a proof of concept tool for this, which can be found at this Github repository. For instance, to verify (1), the relevant Python code is

    a = Variable("a")
    b = Variable("b")
    c = Variable("c")
    assumptions = Assumptions()
    assumptions.can_bound((a * b * c) ** (1 / 3), max(a, b, c))

and the (somewhat verbose) output verifying the inequality is

Checking if we can bound (((a * b) * c) ** 0.3333333333333333) by max(a, b, c) from the given axioms.
We will split into the following cases:
[[b <~ a, c <~ a], [a <~ b, c <~ b], [a <~ c, b <~ c]]
Trying case: ([b <~ a, c <~ a],)
Simplify to proving (((a ** 0.6666666666666667) * (b ** -0.3333333333333333)) * (c ** -0.3333333333333333)) >= 1.
Bound was proven true by multiplying the following hypotheses :
b <~ a raised to power 0.33333333
c <~ a raised to power 0.33333333
Trying case: ([a <~ b, c <~ b],)
Simplify to proving (((b ** 0.6666666666666667) * (a ** -0.3333333333333333)) * (c ** -0.3333333333333333)) >= 1.
Bound was proven true by multiplying the following hypotheses :
a <~ b raised to power 0.33333333
c <~ b raised to power 0.33333333
Trying case: ([a <~ c, b <~ c],)
Simplify to proving (((c ** 0.6666666666666667) * (a ** -0.3333333
333333333)) * (b ** -0.3333333333333333)) >= 1.
Bound was proven true by multiplying the following hypotheses :
a <~ c raised to power 0.33333333
b <~ c raised to power 0.33333333
Bound was proven true in all cases!

This is of course an extremely inelegant proof, but elegance is not the point here; rather, that it is automated. (See also this recent article of Heather Macbeth for how proof writing styles change in the presence of automated tools, such as formal proof assistants.)

The code is close to also being able to handle more complicated estimates such as (3); right now I have not written code to properly handle hypotheses such as ${N_{\max} \sim N_{\mathrm{med}} \sim N}$ that involve complex expressions such as ${N_{\max} = \max(N_1,N_2,N_3)}$ , as opposed to hypotheses that only involve atomic variables such as ${N_1}$ , ${N_2, N_3}$ , but I can at least handle such complex expressions in the left and right-hand sides of the estimate I am trying to verify.

In any event, the code, being a mixture of LLM-generated code and my own rudimentary Python skills, is hardly an exemplar of efficient or elegant coding, and I am sure that there are many expert programmers who could do a much better job. But I think this is proof of concept that a more sophisticated tool of this form could be quite readily created to do more advanced tasks. One such example task was the one I gave in the above MathOverflow question, namely being able to automatically verify a claim such as

$\displaystyle \sum_{d=0}^\infty \frac{2d+1}{2h^2 (1 + \frac{d(d+1)}{h^2}) (1 + \frac{d(d+1)}{h^2m^2})^2} \lesssim 1 + \log(m^2)$

for all ${h,m > 0}$ . Another task would be to automatically verify the ability to estimate some multilinear expression of various functions, in terms of norms of such functions in standard spaces such as Sobolev spaces; this is a task that is particularly prevalent in PDE and harmonic analysis (and can frankly get somewhat tedious to do by hand). As speculated in that MO post, one could eventually hope to also utilize AI to assist in the verification process, for instance by suggesting possible splittings of the various sums or integrals involved, but that would be a long-term objective.

This sort of software development would likely best be performed as a collaborative project, involving both mathematicians and expert programmers. I would be interested to receive advice on how best to proceed with such a project (for instance, would it make sense to incorporate such a tool into an existing platform such as SageMATH), and what features for a general estimate verifier would be most desirable for mathematicians. One thing on my wishlist is the ability to give a tool an expression to estimate (such as a multilinear integral of some unknown functions), as well as a fixed set of tools to bound that integral (e.g., splitting the integral into pieces, integrating by parts, using the Hölder and Sobolev inequalities, etc.), and have the computer do its best to optimize the bound it can produce with those tools (complete with some independently verifiable proof certificate for its output). One could also imagine such tools having the option to output their proof certificates in a formal proof assistant language such as Lean. But perhaps there are other useful features that readers may wish to propose.

Closing the “green gap”: from the mathematics of the landscape function to lower electricity costs for households

23 February, 2025 in expository, math.AP, math.MP, The sciences | Tags: green gap, landscape theory, LED lighting, Svitlana Mayboroda | by Terence Tao | 7 comments

I recently returned from the 2025 Annual Meeting of the “Localization of Waves” collaboration (supported by the Simons Foundation, with additional related support from the NSF), where I learned (from Svitlana Mayboroda, the director of the collaboration as well as one of the principal investigators) of a remarkable statistic: net electricity consumption by residential customers in the US has actually experienced a slight decrease in recent years:

The decrease is almost entirely due to gains in lighting efficiency in households, and particularly the transition from incandescent (and compact fluorescent) light bulbs to LED light bulbs:

Annual energy savings from this switch to consumers in the US were already estimated to be $14.7 billion in 2020 – or several hundred dollars per household – and are projected to increase, even in the current inflationary era, with the cumulative savings across the US estimated to reach $890 billion by 2035.

What I also did not realize before this meeting is the role that recent advances in pure mathematics – and specifically, the development of the “landscape function” that was a primary focus of this collaboration – played in accelerating this transition. This is not to say that this piece of mathematics was solely responsible for these developments; but, as I hope to explain here, it was certainly part of the research and development ecosystem in both academia and industry, spanning multiple STEM disciplines and supported by both private and public funding. This application of the landscape function was already reported upon by Quanta magazine at the very start of this collaboration back in 2017; but it is only in the last few years that the mathematical theory has been incorporated into the latest LED designs and led to actual savings at the consumer end.

LED lights are made from layers of semiconductor material (e.g., Gallium nitride or Indium gallium nitride) arranged in a particular fashion. When enough of a voltage difference is applied to this material, electrons are injected into the “n-type” side of the LED, while holes of electrons are injected into the “p-type” side, creating a current. In the active layer of the LED, these electrons and holes recombine in the quantum wells of the layer, generating radiation (light) via the mechanism of electroluminescence. The brightness of the LED is determined by the current, while the power consumption is the product of the current and the voltage. Thus, to improve energy efficiency, one seeks to design LEDs to require as little voltage as possible to generate a target amount of current.

As it turns out, the efficiency of an LED, as well as the spectral frequencies of light they generate, depend in many subtle ways on the precise geometry of the chemical composition of the semiconductors, the thickness of the layers, the geometry of how the layers are placed atop one another, the temperature of the materials, and the amount of disorder (impurities) introduced into each layer. In particular, in order to create quantum wells that can efficiently trap the electrons and holes together to recombine to create light of a desired frequency, it is useful to introduce a certain amount of disorder into the layers in order to take advantage of the phenomenon of Anderson localization. However, one cannot add too much disorder, lest the electron states become fully bound and the material behaves too much like an insulator to generate appreciable current.

One can of course make empirical experiments to measure the performance of various proposed LED designs by fabricating them and then testing them in a laboratory. But this is an expensive and painstaking process that does not scale well; one cannot test thousands of candidate designs this way to isolate the best performing ones. So, it becomes desirable to perform numerical simulations of these designs instead, which – if they are sufficiently accurate and computationally efficient – can lead to a much shorter and cheaper design cycle. (In the near future one may also hope to accelerate the design cycle further by incorporating machine learning and AI methods; but these techniques, while promising, are still not fully developed at the present time.)

So, how can one perform numerical simulation of an LED? By the semiclassical approximation, the wave function ${\psi_i}$ of an individual electron should solve the time-independent Schrödinger equation

$\displaystyle -\frac{\hbar^2}{2m_e} \Delta \psi_i + E_c \psi_i = E_i \psi_i,$

where ${\psi}$ is the wave function of the electron at this energy level, and ${E_c}$ is the conduction band energy. The behavior of hole wavefunctions follows a similar equation, governed by the valence band energy ${E_v}$ instead of ${E_c}$ . However, there is a complication: these band energies are not solely coming from the semiconductor, but also contain a contribution ${\mp e \varphi}$ that comes from electrostatic effects from the electrons and holes, and more specifically by solving the Poisson equation

$\displaystyle \mathrm{div}( \varepsilon_r \nabla \varphi ) = \frac{e}{\varepsilon_0} (n-p + N_A^+ - N_D^-)$

where ${\varepsilon_r}$ is the dielectric constant of the semiconductor, ${n,p}$ are the carrier densities of electrons and holes respectively, ${N_A^+}$ , ${N_D^-}$ are further densities of ionized acceptor and donor atoms, and ${\hbar, m_e, e, \varepsilon_0}$ are physical constants. This equation looks somewhat complicated, but is mostly determined by the carrier densities ${n,p}$ , which in turn ultimately arise from the probability densities ${|\psi_i|^2}$ associated to the eigenfunctions ${\psi_i}$ via the Born rule, combined with the Fermi-Dirac distribution from statistical mechanics; for instance, the electron carrier density ${n}$ is given by the formula

$\displaystyle n = \sum_i \frac{|\psi_i|^2}{1 + e^{(E_i - E_{Fn})/k_B T}},$

with a similar formula for ${p}$ . In particular, the net potential ${E_c}$ depends on the wave functions ${\psi_i}$ , turning the Schrödinger equation into a nonlinear self-consistent Hartree-type equation. From the wave functions one can also compute the current, determine the amount of recombination between electrons and holes, and therefore also calculate the light intensity and absorption rates. But the main difficulty is to solve for the wave functions ${\psi_i}$ for the different energy levels of the electron (as well as the counterpart for holes).

One could attempt to solve this nonlinear system iteratively, by first proposing an initial candidate for the wave functions ${\psi_i}$ , using this to obtain a first approximation for the conduction band energy ${E_c}$ and valence band energy ${E_v}$ , and then solving the Schrödinger equations to obtain a new approximation for ${\psi_i}$ , and repeating this process until it converges. However, the regularity of the potentials ${E_c, E_v}$ plays an important role in being able to solve the Schrödinger equation. (The Poisson equation, being elliptic, is relatively easy to solve to high accuracy by standard methods, such as finite element methods.) If the potential ${E_c}$ is quite smooth and slowly varying, then one expects the wave functions ${\psi_i}$ to be quite delocalizated, and for traditional approximations such as the WKB approximation to be accurate.

However, in the presence of disorder, such approximations are no longer valid. As a consequence, traditional methods for numerically solving these equations had proven to be too inaccurate to be of practical use in simulating the performance of a LED design, so until recently one had to rely primarily on slower and more expensive empirical testing methods. One real-world consequence of this was the “green gap“; while reasonably efficient LED designs were available in the blue and red portions of the spectrum, there was not a suitable design that gave efficient output in the green spectrum. Given that many applications of LED lighting required white light that was balanced across all visible colors of the spectrum, this was a significant impediment to realizing the energy-saving potential of LEDs.

Here is where the landscape function comes in. This function started as a purely mathematical discovery: when solving a Schrödinger equation such as

$\displaystyle -\Delta \phi + V \phi = E \phi$

(where we have now suppressed all physical constants for simplicity), it turns out that the behavior of the eigenfunctions ${\phi}$ at various energy levels ${E}$ is controlled to a remarkable extent by the landscape function ${u}$ , defined to be the solution to the equation

$\displaystyle -\Delta u + V u = 1.$

As discussed in this previous blog post (discussing a paper on this topic I wrote with some of the members of this collaboration), one reason for this is that the Schrödinger equation can be transformed after some routine calculations to

$\displaystyle -\frac{1}{u^2} \mathrm{div}( u^2 \nabla (\phi/u)) + \frac{1}{u} (\phi/u) = E (\phi/u),$

thus making ${\frac{1}{u}}$ an effective potential for the Schrödinger equation (and ${u^2}$ also being the coefficients of an effective geometry for the equation). In practice, when ${V}$ is a disordered potential, the effective potential ${1/u}$ tends to be behave like a somewhat “smoothed out” or “homogenized” version of ${V}$ that exhibits superior numerical performance. For instance, the classical Weyl law predicts (assuming a smooth confining potential ${V}$ ) that the density of states up to energy ${E}$ – that is to say, the number of bound states up to ${E}$ – should asymptotically behave like ${\frac{1}{(2\pi)^2}|\{ (x,\xi): \xi^2 + V(x) \leq E\}|}$ . This is accurate at very high energies ${E}$ , but when ${V}$ is disordered, it tends to break down at low and medium energies. However, the landscape function makes a prediction ${\frac{1}{(2\pi)^2}|\{ (x,\xi): \xi^2 + 1/u(x) \leq E\}|}$ for this density of states that is significantly more accurate in practice in these regimes, with a mathematical justification (up to multiplicative constants) of this accuracy obtained in this paper of David, Filoche, and Mayboroda. More refined predictions (again with some degree of theoretical support from mathematical analysis) can be made on the local integrated density of states, and with more work one can then also obtain approximations for the carrier density functions ${n,p}$ mentioned previously in terms of the energy band level functions ${E_c}$ , ${E_v}$ . As the landscape function ${u}$ is relatively easy to compute (coming from solving a single elliptic equation), this gives a very practical numerical way to carry out the iterative procedure described previously to model LEDs in a way that has proven to be both numerically accurate, and significantly faster than empirical testing, leading to a significantly more rapid design cycle.

In particular, recent advances in LED technology have largely closed the “green gap” by introducing designs that incorporate “ ${V}$ -defects”: ${V}$ -shaped dents in the semiconductor layers of the LED that create lateral carrier injection pathways and modify the internal electric field, enhancing hole transport into the active layer. The ability to accurately simulate the effects of these defects has allowed researchers to largely close this gap:

My understanding is that the major companies involved in developing LED lighting are now incorporating landscape-based methods into their own proprietary simulation models to achieve similar effects in commercially produced LEDs, which should lead to further energy savings in the near future.

Thanks to Svitlana Mayboroda and Marcel Filoche for detailed discussions, comments, and corrections of the material here.

A possible approach to finite time blowup for the Euler equations

8 September, 2024 in math.AP, math.MP | Tags: finite time blowup, incompressible Euler equations | by Terence Tao | 7 comments

Given a smooth compact Riemannian manifold ${M = (M,g)}$ , the incompressible Euler equations can be written in abstract index notation as

$\displaystyle \partial_t u^\alpha + u^\beta \nabla_\beta u^\alpha = - \nabla^\alpha p$

$\displaystyle \nabla_\alpha u^\alpha = 0$

where ${p: [0,T) \rightarrow C^\infty(M)}$ is a time-dependent scalar field (representing pressure), and ${u: [0,T) \rightarrow \Gamma(TM)}$ is a time-dependent vector field (representing fluid velocity). Here ${\nabla}$ is the Levi-Civita connection associated to ${g}$ . One can recover ${p}$ from ${u}$ (up to constants), so I will abuse notation and refer to the solution to this system as ${u}$ rather than ${(u,p)}$ . Over the last few years I have been interested in the following conjecture:

Conjecture 1 (Finite time blowup) There exists a manifold ${M}$ and a smooth solution ${u}$ to the Euler equations that blows up at some finite time ${T}$ .

This remains open, however there has been progress on rougher versions of this problem. For instance, there is the well-known result of Elgindi (discussed in this previous post) that when ${M = {\bf R}^3}$ and ${\alpha>0}$ is sufficiently small, there exists a ${C^{1,\alpha}}$ solution ${u}$ to the Euler equations on ${{\bf R}^3}$ that blows up in finite time. There has also been progress in establishing various “universality” properties of the Euler flow on manifolds (which informally state that “fluid computers” are possible); see for instance this recent survey of Cardona, Miranda, and Peralta-Salas. Unfortunately, these “fluid computers” do not combine well with scaling symmetries, and so thus far have not been able to produce (finite energy) blowups.

I have been playing with one approach to this conjecture, which reduces to solving a certain underdetermined system of partial differential equations, and then establishing some stability result for the resulting solution. However, I have not been able to make headway on solving this latter system despite its underdetermined nature; so I thought I would record my partial attempt here in case anyone is interested in pursuing it further (and also to contribute to the practice of sharing unsuccessful attempts to solve a problem, which is still quite infrequently done in our community).

To avoid technicalities let us simplify the problem by adding a forcing term ${f: [0,T) \rightarrow \Gamma(TM)}$ :

$\displaystyle \partial_t u^\alpha + u^\beta \nabla_\beta u^\alpha = - \nabla^\alpha p + f^\alpha$

$\displaystyle \nabla_\alpha u^\alpha = 0.$

Standard local well-posedness theory (using the vorticity-stream formulation of the Euler equations) tells us that this problem is well-posed if the initial data ${u_0}$ is divergence free and in ${C^{1,\alpha}(M)}$ for some ${\alpha>0}$ , and the forcing term ${f}$ is in ${L^1_t C^{1,\alpha}(M)}$ . We have the following recent result of Córdoba and Martínez-Zoroa, solving a version of the above conjecture in the presence of a reasonably regular forcing term:

Theorem 2 (Finite time blowup for the forced equation) There exists a smooth solution to the forced Euler equations on ${{\bf R}^3}$ that exhibits finite time blowup, in which the forcing term ${f}$ stays uniformly bounded in ${C^{1,\alpha}({\bf R}^3)}$ for any ${\alpha < 1/2}$ .

Roughly speaking, their argument proceeds by a multiscale construction, in which the solution is set up to eventually have some presence at a spatial scale ${1/N}$ , which is conducive to generating an exponential “stretching” of a small forcing term at a much higher spatial scale ${1/M}$ , which one then introduces to then set up the solution for the next scale.

As a model problem, I tried to reproduce this type of result from a more geometric perspective, trying to aim for a more “self-similar” blowup than a “multi-scale” one, in the hope that this latter type of blowup might be more tractable to analyze and eventually resolve Conjecture 1. I didn’t fully succeed; but I think the approach I outline below is in principle feasible.

The manifold I will work on is a cylinder ${M = {\bf R} \times N}$ , where ${N = (N,h)}$ is a smooth compact manifold, and the metric on ${M}$ is just the sum of the standard metric ${dx_1^2}$ on the first coordinate and ${h}$ :

$\displaystyle dg^2 = dx_1^2 + dh^2.$

(I have experimented with working with more warped metrics to gain more flexibility, but this seems to only influence lower order terms. However, such tricks may be useful to improve the regularity of the forcing term, or perhaps even to eliminate it entirely.) The idea is to try to create a solution that blows up on a slice ${\{0\} \times N}$ on this cylinder, but stays smooth until the blowup time, and also uniformly smooth away from this slice. As such, one can localize the solution away from this slice, and replace the unbounded component ${{\bf R}}$ of ${M}$ by a compact circle ${{\bf R}/L{\bf Z}}$ for some large ${L}$ if desired. However, I prefer to work with the unbounded component ${{\bf R}}$ here in order to scale in this direction.

If we now use Greek indices to only denote coordinates in the “vertical” coordinate ${N}$ , the velocity field ${u}$ now becomes ${(u^1, u^\alpha)}$ , and the Euler equations now split as

$\displaystyle \partial_t u^1 + u^1 \partial_1 u^1 + u^\beta \nabla_\beta u^1 = - \partial_1 p + f^1$

$\displaystyle \partial_t u^\alpha + u^1 \partial_1 u^\alpha + u^\beta \nabla_\beta u^\alpha = - \nabla^\alpha p + f^\alpha$

$\displaystyle \partial_1 u^1 + \nabla_\alpha u^\alpha = 0.$

If the solution is concentrating in a narrow neighborhood of the slice ${\{0\} \times N}$ , we expect the terms involving ${\partial_1}$ to be quite large, and the terms involving ${u_1}$ to be rather small. This suggests that the ${\partial_1 p}$ pressure term is going to be more significant than the ${\nabla^\alpha p}$ term. We therefore select the forcing term to cancel this term by choosing

$\displaystyle f^1 = 0; f^\alpha = \nabla^\alpha p$

leaving us with the simplified equation

$\displaystyle \partial_t u^1 + u^1 \partial_1 u^1 + u^\beta \nabla_\beta u^1 = - \partial_1 p$

$\displaystyle \partial_t u^\alpha + u^1 \partial_1 u^\alpha + u^\beta \nabla_\beta u^\alpha = 0$

$\displaystyle \partial_1 u^1 + \nabla_\alpha u^\alpha = 0.$

The nice thing about this latter equation is that the first equation is basically just a formula for ${p}$ and so can be dropped, leaving us with

$\displaystyle \partial_t u^\alpha + u^1 \partial_1 u^\alpha + u^\beta \nabla_\beta u^\alpha = 0 \ \ \ \ \ (1)$

$\displaystyle \partial_1 u^1 + \nabla_\alpha u^\alpha = 0. \ \ \ \ \ (2)$

This equation now admits a two-parameter family of scale invariances

$\displaystyle u^1(t,x^1,y) \mapsto \frac{\lambda}{\mu} u^1(t/\mu, x/\lambda,y); u^\alpha(t,x^1,y) \mapsto \frac{1}{\mu} u^1(t/\mu, x/\lambda,x^\alpha)$

for ${\lambda, \mu > 0}$ , where we use ${y}$ to ednote the ${N}$ coordinate. It also inherits an energy conservation law from the original Euler equations; the conserved energy is

$\displaystyle \frac{1}{2} \int_{\bf R} \int_N u^\alpha(t,x^1,y) u_\alpha(t,x^1,y)\ dy dx^1$

using the metric on ${N}$ to raise and lower the Greek indices.

It is now tempting to try to set up an approximately scale-invariant blowup solution. It seems that the first step in this is to construct a “soliton” type localized steady state solution, that is a solution ${u_1 \in C^\infty({\bf R} \times N)}$ , ${u^\alpha : {\bf R} \rightarrow \Gamma(TN)}$ to the equation

$\displaystyle u^1 \partial_1 u^\alpha + u^\beta \nabla_\beta u^\alpha = 0$

$\displaystyle \partial_1 u^1 + \nabla_\alpha u^\alpha = 0.$

that decays in the ${x_1}$ variable; one can then hope to do a suitable stability (or instability) analysis of this soliton to perturb it to a blowup solution, as there are many results of this type for other equations that one could use as a model. The energy conservation law does constrain to some extent the nature of the blowup (basically, the two scaling parameters ${\mu,\lambda}$ above become linked by the relation ${\mu \sim \lambda^{1/2}}$ ), but does not seem to otherwise prevent such a blowup from occuring.

Analytically, this is not a particularly pleasant equation to try to solve; one can substitute the second equation into the first to obtain a single equation

$\displaystyle -(\partial_1^{-1} \nabla_\beta u^\beta) \partial_1 u^\alpha + u^\beta \nabla_\beta u^\alpha = 0$

but the inverse derivative ${\partial_1^{-1}}$ is difficult to work with and seems to create ill-posedness (somewhat reminiscent of the ill-posedness of the Prandtl boundary layer equation).

Nevertheless, one can still attempt to solve this equation by separation of variables. If one makes the ansatz

$\displaystyle u^1(x^1,y) = \frac{1}{1+(x^1)^2} \varphi(y)$

$\displaystyle u^\alpha(x^1,y) = \frac{v^\alpha(y) + x^1 w^\alpha(y)}{(1+(x^1)^2)^2}$

for some smooth ${\varphi \in C^\infty(N)}$ and ${v,w \in \Gamma(TN)}$ , some calculation shows that the system now reduces to a system purely on ${N}$ :

$\displaystyle \varphi w^\alpha + v^\beta \nabla_\beta v^\alpha = 0$

$\displaystyle -2 \varphi v^\alpha + v^\beta \nabla_\beta w^\alpha + w^\beta \nabla_\beta v^\alpha = 0$

$\displaystyle -3 \varphi w^\alpha + w^\beta \nabla_\beta w^\beta = 0$

$\displaystyle \nabla_\alpha v^\alpha = 0.$

$\displaystyle \nabla_\alpha w^\alpha = 2\varphi.$

The metric ${h}$ is hidden in this system through the covariant derivative ${\nabla}$ . To eliminate the metric, we can lower indices to write

$\displaystyle \varphi w_\alpha + v^\beta \nabla_\beta v_\alpha = 0$

$\displaystyle -2 \varphi v_\alpha + v^\beta \nabla_\beta w_\alpha + w^\beta \nabla_\beta v_\alpha = 0$

$\displaystyle -3 \varphi w_\alpha + w^\beta \nabla_\beta w_\beta = 0$

$\displaystyle \mathrm{div}_h v = 0.$

$\displaystyle \mathrm{div}_h w = 2\varphi.$

Here the divergence is relative to the volume form induced by ${h}$ , but by a dimension lifting trick (see Section 3 of this paper of mine) one can replace this divergence with a divergence ${\mathrm{div}_{vol}}$ with respect to any other volume form on ${N}$ . We have the identity

$\displaystyle v^\beta \nabla_\beta v_\alpha = v^\beta (\nabla_\beta v_\alpha - \nabla_\alpha v_\beta) + \frac{1}{2} \partial_\alpha (v^\beta v_\beta)$

$\displaystyle v \neg d\theta + \frac{1}{2} d (\theta(v))$

where we have switched to coordinate-free notation, and ${\theta = h \cdot v}$ is the ${1}$ -form associated to ${v}$ . If we similarly let ${\lambda}$ be the ${1}$ -form associated to ${w}$ , and eliminate ${\varphi}$ , the system now becomes

$\displaystyle \frac{1}{2} \lambda \mathrm{div}_{vol} w + v \neg d\theta + \frac{1}{2} d (\theta(v)) = 0 \ \ \ \ \ (3)$

$\displaystyle -\theta \mathrm{div}_{vol} w + v \neg d\lambda + w \neg d\theta + \frac{1}{2} d (\theta(w) + \lambda(v)) = 0 \ \ \ \ \ (4)$

$\displaystyle -\frac{3}{2}\lambda \mathrm{div}_{vol} w + w \neg d\lambda + \frac{1}{2} d (\lambda(w)) = 0 \ \ \ \ \ (5)$

$\displaystyle \mathrm{div}_{vol} v = 0. \ \ \ \ \ (6)$

Also, the fact that ${\lambda,\theta}$ are dual to ${v,w}$ with respect to some unspecified Riemannian metric ${h}$ turns out to essentially be equivalent to the assumption that the Gram matrix is positive definite,

$\displaystyle \begin{pmatrix} \lambda(v) & \lambda(w) \\ \theta(v) & \theta(w) \end{pmatrix} \succ 0; \ \ \ \ \ (7)$

see Section 4 of the aforementioned paper. This looks like a rather strange system; but it is three vector equations (3), (4), (5) in four vector (or one-form) unknowns ${v, w, \theta, \lambda}$ , together with a divergence-free condition (6) and a positive definiteness condition (7), which I view as basically being scalar conditions. Thus, this system becomes underdetermined when the dimension of ${N}$ is large enough (in fact a naive count of degrees of freedom suggests that dimension at least three should be sufficient). It should thus be possible to locate an abundant number of solutions to this system; but to my frustration, the system is just barely complicated enough to prevent me from simplifying it to the point where it becomes evident how to construct solutions; in particular, I have not been able to find a viable further ansatz to transform the system to a more tractable one. Part of the problem is that while the system is technically underdetermined, there are not that many degrees of freedom to spare to ensure this underdetermined nature, so one cannot afford to make an ansatz that sacrifices too many of these degrees of freedom. In my previous paper I was able to use a very symmetric construction (taking ${N}$ to basically be the product of an orthogonal group and a torus) to solve a similar system that was in fact quite overdetermined, but I have not been able to exploit such symmetries here. I have also experimented with other separation of variable ansatzes, but they tend to give similar systems to the ones here (up to some changes in the coefficients).

Remark 3 One can also try to directy create a self-similar blowup to (1), (2), for instance by making the ansatz
$\displaystyle u^1(t,x^1,y) = (t^6+(x^1)^2)^{1/3} \varphi(y)$

$\displaystyle u^\alpha(t,x^1,y) = \frac{v^\alpha(y)}{(t^6+(x^1)^2)^{1/3}} + \frac{w^\alpha(y) + x^1 q^\alpha(y)}{(t^6+(x^1)^2)^{2/3}} + \frac{r^\alpha(y) + x^1 s^\alpha(y)}{(t^6+(x^1)^2)}$
for ${t<0}$ and some fields ${\varphi \in C^\infty(N)}$ and ${v,w,q,r,s \in \Gamma(TN)}$ . This particular ansatz seems consistent with all known conservation laws; however it works out to basically be ten vector equations (plus some additional scalar constraints) on ten vector field unknowns, so is just barely overdetermined. I have not been able to locate a self-similar blowup ansatz that is underdetermined.

The effective potential of an M-matrix

6 January, 2021 in math.AP, math.SP, paper | Tags: effective potential, eigenvectors, landscape theory, localisation, Marcel Filoche, Svitlana Mayboroda | by Terence Tao | 7 comments

Marcel Filoche, Svitlana Mayboroda, and I have just uploaded to the arXiv our preprint “The effective potential of an ${M}$ -matrix“. This paper explores the analogue of the effective potential of Schrödinger operators ${-\Delta + V}$ provided by the “landscape function” ${u}$ , when one works with a certain type of self-adjoint matrix known as an ${M}$ -matrix instead of a Schrödinger operator.

Suppose one has an eigenfunction

$\displaystyle (-\Delta + V) \phi = E \phi$

of a Schrödinger operator ${-\Delta+V}$ , where ${\Delta}$ is the Laplacian on ${{\bf R}^d}$ , ${V: {\bf R}^d \rightarrow {\bf R}}$ is a potential, and ${E}$ is an energy. Where would one expect the eigenfunction ${\phi}$ to be concentrated? If the potential ${V}$ is smooth and slowly varying, the correspondence principle suggests that the eigenfunction ${\phi}$ should be mostly concentrated in the potential energy wells ${\{ x: V(x) \leq E \}}$ , with an exponentially decaying amount of tunnelling between the wells. One way to rigorously establish such an exponential decay is through an argument of Agmon, which we will sketch later in this post, which gives an exponentially decaying upper bound (in an ${L^2}$ sense) of eigenfunctions ${\phi}$ in terms of the distance to the wells ${\{ V \leq E \}}$ in terms of a certain “Agmon metric” on ${{\bf R}^d}$ determined by the potential ${V}$ and energy level ${E}$ (or any upper bound ${\overline{E}}$ on this energy). Similar exponential decay results can also be obtained for discrete Schrödinger matrix models, in which the domain ${{\bf R}^d}$ is replaced with a discrete set such as the lattice ${{\bf Z}^d}$ , and the Laplacian ${\Delta}$ is replaced by a discrete analogue such as a graph Laplacian.

When the potential ${V}$ is very “rough”, as occurs for instance in the random potentials arising in the theory of Anderson localisation, the Agmon bounds, while still true, become very weak because the wells ${\{ V \leq E \}}$ are dispersed in a fairly dense fashion throughout the domain ${{\bf R}^d}$ , and the eigenfunction can tunnel relatively easily between different wells. However, as was first discovered in 2012 by my two coauthors, in these situations one can replace the rough potential ${V}$ by a smoother effective potential ${1/u}$ , with the eigenfunctions typically localised to a single connected component of the effective wells ${\{ 1/u \leq E \}}$ . In fact, a good choice of effective potential comes from locating the landscape function ${u}$ , which is the solution to the equation ${(-\Delta + V) u = 1}$ with reasonable behavior at infinity, and which is non-negative from the maximum principle, and then the reciprocal ${1/u}$ of this landscape function serves as an effective potential.

There are now several explanations for why this particular choice ${1/u}$ is a good effective potential. Perhaps the simplest (as found for instance in this recent paper of Arnold, David, Jerison, and my two coauthors) is the following observation: if ${\phi}$ is an eigenvector for ${-\Delta+V}$ with energy ${E}$ , then ${\phi/u}$ is an eigenvector for ${-\frac{1}{u^2} \mathrm{div}(u^2 \nabla \cdot) + \frac{1}{u}}$ with the same energy ${E}$ , thus the original Schrödinger operator ${-\Delta+V}$ is conjugate to a (variable coefficient, but still in divergence form) Schrödinger operator with potential ${1/u}$ instead of ${V}$ . Closely related to this, we have the integration by parts identity

$\displaystyle \int_{{\bf R}^d} |\nabla f|^2 + V |f|^2\ dx = \int_{{\bf R}^d} u^2 |\nabla(f/u)|^2 + \frac{1}{u} |f|^2\ dx \ \ \ \ \ (1)$

for any reasonable function ${f}$ , thus again highlighting the emergence of the effective potential ${1/u}$ .

These particular explanations seem rather specific to the Schrödinger equation (continuous or discrete); we have for instance not been able to find similar identities to explain an effective potential for the bi-Schrödinger operator ${\Delta^2 + V}$ .

In this paper, we demonstrate the (perhaps surprising) fact that effective potentials continue to exist for operators that bear very little resemblance to Schrödinger operators. Our chosen model is that of an ${M}$ -matrix: self-adjoint positive definite matrices ${A}$ whose off-diagonal entries are negative. This model includes discrete Schrödinger operators (with non-negative potentials) but can allow for significantly more non-local interactions. The analogue of the landscape function would then be the vector ${u := A^{-1} 1}$ , where ${1}$ denotes the vector with all entries ${1}$ . Our main result, roughly speaking, asserts that an eigenvector ${A \phi = E \phi}$ of ${A}$ will then be exponentially localised to the “potential wells” ${K := \{ j: \frac{1}{u_j} \leq E \}}$ , where ${u_j}$ denotes the coordinates of the landscape function ${u}$ . In particular, we establish the inequality

$\displaystyle \sum_k \phi_k^2 e^{2 \rho(k,K) / \sqrt{W}} ( \frac{1}{u_k} - E )_+ \leq W \max_{i,j} |a_{ij}|$

if ${\phi}$ is normalised in ${\ell^2}$ , where the connectivity ${W}$ is the maximum number of non-zero entries of ${A}$ in any row or column, ${a_{ij}}$ are the coefficients of ${A}$ , and ${\rho}$ is a certain moderately complicated but explicit metric function on the spatial domain. Informally, this inequality asserts that the eigenfunction ${\phi_k}$ should decay like ${e^{-\rho(k,K) / \sqrt{W}}}$ or faster. Indeed, our numerics show a very strong log-linear relationship between ${\phi_k}$ and ${\rho(k,K)}$ , although it appears that our exponent ${1/\sqrt{W}}$ is not quite optimal. We also provide an associated localisation result which is technical to state but very roughly asserts that a given eigenvector will in fact be localised to a single connected component of ${K}$ unless there is a resonance between two wells (by which we mean that an eigenvalue for a localisation of ${A}$ associated to one well is extremely close to an eigenvalue for a localisation of ${A}$ associated to another well); such localisation is also strongly supported by numerics. (Analogous results for Schrödinger operators had been previously obtained by the previously mentioned paper of Arnold, David, Jerison, and my two coauthors, and to quantum graphs in a very recent paper of Harrell and Maltsev.)

Our approach is based on Agmon’s methods, which we interpret as a double commutator method, and in particular relying on exploiting the negative definiteness of certain double commutator operators. In the case of Schrödinger operators ${-\Delta+V}$ , this negative definiteness is provided by the identity

$\displaystyle \langle [[-\Delta+V,g],g] u, u \rangle = -2\int_{{\bf R}^d} |\nabla g|^2 |u|^2\ dx \leq 0 \ \ \ \ \ (2)$

for any sufficiently reasonable functions ${u, g: {\bf R}^d \rightarrow {\bf R}}$ , where we view ${g}$ (like ${V}$ ) as a multiplier operator. To exploit this, we use the commutator identity

$\displaystyle \langle g [\psi, -\Delta+V] u, g \psi u \rangle = \frac{1}{2} \langle [[-\Delta+V, g \psi],g\psi] u, u \rangle$

$\displaystyle -\frac{1}{2} \langle [[-\Delta+V, g],g] \psi u, \psi u \rangle$

valid for any ${g,\psi,u: {\bf R}^d \rightarrow {\bf R}}$ after a brief calculation. The double commutator identity then tells us that

$\displaystyle \langle g [\psi, -\Delta+V] u, g \psi u \rangle \leq \int_{{\bf R}^d} |\nabla g|^2 |\psi u|^2\ dx.$

If we choose ${u}$ to be a non-negative weight and let ${\psi := \phi/u}$ for an eigenfunction ${\phi}$ , then we can write

$\displaystyle [\psi, -\Delta+V] u = [\psi, -\Delta+V - E] u = \psi (-\Delta+V - E) u$

and we conclude that

$\displaystyle \int_{{\bf R}^d} \frac{(-\Delta+V-E)u}{u} |g|^2 |\phi|^2\ dx \leq \int_{{\bf R}^d} |\nabla g|^2 |\phi|^2\ dx. \ \ \ \ \ (3)$

We have considerable freedom in this inequality to select the functions ${u,g}$ . If we select ${u=1}$ , we obtain the clean inequality

$\displaystyle \int_{{\bf R}^d} (V-E) |g|^2 |\phi|^2\ dx \leq \int_{{\bf R}^d} |\nabla g|^2 |\phi|^2\ dx.$

If we take ${g}$ to be a function which equals ${1}$ on the wells ${\{ V \leq E \}}$ but increases exponentially away from these wells, in such a way that

$\displaystyle |\nabla g|^2 \leq \frac{1}{2} (V-E) |g|^2$

outside of the wells, we can obtain the estimate

$\displaystyle \int_{V > E} (V-E) |g|^2 |\phi|^2\ dx \leq 2 \int_{V < E} (E-V) |\phi|^2\ dx,$

which then gives an exponential type decay of ${\phi}$ away from the wells. This is basically the classic exponential decay estimate of Agmon; one can basically take ${g}$ to be the distance to the wells ${\{ V \leq E \}}$ with respect to the Euclidean metric conformally weighted by a suitably normalised version of ${V-E}$ . If we instead select ${u}$ to be the landscape function ${u = (-\Delta+V)^{-1} 1}$ , (3) then gives

$\displaystyle \int_{{\bf R}^d} (\frac{1}{u} - E) |g|^2 |\phi|^2\ dx \leq \int_{{\bf R}^d} |\nabla g|^2 |\phi|^2\ dx,$

and by selecting ${g}$ appropriately this gives an exponential decay estimate away from the effective wells ${\{ \frac{1}{u} \leq E \}}$ , using a metric weighted by ${\frac{1}{u}-E}$ .

It turns out that this argument extends without much difficulty to the ${M}$ -matrix setting. The analogue of the crucial double commutator identity (2) is

$\displaystyle \langle [[A,D],D] u, u \rangle = \sum_{i \neq j} a_{ij} u_i u_j (d_{ii} - d_{jj})^2 \leq 0$

for any diagonal matrix ${D = \mathrm{diag}(d_{11},\dots,d_{NN})}$ . The remainder of the Agmon type arguments go through after making the natural modifications.

Numerically we have also found some aspects of the landscape theory to persist beyond the ${M}$ -matrix setting, even though the double commutators cease being negative definite, so this may not yet be the end of the story, but it does at least demonstrate that utility the landscape does not purely rely on identities such as (1).

247B, Notes 3: pseudodifferential operators

2 May, 2020 in 247B - Classical Fourier Analysis, math.AP, math.CA, math.QA | Tags: almost orthogonality, pseudodifferential operators, quantum mechanics, singular integrals | by Terence Tao | 71 comments

In contrast to previous notes, in this set of notes we shall focus exclusively on Fourier analysis in the one-dimensional setting ${d=1}$ for simplicity of notation, although all of the results here have natural extensions to higher dimensions. Depending on the physical context, one can view the physical domain ${{\bf R}}$ as representing either space or time; we will mostly think in terms of the former interpretation, even though the standard terminology of “time-frequency analysis”, which we will make more prominent use of in later notes, clearly originates from the latter.

In previous notes we have often performed various localisations in either physical space or Fourier space ${{\bf R}}$ , for instance in order to take advantage of the uncertainty principle. One can formalise these operations in terms of the functional calculus of two basic operations on Schwartz functions ${{\mathcal S}({\bf R})}$ , the position operator ${X: {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})}$ defined by

$\displaystyle (Xf)(x) := x f(x)$

and the momentum operator ${D: {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})}$ , defined by

$\displaystyle (Df)(x) := \frac{1}{2\pi i} \frac{d}{dx} f(x). \ \ \ \ \ (1)$

(The terminology comes from quantum mechanics, where it is customary to also insert a small constant ${h}$ on the right-hand side of (1) in accordance with de Broglie’s law. Such a normalisation is also used in several branches of mathematics, most notably semiclassical analysis and microlocal analysis, where it becomes profitable to consider the semiclassical limit ${h \rightarrow 0}$ , but we will not emphasise this perspective here.) The momentum operator can be viewed as the counterpart to the position operator, but in frequency space instead of physical space, since we have the standard identity

$\displaystyle \widehat{Df}(\xi) = \xi \hat f(\xi)$

for any ${\xi \in {\bf R}}$ and ${f \in {\mathcal S}({\bf R})}$ . We observe that both operators ${X,D}$ are formally self-adjoint in the sense that

$\displaystyle \langle Xf, g \rangle = \langle f, Xg \rangle; \quad \langle Df, g \rangle = \langle f, Dg \rangle$

for all ${f,g \in {\mathcal S}({\bf R})}$ , where we use the ${L^2({\bf R})}$ Hermitian inner product

$\displaystyle \langle f, g\rangle := \int_{\bf R} f(x) \overline{g(x)}\ dx.$

Clearly, for any polynomial ${P(x)}$ of one real variable ${x}$ (with complex coefficients), the operator ${P(X): {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})}$ is given by the spatial multiplier operator

$\displaystyle (P(X) f)(x) = P(x) f(x)$

and similarly the operator ${P(D): {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})}$ is given by the Fourier multiplier operator

$\displaystyle \widehat{P(D) f}(\xi) = P(\xi) \hat f(\xi).$

Inspired by this, if ${m: {\bf R} \rightarrow {\bf C}}$ is any smooth function that obeys the derivative bounds

$\displaystyle \frac{d^j}{dx^j} m(x) \lesssim_{m,j} \langle x \rangle^{O_{m,j}(1)} \ \ \ \ \ (2)$

for all ${j \geq 0}$ and ${x \in {\bf R}}$ (that is to say, all derivatives of ${m}$ grow at most polynomially), then we can define the spatial multiplier operator ${m(X): {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})}$ by the formula

$\displaystyle (m(X) f)(x) := m(x) f(x);$

one can easily verify from several applications of the Leibniz rule that ${m(X)}$ maps Schwartz functions to Schwartz functions. We refer to ${m(x)}$ as the symbol of this spatial multiplier operator. In a similar fashion, we define the Fourier multiplier operator ${m(D)}$ associated to the symbol ${m(\xi)}$ by the formula

$\displaystyle \widehat{m(D) f}(\xi) := m(\xi) \hat f(\xi).$

For instance, any constant coefficient linear differential operators ${\sum_{k=0}^n c_k \frac{d^k}{dx^k}}$ can be written in this notation as

$\displaystyle \sum_{k=0}^n c_k \frac{d^k}{dx^k} =\sum_{k=0}^n c_k (2\pi i D)^k;$

however there are many Fourier multiplier operators that are not of this form, such as fractional derivative operators ${\langle D \rangle^s = (1- \frac{1}{4\pi^2} \frac{d^2}{dx^2})^{s/2}}$ for non-integer values of ${s}$ , which is a Fourier multiplier operator with symbol ${\langle \xi \rangle^s}$ . It is also very common to use spatial cutoffs ${\psi(X)}$ and Fourier cutoffs ${\psi(D)}$ for various bump functions ${\psi}$ to localise functions in either space or frequency; we have seen several examples of such cutoffs in action in previous notes (often in the higher dimensional setting ${d>1}$ ).

We observe that the maps ${m \mapsto m(X)}$ and ${m \mapsto m(D)}$ are ring homomorphisms, thus for instance

$\displaystyle (m_1 + m_2)(D) = m_1(D) + m_2(D)$

and

$\displaystyle (m_1 m_2)(D) = m_1(D) m_2(D)$

for any ${m_1,m_2}$ obeying the derivative bounds (2); also ${m(D)}$ is formally adjoint to ${\overline{m(D)}}$ in the sense that

$\displaystyle \langle m(D) f, g \rangle = \langle f, \overline{m}(D) g \rangle$

for ${f,g \in {\mathcal S}({\bf R})}$ , and similarly for ${m(X)}$ and ${\overline{m}(X)}$ . One can interpret these facts as part of the functional calculus of the operators ${X,D}$ , which can be interpreted as densely defined self-adjoint operators on ${L^2({\bf R})}$ . However, in this set of notes we will not develop the spectral theory necessary in order to fully set out this functional calculus rigorously.

In the field of PDE and ODE, it is also very common to study variable coefficient linear differential operators

$\displaystyle \sum_{k=0}^n c_k(x) \frac{d^k}{dx^k} \ \ \ \ \ (3)$

where the ${c_0,\dots,c_n}$ are now functions of the spatial variable ${x}$ obeying the derivative bounds (2). A simple example is the quantum harmonic oscillator Hamiltonian ${-\frac{d^2}{dx^2} + x^2}$ . One can rewrite this operator in our notation as

$\displaystyle \sum_{k=0}^n c_k(X) (2\pi i D)^k$

and so it is natural to interpret this operator as a combination ${a(X,D)}$ of both the position operator ${X}$ and the momentum operator ${D}$ , where the symbol ${a: {\bf R} \times {\bf R} \rightarrow {\bf C}}$ this operator is the function

$\displaystyle a(x,\xi) := \sum_{k=0}^n c_k(x) (2\pi i \xi)^k. \ \ \ \ \ (4)$

Indeed, from the Fourier inversion formula

$\displaystyle f(x) = \int_{\bf R} \hat f(\xi) e^{2\pi i x \xi}\ d\xi$

for any ${f \in {\mathcal S}({\bf R})}$ we have

$\displaystyle (2\pi i D)^k f(x) = \int_{\bf R} (2\pi i \xi)^k \hat f(\xi) e^{2\pi i x \xi}\ d\xi$

and hence on multiplying by ${c_k(x)}$ and summing we have

$\displaystyle (\sum_{k=0}^n c_k(X) (2\pi i D)^k) f(x) = \int_{\bf R} a(x,\xi) \hat f(\xi) e^{2\pi i x \xi}\ d\xi.$

Inspired by this, we can introduce the Kohn-Nirenberg quantisation by defining the operator ${a(X,D) = a_{KN}(X,D): {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})}$ by the formula

$\displaystyle a(X,D) f(x) = \int_{\bf R} a(x,\xi) \hat f(\xi) e^{2\pi i x \xi}\ d\xi \ \ \ \ \ (5)$

whenever ${f \in {\mathcal S}({\bf R})}$ and ${a: {\bf R} \times {\bf R} \rightarrow {\bf C}}$ is any smooth function obeying the derivative bounds

$\displaystyle \frac{\partial^j}{\partial x^j} \frac{\partial^l}{\partial \xi^l} a(x,\xi) \lesssim_{a,j,l} \langle x \rangle^{O_{a,j}(1)} \langle \xi \rangle^{O_{a,j,l}(1)} \ \ \ \ \ (6)$

for all ${j,l \geq 0}$ and ${x \in {\bf R}}$ (note carefully that the exponent in ${x}$ on the right-hand side is required to be uniform in ${l}$ ). This quantisation clearly generalises both the spatial multiplier operators ${m(X)}$ and the Fourier multiplier operators ${m(D)}$ defined earlier, which correspond to the cases when the symbol ${a(x,\xi)}$ is a function of ${x}$ only or ${\xi}$ only respectively. Thus we have combined the physical space ${{\bf R} = \{ x: x \in {\bf R}\}}$ and the frequency space ${{\bf R} = \{ \xi: \xi \in {\bf R}\}}$ into a single domain, known as phase space ${{\bf R} \times {\bf R} = \{ (x,\xi): x,\xi \in {\bf R} \}}$ . The term “time-frequency analysis” encompasses analysis based on decompositions and other manipulations of phase space, in much the same way that “Fourier analysis” encompasses analysis based on decompositions and other manipulations of frequency space. We remark that the Kohn-Nirenberg quantization is not the only choice of quantization one could use; see Remark 19 below.

Exercise 1

(i) Show that for ${a}$ obeying (6), that ${a(X,D)}$ does indeed map ${{\mathcal S}({\bf R})}$ to ${{\mathcal S}({\bf R})}$ .

(ii) Show that the symbol ${a}$ is uniquely determined by the operator ${a(X,D)}$ . That is to say, if ${a,b}$ are two functions obeying (6) with ${a(X,D) f = b(X,D) f}$ for all ${f \in {\mathcal S}({\bf R})}$ , then ${a=b}$ . (Hint: apply ${a(X,D)-b(X,D)}$ to a suitable truncation of a plane wave ${x \mapsto e^{2\pi i x \xi}}$ and then take limits.)

In principle, the quantisations ${a(X,D)}$ are potentially very useful for such tasks as inverting variable coefficient linear operators, or to localize a function simultaneously in physical and Fourier space. However, a fundamental difficulty arises: map from symbols ${a}$ to operators ${a(X,D)}$ is now no longer a ring homomorphism, in particular

$\displaystyle (a_1 a_2)(X,D) \neq a_1(X,D) a_2(X,D) \ \ \ \ \ (7)$

in general. Fundamentally, this is due to the fact that pointwise multiplication of symbols is a commutative operation, whereas the composition of operators such as ${X}$ and ${D}$ does not necessarily commute. This lack of commutativity can be measured by introducing the commutator

$\displaystyle [A,B] := AB - BA$

of two operators ${A,B}$ , and noting from the product rule that

$\displaystyle [X,D] = -\frac{1}{2\pi i} \neq 0.$

(In the language of Lie groups and Lie algebras, this tells us that ${X,D}$ are (up to complex constants) the standard Lie algebra generators of the Heisenberg group.) From a quantum mechanical perspective, this lack of commutativity is the root cause of the uncertainty principle that prevents one from simultaneously localizing in both position and momentum past a certain point. Here is one basic way of formalising this principle:

Exercise 2 (Heisenberg uncertainty principle) For any ${x_0, \xi_0 \in {\bf R}}$ and ${f \in \mathcal{S}({\bf R})}$ , show that

$\displaystyle \| (X-x_0) f \|_{L^2({\bf R})} \| (D-\xi_0) f\|_{L^2({\bf R})} \geq \frac{1}{4\pi} \|f\|_{L^2({\bf R})}^2.$

(Hint: evaluate the expression ${\langle [X-x_0, D - \xi_0] f, f \rangle}$ in two different ways and apply the Cauchy-Schwarz inequality.) Informally, this exercise asserts that the spatial uncertainty ${\Delta x}$ and the frequency uncertainty ${\Delta \xi}$ of a function obey the Heisenberg uncertainty relation ${\Delta x \Delta \xi \gtrsim 1}$ .

Nevertheless, one still has the correspondence principle, which asserts that in certain regimes (which, with our choice of normalisations, corresponds to the high-frequency regime), quantum mechanics continues to behave like a commutative theory, and one can sometimes proceed as if the operators ${X,D}$ (and the various operators ${a(X,D)}$ constructed from them) commute up to “lower order” errors. This can be formalised using the pseudodifferential calculus, which we give below the fold, in which we restrict the symbol ${a}$ to certain “symbol classes” of various orders (which then restricts ${a(X,D)}$ to be pseudodifferential operators of various orders), and obtains approximate identities such as

$\displaystyle (a_1 a_2)(X,D) \approx a_1(X,D) a_2(X,D)$

where the error between the left and right-hand sides is of “lower order” and can in fact enjoys a useful asymptotic expansion. As a first approximation to this calculus, one can think of functions ${f \in {\mathcal S}({\bf R})}$ as having some sort of “phase space portrait” ${\tilde f(x,\xi)}$ which somehow combines the physical space representation ${x \mapsto f(x)}$ with its Fourier representation ${\xi \mapsto f(\xi)}$ , and pseudodifferential operators ${a(X,D)}$ behave approximately like “phase space multiplier operators” in this representation in the sense that

$\displaystyle \widetilde{a(X,D) f}(x,\xi) \approx a(x,\xi) \tilde f(x,\xi).$

Unfortunately the uncertainty principle (or the non-commutativity of ${X}$ and ${D}$ ) prevents us from making these approximations perfectly precise, and it is not always clear how to even define a phase space portrait ${\tilde f}$ of a function ${f}$ precisely (although there are certain popular candidates for such a portrait, such as the FBI transform (also known as the Gabor transform in signal processing literature), or the Wigner quasiprobability distribution, each of which have some advantages and disadvantages). Nevertheless even if the concept of a phase space portrait is somewhat fuzzy, it is of great conceptual benefit both within mathematics and outside of it. For instance, the musical score one assigns a piece of music can be viewed as a phase space portrait of the sound waves generated by that music.

To complement the pseudodifferential calculus we have the basic Calderón-Vaillancourt theorem, which asserts that pseudodifferential operators of order zero are Calderón-Zygmund operators and thus bounded on ${L^p({\bf R})}$ for ${1 < p < \infty}$ . The standard proof of this theorem is a classic application of one of the basic techniques in harmonic analysis, namely the exploitation of almost orthogonality; the proof we will give here will achieve this through the elegant device of the Cotlar-Stein lemma.

Pseudodifferential operators (especially when generalised to higher dimensions ${d \geq 1}$ ) are a fundamental tool in the theory of linear PDE, as well as related fields such as semiclassical analysis, microlocal analysis, and geometric quantisation. There is an even wider class of operators that is also of interest, namely the Fourier integral operators, which roughly speaking not only approximately multiply the phase space portrait ${\tilde f(x,\xi)}$ of a function by some multiplier ${a(x,\xi)}$ , but also move the portrait around by a canonical transformation. However, the development of theory of these operators is beyond the scope of these notes; see for instance the texts of Hormander or Eskin.

This set of notes is only the briefest introduction to the theory of pseudodifferential operators. Many texts are available that cover the theory in more detail, for instance this text of Taylor.

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Some recent papers

11 January, 2020 in expository, math.AP, math.CA, math.CO, math.NT | Tags: Chowla conjecture, finite time blowup, Frank Merle, Hong Wang, Igor Rodnianski, Jeremie Szeftel, Larry Guth, Mark Shusterman, nonlinear Schrodinger equation, Pierre Raphael, restriction theorems, Ruixiang Zhang, Sarah Peluse, Szemeredi's theorem, twin primes, Will Sawin | by Terence Tao | 4 comments

Just a brief post to record some notable papers in my fields of interest that appeared on the arXiv recently.

“A sharp square function estimate for the cone in ${\bf R}^3$ “, by Larry Guth, Hong Wang, and Ruixiang Zhang. This paper establishes an optimal (up to epsilon losses) square function estimate for the three-dimensional light cone that was essentially conjectured by Mockenhaupt, Seeger, and Sogge, which has a number of other consequences including Sogge’s local smoothing conjecture for the wave equation in two spatial dimensions, which in turn implies the (already known) Bochner-Riesz, restriction, and Kakeya conjectures in two dimensions. Interestingly, modern techniques such as polynomial partitioning and decoupling estimates are not used in this argument; instead, the authors mostly rely on an induction on scales argument and Kakeya type estimates. Many previous authors (including myself) were able to get weaker estimates of this type by an induction on scales method, but there were always significant inefficiencies in doing so; in particular knowing the sharp square function estimate at smaller scales did not imply the sharp square function estimate at the given larger scale. The authors here get around this issue by finding an even stronger estimate that implies the square function estimate, but behaves significantly better with respect to induction on scales.
“On the Chowla and twin primes conjectures over ${\mathbb F}_q[T]$ “, by Will Sawin and Mark Shusterman. This paper resolves a number of well known open conjectures in analytic number theory, such as the Chowla conjecture and the twin prime conjecture (in the strong form conjectured by Hardy and Littlewood), in the case of function fields where the field is a prime power $q=p^j$ which is fixed (in contrast to a number of existing results in the “large $q$ ” limit) but has a large exponent $j$ . The techniques here are orthogonal to those used in recent progress towards the Chowla conjecture over the integers (e.g., in this previous paper of mine); the starting point is an algebraic observation that in certain function fields, the Mobius function behaves like a quadratic Dirichlet character along certain arithmetic progressions. In principle, this reduces problems such as Chowla’s conjecture to problems about estimating sums of Dirichlet characters, for which more is known; but the task is still far from trivial.
“Bounds for sets with no polynomial progressions“, by Sarah Peluse. This paper can be viewed as part of a larger project to obtain quantitative density Ramsey theorems of Szemeredi type. For instance, Gowers famously established a relatively good quantitative bound for Szemeredi’s theorem that all dense subsets of integers contain arbitrarily long arithmetic progressions $a, a+r, \dots, a+(k-1)r$ . The corresponding question for polynomial progressions $a+P_1(r), \dots, a+P_k(r)$ is considered more difficult for a number of reasons. One of them is that dilation invariance is lost; a dilation of an arithmetic progression is again an arithmetic progression, but a dilation of a polynomial progression will in general not be a polynomial progression with the same polynomials $P_1,\dots,P_k$ . Another issue is that the ranges of the two parameters $a,r$ are now at different scales. Peluse gets around these difficulties in the case when all the polynomials $P_1,\dots,P_k$ have distinct degrees, which is in some sense the opposite case to that considered by Gowers (in particular, she avoids the need to obtain quantitative inverse theorems for high order Gowers norms; which was recently obtained in this integer setting by Manners but with bounds that are probably not strong enough to for the bounds in Peluse’s results, due to a degree lowering argument that is available in this case). To resolve the first difficulty one has to make all the estimates rather uniform in the coefficients of the polynomials $P_j$ , so that one can still run a density increment argument efficiently. To resolve the second difficulty one needs to find a quantitative concatenation theorem for Gowers uniformity norms. Many of these ideas were developed in previous papers of Peluse and Peluse-Prendiville in simpler settings.
“On blow up for the energy super critical defocusing non linear Schrödinger equations“, by Frank Merle, Pierre Raphael, Igor Rodnianski, and Jeremie Szeftel. This paper (when combined with two companion papers) resolves a long-standing problem as to whether finite time blowup occurs for the defocusing supercritical nonlinear Schrödinger equation (at least in certain dimensions and nonlinearities). I had a previous paper establishing a result like this if one “cheated” by replacing the nonlinear Schrodinger equation by a system of such equations, but remarkably they are able to tackle the original equation itself without any such cheating. Given the very analogous situation with Navier-Stokes, where again one can create finite time blowup by “cheating” and modifying the equation, it does raise hope that finite time blowup for the incompressible Navier-Stokes and Euler equations can be established… In fact the connection may not just be at the level of analogy; a surprising key ingredient in the proofs here is the observation that a certain blowup ansatz for the nonlinear Schrodinger equation is governed by solutions to the (compressible) Euler equation, and finite time blowup examples for the latter can be used to construct finite time blowup examples for the former.

Elgindi’s approximation of the Biot-Savart law

26 December, 2019 in math.AP, math.CA, math.MP | Tags: Biot-Savart law, incompressible Euler equations, Tarek Elgindi | by Terence Tao | 11 comments

Let ${u: {\bf R}^3 \rightarrow {\bf R}^3}$ be a divergence-free vector field, thus ${\nabla \cdot u = 0}$ , which we interpret as a velocity field. In this post we will proceed formally, largely ignoring the analytic issues of whether the fields in question have sufficient regularity and decay to justify the calculations. The vorticity field ${\omega: {\bf R}^3 \rightarrow {\bf R}^3}$ is then defined as the curl of the velocity:

$\displaystyle \omega = \nabla \times u.$

(From a differential geometry viewpoint, it would be more accurate (especially in other dimensions than three) to define the vorticity as the exterior derivative ${\omega = d(g \cdot u)}$ of the musical isomorphism ${g \cdot u}$ of the Euclidean metric ${g}$ applied to the velocity field ${u}$ ; see these previous lecture notes. However, we will not need this geometric formalism in this post.)

Assuming suitable regularity and decay hypotheses of the velocity field ${u}$ , it is possible to recover the velocity from the vorticity as follows. From the general vector identity ${\nabla \times \nabla \times X = \nabla(\nabla \cdot X) - \Delta X}$ applied to the velocity field ${u}$ , we see that

$\displaystyle \nabla \times \omega = -\Delta u$

and thus (by the commutativity of all the differential operators involved)

$\displaystyle u = - \nabla \times \Delta^{-1} \omega.$

Using the Newton potential formula

$\displaystyle -\Delta^{-1} \omega(x) := \frac{1}{4\pi} \int_{{\bf R}^3} \frac{\omega(y)}{|x-y|}\ dy$

and formally differentiating under the integral sign, we obtain the Biot-Savart law

$\displaystyle u(x) = \frac{1}{4\pi} \int_{{\bf R}^3} \frac{\omega(y) \times (x-y)}{|x-y|^3}\ dy. \ \ \ \ \ (1)$

This law is of fundamental importance in the study of incompressible fluid equations, such as the Euler equations

$\displaystyle \partial_t u + (u \cdot \nabla) u = -\nabla p; \quad \nabla \cdot u = 0$

since on applying the curl operator one obtains the vorticity equation

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (2)$

and then by substituting (1) one gets an autonomous equation for the vorticity field ${\omega}$ . Unfortunately, this equation is non-local, due to the integration present in (1).

In a recent work, it was observed by Elgindi that in a certain regime, the Biot-Savart law can be approximated by a more “low rank” law, which makes the non-local effects significantly simpler in nature. This simplification was carried out in spherical coordinates, and hinged on a study of the invertibility properties of a certain second order linear differential operator in the latitude variable ${\theta}$ ; however in this post I would like to observe that the approximation can also be seen directly in Cartesian coordinates from the classical Biot-Savart law (1). As a consequence one can also initiate the beginning of Elgindi’s analysis in constructing somewhat regular solutions to the Euler equations that exhibit self-similar blowup in finite time, though I have not attempted to execute the entirety of the analysis in this setting.

Elgindi’s approximation applies under the following hypotheses:

(i) (Axial symmetry without swirl) The velocity field ${u}$ is assumed to take the form
$\displaystyle u(x_1,x_2,x_3) = ( u_r(r,x_3) \frac{x_1}{r}, u_r(r,x_3) \frac{x_2}{r}, u_3(r,x_3) ) \ \ \ \ \ (3)$

for some functions ${u_r, u_3: [0,+\infty) \times {\bf R} \rightarrow {\bf R}}$ of the cylindrical radial variable ${r := \sqrt{x_1^2+x_2^2}}$ and the vertical coordinate ${x_3}$ . As a consequence, the vorticity field ${\omega}$ takes the form

$\displaystyle \omega(x_1,x_2,x_3) = (\omega_{r3}(r,x_3) \frac{x_2}{r}, \omega_{r3}(r,x_3) \frac{-x_1}{r}, 0) \ \ \ \ \ (4)$

where ${\omega_{r3}: [0,+\infty) \times {\bf R} \rightarrow {\bf R}}$ is the field

$\displaystyle \omega_{r3} = \partial_r u_3 - \partial_3 u_r.$
(ii) (Odd symmetry) We assume that ${u_3(r,-x_3) = -u_3(r,x_3)}$ and ${u_r(r,-x_3)=u_r(r,x_3)}$ , so that ${\omega_{r3}(r,-x_3)=\omega_{r3}(r,x_3)}$ .

A model example of a divergence-free vector field obeying these properties (but without good decay at infinity) is the linear vector field

$\displaystyle X(x) = (x_1, x_2, -2x_3) \ \ \ \ \ (5)$

which is of the form (3) with ${u_r(r,x_3) = r}$ and ${u_3(r,x_3) = -2x_3}$ . The associated vorticity ${\omega}$ vanishes.

We can now give an illustration of Elgindi’s approximation:

Proposition 1 (Elgindi’s approximation) Under the above hypotheses (and assuing suitable regularity and decay), we have the pointwise bounds

$\displaystyle u(x) = \frac{1}{2} {\mathcal L}_{12}(\omega)(|x|) X(x) + O( |x| \|\omega\|_{L^\infty({\bf R}^3)} )$

for any ${x \in {\bf R}^3}$ , where ${X}$ is the vector field (5), and ${{\mathcal L}_{12}(\omega): {\bf R}^+ \rightarrow {\bf R}}$ is the scalar function

$\displaystyle {\mathcal L}_{12}(\omega)(\rho) := \frac{3}{4\pi} \int_{|y| \geq \rho} \frac{r y_3}{|y|^5} \omega_{r3}(r,y_3)\ dy.$

Thus under the hypotheses (i), (ii), and assuming that ${\omega}$ is slowly varying, we expect ${u}$ to behave like the linear vector field ${X}$ modulated by a radial scalar function. In applications one needs to control the error in various function spaces instead of pointwise, and with ${\omega}$ similarly controlled in other function space norms than the ${L^\infty}$ norm, but this proposition already gives a flavour of the approximation. If one uses spherical coordinates

$\displaystyle \omega_{r3}( \rho \cos \theta, \rho \sin \theta ) = \Omega( \rho, \theta )$

then we have (using the spherical change of variables formula ${dy = \rho^2 \cos \theta d\rho d\theta d\phi}$ and the odd nature of ${\Omega}$ )

$\displaystyle {\mathcal L}_{12}(\omega) = L_{12}(\Omega),$

where

$\displaystyle L_{12}(\Omega)(\rho) = 3 \int_\rho^\infty \int_0^{\pi/2} \frac{\Omega(r, \theta) \sin(\theta) \cos^2(\theta)}{r}\ d\theta dr$

is the operator introduced in Elgindi’s paper.

Proof: By a limiting argument we may assume that ${x}$ is non-zero, and we may normalise ${\|\omega\|_{L^\infty({\bf R}^3)}=1}$ . From the triangle inequality we have

$\displaystyle \int_{|y| \leq 10|x|} \frac{\omega(y) \times (x-y)}{|x-y|^3}\ dy \leq \int_{|y| \leq 10|x|} \frac{1}{|x-y|^2}\ dy$

$\displaystyle \leq \int_{|z| \leq 11 |x|} \frac{1}{|z|^2}\ dz$

$\displaystyle = O( |x| )$

and hence by (1)

$\displaystyle u(x) = \frac{1}{4\pi} \int_{|y| > 10|x|} \frac{\omega(y) \times (x-y)}{|x-y|^3}\ dy + O(|x|).$

In the regime ${|y| > 2|x|}$ we may perform the Taylor expansion

$\displaystyle \frac{x-y}{|x-y|^3} = \frac{x-y}{|y|^3} (1 - \frac{2 x \cdot y}{|y|^2} + \frac{|x|^2}{|y|^2})^{-3/2}$

$\displaystyle = \frac{x-y}{|y|^3} (1 + \frac{3 x \cdot y}{|y|^2} + O( \frac{|x|^2}{|y|^2} ) )$

$\displaystyle = -\frac{y}{|y|^3} + \frac{x}{|y|^3} - \frac{3 (x \cdot y) y}{|y|^5} + O( \frac{|x|^2}{|y|^4} ).$

Since

$\displaystyle \int_{|y| > 10|x|} \frac{|x|^2}{|y|^4}\ dy = O(|x|)$

we see from the triangle inequality that the error term contributes ${O(|x|)}$ to ${u(x)}$ . We thus have

$\displaystyle u(x) = -A_0(x) + A_1(x) - 3A'_1(x) + O(|x|)$

where ${A_0}$ is the constant term

$\displaystyle A_0 := \int_{|y| > 10|x|} \frac{\omega(y) \times y}{|y|^3}\ dy,$

and ${A_1, A'_1}$ are the linear term

$\displaystyle A_1 := \int_{|y| > 10|x|} \frac{\omega(y) \times x}{|y|^3}\ dy,$

$\displaystyle A'_1 := \int_{|y| > 10|x|} (x \cdot y) \frac{\omega(y) \times y}{|y|^5}\ dy.$

By the hypotheses (i), (ii), we have the symmetries

$\displaystyle \omega(y_1,y_2,-y_3) = - \omega(y_1,y_2,y_3) \ \ \ \ \ (6)$

and

$\displaystyle \omega(-y_1,-y_2,y_3) = - \omega(y_1,y_2,y_3) \ \ \ \ \ (7)$

and hence also

$\displaystyle \omega(-y_1,-y_2,-y_3) = \omega(y_1,y_2,y_3). \ \ \ \ \ (8)$

The even symmetry (8) ensures that the integrand in ${A_0}$ is odd, so ${A_0}$ vanishes. The symmetry (6) or (7) similarly ensures that ${\int_{|y| > 10|x|} \frac{\omega(y)}{|y|^3}\ dy = 0}$ , so ${A_1}$ vanishes. Since ${\int_{|x| < y \leq 10|x|} \frac{|x \cdot y| |y|}{|y|^5}\ dy = O( |x| )}$ , we conclude that

$\displaystyle \omega(x) = -3\int_{|y| \geq |x|} (x \cdot y) \frac{\omega(y) \times y}{|y|^5}\ dy + O(|x|).$

Using (4), the right-hand side is

$\displaystyle -3\int_{|y| \geq |x|} (x_1 y_1 + x_2 y_2 + x_3 y_3) \frac{\omega_{r3}(r,y_3) (-y_1 y_3, -y_2 y_3, y_1^2+y_2^2)}{r|y|^5}\ dy$

$\displaystyle + O(|x|)$

where ${r := \sqrt{y_1^2+y_2^2}}$ . Because of the odd nature of ${\omega_{r3}}$ , only those terms with one factor of ${y_3}$ give a non-vanishing contribution to the integral. Using the rotation symmetry ${(y_1,y_2,y_3) \mapsto (-y_2,y_1,y_3)}$ we also see that any term with a factor of ${y_1 y_2}$ also vanishes. We can thus simplify the above expression as

$\displaystyle -3\int_{|y| \geq |x|} \frac{\omega_{r3}(r,y_3) (-x_1 y_1^2 y_3, -x_2 y_2^2 y_3, x_3 (y_1^2+y_2^2) y_3)}{r|y|^5}\ dy + O(|x|).$

Using the rotation symmetry ${(y_1,y_2,y_3) \mapsto (-y_2,y_1,y_3)}$ again, we see that the term ${y_1^2}$ in the first component can be replaced by ${y_2^2}$ or by ${\frac{1}{2} (y_1^2+y_2^2) = \frac{r^2}{2}}$ , and similarly for the ${y_2^2}$ term in the second component. Thus the above expression is

$\displaystyle \frac{3}{2} \int_{|y| \geq |x|} \frac{\omega_{r3}(r,y_3) (x_1 , x_2, -2x_3) r y_3}{|y|^5}\ dy + O(|x|)$

giving the claim. $\Box$

Example 2 Consider the divergence-free vector field ${u := \nabla \times \psi}$ , where the vector potential ${\psi}$ takes the form

$\displaystyle \psi(x_1,x_2,x_3) := (x_2 x_3, -x_1 x_3, 0) \eta(|x|)$

for some bump function ${\eta: {\bf R} \rightarrow {\bf R}}$ supported in ${(0,+\infty)}$ . We can then calculate

$\displaystyle u(x_1,x_2,x_3) = X(x) \eta(|x|) + (x_1 x_3, x_2 x_3, -x_1^2-x_2^2) \frac{\eta'(|x|) x_3}{|x|}.$

and

$\displaystyle \omega(x_1,x_2,x_3) = (-6x_2 x_3, 6x_1 x_3, 0) \frac{\eta'(|x|)}{|x|} + (-x_2 x_3, x_1 x_3, 0) \eta''(|x|).$

In particular the hypotheses (i), (ii) are satisfied with

$\displaystyle \omega_{r3}(r,x_3) = - 6 \eta'(|x|) \frac{x_3 r}{|x|} - \eta''(|x|) x_3 r.$

One can then calculate

$\displaystyle L_{12}(\omega)(\rho) = -\frac{3}{4\pi} \int_{|y| \geq \rho} (6\frac{\eta'(|y|)}{|y|^6} + \frac{\eta''(|y|)}{|y|^5}) r^2 y_3^2\ dy$

$\displaystyle = -\frac{2}{5} \int_\rho^\infty 6\eta'(s) + s\eta''(s)\ ds$

$\displaystyle = 2\eta(\rho) + \frac{2}{5} \rho \eta'(\rho).$

If we take the specific choice

$\displaystyle \eta(\rho) = \varphi( \rho^\alpha )$

where ${\varphi}$ is a fixed bump function supported some interval ${[c,C] \subset (0,+\infty)}$ and ${\alpha>0}$ is a small parameter (so that ${\eta}$ is spread out over the range ${\rho \in [c^{1/\alpha},C^{1/\alpha}]}$ ), then we see that

$\displaystyle \| \omega \|_{L^\infty} = O( \alpha )$

(with implied constants allowed to depend on ${\varphi}$ ),

$\displaystyle L_{12}(\omega)(\rho) = 2\eta(\rho) + O(\alpha),$

and

$\displaystyle u = X(x) \eta(|x|) + O( \alpha |x| ),$

which is completely consistent with Proposition 1.

One can use this approximation to extract a plausible ansatz for a self-similar blowup to the Euler equations. We let ${\alpha>0}$ be a small parameter and let ${\omega_{rx_3}}$ be a time-dependent vorticity field obeying (i), (ii) of the form

$\displaystyle \omega_{rx_3}(t,r,x_3) \approx \alpha \Omega( t, R ) \mathrm{sgn}(x_3)$

where ${R := |x|^\alpha = (r^2+x_3^2)^{\alpha/2}}$ and ${\Omega: {\bf R} \times [0,+\infty) \rightarrow {\bf R}}$ is a smooth field to be chosen later. Admittedly the signum function ${\mathrm{sgn}}$ is not smooth at ${x_3}$ , but let us ignore this issue for now (to rigorously make an ansatz one will have to smooth out this function a little bit; Elgindi uses the choice ${(|\sin \theta| \cos^2 \theta)^{\alpha/3} \mathrm{sgn}(x_3)}$ , where ${\theta := \mathrm{arctan}(x_3/r)}$ ). With this ansatz one may compute

$\displaystyle {\mathcal L}_{12}(\omega(t))(\rho) \approx \frac{3\alpha}{2\pi} \int_{|y| \geq \rho; y_3 \geq 0} \Omega(t,R) \frac{r y_3}{|y|^5}\ dy$

$\displaystyle = \alpha \int_\rho^\infty \Omega(t, s^\alpha) \frac{ds}{s}$

$\displaystyle = \int_{\rho^\alpha}^\infty \Omega(t,s) \frac{ds}{s}.$

By Proposition 1, we thus expect to have the approximation

$\displaystyle u(t,x) \approx \frac{1}{2} \int_{|x|^\alpha}^\infty \Omega(t,s) \frac{ds}{s} X(x).$

We insert this into the vorticity equation (2). The transport term ${(u \cdot \nabla) \omega}$ will be expected to be negligible because ${R}$ , and hence ${\omega_{rx_3}}$ , is slowly varying (the discontinuity of ${\mathrm{sgn}(x_3)}$ will not be encountered because the vector field ${X}$ is parallel to this singularity). The modulating function ${\frac{1}{2} \int_{|x|^\alpha}^\infty \Omega(t,s) \frac{ds}{s}}$ is similarly slowly varying, so derivatives falling on this function should be lower order. Neglecting such terms, we arrive at the approximation

$\displaystyle (\omega \cdot \nabla) u \approx \frac{1}{2} \int_{|x|^\alpha}^\infty \Omega(t,s) \frac{ds}{s} \omega$

and so in the limit ${\alpha \rightarrow 0}$ we expect obtain a simple model equation for the evolution of the vorticity envelope ${\Omega}$ :

$\displaystyle \partial_t \Omega(t,R) = \frac{1}{2} \int_R^\infty \Omega(t,S) \frac{dS}{S} \Omega(t,R).$

If we write ${L(t,R) := \int_R^\infty \Omega(t,S)\frac{dS}{S}}$ for the logarithmic primitive of ${\Omega}$ , then we have ${\Omega = - R \partial_R L}$ and hence

$\displaystyle \partial_t (R \partial_R L) = \frac{1}{2} L (R \partial_R L)$

which integrates to the Ricatti equation

$\displaystyle \partial_t L = \frac{1}{4} L^2$

which can be explicitly solved as

$\displaystyle L(t,R) = \frac{2}{f(R) - t/2}$

where ${f(R)}$ is any function of ${R}$ that one pleases. (In Elgindi’s work a time dilation is used to remove the unsightly factor of ${1/2}$ appearing here in the denominator.) If for instance we set ${f(R) = 1+R}$ , we obtain the self-similar solution

$\displaystyle L(t,R) = \frac{2}{1+R-t/2}$

and then on applying ${-R \partial_R}$

$\displaystyle \Omega(t,R) = \frac{2R}{(1+R-t/2)^2}.$

Thus, we expect to be able to construct a self-similar blowup to the Euler equations with a vorticity field approximately behaving like

$\displaystyle \omega(t,x) \approx \alpha \frac{2R}{(1+R-t/2)^2} \mathrm{sgn}(x_3) (\frac{x_2}{r}, -\frac{x_1}{r}, 0)$

and velocity field behaving like

$\displaystyle u(t,x) \approx \frac{1}{1+R-t/2} X(x).$

In particular, ${u}$ would be expected to be of regularity ${C^{1,\alpha}}$ (and smooth away from the origin), and blows up in (say) ${L^\infty}$ norm at time ${t/2 = 1}$ , and one has the self-similarity

$\displaystyle u(t,x) = (1-t/2)^{\frac{1}{\alpha}-1} u( 0, \frac{x}{(1-t/2)^{1/\alpha}} )$

and

$\displaystyle \omega(t,x) = (1-t/2)^{-1} \omega( 0, \frac{x}{(1-t/2)^{1/\alpha}} ).$

A self-similar solution of this approximate shape is in fact constructed rigorously in Elgindi’s paper (using spherical coordinates instead of the Cartesian approach adopted here), using a nonlinear stability analysis of the above ansatz. It seems plausible that one could also carry out this stability analysis using this Cartesian coordinate approach, although I have not tried to do this in detail.

Quantitative bounds for critically bounded solutions to the Navier-Stokes equations

15 August, 2019 in math.AP, paper | Tags: hard analysis, Navier-Stokes equations | by Terence Tao | 52 comments

I’ve just uploaded to the arXiv my paper “Quantitative bounds for critically bounded solutions to the Navier-Stokes equations“, submitted to the proceedings of the Linde Hall Inaugural Math Symposium. (I unfortunately had to cancel my physical attendance at this symposium for personal reasons, but was still able to contribute to the proceedings.) In recent years I have been interested in working towards establishing the existence of classical solutions for the Navier-Stokes equations

$\displaystyle \partial_t u + (u \cdot \nabla) u = \Delta u - \nabla p$

$\displaystyle \nabla \cdot u = 0$

that blow up in finite time, but this time for a change I took a look at the other side of the theory, namely the conditional regularity results for this equation. There are several such results that assert that if a certain norm of the solution stays bounded (or grows at a controlled rate), then the solution stays regular; taken in the contrapositive, they assert that if a solution blows up at a certain finite time ${T_*}$ , then certain norms of the solution must also go to infinity. Here are some examples (not an exhaustive list) of such blowup criteria:

(Leray blowup criterion, 1934) If ${u}$ blows up at a finite time ${T_*}$ , and ${3 < p \leq \infty}$ , then ${\liminf_{t \rightarrow T_*} (T_* - t)^{\frac{1}{2}-\frac{3}{2p}} \| u(t) \|_{L^p_x({\bf R}^3)} \geq c}$ for an absolute constant ${c>0}$ .
(Prodi–Serrin–Ladyzhenskaya blowup criterion, 1959-1967) If ${u}$ blows up at a finite time ${T_*}$ , and ${3 < p \leq \infty}$ , then ${\| u \|_{L^q_t L^p_x([0,T_*) \times {\bf R}^3)} =+\infty}$ , where ${\frac{1}{q} := \frac{1}{2} - \frac{3}{2p}}$ .
(Beale-Kato-Majda blowup criterion, 1984) If ${u}$ blows up at a finite time ${T_*}$ , then ${\| \omega \|_{L^1_t L^\infty_x([0,T_*) \times {\bf R}^3)} = +\infty}$ , where ${\omega := \nabla \times u}$ is the vorticity.
(Kato blowup criterion, 1984) If ${u}$ blows up at a finite time ${T_*}$ , then ${\liminf_{t \rightarrow T_*} \|u(t) \|_{L^3_x({\bf R}^3)} \geq c}$ for some absolute constant ${c>0}$ .
(Escauriaza-Seregin-Sverak blowup criterion, 2003) If ${u}$ blows up at a finite time ${T_*}$ , then ${\limsup_{t \rightarrow T_*} \|u(t) \|_{L^3_x({\bf R}^3)} = +\infty}$ .
(Seregin blowup criterion, 2012) If ${u}$ blows up at a finite time ${T_*}$ , then ${\lim_{t \rightarrow T_*} \|u(t) \|_{L^3_x({\bf R}^3)} = +\infty}$ .
(Phuc blowup criterion, 2015) If ${u}$ blows up at a finite time ${T_*}$ , then ${\limsup_{t \rightarrow T_*} \|u(t) \|_{L^{3,q}_x({\bf R}^3)} = +\infty}$ for any ${q < \infty}$ .
(Gallagher-Koch-Planchon blowup criterion, 2016) If ${u}$ blows up at a finite time ${T_*}$ , then ${\limsup_{t \rightarrow T_*} \|u(t) \|_{\dot B_{p,q}^{-1+3/p}({\bf R}^3)} = +\infty}$ for any ${3 < p, q < \infty}$ .
(Albritton blowup criterion, 2016) If ${u}$ blows up at a finite time ${T_*}$ , then ${\lim_{t \rightarrow T_*} \|u(t) \|_{\dot B_{p,q}^{-1+3/p}({\bf R}^3)} = +\infty}$ for any ${3 < p, q < \infty}$ .

My current paper is most closely related to the Escauriaza-Seregin-Sverak blowup criterion, which was the first to show a critical (i.e., scale-invariant, or dimensionless) spatial norm, namely ${L^3_x({\bf R}^3)}$ , had to become large. This result now has many proofs; for instance, many of the subsequent blowup criterion results imply the Escauriaza-Seregin-Sverak one as a special case, and there are also additional proofs by Gallagher-Koch-Planchon (building on ideas of Kenig-Koch), and by Dong-Du. However, all of these proofs rely on some form of a compactness argument: given a finite time blowup, one extracts some suitable family of rescaled solutions that converges in some weak sense to a limiting solution that has some additional good properties (such as almost periodicity modulo symmetries), which one can then rule out using additional qualitative tools, such as unique continuation and backwards uniqueness theorems for parabolic heat equations. In particular, all known proofs use some version of the backwards uniqueness theorem of Escauriaza, Seregin, and Sverak. Because of this reliance on compactness, the existing proofs of the Escauriaza-Seregin-Sverak blowup criterion are qualitative, in that they do not provide any quantitative information on how fast the ${\|u(t)\|_{L^3_x({\bf R}^3)}}$ norm will go to infinity (along a subsequence of times).

On the other hand, it is a general principle that qualitative arguments established using compactness methods ought to have quantitative analogues that replace the use of compactness by more complicated substitutes that give effective bounds; see for instance these previous blog posts for more discussion. I therefore was interested in trying to obtain a quantitative version of this blowup criterion that gave reasonably good effective bounds (in particular, my objective was to avoid truly enormous bounds such as tower-exponential or Ackermann function bounds, which often arise if one “naively” tries to make a compactness argument effective). In particular, I obtained the following triple-exponential quantitative regularity bounds:

Theorem 1 If ${u}$ is a classical solution to Navier-Stokes on ${[0,T) \times {\bf R}^3}$ with

$\displaystyle \| u \|_{L^\infty_t L^3_x([0,T) \times {\bf R}^3)} \leq A \ \ \ \ \ (1)$

for some ${A \geq 2}$ , then

$\displaystyle | \nabla^j u(t,x)| \leq \exp\exp\exp(A^{O(1)}) t^{-\frac{j+1}{2}}$

and

$\displaystyle | \nabla^j \omega(t,x)| \leq \exp\exp\exp(A^{O(1)}) t^{-\frac{j+2}{2}}$

for ${(t,x) \in [0,T) \times {\bf R}^3}$ and ${j=0,1}$ .

As a corollary, one can now improve the Escauriaza-Seregin-Sverak blowup criterion to

$\displaystyle \limsup_{t \rightarrow T_*} \frac{\|u(t) \|_{L^3_x({\bf R}^3)}}{(\log\log\log \frac{1}{T_*-t})^c} = +\infty$

for some absolute constant ${c>0}$ , which to my knowledge is the first (very slightly) supercritical blowup criterion for Navier-Stokes in the literature.

The proof uses many of the same quantitative inputs as previous arguments, most notably the Carleman inequalities used to establish unique continuation and backwards uniqueness theorems for backwards heat equations, but also some additional techniques that make the quantitative bounds more efficient. The proof focuses initially on points of concentration of the solution, which we define as points ${(t_0,x_0)}$ where there is a frequency ${N_0}$ for which one has the bound

$\displaystyle |N_0^{-1} P_{N_0} u(t_0,x_0)| \geq A^{-C_0} \ \ \ \ \ (2)$

for a large absolute constant ${C_0}$ , where ${P_{N_0}}$ is a Littlewood-Paley projection to frequencies ${\sim N_0}$ . (This can be compared with the upper bound of ${O(A)}$ for the quantity on the left-hand side that follows from (1).) The factor of ${N_0^{-1}}$ normalises the left-hand side of (2) to be dimensionless (i.e., critical). The main task is to show that the dimensionless quantity ${t_0 N_0^2}$ cannot get too large; in particular, we end up establishing a bound of the form

$\displaystyle t_0 N_0^2 \lesssim \exp\exp\exp A^{O(C_0^6)}$

from which the above theorem ends up following from a routine adaptation of the local well-posedness and regularity theory for Navier-Stokes.

The strategy is to show that any concentration such as (2) when ${t_0 N_0^2}$ is large must force a significant component of the ${L^3_x}$ norm of ${u(t_0)}$ to also show up at many other locations than ${x_0}$ , which eventually contradicts (1) if one can produce enough such regions of non-trivial ${L^3_x}$ norm. (This can be viewed as a quantitative variant of the “rigidity” theorems in some of the previous proofs of the Escauriaza-Seregin-Sverak theorem that rule out solutions that exhibit too much “compactness” or “almost periodicity” in the ${L^3_x}$ topology.) The chain of causality that leads from a concentration (2) at ${(t_0,x_0)}$ to significant ${L^3_x}$ norm at other regions of the time slice ${t_0 \times {\bf R}^3}$ is somewhat involved (though simpler than the much more convoluted schemes I initially envisaged for this argument):

Firstly, by using Duhamel’s formula, one can show that a concentration (2) can only occur (with ${t_0 N_0^2}$ large) if there was also a preceding concentration
$\displaystyle |N_1^{-1} P_{N_1} u(t_1,x_1)| \geq A^{-C_0} \ \ \ \ \ (3)$

at some slightly previous point ${(t_1,x_1)}$ in spacetime, with ${N_1}$ also close to ${N_0}$ (more precisely, we have ${t_1 = t_0 - A^{-O(C_0^3)} N_0^{-2}}$ , ${N_1 = A^{O(C_0^2)} N_0}$ , and ${x_1 = x_0 + O( A^{O(C_0^4)} N_0^{-1})}$ ). This can be viewed as a sort of contrapositive of a “local regularity theorem”, such as the ones established by Caffarelli, Kohn, and Nirenberg. A key point here is that the lower bound ${A^{-C_0}}$ in the conclusion (3) is precisely the same as the lower bound in (2), so that this backwards propagation of concentration can be iterated.
Iterating the previous step, one can find a sequence of concentration points
$\displaystyle |N_n^{-1} P_{N_n} u(t_n,x_n)| \geq A^{-C_0} \ \ \ \ \ (4)$

with the ${(t_n,x_n)}$ propagating backwards in time; by using estimates ultimately resulting from the dissipative term in the energy identity, one can extract such a sequence in which the ${t_0-t_n}$ increase geometrically with time, the ${N_n}$ are comparable (up to polynomial factors in ${A}$ ) to the natural frequency scale ${(t_0-t_n)^{-1/2}}$ , and one has ${x_n = x_0 + O( |t_0-t_n|^{1/2} )}$ . Using the “epochs of regularity” theory that ultimately dates back to Leray, and tweaking the ${t_n}$ slightly, one can also place the times ${t_n}$ in intervals ${I_n}$ (of length comparable to a small multiple of ${|t_0-t_n|}$ ) in which the solution is quite regular (in particular, ${u, \nabla u, \omega, \nabla \omega}$ enjoy good ${L^\infty_t L^\infty_x}$ bounds on ${I_n \times {\bf R}^3}$ ).
The concentration (4) can be used to establish a lower bound for the ${L^2_t L^2_x}$ norm of the vorticity ${\omega}$ near ${(t_n,x_n)}$ . As is well known, the vorticity obeys the vorticity equation
$\displaystyle \partial_t \omega = \Delta \omega - (u \cdot \nabla) \omega + (\omega \cdot \nabla) u.$

In the epoch of regularity ${I_n \times {\bf R}^3}$ , the coefficients ${u, \nabla u}$ of this equation obey good ${L^\infty_x}$ bounds, allowing the machinery of Carleman estimates to come into play. Using a Carleman estimate that is used to establish unique continuation results for backwards heat equations, one can propagate this lower bound to also give lower ${L^2}$ bounds on the vorticity (and its first derivative) in annuli of the form ${\{ (t,x) \in I_n \times {\bf R}^3: R \leq |x-x_n| \leq R' \}}$ for various radii ${R,R'}$ , although the lower bounds decay at a gaussian rate with ${R}$ .
Meanwhile, using an energy pigeonholing argument of Bourgain (which, in this Navier-Stokes context, is actually an enstrophy pigeonholing argument), one can locate some annuli ${\{ x \in {\bf R}^3: R \leq |x-x_n| \leq R'\}}$ where (a slightly normalised form of) the entrosphy is small at time ${t=t_n}$ ; using a version of the localised enstrophy estimates from a previous paper of mine, one can then propagate this sort of control forward in time, obtaining an “annulus of regularity” of the form ${\{ (t,x) \in [t_n,t_0] \times {\bf R}^3: R_n \leq |x-x_n| \leq R'_n\}}$ in which one has good estimates; in particular, one has ${L^\infty_x}$ type bounds on ${u, \nabla u, \omega, \nabla \omega}$ in this cylindrical annulus.
By intersecting the previous epoch of regularity ${I_n \times {\bf R}^3}$ with the above annulus of regularity, we have some lower bounds on the ${L^2}$ norm of the vorticity (and its first derivative) in the annulus of regularity. Using a Carleman estimate first introduced by Escauriaza, Seregin, and Sverak, as well as a second application of the Carleman estimate used previously, one can then propagate this lower bound back up to time ${t=t_0}$ , establishing a lower bound for the vorticity on the spatial annulus ${\{ (t_0,x): R_n \leq |x-x_n| \leq R'_n \}}$ . By some basic Littlewood-Paley theory one can parlay this lower bound to a lower bound on the ${L^3}$ norm of the velocity ${u}$ ; crucially, this lower bound is uniform in ${n}$ .
If ${t_0 N_0^2}$ is very large (triple exponential in ${A}$ !), one can then find enough scales ${n}$ with disjoint ${\{ (t_0,x): R_n \leq |x-x_n| \leq R'_n \}}$ annuli that the total lower bound on the ${L^3_x}$ norm of ${u(t_0)}$ provided by the above arguments is inconsistent with (1), thus establishing the claim.

The chain of causality is summarised in the following image:

scheme

It seems natural to conjecture that similar triply logarithmic improvements can be made to several of the other blowup criteria listed above, but I have not attempted to pursue this question. It seems difficult to improve the triple logarithmic factor using only the techniques here; the Bourgain pigeonholing argument inevitably costs one exponential, the Carleman inequalities cost a second, and the stacking of scales at the end to contradict the ${L^3}$ upper bound costs the third.

Symmetric functions in a fractional number of variables, and the multilinear Kakeya conjecture

29 June, 2019 in expository, math.AP, math.CA, math.CO | Tags: fractional dimension, heat flow, multilinear Kakeya conjecture, symmetric polynomials | by Terence Tao | 21 comments

Let ${\Omega}$ be some domain (such as the real numbers). For any natural number ${p}$ , let ${L(\Omega^p)_{sym}}$ denote the space of symmetric real-valued functions ${F^{(p)}: \Omega^p \rightarrow {\bf R}}$ on ${p}$ variables ${x_1,\dots,x_p \in \Omega}$ , thus

$\displaystyle F^{(p)}(x_{\sigma(1)},\dots,x_{\sigma(p)}) = F^{(p)}(x_1,\dots,x_p)$

for any permutation ${\sigma: \{1,\dots,p\} \rightarrow \{1,\dots,p\}}$ . For instance, for any natural numbers ${k,p}$ , the elementary symmetric polynomials

$\displaystyle e_k^{(p)}(x_1,\dots,x_p) = \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq p} x_{i_1} \dots x_{i_k}$

will be an element of ${L({\bf R}^p)_{sym}}$ . With the pointwise product operation, ${L(\Omega^p)_{sym}}$ becomes a commutative real algebra. We include the case ${p=0}$ , in which case ${L(\Omega^0)_{sym}}$ consists solely of the real constants.

Given two natural numbers ${k,p}$ , one can “lift” a symmetric function ${F^{(k)} \in L(\Omega^k)_{sym}}$ of ${k}$ variables to a symmetric function ${[F^{(k)}]_{k \rightarrow p} \in L(\Omega^p)_{sym}}$ of ${p}$ variables by the formula

$\displaystyle [F^{(k)}]_{k \rightarrow p}(x_1,\dots,x_p) = \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq p} F^{(k)}(x_{i_1}, \dots, x_{i_k})$

$\displaystyle = \frac{1}{k!} \sum_\pi F^{(k)}( x_{\pi(1)}, \dots, x_{\pi(k)} )$

where ${\pi}$ ranges over all injections from ${\{1,\dots,k\}}$ to ${\{1,\dots,p\}}$ (the latter formula making it clearer that ${[F^{(k)}]_{k \rightarrow p}}$ is symmetric). Thus for instance

$\displaystyle [F^{(1)}(x_1)]_{1 \rightarrow p} = \sum_{i=1}^p F^{(1)}(x_i)$

$\displaystyle [F^{(2)}(x_1,x_2)]_{2 \rightarrow p} = \sum_{1 \leq i < j \leq p} F^{(2)}(x_i,x_j)$

and

$\displaystyle e_k^{(p)}(x_1,\dots,x_p) = [x_1 \dots x_k]_{k \rightarrow p}.$

Also we have

$\displaystyle [1]_{k \rightarrow p} = \binom{p}{k} = \frac{p(p-1)\dots(p-k+1)}{k!}.$

With these conventions, we see that ${[F^{(k)}]_{k \rightarrow p}}$ vanishes for ${p=0,\dots,k-1}$ , and is equal to ${F}$ if ${k=p}$ . We also have the transitivity

$\displaystyle [F^{(k)}]_{k \rightarrow p} = \frac{1}{\binom{p-k}{p-l}} [[F^{(k)}]_{k \rightarrow l}]_{l \rightarrow p}$

if ${k \leq l \leq p}$ .

The lifting map ${[]_{k \rightarrow p}}$ is a linear map from ${L(\Omega^k)_{sym}}$ to ${L(\Omega^p)_{sym}}$ , but it is not a ring homomorphism. For instance, when ${\Omega={\bf R}}$ , one has

$\displaystyle [x_1]_{1 \rightarrow p} [x_1]_{1 \rightarrow p} = (\sum_{i=1}^p x_i)^2 \ \ \ \ \ (1)$

$\displaystyle = \sum_{i=1}^p x_i^2 + 2 \sum_{1 \leq i < j \leq p} x_i x_j$

$\displaystyle = [x_1^2]_{1 \rightarrow p} + 2 [x_1 x_2]_{1 \rightarrow p}$

$\displaystyle \neq [x_1^2]_{1 \rightarrow p}.$

In general, one has the identity

$\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow p} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow p} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (2)$

$\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow p}$

for all natural numbers ${k,l,p}$ and ${F^{(k)} \in L(\Omega^k)_{sym}}$ , ${G^{(l)} \in L(\Omega^l)_{sym}}$ , where ${\pi, \rho}$ range over all injections ${\pi: \{1,\dots,k\} \rightarrow \{1,\dots,m\}}$ , ${\rho: \{1,\dots,l\} \rightarrow \{1,\dots,m\}}$ with ${\pi(\{1,\dots,k\}) \cup \rho(\{1,\dots,l\}) = \{1,\dots,m\}}$ . Combinatorially, the identity (2) follows from the fact that given any injections ${\tilde \pi: \{1,\dots,k\} \rightarrow \{1,\dots,p\}}$ and ${\tilde \rho: \{1,\dots,l\} \rightarrow \{1,\dots,p\}}$ with total image ${\tilde \pi(\{1,\dots,k\}) \cup \tilde \rho(\{1,\dots,l\})}$ of cardinality ${m}$ , one has ${k,l \leq m \leq k+l}$ , and furthermore there exist precisely ${m!}$ triples ${(\pi, \rho, \sigma)}$ of injections ${\pi: \{1,\dots,k\} \rightarrow \{1,\dots,m\}}$ , ${\rho: \{1,\dots,l\} \rightarrow \{1,\dots,m\}}$ , ${\sigma: \{1,\dots,m\} \rightarrow \{1,\dots,p\}}$ such that ${\tilde \pi = \sigma \circ \pi}$ and ${\tilde \rho = \sigma \circ \rho}$ .

Example 1 When ${\Omega = {\bf R}}$ , one has

$\displaystyle [x_1 x_2]_{2 \rightarrow p} [x_1]_{1 \rightarrow p} = [\frac{1}{2! 1!}( 2 x_1^2 x_2 + 2 x_1 x_2^2 )]_{2 \rightarrow p} + [\frac{1}{2! 1!} 6 x_1 x_2 x_3]_{3 \rightarrow p}$

$\displaystyle = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow p} + [3x_1 x_2 x_3]_{3 \rightarrow p}$

which is just a restatement of the identity

$\displaystyle (\sum_{i < j} x_i x_j) (\sum_k x_k) = \sum_{i<j} x_i^2 x_j + x_i x_j^2 + \sum_{i < j < k} 3 x_i x_j x_k.$

Note that the coefficients appearing in (2) do not depend on the final number of variables ${p}$ . We may therefore abstract the role of ${p}$ from the law (2) by introducing the real algebra ${L(\Omega^*)_{sym}}$ of formal sums

$\displaystyle F^{(*)} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}$

where for each ${k}$ , ${F^{(k)}}$ is an element of ${L(\Omega^k)_{sym}}$ (with only finitely many of the ${F^{(k)}}$ being non-zero), and with the formal symbol ${[]_{k \rightarrow *}}$ being formally linear, thus

$\displaystyle [F^{(k)}]_{k \rightarrow *} + [G^{(k)}]_{k \rightarrow *} := [F^{(k)} + G^{(k)}]_{k \rightarrow *}$

and

$\displaystyle c [F^{(k)}]_{k \rightarrow *} := [cF^{(k)}]_{k \rightarrow *}$

for ${F^{(k)}, G^{(k)} \in L(\Omega^k)_{sym}}$ and scalars ${c \in {\bf R}}$ , and with multiplication given by the analogue

$\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow *} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow *} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (3)$

$\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow *}$

of (2). Thus for instance, in this algebra ${L(\Omega^*)_{sym}}$ we have

$\displaystyle [x_1]_{1 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2]_{1 \rightarrow *} + 2 [x_1 x_2]_{2 \rightarrow *}$

and

$\displaystyle [x_1 x_2]_{2 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow *} + [3 x_1 x_2 x_3]_{3 \rightarrow *}.$

Informally, ${L(\Omega^*)_{sym}}$ is an abstraction (or “inverse limit”) of the concept of a symmetric function of an unspecified number of variables, which are formed by summing terms that each involve only a bounded number of these variables at a time. One can check (somewhat tediously) that ${L(\Omega^*)_{sym}}$ is indeed a commutative real algebra, with a unit ${[1]_{0 \rightarrow *}}$ . (I do not know if this algebra has previously been studied in the literature; it is somewhat analogous to the abstract algebra of finite linear combinations of Schur polynomials, with multiplication given by a Littlewood-Richardson rule. )

For natural numbers ${p}$ , there is an obvious specialisation map ${[]_{* \rightarrow p}}$ from ${L(\Omega^*)_{sym}}$ to ${L(\Omega^p)_{sym}}$ , defined by the formula

$\displaystyle [\sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}]_{* \rightarrow p} := \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}.$

Thus, for instance, ${[]_{* \rightarrow p}}$ maps ${[x_1]_{1 \rightarrow *}}$ to ${[x_1]_{1 \rightarrow p}}$ and ${[x_1 x_2]_{2 \rightarrow *}}$ to ${[x_1 x_2]_{2 \rightarrow p}}$ . From (2) and (3) we see that this map ${[]_{* \rightarrow p}: L(\Omega^*)_{sym} \rightarrow L(\Omega^p)_{sym}}$ is an algebra homomorphism, even though the maps ${[]_{k \rightarrow *}: L(\Omega^k)_{sym} \rightarrow L(\Omega^*)_{sym}}$ and ${[]_{k \rightarrow p}: L(\Omega^k)_{sym} \rightarrow L(\Omega^p)_{sym}}$ are not homomorphisms. By inspecting the ${p^{th}}$ component of ${L(\Omega^*)_{sym}}$ we see that the homomorphism ${[]_{* \rightarrow p}}$ is in fact surjective.

Now suppose that we have a measure ${\mu}$ on the space ${\Omega}$ , which then induces a product measure ${\mu^p}$ on every product space ${\Omega^p}$ . To avoid degeneracies we will assume that the integral ${\int_\Omega \mu}$ is strictly positive. Assuming suitable measurability and integrability hypotheses, a function ${F \in L(\Omega^p)_{sym}}$ can then be integrated against this product measure to produce a number

$\displaystyle \int_{\Omega^p} F\ d\mu^p.$

In the event that ${F}$ arises as a lift ${[F^{(k)}]_{k \rightarrow p}}$ of another function ${F^{(k)} \in L(\Omega^k)_{sym}}$ , then from Fubini’s theorem we obtain the formula

$\displaystyle \int_{\Omega^p} F\ d\mu^p = \binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu^k) (\int_\Omega\ d\mu)^{p-k}.$

Thus for instance, if ${\Omega={\bf R}}$ ,

$\displaystyle \int_{{\bf R}^p} [x_1]_{1 \rightarrow p}\ d\mu^p = p (\int_{\bf R} x\ d\mu(x)) (\int_{\bf R} \mu)^{p-1} \ \ \ \ \ (4)$

and

$\displaystyle \int_{{\bf R}^p} [x_1 x_2]_{2 \rightarrow p}\ d\mu^p = \binom{p}{2} (\int_{{\bf R}^2} x_1 x_2\ d\mu(x_1) d\mu(x_2)) (\int_{\bf R} \mu)^{p-2}. \ \ \ \ \ (5)$

On summing, we see that if

$\displaystyle F^{(*)} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}$

is an element of the formal algebra ${L(\Omega^*)_{sym}}$ , then

$\displaystyle \int_{\Omega^p} [F^{(*)}]_{* \rightarrow p}\ d\mu^p = \sum_{k=0}^\infty \binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu^k) (\int_\Omega\ d\mu)^{p-k}. \ \ \ \ \ (6)$

Note that by hypothesis, only finitely many terms on the right-hand side are non-zero.

Now for a key observation: whereas the left-hand side of (6) only makes sense when ${p}$ is a natural number, the right-hand side is meaningful when ${p}$ takes a fractional value (or even when it takes negative or complex values!), interpreting the binomial coefficient ${\binom{p}{k}}$ as a polynomial ${\frac{p(p-1) \dots (p-k+1)}{k!}}$ in ${p}$ . As such, this suggests a way to introduce a “virtual” concept of a symmetric function on a fractional power space ${\Omega^p}$ for such values of ${p}$ , and even to integrate such functions against product measures ${\mu^p}$ , even if the fractional power ${\Omega^p}$ does not exist in the usual set-theoretic sense (and ${\mu^p}$ similarly does not exist in the usual measure-theoretic sense). More precisely, for arbitrary real or complex ${p}$ , we now define ${L(\Omega^p)_{sym}}$ to be the space of abstract objects

$\displaystyle F^{(p)} = [F^{(*)}]_{* \rightarrow p} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}$

with ${F^{(*)} \in L(\Omega^*)_{sym}}$ and ${[]_{* \rightarrow p}}$ (and ${[]_{k \rightarrow p}}$ now interpreted as formal symbols, with the structure of a commutative real algebra inherited from ${L(\Omega^*)_{sym}}$ , thus

$\displaystyle [F^{(*)}]_{* \rightarrow p} + [G^{(*)}]_{* \rightarrow p} := [F^{(*)} + G^{(*)}]_{* \rightarrow p}$

$\displaystyle c [F^{(*)}]_{* \rightarrow p} := [c F^{(*)}]_{* \rightarrow p}$

$\displaystyle [F^{(*)}]_{* \rightarrow p} [G^{(*)}]_{* \rightarrow p} := [F^{(*)} G^{(*)}]_{* \rightarrow p}.$

In particular, the multiplication law (2) continues to hold for such values of ${p}$ , thanks to (3). Given any measure ${\mu}$ on ${\Omega}$ , we formally define a measure ${\mu^p}$ on ${\Omega^p}$ with regards to which we can integrate elements ${F^{(p)}}$ of ${L(\Omega^p)_{sym}}$ by the formula (6) (providing one has sufficient measurability and integrability to make sense of this formula), thus providing a sort of “fractional dimensional integral” for symmetric functions. Thus, for instance, with this formalism the identities (4), (5) now hold for fractional values of ${p}$ , even though the formal space ${{\bf R}^p}$ no longer makes sense as a set, and the formal measure ${\mu^p}$ no longer makes sense as a measure. (The formalism here is somewhat reminiscent of the technique of dimensional regularisation employed in the physical literature in order to assign values to otherwise divergent integrals. See also this post for an unrelated abstraction of the integration concept involving integration over supercommutative variables (and in particular over fermionic variables).)

Example 2 Suppose ${\mu}$ is a probability measure on ${\Omega}$ , and ${X: \Omega \rightarrow {\bf R}}$ is a random variable; on any power ${\Omega^k}$ , we let ${X_1,\dots,X_k: \Omega^k \rightarrow {\bf R}}$ be the usual independent copies of ${X}$ on ${\Omega^k}$ , thus ${X_j(\omega_1,\dots,\omega_k) := X(\omega_j)}$ for ${(\omega_1,\dots,\omega_k) \in \Omega^k}$ . Then for any real or complex ${p}$ , the formal integral

$\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p$

can be evaluated by first using the identity

$\displaystyle [X_1]_{1 \rightarrow p}^2 = [X_1^2]_{1 \rightarrow p} + 2[X_1 X_2]_{2 \rightarrow p}$

(cf. (1)) and then using (6) and the probability measure hypothesis ${\int_\Omega\ d\mu = 1}$ to conclude that

$\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = \binom{p}{1} \int_{\Omega} X^2\ d\mu + 2 \binom{p}{2} \int_{\Omega^2} X_1 X_2\ d\mu^2$

$\displaystyle = p (\int_\Omega X^2\ d\mu - (\int_\Omega X\ d\mu)^2) + p^2 (\int_\Omega X\ d\mu)^2$

or in probabilistic notation

$\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = p \mathbf{Var}(X) + p^2 \mathbf{E}(X)^2. \ \ \ \ \ (7)$

For ${p}$ a natural number, this identity has the probabilistic interpretation

$\displaystyle \mathbf{E}( X_1 + \dots + X_p)^2 = p \mathbf{Var}(X) + p^2 \mathbf{E}(X)^2 \ \ \ \ \ (8)$

whenever ${X_1,\dots,X_p}$ are jointly independent copies of ${X}$ , which reflects the well known fact that the sum ${X_1 + \dots + X_p}$ has expectation ${p \mathbf{E} X}$ and variance ${p \mathbf{Var}(X)}$ . One can thus view (7) as an abstract generalisation of (8) to the case when ${p}$ is fractional, negative, or even complex, despite the fact that there is no sensible way in this case to talk about ${p}$ independent copies ${X_1,\dots,X_p}$ of ${X}$ in the standard framework of probability theory.

In this particular case, the quantity (7) is non-negative for every nonnegative ${p}$ , which looks plausible given the form of the left-hand side. Unfortunately, this sort of non-negativity does not always hold; for instance, if ${X}$ has mean zero, one can check that

$\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^4\ d\mu^p = p \mathbf{Var}(X^2) + p(3p-2) (\mathbf{E}(X^2))^2$

and the right-hand side can become negative for ${p < 2/3}$ . This is a shame, because otherwise one could hope to start endowing ${L(X^p)_{sym}}$ with some sort of commutative von Neumann algebra type structure (or the abstract probability structure discussed in this previous post) and then interpret it as a genuine measure space rather than as a virtual one. (This failure of positivity is related to the fact that the characteristic function of a random variable, when raised to the ${p^{th}}$ power, need not be a characteristic function of any random variable once ${p}$ is no longer a natural number: “fractional convolution” does not preserve positivity!) However, one vestige of positivity remains: if ${F: \Omega \rightarrow {\bf R}}$ is non-negative, then so is

$\displaystyle \int_{\Omega^p} [F]_{1 \rightarrow p}\ d\mu^p = p (\int_\Omega F\ d\mu) (\int_\Omega\ d\mu)^{p-1}.$

One can wonder what the point is to all of this abstract formalism and how it relates to the rest of mathematics. For me, this formalism originated implicitly in an old paper I wrote with Jon Bennett and Tony Carbery on the multilinear restriction and Kakeya conjectures, though we did not have a good language for working with it at the time, instead working first with the case of natural number exponents ${p}$ and appealing to a general extrapolation theorem to then obtain various identities in the fractional ${p}$ case. The connection between these fractional dimensional integrals and more traditional integrals ultimately arises from the simple identity

$\displaystyle (\int_\Omega\ d\mu)^p = \int_{\Omega^p}\ d\mu^p$

(where the right-hand side should be viewed as the fractional dimensional integral of the unit ${[1]_{0 \rightarrow p}}$ against ${\mu^p}$ ). As such, one can manipulate ${p^{th}}$ powers of ordinary integrals using the machinery of fractional dimensional integrals. A key lemma in this regard is

Lemma 3 (Differentiation formula) Suppose that a positive measure ${\mu = \mu(t)}$ on ${\Omega}$ depends on some parameter ${t}$ and varies by the formula

$\displaystyle \frac{d}{dt} \mu(t) = a(t) \mu(t) \ \ \ \ \ (9)$

for some function ${a(t): \Omega \rightarrow {\bf R}}$ . Let ${p}$ be any real or complex number. Then, assuming sufficient smoothness and integrability of all quantities involved, we have

$\displaystyle \frac{d}{dt} \int_{\Omega^p} F^{(p)}\ d\mu(t)^p = \int_{\Omega^p} F^{(p)} [a(t)]_{1 \rightarrow p}\ d\mu(t)^p \ \ \ \ \ (10)$

for all ${F^{(p)} \in L(\Omega^p)_{sym}}$ that are independent of ${t}$ . If we allow ${F^{(p)}(t)}$ to now depend on ${t}$ also, then we have the more general total derivative formula

$\displaystyle \frac{d}{dt} \int_{\Omega^p} F^{(p)}(t)\ d\mu(t)^p \ \ \ \ \ (11)$

$\displaystyle = \int_{\Omega^p} \frac{d}{dt} F^{(p)}(t) + F^{(p)}(t) [a(t)]_{1 \rightarrow p}\ d\mu(t)^p,$

again assuming sufficient amounts of smoothness and regularity.

Proof: We just prove (10), as (11) then follows by same argument used to prove the usual product rule. By linearity it suffices to verify this identity in the case ${F^{(p)} = [F^{(k)}]_{k \rightarrow p}}$ for some symmetric function ${F^{(k)} \in L(\Omega^k)_{sym}}$ for a natural number ${k}$ . By (6), the left-hand side of (10) is then

$\displaystyle \frac{d}{dt} [\binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu(t)^k) (\int_\Omega\ d\mu(t))^{p-k}]. \ \ \ \ \ (12)$

Differentiating under the integral sign using (9) we have

$\displaystyle \frac{d}{dt} \int_\Omega\ d\mu(t) = \int_\Omega\ a(t)\ d\mu(t)$

and similarly

$\displaystyle \frac{d}{dt} \int_{\Omega^k} F^{(k)}\ d\mu(t)^k = \int_{\Omega^k} F^{(k)}(a_1+\dots+a_k)\ d\mu(t)^k$

where ${a_1,\dots,a_k}$ are the standard ${k}$ copies of ${a = a(t)}$ on ${\Omega^k}$ :

$\displaystyle a_j(\omega_1,\dots,\omega_k) := a(\omega_j).$

By the product rule, we can thus expand (12) as

$\displaystyle \binom{p}{k} (\int_{\Omega^k} F^{(k)}(a_1+\dots+a_k)\ d\mu^k ) (\int_\Omega\ d\mu)^{p-k}$

$\displaystyle + \binom{p}{k} (p-k) (\int_{\Omega^k} F^{(k)}\ d\mu^k) (\int_\Omega\ a\ d\mu) (\int_\Omega\ d\mu)^{p-k-1}$

where we have suppressed the dependence on ${t}$ for brevity. Since ${\binom{p}{k} (p-k) = \binom{p}{k+1} (k+1)}$ , we can write this expression using (6) as

$\displaystyle \int_{\Omega^p} [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p}\ d\mu^p$

where ${F^{(k)} \ast a \in L(\Omega^{k+1})_{sym}}$ is the symmetric function

$\displaystyle F^{(k)} \ast a(\omega_1,\dots,\omega_{k+1}) := \sum_{j=1}^{k+1} F^{(k)}(\omega_1,\dots,\omega_{j-1},\omega_{j+1} \dots \omega_{k+1}) a(\omega_j).$

But from (2) one has

$\displaystyle [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p} = [F^{(k)}]_{k \rightarrow p} [a]_{1 \rightarrow p}$

and the claim follows. $\Box$

Remark 4 It is also instructive to prove this lemma in the special case when ${p}$ is a natural number, in which case the fractional dimensional integral ${\int_{\Omega^p} F^{(p)}\ d\mu(t)^p}$ can be interpreted as a classical integral. In this case, the identity (10) is immediate from applying the product rule to (9) to conclude that

$\displaystyle \frac{d}{dt} d\mu(t)^p = [a(t)]_{1 \rightarrow p} d\mu(t)^p.$

One could in fact derive (10) for arbitrary real or complex ${p}$ from the case when ${p}$ is a natural number by an extrapolation argument; see the appendix of my paper with Bennett and Carbery for details.

Let us give a simple PDE application of this lemma as illustration:

Proposition 5 (Heat flow monotonicity) Let ${u: [0,+\infty) \times {\bf R}^d \rightarrow {\bf R}}$ be a solution to the heat equation ${u_t = \Delta u}$ with initial data ${\mu_0}$ a rapidly decreasing finite non-negative Radon measure, or more explicitly

$\displaystyle u(t,x) = \frac{1}{(4\pi t)^{d/2}} \int_{{\bf R}^d} e^{-|x-y|^2/4t}\ d\mu_0(y)$

for al ${t>0}$ . Then for any ${p>0}$ , the quantity

$\displaystyle Q_p(t) := t^{\frac{d}{2} (p-1)} \int_{{\bf R}^d} u(t,x)^p\ dx$

is monotone non-decreasing in ${t \in (0,+\infty)}$ for ${1 < p < \infty}$ , constant for ${p=1}$ , and monotone non-increasing for ${0 < p < 1}$ .

Proof: By a limiting argument we may assume that ${d\mu_0}$ is absolutely continuous, with Radon-Nikodym derivative a test function; this is more than enough regularity to justify the arguments below.

For any ${(t,x) \in (0,+\infty) \times {\bf R}^d}$ , let ${\mu(t,x)}$ denote the Radon measure

$\displaystyle d\mu(t,x)(y) := \frac{1}{(4\pi)^{d/2}} e^{-|x-y|^2/4t}\ d\mu_0(y).$

Then the quantity ${Q_p(t)}$ can be written as a fractional dimensional integral

$\displaystyle Q_p(t) = t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p}\ d\mu(t,x)^p\ dx.$

Observe that

$\displaystyle \frac{\partial}{\partial t} d\mu(t,x) = \frac{|x-y|^2}{4t^2} d\mu(t,x)$

and thus by Lemma 3 and the product rule

$\displaystyle \frac{d}{dt} Q_p(t) = -\frac{d}{2t} Q_p(t) + t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [\frac{|x-y|^2}{4t^2}]_{1 \rightarrow p} d\mu(t,x)^p\ dx \ \ \ \ \ (13)$

where we use ${y}$ for the variable of integration in the factor space ${{\bf R}^d}$ of ${({\bf R}^d)^p}$ .

To simplify this expression we will take advantage of integration by parts in the ${x}$ variable. Specifically, in any direction ${x_j}$ , we have

$\displaystyle \frac{\partial}{\partial x_j} d\mu(t,x) = -\frac{x_j-y_j}{2t} d\mu(t,x)$

and hence by Lemma 3

$\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}\ d\mu(t,x)^p\ dx = - \int_{({\bf R}^d)^p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.$

Multiplying by ${x_j}$ and integrating by parts, we see that

$\displaystyle d Q_p(t) = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx$

$\displaystyle = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx$

where we use the Einstein summation convention in ${j}$ . Similarly, if ${F_j(y)}$ is any reasonable function depending only on ${y}$ , we have

$\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}[F_j(y)]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx$

$\displaystyle = - \int_{({\bf R}^d)^p} [F_j(y)]_{1 \rightarrow p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx$

and hence on integration by parts

$\displaystyle 0 = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [F_j(y) \frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.$

We conclude that

$\displaystyle \frac{d}{2t} Q_p(t) = t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} (x_j - [F_j(y)]_{1 \rightarrow p}) [\frac{(x_j-y_j)}{4t}]_{1 \rightarrow p} d\mu(t,x)^p\ dx$

and thus by (13)

$\displaystyle \frac{d}{dt} Q_p(t) = \frac{1}{4t^{\frac{d}{2}+2}} \int_{{\bf R}^d} \int_{({\bf R}^d)^p}$

$\displaystyle [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p} - (x_j - [F_j(y)]_{1 \rightarrow p}) [x_j - y_j]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.$

The choice of ${F_j}$ that then achieves the most cancellation turns out to be ${F_j(y) = \frac{1}{p} y_j}$ (this cancels the terms that are linear or quadratic in the ${x_j}$ ), so that ${x_j - [F_j(y)]_{1 \rightarrow p} = \frac{1}{p} [x_j - y_j]_{1 \rightarrow p}}$ . Repeating the calculations establishing (7), one has

$\displaystyle \int_{({\bf R}^d)^p} [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p}\ d\mu^p = p \mathop{\bf E} |x-Y|^2 (\int_{{\bf R}^d}\ d\mu)^{p}$

and

$\displaystyle \int_{({\bf R}^d)^p} [x_j-y_j]_{1 \rightarrow p} [x_j-y_j]_{1 \rightarrow p}\ d\mu^p$

$\displaystyle = (p \mathbf{Var}(x-Y) + p^2 |\mathop{\bf E} x-Y|^2) (\int_{{\bf R}^d}\ d\mu)^{p}$

where ${Y}$ is the random variable drawn from ${{\bf R}^d}$ with the normalised probability measure ${\mu / \int_{{\bf R}^d}\ d\mu}$ . Since ${\mathop{\bf E} |x-Y|^2 = \mathbf{Var}(x-Y) + |\mathop{\bf E} x-Y|^2}$ , one thus has

$\displaystyle \frac{d}{dt} Q_p(t) = (p-1) \frac{1}{4t^{\frac{d}{2}+2}} \int_{{\bf R}^d} \mathbf{Var}(x-Y) (\int_{{\bf R}^d}\ d\mu)^{p}\ dx. \ \ \ \ \ (14)$

This expression is clearly non-negative for ${p>1}$ , equal to zero for ${p=1}$ , and positive for ${0 < p < 1}$ , giving the claim. (One could simplify ${\mathbf{Var}(x-Y)}$ here as ${\mathbf{Var}(Y)}$ if desired, though it is not strictly necessary to do so for the proof.) $\Box$

Remark 6 As with Remark 4, one can also establish the identity (14) first for natural numbers ${p}$ by direct computation avoiding the theory of fractional dimensional integrals, and then extrapolate to the case of more general values of ${p}$ . This particular identity is also simple enough that it can be directly established by integration by parts without much difficulty, even for fractional values of ${p}$ .

A more complicated version of this argument establishes the non-endpoint multilinear Kakeya inequality (without any logarithmic loss in a scale parameter ${R}$ ); this was established in my previous paper with Jon Bennett and Tony Carbery, but using the “natural number ${p}$ first” approach rather than using the current formalism of fractional dimensional integration. However, the arguments can be translated into this formalism without much difficulty; we do so below the fold. (To simplify the exposition slightly we will not address issues of establishing enough regularity and integrability to justify all the manipulations, though in practice this can be done by standard limiting arguments.)

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Searching for singularities in the Navier–Stokes equations

31 May, 2019 in expository, math.AP, math.MP, paper | Tags: Navier-Stokes equations | by Terence Tao | 31 comments

I was recently asked to contribute a short comment to Nature Reviews Physics, as part of a series of articles on fluid dynamics on the occasion of the 200th anniversary (this August) of the birthday of George Stokes. My contribution is now online as “Searching for singularities in the Navier–Stokes equations“, where I discuss the global regularity problem for Navier-Stokes and my thoughts on how one could try to construct a solution that blows up in finite time via an approximately discretely self-similar “fluid computer”. (The rest of the series does not currently seem to be available online, but I expect they will become so shortly.)

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Category Archive

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Quantitative bounds for critically bounded solutions to the Navier-Stokes equations

Symmetric functions in a fractional number of variables, and the multilinear Kakeya conjecture

Searching for singularities in the Navier–Stokes equations

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