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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).

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.

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.

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:

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.

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 ${c>0}$.
• (ProdiSerrinLadyzhenskaya 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):

1. 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.

2. 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 ${I_n \times {\bf R}^3}$).

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 ${R}$.

4. 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.
5. 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 ${n}$.
6. 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:

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.

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

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.)

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.)

I was pleased to learn this week that the 2019 Abel Prize was awarded to Karen Uhlenbeck. Uhlenbeck laid much of the foundations of modern geometric PDE. One of the few papers I have in this area is in fact a joint paper with Gang Tian extending a famous singularity removal theorem of Uhlenbeck for four-dimensional Yang-Mills connections to higher dimensions. In both these papers, it is crucial to be able to construct “Coulomb gauges” for various connections, and there is a clever trick of Uhlenbeck for doing so, introduced in another important paper of hers, which is absolutely critical in my own paper with Tian. Nowadays it would be considered a standard technique, but it was definitely not so at the time that Uhlenbeck introduced it.

Suppose one has a smooth connection ${A}$ on a (closed) unit ball ${B(0,1)}$ in ${{\bf R}^n}$ for some ${n \geq 1}$, taking values in some Lie algebra ${{\mathfrak g}}$ associated to a compact Lie group ${G}$. This connection then has a curvature ${F(A)}$, defined in coordinates by the usual formula

$\displaystyle F(A)_{\alpha \beta} = \partial_\alpha A_\beta - \partial_\beta A_\alpha + [A_\alpha,A_\beta]. \ \ \ \ \ (1)$

It is natural to place the curvature in a scale-invariant space such as ${L^{n/2}(B(0,1))}$, and then the natural space for the connection would be the Sobolev space ${W^{n/2,1}(B(0,1))}$. It is easy to see from (1) and Sobolev embedding that if ${A}$ is bounded in ${W^{n/2,1}(B(0,1))}$, then ${F(A)}$ will be bounded in ${L^{n/2}(B(0,1))}$. One can then ask the converse question: if ${F(A)}$ is bounded in ${L^{n/2}(B(0,1))}$, is ${A}$ bounded in ${W^{n/2,1}(B(0,1))}$? This can be viewed as asking whether the curvature equation (1) enjoys “elliptic regularity”.

There is a basic obstruction provided by gauge invariance. For any smooth gauge ${U: B(0,1) \rightarrow G}$ taking values in the Lie group, one can gauge transform ${A}$ to

$\displaystyle A^U_\alpha := U^{-1} \partial_\alpha U + U^{-1} A_\alpha U$

and then a brief calculation shows that the curvature is conjugated to

$\displaystyle F(A^U)_{\alpha \beta} = U^{-1} F_{\alpha \beta} U.$

This gauge symmetry does not affect the ${L^{n/2}(B(0,1))}$ norm of the curvature tensor ${F(A)}$, but can make the connection ${A}$ extremely large in ${W^{n/2,1}(B(0,1))}$, since there is no control on how wildly ${U}$ can oscillate in space.

However, one can hope to overcome this problem by gauge fixing: perhaps if ${F(A)}$ is bounded in ${L^{n/2}(B(0,1))}$, then one can make ${A}$ bounded in ${W^{n/2,1}(B(0,1))}$ after applying a gauge transformation. The basic and useful result of Uhlenbeck is that this can be done if the ${L^{n/2}}$ norm of ${F(A)}$ is sufficiently small (and then the conclusion is that ${A}$ is small in ${W^{n/2,1}}$). (For large connections there is a serious issue related to the Gribov ambiguity.) In my (much) later paper with Tian, we adapted this argument, replacing Lebesgue spaces by Morrey space counterparts. (This result was also independently obtained at about the same time by Meyer and Riviére.)

To make the problem elliptic, one can try to impose the Coulomb gauge condition

$\displaystyle \partial^\alpha A_\alpha = 0 \ \ \ \ \ (2)$

(also known as the Lorenz gauge or Hodge gauge in various papers), together with a natural boundary condition on ${\partial B(0,1)}$ that will not be discussed further here. This turns (1), (2) into a divergence-curl system that is elliptic at the linear level at least. Indeed if one takes the divergence of (1) using (2) one sees that

$\displaystyle \partial^\alpha F(A)_{\alpha \beta} = \Delta A_\beta + \partial^\alpha [A_\alpha,A_\beta] \ \ \ \ \ (3)$

and if one could somehow ignore the nonlinear term ${\partial^\alpha [A_\alpha,A_\beta]}$ then we would get the required regularity on ${A}$ by standard elliptic regularity estimates.

The problem is then how to handle the nonlinear term. If we already knew that ${A}$ was small in the right norm ${W^{n/2,1}(B(0,1))}$ then one can use Sobolev embedding, Hölder’s inequality, and elliptic regularity to show that the second term in (3) is small compared to the first term, and so one could then hope to eliminate it by perturbative analysis. However, proving that ${A}$ is small in this norm is exactly what we are trying to prove! So this approach seems circular.

Uhlenbeck’s clever way out of this circularity is a textbook example of what is now known as a “continuity” argument. Instead of trying to work just with the original connection ${A}$, one works with the rescaled connections ${A^{(t)}_\alpha(x) := t A_\alpha(tx)}$ for ${0 \leq t \leq 1}$, with associated rescaled curvatures ${F(A^{(t)})_\alpha = t^2 F(A)_{\alpha \beta}(tx)}$. If the original curvature ${F(A)}$ is small in ${L^{n/2}}$ norm (e.g. bounded by some small ${\varepsilon>0}$), then so are all the rescaled curvatures ${F(A^{(t)})}$. We want to obtain a Coulomb gauge at time ${t=1}$; this is difficult to do directly, but it is trivial to obtain a Coulomb gauge at time ${t=0}$, because the connection vanishes at this time. On the other hand, once one has successfully obtained a Coulomb gauge at some time ${t \in [0,1]}$ with ${A^{(t)}}$ small in the natural norm ${W^{n/2,1}}$ (say bounded by ${C \varepsilon}$ for some constant ${C}$ which is large in absolute terms, but not so large compared with say ${1/\varepsilon}$), the perturbative argument mentioned earlier (combined with the qualitative hypothesis that ${A}$ is smooth) actually works to show that a Coulomb gauge can also be constructed and be small for all sufficiently close nearby times ${t' \in [0,1]}$ to ${t}$; furthermore, the perturbative analysis actually shows that the nearby gauges enjoy a slightly better bound on the ${W^{n/2,1}}$ norm, say ${C\varepsilon/2}$ rather than ${C\varepsilon}$. As a consequence of this, the set of times ${t}$ for which one has a good Coulomb gauge obeying the claimed estimates is both open and closed in ${[0,1]}$, and also contains ${t=0}$. Since the unit interval ${[0,1]}$ is connected, it must then also contain ${t=1}$. This concludes the proof.

One of the lessons I drew from this example is to not be deterred (especially in PDE) by an argument seeming to be circular; if the argument is still sufficiently “nontrivial” in nature, it can often be modified into a usefully non-circular argument that achieves what one wants (possibly under an additional qualitative hypothesis, such as a continuity or smoothness hypothesis).

I have just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows“, submitted to Pure and Applied Functional Analysis. This paper continues my attempts to establish “universality” properties of the Euler equations on Riemannian manifolds ${(M,g)}$, as I conjecture that the freedom to set the metric ${g}$ ought to allow one to “program” such Euler flows to exhibit a wide range of behaviour, and in particular to achieve finite time blowup (if the dimension is sufficiently large, at least).

In coordinates, the Euler equations read

$\displaystyle \partial_t u^k + u^j \nabla_j u^k = - \nabla^k p \ \ \ \ \ (1)$

$\displaystyle \nabla_k u^k = 0$

where ${p: [0,T] \rightarrow C^\infty(M)}$ is the pressure field and ${u: [0,T] \rightarrow \Gamma(TM)}$ is the velocity field, and ${\nabla}$ denotes the Levi-Civita connection with the usual Penrose abstract index notation conventions; we restrict attention here to the case where ${u,p}$ are smooth and ${M}$ is compact, smooth, orientable, connected, and without boundary. Let’s call ${u}$ an Euler flow on ${M}$ (for the time interval ${[0,T]}$) if it solves the above system of equations for some pressure ${p}$, and an incompressible flow if it just obeys the divergence-free relation ${\nabla_k u^k=0}$. Thus every Euler flow is an incompressible flow, but the converse is certainly not true; for instance the various conservation laws of the Euler equation, such as conservation of energy, will already block most incompressible flows from being an Euler flow, or even being approximated in a reasonably strong topology by such Euler flows.

However, one can ask if an incompressible flow can be extended to an Euler flow by adding some additional dimensions to ${M}$. In my paper, I formalise this by considering warped products ${\tilde M}$ of ${M}$ which (as a smooth manifold) are products ${\tilde M = M \times ({\bf R}/{\bf Z})^m}$ of ${M}$ with a torus, with a metric ${\tilde g}$ given by

$\displaystyle d \tilde g^2 = g_{ij}(x) dx^i dx^j + \sum_{s=1}^m \tilde g_{ss}(x) (d\theta^s)^2$

for ${(x,\theta) \in \tilde M}$, where ${\theta^1,\dots,\theta^m}$ are the coordinates of the torus ${({\bf R}/{\bf Z})^m}$, and ${\tilde g_{ss}: M \rightarrow {\bf R}^+}$ are smooth positive coefficients for ${s=1,\dots,m}$; in order to preserve the incompressibility condition, we also require the volume preservation property

$\displaystyle \prod_{s=1}^m \tilde g_{ss}(x) = 1 \ \ \ \ \ (2)$

though in practice we can quickly dispose of this condition by adding one further “dummy” dimension to the torus ${({\bf R}/{\bf Z})^m}$. We say that an incompressible flow ${u}$ is extendible to an Euler flow if there exists a warped product ${\tilde M}$ extending ${M}$, and an Euler flow ${\tilde u}$ on ${\tilde M}$ of the form

$\displaystyle \tilde u(t,(x,\theta)) = u^i(t,x) \frac{d}{dx^i} + \sum_{s=1}^m \tilde u^s(t,x) \frac{d}{d\theta^s}$

for some “swirl” fields ${\tilde u^s: [0,T] \times M \rightarrow {\bf R}}$. The situation here is motivated by the familiar situation of studying axisymmetric Euler flows ${\tilde u}$ on ${{\bf R}^3}$, which in cylindrical coordinates take the form

$\displaystyle \tilde u(t,(r,z,\theta)) = u^r(t,r,z) \frac{d}{dr} + u^z(t,r,z) \frac{d}{dz} + \tilde u^\theta(t,r,z) \frac{d}{d\theta}.$

The base component

$\displaystyle u^r(t,r,z) \frac{d}{dr} + u^z(t,r,z) \frac{d}{dz}$

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

On a fixed ${d}$-dimensional manifold ${(M,g)}$, let ${{\mathcal F}}$ denote the space of incompressible flows ${u: [0,T] \rightarrow \Gamma(TM)}$, equipped with the smooth topology (in spacetime), and let ${{\mathcal E} \subset {\mathcal F}}$ denote the space of such flows that are extendible to Euler flows. Our main theorem is

Theorem 1

• (i) (Generic inextendibility) Assume ${d \geq 3}$. Then ${{\mathcal E}}$ is of the first category in ${{\mathcal F}}$ (the countable union of nowhere dense sets in ${{\mathcal F}}$).
• (ii) (Non-rigidity) Assume ${M = ({\bf R}/{\bf Z})^d}$ (with an arbitrary metric ${g}$). Then ${{\mathcal E}}$ is somewhere dense in ${{\mathcal F}}$ (that is, the closure of ${{\mathcal E}}$ has non-empty interior).

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

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

The proof of part (i) of the theorem basically proceeds by a dimension counting argument (similar to that in the proof of Proposition 9 of these recent lecture notes of mine). Heuristically, the point is that an arbitrary incompressible flow ${u}$ is essentially determined by ${d-1}$ independent functions of space and time, whereas the warping factors ${\tilde g_{ss}}$ are functions of space only, the pressure field is one function of space and time, and the swirl fields ${u^s}$ are technically functions of both space and time, but have the same number of degrees of freedom as a function just of space, because they solve an evolution equation. When ${d>2}$, this means that there are fewer unknown functions of space and time than prescribed functions of space and time, which is the source of the generic inextendibility. This simple argument breaks down when ${d=2}$, but we do not know whether the claim is actually false in this case.

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

We consider the incompressible Euler equations on the (Eulerian) torus ${\mathbf{T}_E := ({\bf R}/{\bf Z})^d}$, which we write in divergence form as

$\displaystyle \partial_t u^i + \partial_j(u^j u^i) = - \eta^{ij} \partial_j p \ \ \ \ \ (1)$

$\displaystyle \partial_i u^i = 0, \ \ \ \ \ (2)$

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

As noted previously, the kinetic energy

$\displaystyle \frac{1}{2} \int_{\mathbf{T}_E} |u(t,x)|^2\ dx = \frac{1}{2} \int_{\mathbf{T}_E} \eta_{ij} u^i(t,x) u^j(t,x)\ dx$

is formally conserved by the flow, where ${\eta_{ij}}$ is the Euclidean metric. Indeed, if one assumes that ${u,p}$ are continuously differentiable in both space and time on ${[0,T] \times \mathbf{T}}$, then one can multiply the equation (1) by ${u^l}$ and contract against ${\eta_{il}}$ to obtain

$\displaystyle \eta_{il} u^l \partial_t u^i + \eta_{il} u^l \partial_j (u^j u^i) = - \eta_{il} u^l \eta^{ij} \partial_j p = 0$

which rearranges using (2) and the product rule to

$\displaystyle \partial_t (\frac{1}{2} \eta_{ij} u^i u^j) + \partial_j( \frac{1}{2} \eta_{il} u^i u^j u^l ) + \partial_j (u^j p)$

and then if one integrates this identity on ${[0,T] \times \mathbf{T}_E}$ and uses Stokes’ theorem, one obtains the required energy conservation law

$\displaystyle \frac{1}{2} \int_{\mathbf{T}_E} \eta_{ij} u^i(T,x) u^j(T,x)\ dx = \frac{1}{2} \int_{\mathbf{T}_E} \eta_{ij} u^i(0,x) u^j(0,x)\ dx. \ \ \ \ \ (3)$

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

But what is the precise level of regularity needed in order to for this anomalous energy dissipation to occur? To make this question precise, we need a quantitative notion of regularity. One such measure is given by the Hölder space ${C^{0,\alpha}(\mathbf{T}_E \rightarrow {\bf R})}$ for ${0 < \alpha < 1}$, defined as the space of continuous functions ${f: \mathbf{T}_E \rightarrow {\bf R}}$ whose norm

$\displaystyle \| f \|_{C^{0,\alpha}(\mathbf{T}_E \rightarrow {\bf R})} := \sup_{x \in \mathbf{T}_E} |f(x)| + \sup_{x,y \in \mathbf{T}_E: x \neq y} \frac{|f(x)-f(y)|}{|x-y|^\alpha}$

is finite. The space ${C^{0,\alpha}}$ lies between the space ${C^0}$ of continuous functions and the space ${C^1}$ of continuously differentiable functions, and informally describes a space of functions that is “${\alpha}$ times differentiable” in some sense. The above derivation of the energy conservation law involved the integral

$\displaystyle \int_{\mathbf{T}_E} \eta_{ik} u^k \partial_j (u^j u^i)\ dx$

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

$\displaystyle \int_{\mathbf{T}} \nabla^{1/3} u \nabla^{1/3} u \nabla^{1/3} u\ dx$

which suggests that the integral can be made sense of for ${u \in C^0_t C^{0,\alpha}_x}$ once ${\alpha > 1/3}$. More precisely, one can make

Conjecture 1 (Onsager’s conjecture) Let ${0 < \alpha < 1}$ and ${d \geq 2}$, and let ${0 < T < \infty}$.
• (i) If ${\alpha > 1/3}$, then any weak solution ${u \in C^0_t C^{0,\alpha}([0,T] \times \mathbf{T} \rightarrow {\bf R})}$ to the Euler equations (in the Leray form ${\partial_t u + \partial_j {\mathbb P} (u^j u) = u_0(x) \delta_0(t)}$) obeys the energy conservation law (3).
• (ii) If ${\alpha \leq 1/3}$, then there exist weak solutions ${u \in C^0_t C^{0,\alpha}([0,T] \times \mathbf{T} \rightarrow {\bf R})}$ to the Euler equations (in Leray form) which do not obey energy conservation.

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

The positive part (i) of Onsager conjecture was established by Constantin, E, and Titi, building upon earlier partial results by Eyink; the proof is a relatively straightforward application of Littlewood-Paley theory, and they were also able to work in larger function spaces than ${C^0_t C^{0,\alpha}_x}$ (using ${L^3_x}$-based Besov spaces instead of Hölder spaces, see Exercise 3 below). The negative part (ii) is harder. Discontinuous weak solutions to the Euler equations that did not conserve energy were first constructed by Sheffer, with an alternate construction later given by Shnirelman. De Lellis and Szekelyhidi noticed the resemblance of this problem to that of the Nash-Kuiper theorem in the isometric embedding problem, and began adapting the convex integration technique used in that theorem to construct weak solutions of the Euler equations. This began a long series of papers in which increasingly regular weak solutions that failed to conserve energy were constructed, culminating in a recent paper of Isett establishing part (ii) of the Onsager conjecture in the non-endpoint case ${\alpha < 1/3}$ in three and higher dimensions ${d \geq 3}$; the endpoint ${\alpha = 1/3}$ remains open. (In two dimensions it may be the case that the positive results extend to a larger range than Onsager’s conjecture predicts; see this paper of Cheskidov, Lopes Filho, Nussenzveig Lopes, and Shvydkoy for more discussion.) Further work continues into several variations of the Onsager conjecture, in which one looks at other differential equations, other function spaces, or other criteria for bad behavior than breakdown of energy conservation. See this recent survey of de Lellis and Szekelyhidi for more discussion.

In these notes we will first establish (i), then discuss the convex integration method in the original context of the Nash-Kuiper embedding theorem. Before tackling the Onsager conjecture (ii) directly, we discuss a related construction of high-dimensional weak solutions in the Sobolev space ${L^2_t H^s_x}$ for ${s}$ close to ${1/2}$, which is slightly easier to establish, though still rather intricate. Finally, we discuss the modifications of that construction needed to establish (ii), though we shall stop short of a full proof of that part of the conjecture.

We thank Phil Isett for some comments and corrections.