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I’ve just posted to the arXiv my paper “Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation“. This paper is loosely in the spirit of other recent papers of mine in which I explore how close one can get to supercritical PDE of physical interest (such as the Euler and Navier-Stokes equations), while still being able to rigorously demonstrate finite time blowup for at least some choices of initial data. Here, the PDE we are trying to get close to is the incompressible inviscid Euler equations

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

$\displaystyle \nabla \cdot u = 0$

in three spatial dimensions, where ${u}$ is the velocity vector field and ${p}$ is the pressure field. In vorticity form, and viewing the vorticity ${\omega}$ as a ${2}$-form (rather than a vector), we can rewrite this system using the language of differential geometry as

$\displaystyle \partial_t \omega + {\mathcal L}_u \omega = 0$

$\displaystyle u = \delta \tilde \eta^{-1} \Delta^{-1} \omega$

where ${{\mathcal L}_u}$ is the Lie derivative along ${u}$, ${\delta}$ is the codifferential (the adjoint of the differential ${d}$, or equivalently the negative of the divergence operator) that sends ${k+1}$-vector fields to ${k}$-vector fields, ${\Delta}$ is the Hodge Laplacian, and ${\tilde \eta}$ is the identification of ${k}$-vector fields with ${k}$-forms induced by the Euclidean metric ${\tilde \eta}$. The equation${u = \delta \tilde \eta^{-1} \Delta^{-1} \omega}$ can be viewed as the Biot-Savart law recovering velocity from vorticity, expressed in the language of differential geometry.

One can then generalise this system by replacing the operator ${\tilde \eta^{-1} \Delta^{-1}}$ by a more general operator ${A}$ from ${2}$-forms to ${2}$-vector fields, giving rise to what I call the generalised Euler equations

$\displaystyle \partial_t \omega + {\mathcal L}_u \omega = 0$

$\displaystyle u = \delta A \omega.$

For example, the surface quasi-geostrophic (SQG) equations can be written in this form, as discussed in this previous post. One can view ${A \omega}$ (up to Hodge duality) as a vector potential for the velocity ${u}$, so it is natural to refer to ${A}$ as a vector potential operator.

The generalised Euler equations carry much of the same geometric structure as the true Euler equations. For instance, the transport equation ${\partial_t \omega + {\mathcal L}_u \omega = 0}$ is equivalent to the Kelvin circulation theorem, which in three dimensions also implies the transport of vortex streamlines and the conservation of helicity. If ${A}$ is self-adjoint and positive definite, then the famous Euler-Poincaré interpretation of the true Euler equations as geodesic flow on an infinite dimensional Riemannian manifold of volume preserving diffeomorphisms (as discussed in this previous post) extends to the generalised Euler equations (with the operator ${A}$ determining the new Riemannian metric to place on this manifold). In particular, the generalised Euler equations have a Lagrangian formulation, and so by Noether’s theorem we expect any continuous symmetry of the Lagrangian to lead to conserved quantities. Indeed, we have a conserved Hamiltonian ${\frac{1}{2} \int \langle \omega, A \omega \rangle}$, and any spatial symmetry of ${A}$ leads to a conserved impulse (e.g. translation invariance leads to a conserved momentum, and rotation invariance leads to a conserved angular momentum). If ${A}$ behaves like a pseudodifferential operator of order ${-2}$ (as is the case with the true vector potential operator ${\tilde \eta^{-1} \Delta^{-1}}$), then it turns out that one can use energy methods to recover the same sort of classical local existence theory as for the true Euler equations (up to and including the famous Beale-Kato-Majda criterion for blowup).

The true Euler equations are suspected of admitting smooth localised solutions which blow up in finite time; there is now substantial numerical evidence for this blowup, but it has not been proven rigorously. The main purpose of this paper is to show that such finite time blowup can at least be established for certain generalised Euler equations that are somewhat close to the true Euler equations. This is similar in spirit to my previous paper on finite time blowup on averaged Navier-Stokes equations, with the main new feature here being that the modified equation continues to have a Lagrangian structure and a vorticity formulation, which was not the case with the averaged Navier-Stokes equation. On the other hand, the arguments here are not able to handle the presence of viscosity (basically because they rely crucially on the Kelvin circulation theorem, which is not available in the viscous case).

In fact, three different blowup constructions are presented (for three different choices of vector potential operator ${A}$). The first is a variant of one discussed previously on this blog, in which a “neck pinch” singularity for a vortex tube is created by using a non-self-adjoint vector potential operator, in which the velocity at the neck of the vortex tube is determined by the circulation of the vorticity somewhat further away from that neck, which when combined with conservation of circulation is enough to guarantee finite time blowup. This is a relatively easy construction of finite time blowup, and has the advantage of being rather stable (any initial data flowing through a narrow tube with a large positive circulation will blow up in finite time). On the other hand, it is not so surprising in the non-self-adjoint case that finite blowup can occur, as there is no conserved energy.

The second blowup construction is based on a connection between the two-dimensional SQG equation and the three-dimensional generalised Euler equations, discussed in this previous post. Namely, any solution to the former can be lifted to a “two and a half-dimensional” solution to the latter, in which the velocity and vorticity are translation-invariant in the vertical direction (but the velocity is still allowed to contain vertical components, so the flow is not completely horizontal). The same embedding also works to lift solutions to generalised SQG equations in two dimensions to solutions to generalised Euler equations in three dimensions. Conveniently, even if the vector potential operator for the generalised SQG equation fails to be self-adjoint, one can ensure that the three-dimensional vector potential operator is self-adjoint. Using this trick, together with a two-dimensional version of the first blowup construction, one can then construct a generalised Euler equation in three dimensions with a vector potential that is both self-adjoint and positive definite, and still admits solutions that blow up in finite time, though now the blowup is now a vortex sheet creasing at on a line, rather than a vortex tube pinching at a point.

This eliminates the main defect of the first blowup construction, but introduces two others. Firstly, the blowup is less stable, as it relies crucially on the initial data being translation-invariant in the vertical direction. Secondly, the solution is not spatially localised in the vertical direction (though it can be viewed as a compactly supported solution on the manifold ${{\bf R}^2 \times {\bf R}/{\bf Z}}$, rather than ${{\bf R}^3}$). The third and final blowup construction of the paper addresses the final defect, by replacing vertical translation symmetry with axial rotation symmetry around the vertical axis (basically, replacing Cartesian coordinates with cylindrical coordinates). It turns out that there is a more complicated way to embed two-dimensional generalised SQG equations into three-dimensional generalised Euler equations in which the solutions to the latter are now axially symmetric (but are allowed to “swirl” in the sense that the velocity field can have a non-zero angular component), while still keeping the vector potential operator self-adjoint and positive definite; the blowup is now that of a vortex ring creasing on a circle.

As with the previous papers in this series, these blowup constructions do not directly imply finite time blowup for the true Euler equations, but they do at least provide a barrier to establishing global regularity for these latter equations, in that one is forced to use some property of the true Euler equations that are not shared by these generalisations. They also suggest some possible blowup mechanisms for the true Euler equations (although unfortunately these mechanisms do not seem compatible with the addition of viscosity, so they do not seem to suggest a viable Navier-Stokes blowup mechanism).

Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise.

A (real) manifold ${M}$ can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space such as a Euclidean space ${{\bf R}^d}$. On the other hand, one can define the manifold intrinsically, as a topological space equipped with an atlas of coordinate charts. The fundamental embedding theorems show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds (up to isomorphism in various categories). For instance, we have the following (special case of) the Whitney embedding theorem:

Theorem 1 (Whitney embedding theorem) Let ${M}$ be a compact manifold. Then there exists an embedding ${u: M \rightarrow {\bf R}^d}$ from ${M}$ to a Euclidean space ${{\bf R}^d}$.

In fact, if ${M}$ is ${n}$-dimensional, one can take ${d}$ to equal ${2n}$, which is often best possible (easy examples include the circle ${{\bf R}/{\bf Z}}$ which embeds into ${{\bf R}^2}$ but not ${{\bf R}^1}$, or the Klein bottle that embeds into ${{\bf R}^4}$ but not ${{\bf R}^3}$). One can also relax the compactness hypothesis on ${M}$ to second countability, but we will not pursue this extension here. We give a “cheap” proof of this theorem below the fold which allows one to take ${d}$ equal to ${2n+1}$.

A significant strengthening of the Whitney embedding theorem is (a special case of) the Nash embedding theorem:

Theorem 2 (Nash embedding theorem) Let ${(M,g)}$ be a compact Riemannian manifold. Then there exists a isometric embedding ${u: M \rightarrow {\bf R}^d}$ from ${M}$ to a Euclidean space ${{\bf R}^d}$.

In order to obtain the isometric embedding, the dimension ${d}$ has to be a bit larger than what is needed for the Whitney embedding theorem; in this article of Gunther the bound

$\displaystyle d = \max( n(n+5)/2, n(n+3)/2 + 5) \ \ \ \ \ (1)$

is attained, which I believe is still the record for large ${n}$. (In the converse direction, one cannot do better than ${d = \frac{n(n+1)}{2}}$, basically because this is the number of degrees of freedom in the Riemannian metric ${g}$.) Nash’s original proof of theorem used what is now known as Nash-Moser inverse function theorem, but a subsequent simplification of Gunther allowed one to proceed using just the ordinary inverse function theorem (in Banach spaces).

I recently had the need to invoke the Nash embedding theorem to establish a blowup result for a nonlinear wave equation, which motivated me to go through the proof of the theorem more carefully. Below the fold I give a proof of the theorem that does not attempt to give an optimal value of ${d}$, but which hopefully isolates the main ideas of the argument (as simplified by Gunther). One advantage of not optimising in ${d}$ is that it allows one to freely exploit the very useful tool of pairing together two maps ${u_1: M \rightarrow {\bf R}^{d_1}}$, ${u_2: M \rightarrow {\bf R}^{d_2}}$ to form a combined map ${(u_1,u_2): M \rightarrow {\bf R}^{d_1+d_2}}$ that can be closer to an embedding or an isometric embedding than the original maps ${u_1,u_2}$. This lets one perform a “divide and conquer” strategy in which one first starts with the simpler problem of constructing some “partial” embeddings of ${M}$ and then pairs them together to form a “better” embedding.

In preparing these notes, I found the articles of Deane Yang and of Siyuan Lu to be helpful.

I’ve just uploaded to the arXiv my paper Finite time blowup for high dimensional nonlinear wave systems with bounded smooth nonlinearity, submitted to Comm. PDE. This paper is in the same spirit as (though not directly related to) my previous paper on finite time blowup of supercritical NLW systems, and was inspired by a question posed to me some time ago by Jeffrey Rauch. Here, instead of looking at supercritical equations, we look at an extremely subcritical equation, namely a system of the form

$\displaystyle \Box u = f(u) \ \ \ \ \ (1)$

where ${u: {\bf R}^{1+d} \rightarrow {\bf R}^m}$ is the unknown field, and ${f: {\bf R}^m \rightarrow {\bf R}^m}$ is the nonlinearity, which we assume to have all derivatives bounded. A typical example of such an equation is the higher-dimensional sine-Gordon equation

$\displaystyle \Box u = \sin u$

for a scalar field ${u: {\bf R}^{1+d} \rightarrow {\bf R}}$. Here ${\Box = -\partial_t^2 + \Delta}$ is the d’Alembertian operator. We restrict attention here to classical (i.e. smooth) solutions to (1).

We do not assume any Hamiltonian structure, so we do not require ${f}$ to be a gradient ${f = \nabla F}$ of a potential ${F: {\bf R}^m \rightarrow {\bf R}}$. But even without such Hamiltonian structure, the equation (1) is very well behaved, with many a priori bounds available. For instance, if the initial position ${u_0(x) = u(0,x)}$ and initial velocity ${u_1(x) = \partial_t u(0,x)}$ are smooth and compactly supported, then from finite speed of propagation ${u(t)}$ has uniformly bounded compact support for all ${t}$ in a bounded interval. As the nonlinearity ${f}$ is bounded, this immediately places ${f(u)}$ in ${L^\infty_t L^2_x}$ in any bounded time interval, which by the energy inequality gives an a priori ${L^\infty_t H^1_x}$ bound on ${u}$ in this time interval. Next, from the chain rule we have

$\displaystyle \nabla f(u) = (\nabla_{{\bf R}^m} f)(u) \nabla u$

which (from the assumption that ${\nabla_{{\bf R}^m} f}$ is bounded) shows that ${f(u)}$ is in ${L^\infty_t H^1_x}$, which by the energy inequality again now gives an a priori ${L^\infty_t H^2_x}$ bound on ${u}$.

One might expect that one could keep iterating this and obtain a priori bounds on ${u}$ in arbitrarily smooth norms. In low dimensions such as ${d \leq 3}$, this is a fairly easy task, since the above estimates and Sobolev embedding already place one in ${L^\infty_t L^\infty_x}$, and the nonlinear map ${f}$ is easily verified to preserve the space ${L^\infty_t H^k_x \cap L^\infty_t L^\infty_x}$ for any natural number ${k}$, from which one obtains a priori bounds in any Sobolev space; from this and standard energy methods, one can then establish global regularity for this equation (that is to say, any smooth choice of initial data generates a global smooth solution). However, one starts running into trouble in higher dimensions, in which no ${L^\infty_x}$ bound is available. The main problem is that even a really nice nonlinearity such as ${u \mapsto \sin u}$ is unbounded in higher Sobolev norms. The estimates

$\displaystyle |\sin u| \leq |u|$

and

$\displaystyle |\nabla(\sin u)| \leq |\nabla u|$

ensure that the map ${u \mapsto \sin u}$ is bounded in low regularity spaces like ${L^2_x}$ or ${H^1_x}$, but one already runs into trouble with the second derivative

$\displaystyle \nabla^2(\sin u) = (\cos u) \nabla^2 u - (\sin u) \nabla u \nabla u$

where there is a troublesome lower order term of size ${O( |\nabla u|^2 )}$ which becomes difficult to control in higher dimensions, preventing the map ${u \mapsto \sin u}$ to be bounded in ${H^2_x}$. Ultimately, the issue here is that when ${u}$ is not controlled in ${L^\infty}$, the function ${\sin u}$ can oscillate at a much higher frequency than ${u}$; for instance, if ${u}$ is the one-dimensional wave ${u = A \sin(kx)}$for some ${k > 0}$ and ${A>1}$, then ${u}$ oscillates at frequency ${k}$, but the function ${\sin(u)= \sin(A \sin(kx))}$ more or less oscillates at the larger frequency ${Ak}$.

In medium dimensions, it is possible to use dispersive estimates for the wave equation (such as the famous Strichartz estimates) to overcome these problems. This line of inquiry was pursued (albeit for slightly different classes of nonlinearity ${f}$ than those considered here) by Heinz-von Wahl, Pecher (in a series of papers), Brenner, and Brenner-von Wahl; to cut a long story short, one of the conclusions of these papers was that one had global regularity for equations such as (1) in dimensions ${d \leq 9}$. (I reprove this result using modern Strichartz estimate and Littlewood-Paley techniques in an appendix to my paper. The references given also allow for some growth in the nonlinearity ${f}$, but we will not detail the precise hypotheses used in these papers here.)

In my paper, I complement these positive results with an almost matching negative result:

Theorem 1 If ${d \geq 11}$ and ${m \geq 2}$, then there exists a nonlinearity ${f: {\bf R}^m \rightarrow {\bf R}^m}$ with all derivatives bounded, and a solution ${u}$ to (1) that is smooth at time zero, but develops a singularity in finite time.

The construction crucially relies on the ability to choose the nonlinearity ${f}$, and also needs some injectivity properties on the solution ${u: {\bf R}^{1+d} \rightarrow {\bf R}^m}$ (after making a symmetry reduction using an assumption of spherical symmetry to view ${u}$ as a function of ${1+1}$ variables rather than ${1+d}$) which restricts our counterexample to the ${m \geq 2}$ case. Thus the model case of the higher-dimensional sine-Gordon equation ${\Box u =\sin u}$ is not covered by our arguments. Nevertheless (as with previous finite-time blowup results discussed on this blog), one can view this result as a barrier to trying to prove regularity for equations such as ${\Box u = \sin u}$ in eleven and higher dimensions, as any such argument must somehow use a property of that equation that is not applicable to the more general system (1).

Let us first give some back-of-the-envelope calculations suggesting why there could be finite time blowup in eleven and higher dimensions. For sake of this discussion let us restrict attention to the sine-Gordon equation ${\Box u = \sin u}$. The blowup ansatz we will use is as follows: for each frequency ${N_j}$ in a sequence ${1 < N_1 < N_2 < N_3 < \dots}$ of large quantities going to infinity, there will be a spacetime “cube” ${Q_j = \{ (t,x): t \sim \frac{1}{N_j}; x = O(\frac{1}{N_j})\}}$ on which the solution ${u}$ oscillates with “amplitude” ${N_j^\alpha}$ and “frequency” ${N_j}$, where ${\alpha>0}$ is an exponent to be chosen later; this ansatz is of course compatible with the uncertainty principle. Since ${N_j^\alpha \rightarrow \infty}$ as ${j \rightarrow \infty}$, this will create a singularity at the spacetime origin ${(0,0)}$. To make this ansatz plausible, we wish to make the oscillation of ${u}$ on ${Q_j}$ driven primarily by the forcing term ${\sin u}$ at ${Q_{j-1}}$. Thus, by Duhamel’s formula, we expect a relation roughly of the form

$\displaystyle u(t,x) \approx \int \frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}} \sin(1_{Q_{j-1}} u(s)) (x)\ ds$

on ${Q_j}$, where ${\frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}}}$ is the usual free wave propagator, and ${1_{Q_{j-1}}}$ is the indicator function of ${Q_{j-1}}$.

On ${Q_{j-1}}$, ${u}$ oscillates with amplitude ${N_{j-1}^\alpha}$ and frequency ${N_{j-1}}$, we expect the derivative ${\nabla_{t,x} u}$ to be of size about ${N_{j-1}^{\alpha+1}}$, and so from the principle of stationary phase we expect ${\sin(u)}$ to oscillate at frequency about ${N_{j-1}^{\alpha+1}}$. Since the wave propagator ${\frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}}}$ preserves frequencies, and ${u}$ is supposed to be of frequency ${N_j}$ on ${Q_j}$ we are thus led to the requirement

$\displaystyle N_j \approx N_{j-1}^{\alpha+1}. \ \ \ \ \ (2)$

Next, when restricted to frequencies of order ${N_{j}}$, the propagator ${\frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}}}$ “behaves like” ${N_{j}^{\frac{d-3}{2}} (s-t)^{\frac{d-1}{2}} A_{s-t}}$, where ${A_{s-t}}$ is the spherical averaging operator

$\displaystyle A_{s-t} f(x) := \frac{1}{\omega_{d-1}} \int_{S^{d-1}} f(x + (s-t)\theta)\ d\theta$

where ${d\theta}$ is surface measure on the unit sphere ${S^{d-1}}$, and ${\omega_{d-1}}$ is the volume of that sphere. In our setting, ${s-t}$ is comparable to ${1/N_{j-1}}$, and so we have the informal approximation

$\displaystyle u(t,x) \approx N_j^{\frac{d-3}{2}} N_{j-1}^{-\frac{d-1}{2}} \int_{s \sim 1/N_{j-1}} A_{s-t} \sin(u(s))(x)\ ds$

on ${Q_j}$.

Since ${\sin(u(s))}$ is bounded, ${A_{s-t} \sin(u(s))}$ is bounded as well. This gives a (non-rigorous) upper bound

$\displaystyle u(t,x) \lessapprox N_j^{\frac{d-3}{2}} N_{j-1}^{-\frac{d-1}{2}} \frac{1}{N_{j-1}}$

which when combined with our ansatz that ${u}$ has ampitude about ${N_j^\alpha}$ on ${Q_j}$, gives the constraint

$\displaystyle N_j^\alpha \lessapprox N_j^{\frac{d-3}{2}} N_{j-1}^{-\frac{d-1}{2}} \frac{1}{N_{j-1}}$

which on applying (2) gives the further constraint

$\displaystyle \alpha(\alpha+1) \leq \frac{d-3}{2} (\alpha+1) - \frac{d-1}{2} - 1$

which can be rearranged as

$\displaystyle \left(\alpha - \frac{d-5}{4}\right)^2 \leq \frac{d^2-10d-7}{16}.$

It is now clear that the optimal choice of ${\alpha}$ is

$\displaystyle \alpha = \frac{d-5}{4},$

and this blowup ansatz is only self-consistent when

$\displaystyle \frac{d^2-10d-7}{16} \geq 0$

or equivalently if ${d \geq 11}$.

To turn this ansatz into an actual blowup example, we will construct ${u}$ as the sum of various functions ${u_j}$ that solve the wave equation with forcing term in ${Q_{j+1}}$, and which concentrate in ${Q_j}$ with the amplitude and frequency indicated by the above heuristic analysis. The remaining task is to show that ${\Box u}$ can be written in the form ${f(u)}$ for some ${f}$ with all derivatives bounded. For this one needs some injectivity properties of ${u}$ (after imposing spherical symmetry to impose a dimensional reduction on the domain of ${u}$ from ${d+1}$ dimensions to ${1+1}$). This requires one to construct some solutions to the free wave equation that have some unusual restrictions on the range (for instance, we will need a solution taking values in the plane ${{\bf R}^2}$ that avoid one quadrant of that plane). In order to do this we take advantage of the very explicit nature of the fundamental solution to the wave equation in odd dimensions (such as ${d=11}$), particularly under the assumption of spherical symmetry. Specifically, one can show that in odd dimension ${d}$, any spherically symmetric function ${u(t,x) = u(t,r)}$ of the form

$\displaystyle u(t,r) = \left(\frac{1}{r} \partial_r\right)^{\frac{d-1}{2}} (g(t+r) + g(t-r))$

for an arbitrary smooth function ${g: {\bf R} \rightarrow {\bf R}^m}$, will solve the free wave equation; this is ultimately due to iterating the “ladder operator” identity

$\displaystyle \left( \partial_{tt} + \partial_{rr} + \frac{d-1}{r} \partial_r \right) \frac{1}{r} \partial_r = \frac{1}{r} \partial_r \left( \partial_{tt} + \partial_{rr} + \frac{d-3}{r} \partial_r \right).$

This precise and relatively simple formula for ${u}$ allows one to create “bespoke” solutions ${u}$ that obey various unusual properties, without too much difficulty.

It is not clear to me what to conjecture for ${d=10}$. The blowup ansatz given above is a little inefficient, in that the frequency ${N_{j+1}}$ component of the solution is only generated from a portion of the ${N_j}$ component, namely the portion close to a certain light cone. In particular, the solution does not saturate the Strichartz estimates that are used to establish the positive results for ${d \leq 9}$, which helps explain the slight gap between the positive and negative results. It may be that a more complicated ansatz could work to give a negative result in ten dimensions; conversely, it is also possible that one could use more advanced estimates than the Strichartz estimate (that somehow capture the “thinness” of the fundamental solution, and not just its dispersive properties) to stretch the positive results to ten dimensions. Which side the ${d=10}$ case falls in all come down to some rather delicate numerology.

I’ve just uploaded to the arXiv my paper Finite time blowup for a supercritical defocusing nonlinear wave system, submitted to Analysis and PDE. This paper was inspired by a question asked of me by Sergiu Klainerman recently, regarding whether there were any analogues of my blowup example for Navier-Stokes type equations in the setting of nonlinear wave equations.

Recall that the defocusing nonlinear wave (NLW) equation reads

$\displaystyle \Box u = |u|^{p-1} u \ \ \ \ \ (1)$

where ${u: {\bf R}^{1+d} \rightarrow {\bf R}}$ is the unknown scalar field, ${\Box = -\partial_t^2 + \Delta}$ is the d’Alambertian operator, and ${p>1}$ is an exponent. We can generalise this equation to the defocusing nonlinear wave system

$\displaystyle \Box u = (\nabla F)(u) \ \ \ \ \ (2)$

where ${u: {\bf R}^{1+d} \rightarrow {\bf R}^m}$ is now a system of scalar fields, and ${F: {\bf R}^m \rightarrow {\bf R}}$ is a potential which is homogeneous of degree ${p+1}$ and strictly positive away from the origin; the scalar equation corresponds to the case where ${m=1}$ and ${F(u) = \frac{1}{p+1} |u|^{p+1}}$. We will be interested in smooth solutions ${u}$ to (2). It is only natural to restrict to the smooth category when the potential ${F}$ is also smooth; unfortunately, if one requires ${F}$ to be homogeneous of order ${p+1}$ all the way down to the origin, then ${F}$ cannot be smooth unless it is identically zero or ${p+1}$ is an odd integer. This is too restrictive for us, so we will only require that ${F}$ be homogeneous away from the origin (e.g. outside the unit ball). In any event it is the behaviour of ${F(u)}$ for large ${u}$ which will be decisive in understanding regularity or blowup for the equation (2).

Formally, solutions to the equation (2) enjoy a conserved energy

$\displaystyle E[u] = \int_{{\bf R}^d} \frac{1}{2} \|\partial_t u \|^2 + \frac{1}{2} \| \nabla_x u \|^2 + F(u)\ dx.$

Using this conserved energy, it is possible to establish global regularity for the Cauchy problem (2) in the energy-subcritical case when ${d \leq 2}$, or when ${d \geq 3}$ and ${p < 1+\frac{4}{d-2}}$. This means that for any smooth initial position ${u_0: {\bf R}^d \rightarrow {\bf R}^m}$ and initial velocity ${u_1: {\bf R}^d \rightarrow {\bf R}^m}$, there exists a (unique) smooth global solution ${u: {\bf R}^{1+d} \rightarrow {\bf R}^m}$ to the equation (2) with ${u(0,x) = u_0(x)}$ and ${\partial_t u(0,x) = u_1(x)}$. These classical global regularity results (essentially due to Jörgens) were famously extended to the energy-critical case when ${d \geq 3}$ and ${p = 1 + \frac{4}{d-2}}$ by Grillakis, Struwe, and Shatah-Struwe (though for various technical reasons, the global regularity component of these results was limited to the range ${3 \leq d \leq 7}$). A key tool used in the energy-critical theory is the Morawetz estimate

$\displaystyle \int_0^T \int_{{\bf R}^d} \frac{|u(t,x)|^{p+1}}{|x|}\ dx dt \lesssim E[u]$

which can be proven by manipulating the properties of the stress-energy tensor

$\displaystyle T_{\alpha \beta} = \langle \partial_\alpha u, \partial_\beta u \rangle - \frac{1}{2} \eta_{\alpha \beta} (\langle \partial^\gamma u, \partial_\gamma u \rangle + F(u))$

(with the usual summation conventions involving the Minkowski metric ${\eta_{\alpha \beta} dx^\alpha dx^\beta = -dt^2 + |dx|^2}$) and in particular exploiting the divergence-free nature of this tensor: ${\partial^\beta T_{\alpha \beta}}$ See for instance the text of Shatah-Struwe, or my own PDE book, for more details. The energy-critical regularity results have also been extended to slightly supercritical settings in which the potential grows by a logarithmic factor or so faster than the critical rate; see the results of myself and of Roy.

This leaves the question of global regularity for the energy supercritical case when ${d \geq 3}$ and ${p > 1+\frac{4}{d-2}}$. On the one hand, global smooth solutions are known for small data (if ${F}$ vanishes to sufficiently high order at the origin, see e.g. the work of Lindblad and Sogge), and global weak solutions for large data were constructed long ago by Segal. On the other hand, the solution map, if it exists, is known to be extremely unstable, particularly at high frequencies; see for instance this paper of Lebeau, this paper of Christ, Colliander, and myself, this paper of Brenner and Kumlin, or this paper of Ibrahim, Majdoub, and Masmoudi for various formulations of this instability. In the case of the focusing NLW ${-\partial_{tt} u + \Delta u = - |u|^{p-1} u}$, one can easily create solutions that blow up in finite time by ODE constructions, for instance one can take ${u(t,x) = c (1-t)^{-\frac{2}{p-1}}}$ with ${c = (\frac{2(p+1)}{(p-1)^2})^{\frac{1}{p-1}}}$, which blows up as ${t}$ approaches ${1}$. However the situation in the defocusing supercritical case is less clear. The strongest positive results are of Kenig-Merle and Killip-Visan, which show (under some additional technical hypotheses) that global regularity for such equations holds under the additional assumption that the critical Sobolev norm of the solution stays bounded. Roughly speaking, this shows that “Type II blowup” cannot occur for (2).

Our main result is that finite time blowup can in fact occur, at least for three-dimensional systems where the number ${m}$ of degrees of freedom is sufficiently large:

Theorem 1 Let ${d=3}$, ${p > 5}$, and ${m \geq 76}$. Then there exists a smooth potential ${F: {\bf R}^m \rightarrow {\bf R}}$, positive and homogeneous of degree ${p+1}$ away from the origin, and a solution to (2) with smooth initial data that develops a singularity in finite time.

The rather large lower bound of ${76}$ on ${m}$ here is primarily due to our use of the Nash embedding theorem (which is the first time I have actually had to use this theorem in an application!). It can certainly be lowered, but unfortunately our methods do not seem to be able to bring ${m}$ all the way down to ${1}$, so we do not directly exhibit finite time blowup for the scalar supercritical defocusing NLW. Nevertheless, this result presents a barrier to any attempt to prove global regularity for that equation, in that it must somehow use a property of the scalar equation which is not available for systems. It is likely that the methods can be adapted to higher dimensions than three, but we take advantage of some special structure to the equations in three dimensions (related to the strong Huygens principle) which does not seem to be available in higher dimensions.

The blowup will in fact be of discrete self-similar type in a backwards light cone, thus ${u}$ will obey a relation of the form

$\displaystyle u(e^S t, e^S x) = e^{-\frac{2}{p-1} S} u(t,x)$

for some fixed ${S>0}$ (the exponent ${-\frac{2}{p-1}}$ is mandated by dimensional analysis considerations). It would be natural to consider continuously self-similar solutions (in which the above relation holds for all ${S}$, not just one ${S}$). And rough self-similar solutions have been constructed in the literature by perturbative methods (see this paper of Planchon, or this paper of Ribaud and Youssfi). However, it turns out that continuously self-similar solutions to a defocusing equation have to obey an additional monotonicity formula which causes them to not exist in three spatial dimensions; this argument is given in my paper. So we have to work just with discretely self-similar solutions.

Because of the discrete self-similarity, the finite time blowup solution will be “locally Type II” in the sense that scale-invariant norms inside the backwards light cone stay bounded as one approaches the singularity. But it will not be “globally Type II” in that scale-invariant norms stay bounded outside the light cone as well; indeed energy will leak from the light cone at every scale. This is consistent with the results of Kenig-Merle and Killip-Visan which preclude “globally Type II” blowup solutions to these equations in many cases.

We now sketch the arguments used to prove this theorem. Usually when studying the NLW, we think of the potential ${F}$ (and the initial data ${u_0,u_1}$) as being given in advance, and then try to solve for ${u}$ as an unknown field. However, in this problem we have the freedom to select ${F}$. So we can look at this problem from a “backwards” direction: we first choose the field ${u}$, and then fit the potential ${F}$ (and the initial data) to match that field.

Now, one cannot write down a completely arbitrary field ${u}$ and hope to find a potential ${F}$ obeying (2), as there are some constraints coming from the homogeneity of ${F}$. Namely, from the Euler identity

$\displaystyle \langle u, (\nabla F)(u) \rangle = (p+1) F(u)$

we see that ${F(u)}$ can be recovered from (2) by the formula

$\displaystyle F(u) = \frac{1}{p+1} \langle u, \Box u \rangle \ \ \ \ \ (3)$

so the defocusing nature of ${F}$ imposes a constraint

$\displaystyle \langle u, \Box u \rangle > 0.$

Furthermore, taking a derivative of (3) we obtain another constraining equation

$\displaystyle \langle \partial_\alpha u, \Box u \rangle = \frac{1}{p+1} \partial_\alpha \langle u, \Box u \rangle$

that does not explicitly involve the potential ${F}$. Actually, one can write this equation in the more familiar form

$\displaystyle \partial^\beta T_{\alpha \beta} = 0$

where ${T_{\alpha \beta}}$ is the stress-energy tensor

$\displaystyle T_{\alpha \beta} = \langle \partial_\alpha u, \partial_\beta u \rangle - \frac{1}{2} \eta_{\alpha \beta} (\langle \partial^\gamma u, \partial_\gamma u \rangle + \frac{1}{p+1} \langle u, \Box u \rangle),$

now written in a manner that does not explicitly involve ${F}$.

With this reformulation, this suggests a strategy for locating ${u}$: first one selects a stress-energy tensor ${T_{\alpha \beta}}$ that is divergence-free and obeys suitable positive definiteness and self-similarity properties, and then locates a self-similar map ${u}$ from the backwards light cone to ${{\bf R}^m}$ that has that stress-energy tensor (one also needs the map ${u}$ (or more precisely the direction component ${u/\|u\|}$ of that map) injective up to the discrete self-similarity, in order to define ${F(u)}$ consistently). If the stress-energy tensor was replaced by the simpler “energy tensor”

$\displaystyle E_{\alpha \beta} = \langle \partial_\alpha u, \partial_\beta u \rangle$

then the question of constructing an (injective) map ${u}$ with the specified energy tensor is precisely the embedding problem that was famously solved by Nash (viewing ${E_{\alpha \beta}}$ as a Riemannian metric on the domain of ${u}$, which in this case is a backwards light cone quotiented by a discrete self-similarity to make it compact). It turns out that one can adapt the Nash embedding theorem to also work with the stress-energy tensor as well (as long as one also specifies the mass density ${M = \|u\|^2}$, and as long as a certain positive definiteness property, related to the positive semi-definiteness of Gram matrices, is obeyed). Here is where the dimension ${76}$ shows up:

Proposition 2 Let ${M}$ be a smooth compact Riemannian ${4}$-manifold, and let ${m \geq 76}$. Then ${M}$ smoothly isometrically embeds into the sphere ${S^{m-1}}$.

Proof: The Nash embedding theorem (in the form given in this ICM lecture of Gunther) shows that ${M}$ can be smoothly isometrically embedded into ${{\bf R}^{19}}$, and thus in ${[-R,R]^{19}}$ for some large ${R}$. Using an irrational slope, the interval ${[-R,R]}$ can be smoothly isometrically embedded into the ${2}$-torus ${\frac{1}{\sqrt{38}} (S^1 \times S^1)}$, and so ${[-R,R]^{19}}$ and hence ${M}$ can be smoothly embedded in ${\frac{1}{\sqrt{38}} (S^1)^{38}}$. But from Pythagoras’ theorem, ${\frac{1}{\sqrt{38}} (S^1)^{38}}$ can be identified with a subset of ${S^{m-1}}$ for any ${m \geq 76}$, and the claim follows. $\Box$

One can presumably improve upon the bound ${76}$ by being more efficient with the embeddings (e.g. by modifying the proof of Nash embedding to embed directly into a round sphere), but I did not try to optimise the bound here.

The remaining task is to construct the stress-energy tensor ${T_{\alpha \beta}}$. One can reduce to tensors that are invariant with respect to rotations around the spatial origin, but this still leaves a fair amount of degrees of freedom (it turns out that there are four fields that need to be specified, which are denoted ${M, E_{tt}, E_{tr}, E_{rr}}$ in my paper). However a small miracle occurs in three spatial dimensions, in that the divergence-free condition involves only two of the four degrees of freedom (or three out of four, depending on whether one considers a function that is even or odd in ${r}$ to only be half a degree of freedom). This is easiest to illustrate with the scalar NLW (1). Assuming spherical symmetry, this equation becomes

$\displaystyle - \partial_{tt} u + \partial_{rr} u + \frac{2}{r} \partial_r u = |u|^{p-1} u.$

Making the substitution ${\phi := ru}$, we can eliminate the lower order term ${\frac{2}{r} \partial_r}$ completely to obtain

$\displaystyle - \partial_{tt} \phi + \partial_{rr} \phi= \frac{1}{r^{p-1}} |\phi|^{p-1} \phi.$

(This can be compared with the situation in higher dimensions, in which an undesirable zeroth order term ${\frac{(d-1)(d-3)}{r^2} \phi}$ shows up.) In particular, if one introduces the null energy density

$\displaystyle e_+ := \frac{1}{2} |\partial_t \phi + \partial_r \phi|^2$

and the potential energy density

$\displaystyle V := \frac{|\phi|^{p+1}}{(p+1) r^{p-1}}$

then one can verify the equation

$\displaystyle (\partial_t - \partial_r) e_+ + (\partial_t + \partial_r) V = - \frac{p-1}{r} V$

which can be viewed as a transport equation for ${e_+}$ with forcing term depending on ${V}$ (or vice versa), and is thus quite easy to solve explicitly by choosing one of these fields and then solving for the other. As it turns out, once one is in the supercritical regime ${p>5}$, one can solve this equation while giving ${e_+}$ and ${V}$ the right homogeneity (they have to be homogeneous of order ${-\frac{4}{p-1}}$, which is greater than ${-1}$ in the supercritical case) and positivity properties, and from this it is possible to prescribe all the other fields one needs to satisfy the conclusions of the main theorem. (It turns out that ${e_+}$ and ${V}$ will be concentrated near the boundary of the light cone, so this is how the solution ${u}$ will concentrate also.)

I’ve been meaning to return to fluids for some time now, in order to build upon my construction two years ago of a solution to an averaged Navier-Stokes equation that exhibited finite time blowup. (I recently spoke on this work in the recent conference in Princeton in honour of Sergiu Klainerman; my slides for that talk are here.)

One of the biggest deficiencies with my previous result is the fact that the averaged Navier-Stokes equation does not enjoy any good equation for the vorticity ${\omega = \nabla \times u}$, in contrast to the true Navier-Stokes equations which, when written in vorticity-stream formulation, become

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

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

(Throughout this post we will be working in three spatial dimensions ${{\bf R}^3}$.) So one of my main near-term goals in this area is to exhibit an equation resembling Navier-Stokes as much as possible which enjoys a vorticity equation, and for which there is finite time blowup.

Heuristically, this task should be easier for the Euler equations (i.e. the zero viscosity case ${\nu=0}$ of Navier-Stokes) than the viscous Navier-Stokes equation, as one expects the viscosity to only make it easier for the solution to stay regular. Indeed, morally speaking, the assertion that finite time blowup solutions of Navier-Stokes exist should be roughly equivalent to the assertion that finite time blowup solutions of Euler exist which are “Type I” in the sense that all Navier-Stokes-critical and Navier-Stokes-subcritical norms of this solution go to infinity (which, as explained in the above slides, heuristically means that the effects of viscosity are negligible when compared against the nonlinear components of the equation). In vorticity-stream formulation, the Euler equations can be written as

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

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

As discussed in this previous blog post, a natural generalisation of this system of equations is the system

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (1)$

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

where ${T}$ is a linear operator on divergence-free vector fields that is “zeroth order” in some sense; ideally it should also be invertible, self-adjoint, and positive definite (in order to have a Hamiltonian that is comparable to the kinetic energy ${\frac{1}{2} \int_{{\bf R}^3} |u|^2}$). (In the previous blog post, it was observed that the surface quasi-geostrophic (SQG) equation could be embedded in a system of the form (1).) The system (1) has many features in common with the Euler equations; for instance vortex lines are transported by the velocity field ${u}$, and Kelvin’s circulation theorem is still valid.

So far, I have not been able to fully achieve this goal. However, I have the following partial result, stated somewhat informally:

Theorem 1 There is a “zeroth order” linear operator ${T}$ (which, unfortunately, is not invertible, self-adjoint, or positive definite) for which the system (1) exhibits smooth solutions that blowup in finite time.

The operator ${T}$ constructed is not quite a zeroth-order pseudodifferential operator; it is instead merely in the “forbidden” symbol class ${S^0_{1,1}}$, and more precisely it takes the form

$\displaystyle T v = \sum_{j \in {\bf Z}} 2^{3j} \langle v, \phi_j \rangle \psi_j \ \ \ \ \ (2)$

for some compactly supported divergence-free ${\phi,\psi}$ of mean zero with

$\displaystyle \phi_j(x) := \phi(2^j x); \quad \psi_j(x) := \psi(2^j x)$

being ${L^2}$ rescalings of ${\phi,\psi}$. This operator is still bounded on all ${L^p({\bf R}^3)}$ spaces ${1 < p < \infty}$, and so is arguably still a zeroth order operator, though not as convincingly as I would like. Another, less significant, issue with the result is that the solution constructed does not have good spatial decay properties, but this is mostly for convenience and it is likely that the construction can be localised to give solutions that have reasonable decay in space. But the biggest drawback of this theorem is the fact that ${T}$ is not invertible, self-adjoint, or positive definite, so in particular there is no non-negative Hamiltonian for this equation. It may be that some modification of the arguments below can fix these issues, but I have so far been unable to do so. Still, the construction does show that the circulation theorem is insufficient by itself to prevent blowup.

We sketch the proof of the above theorem as follows. We use the barrier method, introducing the time-varying hyperboloid domains

$\displaystyle \Omega(t) := \{ (r,\theta,z): r^2 \leq 1-t + z^2 \}$

for ${t>0}$ (expressed in cylindrical coordinates ${(r,\theta,z)}$). We will select initial data ${\omega(0)}$ to be ${\omega(0,r,\theta,z) = (0,0,\eta(r))}$ for some non-negative even bump function ${\eta}$ supported on ${[-1,1]}$, normalised so that

$\displaystyle \int\int \eta(r)\ r dr d\theta = 1;$

in particular ${\omega(0)}$ is divergence-free supported in ${\Omega(0)}$, with vortex lines connecting ${z=-\infty}$ to ${z=+\infty}$. Suppose for contradiction that we have a smooth solution ${\omega}$ to (1) with this initial data; to simplify the discussion we assume that the solution behaves well at spatial infinity (this can be justified with the choice (2) of vorticity-stream operator, but we will not do so here). Since the domains ${\Omega(t)}$ disconnect ${z=-\infty}$ from ${z=+\infty}$ at time ${t=1}$, there must exist a time ${0 < T_* < 1}$ which is the first time where the support of ${\omega(T_*)}$ touches the boundary of ${\Omega(T_*)}$, with ${\omega(t)}$ supported in ${\Omega(t)}$.

From (1) we see that the support of ${\omega(t)}$ is transported by the velocity field ${u(t)}$. Thus, at the point of contact of the support of ${\omega(T_*)}$ with the boundary of ${\Omega(T_*)}$, the inward component of the velocity field ${u(T_*)}$ cannot exceed the inward velocity of ${\Omega(T_*)}$. We will construct the functions ${\phi,\psi}$ so that this is not the case, leading to the desired contradiction. (Geometrically, what is going on here is that the operator ${T}$ is pinching the flow to pass through the narrow cylinder ${\{ z, r = O( \sqrt{1-t} )\}}$, leading to a singularity by time ${t=1}$ at the latest.)

First we observe from conservation of circulation, and from the fact that ${\omega(t)}$ is supported in ${\Omega(t)}$, that the integrals

$\displaystyle \int\int \omega_z(t,r,\theta,z) \ r dr d\theta$

are constant in both space and time for ${0 \leq t \leq T_*}$. From the choice of initial data we thus have

$\displaystyle \int\int \omega_z(t,r,\theta,z) \ r dr d\theta = 1$

for all ${t \leq T_*}$ and all ${z}$. On the other hand, if ${T}$ is of the form (2) with ${\phi = \nabla \times \eta}$ for some bump function ${\eta = (0,0,\eta_z)}$ that only has ${z}$-components, then ${\phi}$ is divergence-free with mean zero, and

$\displaystyle \langle (-\Delta) (\nabla \times \omega), \phi_j \rangle = 2^{-j} \langle (-\Delta) (\nabla \times \omega), \nabla \times \eta_j \rangle$

$\displaystyle = 2^{-j} \langle \omega, \eta_j \rangle$

$\displaystyle = 2^{-j} \int\int\int \omega_z(t,r,\theta,z) \eta_z(2^j r, \theta, 2^j z)\ r dr d\theta dz,$

where ${\eta_j(x) := \eta(2^j x)}$. We choose ${\eta_z}$ to be supported in the slab ${\{ C \leq z \leq 2C\}}$ for some large constant ${C}$, and to equal a function ${f(z)}$ depending only on ${z}$ on the cylinder ${\{ C \leq z \leq 2C; r \leq 10C \}}$, normalised so that ${\int f(z)\ dz = 1}$. If ${C/2^j \geq (1-t)^{1/2}}$, then ${\Omega(t)}$ passes through this cylinder, and we conclude that

$\displaystyle \langle (-\Delta) (\nabla \times \omega), \phi_j \rangle = -2^{-j} \int f(2^j z)\ dz$

$\displaystyle = 2^{-2j}.$

Inserting ths into (2), (1) we conclude that

$\displaystyle u = \sum_{j: C/2^j \geq (1-t)^{1/2}} 2^j \psi_j + \sum_{j: C/2^j < (1-t)^{1/2}} c_j(t) \psi_j$

for some coefficients ${c_j(t)}$. We will not be able to control these coefficients ${c_j(t)}$, but fortunately we only need to understand ${u}$ on the boundary ${\partial \Omega(t)}$, for which ${r+|z| \gg (1-t)^{1/2}}$. So, if ${\psi}$ happens to be supported on an annulus ${1 \ll r+|z| \ll 1}$, then ${\psi_j}$ vanishes on ${\partial \Omega(t)}$ if ${C}$ is large enough. We then have

$\displaystyle u = \sum_j 2^j \psi_j$

on the boundary of ${\partial \Omega(t)}$.

Let ${\Phi(r,\theta,z)}$ be a function of the form

$\displaystyle \Phi(r,\theta,z) = C z \varphi(z/r)$

where ${\varphi}$ is a bump function supported on ${[-2,2]}$ that equals ${1}$ on ${[-1,1]}$. We can perform a dyadic decomposition ${\Phi = \sum_j \Psi_j}$ where

$\displaystyle \Psi_j(r,\theta,z) = \Phi(r,\theta,z) a(2^j r)$

where ${a}$ is a bump function supported on ${[1/2,2]}$ with ${\sum_j a(2^j r) = 1}$. If we then set

$\displaystyle \psi_j = \frac{2^{-j}}{r} (-\partial_z \Psi_j, 0, \partial_r \Psi_j)$

then one can check that ${\psi_j(x) = \psi(2^j x)}$ for a function ${\psi}$ that is divergence-free and mean zero, and supported on the annulus ${1 \ll r+|z| \ll 1}$, and

$\displaystyle \sum_j 2^j \psi_j = \frac{1}{r} (-\partial_z \Phi, 0, \partial_r \Phi)$

so on ${\partial \Omega(t)}$ (where ${|z| \leq r}$) we have

$\displaystyle u = (-\frac{C}{r}, 0, 0 ).$

One can manually check that the inward velocity of this vector on ${\partial \Omega(t)}$ exceeds the inward velocity of ${\Omega(t)}$ if ${C}$ is large enough, and the claim follows.

Remark 2 The type of blowup suggested by this construction, where a unit amount of circulation is squeezed into a narrow cylinder, is of “Type II” with respect to the Navier-Stokes scaling, because Navier-Stokes-critical norms such ${L^3({\bf R}^3)}$ (or at least ${L^{3,\infty}({\bf R}^3)}$) look like they stay bounded during this squeezing procedure (the velocity field is of size about ${2^j}$ in cylinders of radius and length about ${2^j}$). So even if the various issues with ${T}$ are repaired, it does not seem likely that this construction can be directly adapted to obtain a corresponding blowup for a Navier-Stokes type equation. To get a “Type I” blowup that is consistent with Kelvin’s circulation theorem, it seems that one needs to coil the vortex lines around a loop multiple times in order to get increased circulation in a small space. This seems possible to pull off to me – there don’t appear to be any unavoidable obstructions coming from topology, scaling, or conservation laws – but would require a more complicated construction than the one given above.

The Poincaré upper half-plane ${{\mathbf H} := \{ z: \hbox{Im}(z) > 0 \}}$ (with a boundary consisting of the real line ${{\bf R}}$ together with the point at infinity ${\infty}$) carries an action of the projective special linear group

$\displaystyle \hbox{PSL}_2({\bf R}) := \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix}: a,b,c,d \in {\bf R}: ad-bc = 1 \} / \{\pm 1\}$

via fractional linear transformations:

$\displaystyle \begin{pmatrix} a & b \\ c & d \end{pmatrix} z := \frac{az+b}{cz+d}. \ \ \ \ \ (1)$

Here and in the rest of the post we will abuse notation by identifying elements ${\begin{pmatrix} a & b \\ c & d \end{pmatrix}}$ of the special linear group ${\hbox{SL}_2({\bf R})}$ with their equivalence class ${\{ \pm \begin{pmatrix} a & b \\ c & d \end{pmatrix} \}}$ in ${\hbox{PSL}_2({\bf R})}$; this will occasionally create or remove a factor of two in our formulae, but otherwise has very little effect, though one has to check that various definitions and expressions (such as (1)) are unaffected if one replaces a matrix ${\begin{pmatrix} a & b \\ c & d \end{pmatrix}}$ by its negation ${\begin{pmatrix} -a & -b \\ -c & -d \end{pmatrix}}$. In particular, we recommend that the reader ignore the signs ${\pm}$ that appear from time to time in the discussion below.

As the action of ${\hbox{PSL}_2({\bf R})}$ on ${{\mathbf H}}$ is transitive, and any given point in ${{\mathbf H}}$ (e.g. ${i}$) has a stabiliser isomorphic to the projective rotation group ${\hbox{PSO}_2({\bf R})}$, we can view the Poincaré upper half-plane ${{\mathbf H}}$ as a homogeneous space for ${\hbox{PSL}_2({\bf R})}$, and more specifically the quotient space of ${\hbox{PSL}_2({\bf R})}$ of a maximal compact subgroup ${\hbox{PSO}_2({\bf R})}$. In fact, we can make the half-plane a symmetric space for ${\hbox{PSL}_2({\bf R})}$, by endowing ${{\mathbf H}}$ with the Riemannian metric

$\displaystyle dg^2 := \frac{dx^2 + dy^2}{y^2}$

(using Cartesian coordinates ${z=x+iy}$), which is invariant with respect to the ${\hbox{PSL}_2({\bf R})}$ action. Like any other Riemannian metric, the metric on ${{\mathbf H}}$ generates a number of other important geometric objects on ${{\mathbf H}}$, such as the distance function ${d(z,w)}$ which can be computed to be given by the formula

$\displaystyle 2(\cosh(d(z_1,z_2))-1) = \frac{|z_1-z_2|^2}{\hbox{Im}(z_1) \hbox{Im}(z_2)}, \ \ \ \ \ (2)$

the volume measure ${\mu = \mu_{\mathbf H}}$, which can be computed to be

$\displaystyle d\mu = \frac{dx dy}{y^2},$

and the Laplace-Beltrami operator, which can be computed to be ${\Delta = y^2 (\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2})}$ (here we use the negative definite sign convention for ${\Delta}$). As the metric ${dg}$ was ${\hbox{PSL}_2({\bf R})}$-invariant, all of these quantities arising from the metric are similarly ${\hbox{PSL}_2({\bf R})}$-invariant in the appropriate sense.

The Gauss curvature of the Poincaré half-plane can be computed to be the constant ${-1}$, thus ${{\mathbf H}}$ is a model for two-dimensional hyperbolic geometry, in much the same way that the unit sphere ${S^2}$ in ${{\bf R}^3}$ is a model for two-dimensional spherical geometry (or ${{\bf R}^2}$ is a model for two-dimensional Euclidean geometry). (Indeed, ${{\mathbf H}}$ is isomorphic (via projection to a null hyperplane) to the upper unit hyperboloid ${\{ (x,t) \in {\bf R}^{2+1}: t = \sqrt{1+|x|^2}\}}$ in the Minkowski spacetime ${{\bf R}^{2+1}}$, which is the direct analogue of the unit sphere in Euclidean spacetime ${{\bf R}^3}$ or the plane ${{\bf R}^2}$ in Galilean spacetime ${{\bf R}^2 \times {\bf R}}$.)

One can inject arithmetic into this geometric structure by passing from the Lie group ${\hbox{PSL}_2({\bf R})}$ to the full modular group

$\displaystyle \hbox{PSL}_2({\bf Z}) := \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix}: a,b,c,d \in {\bf Z}: ad-bc = 1 \} / \{\pm 1\}$

or congruence subgroups such as

$\displaystyle \Gamma_0(q) := \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \hbox{PSL}_2({\bf Z}): c = 0\ (q) \} / \{ \pm 1 \} \ \ \ \ \ (3)$

for natural number ${q}$, or to the discrete stabiliser ${\Gamma_\infty}$ of the point at infinity:

$\displaystyle \Gamma_\infty := \{ \pm \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix}: b \in {\bf Z} \} / \{\pm 1\}. \ \ \ \ \ (4)$

These are discrete subgroups of ${\hbox{PSL}_2({\bf R})}$, nested by the subgroup inclusions

$\displaystyle \Gamma_\infty \leq \Gamma_0(q) \leq \Gamma_0(1)=\hbox{PSL}_2({\bf Z}) \leq \hbox{PSL}_2({\bf R}).$

There are many further discrete subgroups of ${\hbox{PSL}_2({\bf R})}$ (known collectively as Fuchsian groups) that one could consider, but we will focus attention on these three groups in this post.

Any discrete subgroup ${\Gamma}$ of ${\hbox{PSL}_2({\bf R})}$ generates a quotient space ${\Gamma \backslash {\mathbf H}}$, which in general will be a non-compact two-dimensional orbifold. One can understand such a quotient space by working with a fundamental domain ${\hbox{Fund}( \Gamma \backslash {\mathbf H})}$ – a set consisting of a single representative of each of the orbits ${\Gamma z}$ of ${\Gamma}$ in ${{\mathbf H}}$. This fundamental domain is by no means uniquely defined, but if the fundamental domain is chosen with some reasonable amount of regularity, one can view ${\Gamma \backslash {\mathbf H}}$ as the fundamental domain with the boundaries glued together in an appropriate sense. Among other things, fundamental domains can be used to induce a volume measure ${\mu = \mu_{\Gamma \backslash {\mathbf H}}}$ on ${\Gamma \backslash {\mathbf H}}$ from the volume measure ${\mu = \mu_{\mathbf H}}$ on ${{\mathbf H}}$ (restricted to a fundamental domain). By abuse of notation we will refer to both measures simply as ${\mu}$ when there is no chance of confusion.

For instance, a fundamental domain for ${\Gamma_\infty \backslash {\mathbf H}}$ is given (up to null sets) by the strip ${\{ z \in {\mathbf H}: |\hbox{Re}(z)| < \frac{1}{2} \}}$, with ${\Gamma_\infty \backslash {\mathbf H}}$ identifiable with the cylinder formed by gluing together the two sides of the strip. A fundamental domain for ${\hbox{PSL}_2({\bf Z}) \backslash {\mathbf H}}$ is famously given (again up to null sets) by an upper portion ${\{ z \in {\mathbf H}: |\hbox{Re}(z)| < \frac{1}{2}; |z| > 1 \}}$, with the left and right sides again glued to each other, and the left and right halves of the circular boundary glued to itself. A fundamental domain for ${\Gamma_0(q) \backslash {\mathbf H}}$ can be formed by gluing together

$\displaystyle [\hbox{PSL}_2({\bf Z}) : \Gamma_0(q)] = q \prod_{p|q} (1 + \frac{1}{p}) = q^{1+o(1)}$

copies of a fundamental domain for ${\hbox{PSL}_2({\bf Z}) \backslash {\mathbf H}}$ in a rather complicated but interesting fashion.

While fundamental domains can be a convenient choice of coordinates to work with for some computations (as well as for drawing appropriate pictures), it is geometrically more natural to avoid working explicitly on such domains, and instead work directly on the quotient spaces ${\Gamma \backslash {\mathbf H}}$. In order to analyse functions ${f: \Gamma \backslash {\mathbf H} \rightarrow {\bf C}}$ on such orbifolds, it is convenient to lift such functions back up to ${{\mathbf H}}$ and identify them with functions ${f: {\mathbf H} \rightarrow {\bf C}}$ which are ${\Gamma}$-automorphic in the sense that ${f( \gamma z ) = f(z)}$ for all ${z \in {\mathbf H}}$ and ${\gamma \in \Gamma}$. Such functions will be referred to as ${\Gamma}$-automorphic forms, or automorphic forms for short (we always implicitly assume all such functions to be measurable). (Strictly speaking, these are the automorphic forms with trivial factor of automorphy; one can certainly consider other factors of automorphy, particularly when working with holomorphic modular forms, which corresponds to sections of a more non-trivial line bundle over ${\Gamma \backslash {\mathbf H}}$ than the trivial bundle ${(\Gamma \backslash {\mathbf H}) \times {\bf C}}$ that is implicitly present when analysing scalar functions ${f: {\mathbf H} \rightarrow {\bf C}}$. However, we will not discuss this (important) more general situation here.)

An important way to create a ${\Gamma}$-automorphic form is to start with a non-automorphic function ${f: {\mathbf H} \rightarrow {\bf C}}$ obeying suitable decay conditions (e.g. bounded with compact support will suffice) and form the Poincaré series ${P_\Gamma[f]: {\mathbf H} \rightarrow {\bf C}}$ defined by

$\displaystyle P_{\Gamma}[f](z) = \sum_{\gamma \in \Gamma} f(\gamma z),$

which is clearly ${\Gamma}$-automorphic. (One could equivalently write ${f(\gamma^{-1} z)}$ in place of ${f(\gamma z)}$ here; there are good argument for both conventions, but I have ultimately decided to use the ${f(\gamma z)}$ convention, which makes explicit computations a little neater at the cost of making the group actions work in the opposite order.) Thus we naturally see sums over ${\Gamma}$ associated with ${\Gamma}$-automorphic forms. A little more generally, given a subgroup ${\Gamma_\infty}$ of ${\Gamma}$ and a ${\Gamma_\infty}$-automorphic function ${f: {\mathbf H} \rightarrow {\bf C}}$ of suitable decay, we can form a relative Poincaré series ${P_{\Gamma_\infty \backslash \Gamma}[f]: {\mathbf H} \rightarrow {\bf C}}$ by

$\displaystyle P_{\Gamma_\infty \backslash \Gamma}[f](z) = \sum_{\gamma \in \hbox{Fund}(\Gamma_\infty \backslash \Gamma)} f(\gamma z)$

where ${\hbox{Fund}(\Gamma_\infty \backslash \Gamma)}$ is any fundamental domain for ${\Gamma_\infty \backslash \Gamma}$, that is to say a subset of ${\Gamma}$ consisting of exactly one representative for each right coset of ${\Gamma_\infty}$. As ${f}$ is ${\Gamma_\infty}$-automorphic, we see (if ${f}$ has suitable decay) that ${P_{\Gamma_\infty \backslash \Gamma}[f]}$ does not depend on the precise choice of fundamental domain, and is ${\Gamma}$-automorphic. These operations are all compatible with each other, for instance ${P_\Gamma = P_{\Gamma_\infty \backslash \Gamma} \circ P_{\Gamma_\infty}}$. A key example of Poincaré series are the Eisenstein series, although there are of course many other Poincaré series one can consider by varying the test function ${f}$.

For future reference we record the basic but fundamental unfolding identities

$\displaystyle \int_{\Gamma \backslash {\mathbf H}} P_\Gamma[f] g\ d\mu_{\Gamma \backslash {\mathbf H}} = \int_{\mathbf H} f g\ d\mu_{\mathbf H} \ \ \ \ \ (5)$

for any function ${f: {\mathbf H} \rightarrow {\bf C}}$ with sufficient decay, and any ${\Gamma}$-automorphic function ${g}$ of reasonable growth (e.g. ${f}$ bounded and compact support, and ${g}$ bounded, will suffice). Note that ${g}$ is viewed as a function on ${\Gamma \backslash {\mathbf H}}$ on the left-hand side, and as a ${\Gamma}$-automorphic function on ${{\mathbf H}}$ on the right-hand side. More generally, one has

$\displaystyle \int_{\Gamma \backslash {\mathbf H}} P_{\Gamma_\infty \backslash \Gamma}[f] g\ d\mu_{\Gamma \backslash {\mathbf H}} = \int_{\Gamma_\infty \backslash {\mathbf H}} f g\ d\mu_{\Gamma_\infty \backslash {\mathbf H}} \ \ \ \ \ (6)$

whenever ${\Gamma_\infty \leq \Gamma}$ are discrete subgroups of ${\hbox{PSL}_2({\bf R})}$, ${f}$ is a ${\Gamma_\infty}$-automorphic function with sufficient decay on ${\Gamma_\infty \backslash {\mathbf H}}$, and ${g}$ is a ${\Gamma}$-automorphic (and thus also ${\Gamma_\infty}$-automorphic) function of reasonable growth. These identities will allow us to move fairly freely between the three domains ${{\mathbf H}}$, ${\Gamma_\infty \backslash {\mathbf H}}$, and ${\Gamma \backslash {\mathbf H}}$ in our analysis.

When computing various statistics of a Poincaré series ${P_\Gamma[f]}$, such as its values ${P_\Gamma[f](z)}$ at special points ${z}$, or the ${L^2}$ quantity ${\int_{\Gamma \backslash {\mathbf H}} |P_\Gamma[f]|^2\ d\mu}$, expressions of interest to analytic number theory naturally emerge. We list three basic examples of this below, discussed somewhat informally in order to highlight the main ideas rather than the technical details.

The first example we will give concerns the problem of estimating the sum

$\displaystyle \sum_{n \leq x} \tau(n) \tau(n+1), \ \ \ \ \ (7)$

where ${\tau(n) := \sum_{d|n} 1}$ is the divisor function. This can be rewritten (by factoring ${n=bc}$ and ${n+1=ad}$) as

$\displaystyle \sum_{ a,b,c,d \in {\bf N}: ad-bc = 1} 1_{bc \leq x} \ \ \ \ \ (8)$

which is basically a sum over the full modular group ${\hbox{PSL}_2({\bf Z})}$. At this point we will “cheat” a little by moving to the related, but different, sum

$\displaystyle \sum_{a,b,c,d \in {\bf Z}: ad-bc = 1} 1_{a^2+b^2+c^2+d^2 \leq x}. \ \ \ \ \ (9)$

This sum is not exactly the same as (8), but will be a little easier to handle, and it is plausible that the methods used to handle this sum can be modified to handle (8). Observe from (2) and some calculation that the distance between ${i}$ and ${\begin{pmatrix} a & b \\ c & d \end{pmatrix} i = \frac{ai+b}{ci+d}}$ is given by the formula

$\displaystyle 2(\cosh(d(i,\begin{pmatrix} a & b \\ c & d \end{pmatrix} i))-1) = a^2+b^2+c^2+d^2 - 2$

and so one can express the above sum as

$\displaystyle 2 \sum_{\gamma \in \hbox{PSL}_2({\bf Z})} 1_{d(i,\gamma i) \leq \hbox{cosh}^{-1}(x/2)}$

(the factor of ${2}$ coming from the quotient by ${\{\pm 1\}}$ in the projective special linear group); one can express this as ${P_\Gamma[f](i)}$, where ${\Gamma = \hbox{PSL}_2({\bf Z})}$ and ${f}$ is the indicator function of the ball ${B(i, \hbox{cosh}^{-1}(x/2))}$. Thus we see that expressions such as (7) are related to evaluations of Poincaré series. (In practice, it is much better to use smoothed out versions of indicator functions in order to obtain good control on sums such as (7) or (9), but we gloss over this technical detail here.)

The second example concerns the relative

$\displaystyle \sum_{n \leq x} \tau(n^2+1) \ \ \ \ \ (10)$

of the sum (7). Note from multiplicativity that (7) can be written as ${\sum_{n \leq x} \tau(n^2+n)}$, which is superficially very similar to (10), but with the key difference that the polynomial ${n^2+1}$ is irreducible over the integers.

As with (7), we may expand (10) as

$\displaystyle \sum_{A,B,C \in {\bf N}: B^2 - AC = -1} 1_{B \leq x}.$

At first glance this does not look like a sum over a modular group, but one can manipulate this expression into such a form in one of two (closely related) ways. First, observe that any factorisation ${B + i = (a-bi) (c+di)}$ of ${B+i}$ into Gaussian integers ${a-bi, c+di}$ gives rise (upon taking norms) to an identity of the form ${B^2 - AC = -1}$, where ${A = a^2+b^2}$ and ${C = c^2+d^2}$. Conversely, by using the unique factorisation of the Gaussian integers, every identity of the form ${B^2-AC=-1}$ gives rise to a factorisation of the form ${B+i = (a-bi) (c+di)}$, essentially uniquely up to units. Now note that ${(a-bi)(c+di)}$ is of the form ${B+i}$ if and only if ${ad-bc=1}$, in which case ${B = ac+bd}$. Thus we can essentially write the above sum as something like

$\displaystyle \sum_{a,b,c,d: ad-bc = 1} 1_{|ac+bd| \leq x} \ \ \ \ \ (11)$

and one the modular group ${\hbox{PSL}_2({\bf Z})}$ is now manifest. An equivalent way to see these manipulations is as follows. A triple ${A,B,C}$ of natural numbers with ${B^2-AC=1}$ gives rise to a positive quadratic form ${Ax^2+2Bxy+Cy^2}$ of normalised discriminant ${B^2-AC}$ equal to ${-1}$ with integer coefficients (it is natural here to allow ${B}$ to take integer values rather than just natural number values by essentially doubling the sum). The group ${\hbox{PSL}_2({\bf Z})}$ acts on the space of such quadratic forms in a natural fashion (by composing the quadratic form with the inverse ${\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}}$ of an element ${\begin{pmatrix} a & b \\ c & d \end{pmatrix}}$ of ${\hbox{SL}_2({\bf Z})}$). Because the discriminant ${-1}$ has class number one (this fact is equivalent to the unique factorisation of the gaussian integers, as discussed in this previous post), every form ${Ax^2 + 2Bxy + Cy^2}$ in this space is equivalent (under the action of some element of ${\hbox{PSL}_2({\bf Z})}$) with the standard quadratic form ${x^2+y^2}$. In other words, one has

$\displaystyle Ax^2 + 2Bxy + Cy^2 = (dx-by)^2 + (-cx+ay)^2$

which (up to a harmless sign) is exactly the representation ${B = ac+bd}$, ${A = c^2+d^2}$, ${C = a^2+b^2}$ introduced earlier, and leads to the same reformulation of the sum (10) in terms of expressions like (11). Similar considerations also apply if the quadratic polynomial ${n^2+1}$ is replaced by another quadratic, although one has to account for the fact that the class number may now exceed one (so that unique factorisation in the associated quadratic ring of integers breaks down), and in the positive discriminant case the fact that the group of units might be infinite presents another significant technical problem.

Note that ${\begin{pmatrix} a & b \\ c & d \end{pmatrix} i = \frac{ai+b}{ci+d}}$ has real part ${\frac{ac+bd}{c^2+d^2}}$ and imaginary part ${\frac{1}{c^2+d^2}}$. Thus (11) is (up to a factor of two) the Poincaré series ${P_\Gamma[f](i)}$ as in the preceding example, except that ${f}$ is now the indicator of the sector ${\{ z: |\hbox{Re} z| \leq x |\hbox{Im} z| \}}$.

Sums involving subgroups of the full modular group, such as ${\Gamma_0(q)}$, often arise when imposing congruence conditions on sums such as (10), for instance when trying to estimate the expression ${\sum_{n \leq x: q|n} \tau(n^2+1)}$ when ${q}$ and ${x}$ are large. As before, one then soon arrives at the problem of evaluating a Poincaré series at one or more special points, where the series is now over ${\Gamma_0(q)}$ rather than ${\hbox{PSL}_2({\bf Z})}$.

The third and final example concerns averages of Kloosterman sums

$\displaystyle S(m,n;c) := \sum_{x \in ({\bf Z}/c{\bf Z})^\times} e( \frac{mx + n\overline{x}}{c} ) \ \ \ \ \ (12)$

where ${e(\theta) := e^{2p\i i\theta}}$ and ${\overline{x}}$ is the inverse of ${x}$ in the multiplicative group ${({\bf Z}/c{\bf Z})^\times}$. It turns out that the ${L^2}$ norms of Poincaré series ${P_\Gamma[f]}$ or ${P_{\Gamma_\infty \backslash \Gamma}[f]}$ are closely tied to such averages. Consider for instance the quantity

$\displaystyle \int_{\Gamma_0(q) \backslash {\mathbf H}} |P_{\Gamma_\infty \backslash \Gamma_0(q)}[f]|^2\ d\mu_{\Gamma \backslash {\mathbf H}} \ \ \ \ \ (13)$

where ${q}$ is a natural number and ${f}$ is a ${\Gamma_\infty}$-automorphic form that is of the form

$\displaystyle f(x+iy) = F(my) e(m x)$

for some integer ${m}$ and some test function ${f: (0,+\infty) \rightarrow {\bf C}}$, which for sake of discussion we will take to be smooth and compactly supported. Using the unfolding formula (6), we may rewrite (13) as

$\displaystyle \int_{\Gamma_\infty \backslash {\mathbf H}} \overline{f} P_{\Gamma_\infty \backslash \Gamma_0(q)}[f]\ d\mu_{\Gamma_\infty \backslash {\mathbf H}}.$

To compute this, we use the double coset decomposition

$\displaystyle \Gamma_0(q) = \Gamma_\infty \cup \bigcup_{c \in {\mathbf N}: q|c} \bigcup_{1 \leq d \leq c: (d,c)=1} \Gamma_\infty \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Gamma_\infty,$

where for each ${c,d}$, ${a,b}$ are arbitrarily chosen integers such that ${ad-bc=1}$. To see this decomposition, observe that every element ${\begin{pmatrix} a & b \\ c & d \end{pmatrix}}$ in ${\Gamma_0(q)}$ outside of ${\Gamma_\infty}$ can be assumed to have ${c>0}$ by applying a sign ${\pm}$, and then using the row and column operations coming from left and right multiplication by ${\Gamma_\infty}$ (that is, shifting the top row by an integer multiple of the bottom row, and shifting the right column by an integer multiple of the left column) one can place ${d}$ in the interval ${[1,c]}$ and ${(a,b)}$ to be any specified integer pair with ${ad-bc=1}$. From this we see that

$\displaystyle P_{\Gamma_\infty \backslash \Gamma_0(q)}[f] = f + \sum_{c \in {\mathbf N}: q|c} \sum_{1 \leq d \leq c: (d,c)=1} P_{\Gamma_\infty}[ f( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot ) ]$

and so from further use of the unfolding formula (5) we may expand (13) as

$\displaystyle \int_{\Gamma_\infty \backslash {\mathbf H}} |f|^2\ d\mu_{\Gamma_\infty \backslash {\mathbf H}}$

$\displaystyle + \sum_{c \in {\mathbf N}} \sum_{1 \leq d \leq c: (d,c)=1} \int_{\mathbf H} \overline{f}(z) f( \begin{pmatrix} a & b \\ c & d \end{pmatrix} z)\ d\mu_{\mathbf H}.$

The first integral is just ${m \int_0^\infty |F(y)|^2 \frac{dy}{y^2}}$. The second expression is more interesting. We have

$\displaystyle \begin{pmatrix} a & b \\ c & d \end{pmatrix} z = \frac{az+b}{cz+d} = \frac{a}{c} - \frac{1}{c(cz+d)}$

$\displaystyle = \frac{a}{c} - \frac{cx+d}{c((cx+d)^2+c^2y^2)} + \frac{iy}{(cx+d)^2 + c^2y^2}$

so we can write

$\displaystyle \int_{\mathbf H} \overline{f}(z) f( \begin{pmatrix} a & b \\ c & d \end{pmatrix} z)\ d\mu_{\mathbf H}$

as

$\displaystyle \int_0^\infty \int_{\bf R} \overline{F}(my) F(\frac{imy}{(cx+d)^2 + c^2y^2}) e( -mx + \frac{ma}{c} - m \frac{cx+d}{c((cx+d)^2+c^2y^2)} )$

$\displaystyle \frac{dx dy}{y^2}$

which on shifting ${x}$ by ${d/c}$ simplifies a little to

$\displaystyle e( \frac{ma}{c} + \frac{md}{c} ) \int_0^\infty \int_{\bf R} F(my) \bar{F}(\frac{imy}{c^2(x^2 + y^2)}) e(- mx - m \frac{x}{c^2(x^2+y^2)} )$

$\displaystyle \frac{dx dy}{y^2}$

and then on scaling ${x,y}$ by ${m}$ simplifies a little further to

$\displaystyle e( \frac{ma}{c} + \frac{md}{c} ) \int_0^\infty \int_{\bf R} F(y) \bar{F}(\frac{m^2}{c^2} \frac{iy}{x^2 + y^2}) e(- x - \frac{m^2}{c^2} \frac{x}{x^2+y^2} )\ \frac{dx dy}{y^2}.$

Note that as ${ad-bc=1}$, we have ${a = \overline{d}}$ modulo ${c}$. Comparing the above calculations with (12), we can thus write (13) as

$\displaystyle m (\int_0^\infty |F(y)|^2 \frac{dy}{y^2} + \sum_{q|c} \frac{S(m,m;c)}{c} V(\frac{m}{c})) \ \ \ \ \ (14)$

where

$\displaystyle V(u) := \frac{1}{u} \int_0^\infty \int_{\bf R} F(y) \bar{F}(u^2 \frac{y}{x^2 + y^2}) e(- x - u^2 \frac{x}{x^2+y^2} )\ \frac{dx dy}{y^2}$

is a certain integral involving ${F}$ and a parameter ${u}$, but which does not depend explicitly on parameters such as ${m,c,d}$. Thus we have indeed expressed the ${L^2}$ expression (13) in terms of Kloosterman sums. It is possible to invert this analysis and express varius weighted sums of Kloosterman sums in terms of ${L^2}$ expressions (possibly involving inner products instead of norms) of Poincaré series, but we will not do so here; see Chapter 16 of Iwaniec and Kowalski for further details.

Traditionally, automorphic forms have been analysed using the spectral theory of the Laplace-Beltrami operator ${-\Delta}$ on spaces such as ${\Gamma\backslash {\mathbf H}}$ or ${\Gamma_\infty \backslash {\mathbf H}}$, so that a Poincaré series such as ${P_\Gamma[f]}$ might be expanded out using inner products of ${P_\Gamma[f]}$ (or, by the unfolding identities, ${f}$) with various generalised eigenfunctions of ${-\Delta}$ (such as cuspidal eigenforms, or Eisenstein series). With this approach, special functions, and specifically the modified Bessel functions ${K_{it}}$ of the second kind, play a prominent role, basically because the ${\Gamma_\infty}$-automorphic functions

$\displaystyle x+iy \mapsto y^{1/2} K_{it}(2\pi |m| y) e(mx)$

for ${t \in {\bf R}}$ and ${m \in {\bf Z}}$ non-zero are generalised eigenfunctions of ${-\Delta}$ (with eigenvalue ${\frac{1}{4}+t^2}$), and are almost square-integrable on ${\Gamma_\infty \backslash {\mathbf H}}$ (the ${L^2}$ norm diverges only logarithmically at one end ${y \rightarrow 0^+}$ of the cylinder ${\Gamma_\infty \backslash {\mathbf H}}$, while decaying exponentially fast at the other end ${y \rightarrow +\infty}$).

However, as discussed in this previous post, the spectral theory of an essentially self-adjoint operator such as ${-\Delta}$ is basically equivalent to the theory of various solution operators associated to partial differential equations involving that operator, such as the Helmholtz equation ${(-\Delta + k^2) u = f}$, the heat equation ${\partial_t u = \Delta u}$, the Schrödinger equation ${i\partial_t u + \Delta u = 0}$, or the wave equation ${\partial_{tt} u = \Delta u}$. Thus, one can hope to rephrase many arguments that involve spectral data of ${-\Delta}$ into arguments that instead involve resolvents ${(-\Delta + k^2)^{-1}}$, heat kernels ${e^{t\Delta}}$, Schrödinger propagators ${e^{it\Delta}}$, or wave propagators ${e^{\pm it\sqrt{-\Delta}}}$, or involve the PDE more directly (e.g. applying integration by parts and energy methods to solutions of such PDE). This is certainly done to some extent in the existing literature; resolvents and heat kernels, for instance, are often utilised. In this post, I would like to explore the possibility of reformulating spectral arguments instead using the inhomogeneous wave equation

$\displaystyle \partial_{tt} u - \Delta u = F.$

Actually it will be a bit more convenient to normalise the Laplacian by ${\frac{1}{4}}$, and look instead at the automorphic wave equation

$\displaystyle \partial_{tt} u + (-\Delta - \frac{1}{4}) u = F. \ \ \ \ \ (15)$

This equation somewhat resembles a “Klein-Gordon” type equation, except that the mass is imaginary! This would lead to pathological behaviour were it not for the negative curvature, which in principle creates a spectral gap of ${\frac{1}{4}}$ that cancels out this factor.

The point is that the wave equation approach gives access to some nice PDE techniques, such as energy methods, Sobolev inequalities and finite speed of propagation, which are somewhat submerged in the spectral framework. The wave equation also interacts well with Poincaré series; if for instance ${u}$ and ${F}$ are ${\Gamma_\infty}$-automorphic solutions to (15) obeying suitable decay conditions, then their Poincaré series ${P_{\Gamma_\infty \backslash \Gamma}[u]}$ and ${P_{\Gamma_\infty \backslash \Gamma}[F]}$ will be ${\Gamma}$-automorphic solutions to the same equation (15), basically because the Laplace-Beltrami operator commutes with translations. Because of these facts, it is possible to replicate several standard spectral theory arguments in the wave equation framework, without having to deal directly with things like the asymptotics of modified Bessel functions. The wave equation approach to automorphic theory was introduced by Faddeev and Pavlov (using the Lax-Phillips scattering theory), and developed further by by Lax and Phillips, to recover many spectral facts about the Laplacian on modular curves, such as the Weyl law and the Selberg trace formula. Here, I will illustrate this by deriving three basic applications of automorphic methods in a wave equation framework, namely

• Using the Weil bound on Kloosterman sums to derive Selberg’s 3/16 theorem on the least non-trivial eigenvalue for ${-\Delta}$ on ${\Gamma_0(q) \backslash {\mathbf H}}$ (discussed previously here);
• Conversely, showing that Selberg’s eigenvalue conjecture (improving Selberg’s ${3/16}$ bound to the optimal ${1/4}$) implies an optimal bound on (smoothed) sums of Kloosterman sums; and
• Using the same bound to obtain pointwise bounds on Poincaré series similar to the ones discussed above. (Actually, the argument here does not use the wave equation, instead it just uses the Sobolev inequality.)

This post originated from an attempt to finally learn this part of analytic number theory properly, and to see if I could use a PDE-based perspective to understand it better. Ultimately, this is not that dramatic a depature from the standard approach to this subject, but I found it useful to think of things in this fashion, probably due to my existing background in PDE.

I thank Bill Duke and Ben Green for helpful discussions. My primary reference for this theory was Chapters 15, 16, and 21 of Iwaniec and Kowalski.

The Euler equations for three-dimensional incompressible inviscid fluid flow are

$\displaystyle \partial_t u + (u \cdot \nabla) u = - \nabla p \ \ \ \ \ (1)$

$\displaystyle \nabla \cdot u = 0$

where ${u: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}^3}$ is the velocity field, and ${p: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}}$ is the pressure field. For the purposes of this post, we will ignore all issues of decay or regularity of the fields in question, assuming that they are as smooth and rapidly decreasing as needed to justify all the formal calculations here; in particular, we will apply inverse operators such as ${(-\Delta)^{-1}}$ or ${|\nabla|^{-1} := (-\Delta)^{-1/2}}$ formally, assuming that these inverses are well defined on the functions they are applied to.

Meanwhile, the surface quasi-geostrophic (SQG) equation is given by

$\displaystyle \partial_t \theta + (u \cdot \nabla) \theta = 0 \ \ \ \ \ (2)$

$\displaystyle u = ( -\partial_y |\nabla|^{-1}, \partial_x |\nabla|^{-1} ) \theta \ \ \ \ \ (3)$

where ${\theta: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}}$ is the active scalar, and ${u: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}^2}$ is the velocity field. The SQG equations are often used as a toy model for the 3D Euler equations, as they share many of the same features (e.g. vortex stretching); see this paper of Constantin, Majda, and Tabak for more discussion (or this previous blog post).

I recently found a more direct way to connect the two equations. We first recall that the Euler equations can be placed in vorticity-stream form by focusing on the vorticity ${\omega := \nabla \times u}$. Indeed, taking the curl of (1), we obtain the vorticity equation

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (4)$

while the velocity ${u}$ can be recovered from the vorticity via the Biot-Savart law

$\displaystyle u = (-\Delta)^{-1} \nabla \times \omega. \ \ \ \ \ (5)$

The system (4), (5) has some features in common with the system (2), (3); in (2) it is a scalar field ${\theta}$ that is being transported by a divergence-free vector field ${u}$, which is a linear function of the scalar field as per (3), whereas in (4) it is a vector field ${\omega}$ that is being transported (in the Lie derivative sense) by a divergence-free vector field ${u}$, which is a linear function of the vector field as per (5). However, the system (4), (5) is in three dimensions whilst (2), (3) is in two spatial dimensions, the dynamical field is a scalar field ${\theta}$ for SQG and a vector field ${\omega}$ for Euler, and the relationship between the velocity field and the dynamical field is given by a zeroth order Fourier multiplier in (3) and a ${-1^{th}}$ order operator in (5).

However, we can make the two equations more closely resemble each other as follows. We first consider the generalisation

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (6)$

$\displaystyle u = T (-\Delta)^{-1} \nabla \times \omega \ \ \ \ \ (7)$

where ${T}$ is an invertible, self-adjoint, positive-definite zeroth order Fourier multiplier that maps divergence-free vector fields to divergence-free vector fields. The Euler equations then correspond to the case when ${T}$ is the identity operator. As discussed in this previous blog post (which used ${A}$ to denote the inverse of the operator denoted here as ${T}$), this generalised Euler system has many of the same features as the original Euler equation, such as a conserved Hamiltonian

$\displaystyle \frac{1}{2} \int_{{\bf R}^3} u \cdot T^{-1} u,$

the Kelvin circulation theorem, and conservation of helicity

$\displaystyle \int_{{\bf R}^3} \omega \cdot T^{-1} u.$

Also, if we require ${\omega}$ to be divergence-free at time zero, it remains divergence-free at all later times.

Let us consider “two-and-a-half-dimensional” solutions to the system (6), (7), in which ${u,\omega}$ do not depend on the vertical coordinate ${z}$, thus

$\displaystyle \omega(t,x,y,z) = \omega(t,x,y)$

and

$\displaystyle u(t,x,y,z) = u(t,x,y)$

but we allow the vertical components ${u_z, \omega_z}$ to be non-zero. For this to be consistent, we also require ${T}$ to commute with translations in the ${z}$ direction. As all derivatives in the ${z}$ direction now vanish, we can simplify (6) to

$\displaystyle D_t \omega = (\omega_x \partial_x + \omega_y \partial_y) u \ \ \ \ \ (8)$

where ${D_t}$ is the two-dimensional material derivative

$\displaystyle D_t := \partial_t + u_x \partial_x + u_y \partial_y.$

Also, divergence-free nature of ${\omega,u}$ then becomes

$\displaystyle \partial_x \omega_x + \partial_y \omega_y = 0$

and

$\displaystyle \partial_x u_x + \partial_y u_y = 0. \ \ \ \ \ (9)$

In particular, we may (formally, at least) write

$\displaystyle (\omega_x, \omega_y) = (\partial_y \theta, -\partial_x \theta)$

for some scalar field ${\theta(t,x,y,z) = \theta(t,x,y)}$, so that (7) becomes

$\displaystyle u = T ( (- \Delta)^{-1} \partial_y \omega_z, - (-\Delta^{-1}) \partial_x \omega_z, \theta ). \ \ \ \ \ (10)$

The first two components of (8) become

$\displaystyle D_t \partial_y \theta = \partial_y \theta \partial_x u_x - \partial_x \theta \partial_y u_x$

$\displaystyle - D_t \partial_x \theta = \partial_y \theta \partial_x u_y - \partial_x \theta \partial_y u_y$

which rearranges using (9) to

$\displaystyle \partial_y D_t \theta = \partial_x D_t \theta = 0.$

Formally, we may integrate this system to obtain the transport equation

$\displaystyle D_t \theta = 0, \ \ \ \ \ (11)$

Finally, the last component of (8) is

$\displaystyle D_t \omega_z = \partial_y \theta \partial_x u_z - \partial_x \theta \partial_y u_z. \ \ \ \ \ (12)$

At this point, we make the following choice for ${T}$:

$\displaystyle T ( U_x, U_y, \theta ) = \alpha (U_x, U_y, \theta) + (-\partial_y |\nabla|^{-1} \theta, \partial_x |\nabla|^{-1} \theta, 0) \ \ \ \ \ (13)$

$\displaystyle + P( 0, 0, |\nabla|^{-1} (\partial_y U_x - \partial_x U_y) )$

where ${\alpha > 0}$ is a real constant and ${Pu := (-\Delta)^{-1} (\nabla \times (\nabla \times u))}$ is the Leray projection onto divergence-free vector fields. One can verify that for large enough ${\alpha}$, ${T}$ is a self-adjoint positive definite zeroth order Fourier multiplier from divergence free vector fields to divergence-free vector fields. With this choice, we see from (10) that

$\displaystyle u_z = \alpha \theta - |\nabla|^{-1} \omega_z$

so that (12) simplifies to

$\displaystyle D_t \omega_z = - \partial_y \theta \partial_x |\nabla|^{-1} \omega_z + \partial_x \theta \partial_y |\nabla|^{-1} \omega_z.$

This implies (formally at least) that if ${\omega_z}$ vanishes at time zero, then it vanishes for all time. Setting ${\omega_z=0}$, we then have from (10) that

$\displaystyle (u_x,u_y,u_z) = (-\partial_y |\nabla|^{-1} \theta, \partial_x |\nabla|^{-1} \theta, \alpha \theta )$

and from (11) we then recover the SQG system (2), (3). To put it another way, if ${\theta(t,x,y)}$ and ${u(t,x,y)}$ solve the SQG system, then by setting

$\displaystyle \omega(t,x,y,z) := ( \partial_y \theta(t,x,y), -\partial_x \theta(t,x,y), 0 )$

$\displaystyle \tilde u(t,x,y,z) := ( u_x(t,x,y), u_y(t,x,y), \alpha \theta(t,x,y) )$

then ${\omega,\tilde u}$ solve the modified Euler system (6), (7) with ${T}$ given by (13).

We have ${T^{-1} \tilde u = (0, 0, \theta)}$, so the Hamiltonian ${\frac{1}{2} \int_{{\bf R}^3} \tilde u \cdot T^{-1} \tilde u}$ for the modified Euler system in this case is formally a scalar multiple of the conserved quantity ${\int_{{\bf R}^2} \theta^2}$. The momentum ${\int_{{\bf R}^3} x \cdot \tilde u}$ for the modified Euler system is formally a scalar multiple of the conserved quantity ${\int_{{\bf R}^2} \theta}$, while the vortex stream lines that are preserved by the modified Euler flow become the level sets of the active scalar that are preserved by the SQG flow. On the other hand, the helicity ${\int_{{\bf R}^3} \omega \cdot T^{-1} \tilde u}$ vanishes, and other conserved quantities for SQG (such as the Hamiltonian ${\int_{{\bf R}^2} \theta |\nabla|^{-1} \theta}$) do not seem to correspond to conserved quantities of the modified Euler system. This is not terribly surprising; a low-dimensional flow may well have a richer family of conservation laws than the higher-dimensional system that it is embedded in.

The wave equation is usually expressed in the form

$\displaystyle \partial_{tt} u - \Delta u = 0$

where ${u \colon {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ is a function of both time ${t \in {\bf R}}$ and space ${x \in {\bf R}^d}$, with ${\Delta}$ being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain ${{\bf R}^d}$ with some other manifold and replacing the Laplacian ${\Delta}$ with the Laplace-Beltrami operator or adding lower order terms (such as a potential, or a coupling with a magnetic field). But for sake of discussion let us work with the classical wave equation on ${{\bf R}^d}$. We will work formally in this post, being unconcerned with issues of convergence, justifying interchange of integrals, derivatives, or limits, etc.. One then has a conserved energy

$\displaystyle \int_{{\bf R}^d} \frac{1}{2} |\nabla u(t,x)|^2 + \frac{1}{2} |\partial_t u(t,x)|^2\ dx$

which we can rewrite using integration by parts and the ${L^2}$ inner product ${\langle, \rangle}$ on ${{\bf R}^d}$ as

$\displaystyle \frac{1}{2} \langle -\Delta u(t), u(t) \rangle + \frac{1}{2} \langle \partial_t u(t), \partial_t u(t) \rangle.$

A key feature of the wave equation is finite speed of propagation: if, at time ${t=0}$ (say), the initial position ${u(0)}$ and initial velocity ${\partial_t u(0)}$ are both supported in a ball ${B(x_0,R) := \{ x \in {\bf R}^d: |x-x_0| \leq R \}}$, then at any later time ${t>0}$, the position ${u(t)}$ and velocity ${\partial_t u(t)}$ are supported in the larger ball ${B(x_0,R+t)}$. This can be seen for instance (formally, at least) by inspecting the exterior energy

$\displaystyle \int_{|x-x_0| > R+t} \frac{1}{2} |\nabla u(t,x)|^2 + \frac{1}{2} |\partial_t u(t,x)|^2\ dx$

and observing (after some integration by parts and differentiation under the integral sign) that it is non-increasing in time, non-negative, and vanishing at time ${t=0}$.

The wave equation is second order in time, but one can turn it into a first order system by working with the pair ${(u(t),v(t))}$ rather than just the single field ${u(t)}$, where ${v(t) := \partial_t u(t)}$ is the velocity field. The system is then

$\displaystyle \partial_t u(t) = v(t)$

$\displaystyle \partial_t v(t) = \Delta u(t)$

and the conserved energy is now

$\displaystyle \frac{1}{2} \langle -\Delta u(t), u(t) \rangle + \frac{1}{2} \langle v(t), v(t) \rangle. \ \ \ \ \ (1)$

Finite speed of propagation then tells us that if ${u(0),v(0)}$ are both supported on ${B(x_0,R)}$, then ${u(t),v(t)}$ are supported on ${B(x_0,R+t)}$ for all ${t>0}$. One also has time reversal symmetry: if ${t \mapsto (u(t),v(t))}$ is a solution, then ${t \mapsto (u(-t), -v(-t))}$ is a solution also, thus for instance one can establish an analogue of finite speed of propagation for negative times ${t<0}$ using this symmetry.

If one has an eigenfunction

$\displaystyle -\Delta \phi = \lambda^2 \phi$

of the Laplacian, then we have the explicit solutions

$\displaystyle u(t) = e^{\pm it \lambda} \phi$

$\displaystyle v(t) = \pm i \lambda e^{\pm it \lambda} \phi$

of the wave equation, which formally can be used to construct all other solutions via the principle of superposition.

When one has vanishing initial velocity ${v(0)=0}$, the solution ${u(t)}$ is given via functional calculus by

$\displaystyle u(t) = \cos(t \sqrt{-\Delta}) u(0)$

and the propagator ${\cos(t \sqrt{-\Delta})}$ can be expressed as the average of half-wave operators:

$\displaystyle \cos(t \sqrt{-\Delta}) = \frac{1}{2} ( e^{it\sqrt{-\Delta}} + e^{-it\sqrt{-\Delta}} ).$

One can view ${\cos(t \sqrt{-\Delta} )}$ as a minor of the full wave propagator

$\displaystyle U(t) := \exp \begin{pmatrix} 0 & t \\ t\Delta & 0 \end{pmatrix}$

$\displaystyle = \begin{pmatrix} \cos(t \sqrt{-\Delta}) & \frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}} \\ \sin(t\sqrt{-\Delta}) \sqrt{-\Delta} & \cos(t \sqrt{-\Delta} ) \end{pmatrix}$

which is unitary with respect to the energy form (1), and is the fundamental solution to the wave equation in the sense that

$\displaystyle \begin{pmatrix} u(t) \\ v(t) \end{pmatrix} = U(t) \begin{pmatrix} u(0) \\ v(0) \end{pmatrix}. \ \ \ \ \ (2)$

Viewing the contraction ${\cos(t\sqrt{-\Delta})}$ as a minor of a unitary operator is an instance of the “dilation trick“.

It turns out (as I learned from Yuval Peres) that there is a useful discrete analogue of the wave equation (and of all of the above facts), in which the time variable ${t}$ now lives on the integers ${{\bf Z}}$ rather than on ${{\bf R}}$, and the spatial domain can be replaced by discrete domains also (such as graphs). Formally, the system is now of the form

$\displaystyle u(t+1) = P u(t) + v(t) \ \ \ \ \ (3)$

$\displaystyle v(t+1) = P v(t) - (1-P^2) u(t)$

where ${t}$ is now an integer, ${u(t), v(t)}$ take values in some Hilbert space (e.g. ${\ell^2}$ functions on a graph ${G}$), and ${P}$ is some operator on that Hilbert space (which in applications will usually be a self-adjoint contraction). To connect this with the classical wave equation, let us first consider a rescaling of this system

$\displaystyle u(t+\varepsilon) = P_\varepsilon u(t) + \varepsilon v(t)$

$\displaystyle v(t+\varepsilon) = P_\varepsilon v(t) - \frac{1}{\varepsilon} (1-P_\varepsilon^2) u(t)$

where ${\varepsilon>0}$ is a small parameter (representing the discretised time step), ${t}$ now takes values in the integer multiples ${\varepsilon {\bf Z}}$ of ${\varepsilon}$, and ${P_\varepsilon}$ is the wave propagator operator ${P_\varepsilon := \cos( \varepsilon \sqrt{-\Delta} )}$ or the heat propagator ${P_\varepsilon := \exp( - \varepsilon^2 \Delta/2 )}$ (the two operators are different, but agree to fourth order in ${\varepsilon}$). One can then formally verify that the wave equation emerges from this rescaled system in the limit ${\varepsilon \rightarrow 0}$. (Thus, ${P}$ is not exactly the direct analogue of the Laplacian ${\Delta}$, but can be viewed as something like ${P_\varepsilon = 1 - \frac{\varepsilon^2}{2} \Delta + O( \varepsilon^4 )}$ in the case of small ${\varepsilon}$, or ${P = 1 - \frac{1}{2}\Delta + O(\Delta^2)}$ if we are not rescaling to the small ${\varepsilon}$ case. The operator ${P}$ is sometimes known as the diffusion operator)

Assuming ${P}$ is self-adjoint, solutions to the system (3) formally conserve the energy

$\displaystyle \frac{1}{2} \langle (1-P^2) u(t), u(t) \rangle + \frac{1}{2} \langle v(t), v(t) \rangle. \ \ \ \ \ (4)$

This energy is positive semi-definite if ${P}$ is a contraction. We have the same time reversal symmetry as before: if ${t \mapsto (u(t),v(t))}$ solves the system (3), then so does ${t \mapsto (u(-t), -v(-t))}$. If one has an eigenfunction

$\displaystyle P \phi = \cos(\lambda) \phi$

to the operator ${P}$, then one has an explicit solution

$\displaystyle u(t) = e^{\pm it \lambda} \phi$

$\displaystyle v(t) = \pm i \sin(\lambda) e^{\pm it \lambda} \phi$

to (3), and (in principle at least) this generates all other solutions via the principle of superposition.

Finite speed of propagation is a lot easier in the discrete setting, though one has to offset the support of the “velocity” field ${v}$ by one unit. Suppose we know that ${P}$ has unit speed in the sense that whenever ${f}$ is supported in a ball ${B(x,R)}$, then ${Pf}$ is supported in the ball ${B(x,R+1)}$. Then an easy induction shows that if ${u(0), v(0)}$ are supported in ${B(x_0,R), B(x_0,R+1)}$ respectively, then ${u(t), v(t)}$ are supported in ${B(x_0,R+t), B(x_0, R+t+1)}$.

The fundamental solution ${U(t) = U^t}$ to the discretised wave equation (3), in the sense of (2), is given by the formula

$\displaystyle U(t) = U^t = \begin{pmatrix} P & 1 \\ P^2-1 & P \end{pmatrix}^t$

$\displaystyle = \begin{pmatrix} T_t(P) & U_{t-1}(P) \\ (P^2-1) U_{t-1}(P) & T_t(P) \end{pmatrix}$

where ${T_t}$ and ${U_t}$ are the Chebyshev polynomials of the first and second kind, thus

$\displaystyle T_t( \cos \theta ) = \cos(t\theta)$

and

$\displaystyle U_t( \cos \theta ) = \frac{\sin((t+1)\theta)}{\sin \theta}.$

In particular, ${P}$ is now a minor of ${U(1) = U}$, and can also be viewed as an average of ${U}$ with its inverse ${U^{-1}}$:

$\displaystyle \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} = \frac{1}{2} (U + U^{-1}). \ \ \ \ \ (5)$

As before, ${U}$ is unitary with respect to the energy form (4), so this is another instance of the dilation trick in action. The powers ${P^n}$ and ${U^n}$ are discrete analogues of the heat propagators ${e^{t\Delta/2}}$ and wave propagators ${U(t)}$ respectively.

One nice application of all this formalism, which I learned from Yuval Peres, is the Varopoulos-Carne inequality:

Theorem 1 (Varopoulos-Carne inequality) Let ${G}$ be a (possibly infinite) regular graph, let ${n \geq 1}$, and let ${x, y}$ be vertices in ${G}$. Then the probability that the simple random walk at ${x}$ lands at ${y}$ at time ${n}$ is at most ${2 \exp( - d(x,y)^2 / 2n )}$, where ${d}$ is the graph distance.

This general inequality is quite sharp, as one can see using the standard Cayley graph on the integers ${{\bf Z}}$. Very roughly speaking, it asserts that on a regular graph of reasonably controlled growth (e.g. polynomial growth), random walks of length ${n}$ concentrate on the ball of radius ${O(\sqrt{n})}$ or so centred at the origin of the random walk.

Proof: Let ${P \colon \ell^2(G) \rightarrow \ell^2(G)}$ be the graph Laplacian, thus

$\displaystyle Pf(x) = \frac{1}{D} \sum_{y \sim x} f(y)$

for any ${f \in \ell^2(G)}$, where ${D}$ is the degree of the regular graph and sum is over the ${D}$ vertices ${y}$ that are adjacent to ${x}$. This is a contraction of unit speed, and the probability that the random walk at ${x}$ lands at ${y}$ at time ${n}$ is

$\displaystyle \langle P^n \delta_x, \delta_y \rangle$

where ${\delta_x, \delta_y}$ are the Dirac deltas at ${x,y}$. Using (5), we can rewrite this as

$\displaystyle \langle (\frac{1}{2} (U + U^{-1}))^n \begin{pmatrix} 0 \\ \delta_x\end{pmatrix}, \begin{pmatrix} 0 \\ \delta_y\end{pmatrix} \rangle$

where we are now using the energy form (4). We can write

$\displaystyle (\frac{1}{2} (U + U^{-1}))^n = {\bf E} U^{S_n}$

where ${S_n}$ is the simple random walk of length ${n}$ on the integers, that is to say ${S_n = \xi_1 + \dots + \xi_n}$ where ${\xi_1,\dots,\xi_n = \pm 1}$ are independent uniform Bernoulli signs. Thus we wish to show that

$\displaystyle {\bf E} \langle U^{S_n} \begin{pmatrix} 0 \\ \delta_x\end{pmatrix}, \begin{pmatrix} 0 \\ \delta_y\end{pmatrix} \rangle \leq 2 \exp(-d(x,y)^2 / 2n ).$

By finite speed of propagation, the inner product here vanishes if ${|S_n| < d(x,y)}$. For ${|S_n| \geq d(x,y)}$ we can use Cauchy-Schwarz and the unitary nature of ${U}$ to bound the inner product by ${1}$. Thus the left-hand side may be upper bounded by

$\displaystyle {\bf P}( |S_n| \geq d(x,y) )$

and the claim now follows from the Chernoff inequality. $\Box$

This inequality has many applications, particularly with regards to relating the entropy, mixing time, and concentration of random walks with volume growth of balls; see this text of Lyons and Peres for some examples.

For sake of comparison, here is a continuous counterpart to the Varopoulos-Carne inequality:

Theorem 2 (Continuous Varopoulos-Carne inequality) Let ${t > 0}$, and let ${f,g \in L^2({\bf R}^d)}$ be supported on compact sets ${F,G}$ respectively. Then

$\displaystyle |\langle e^{t\Delta/2} f, g \rangle| \leq \sqrt{\frac{2t}{\pi d(F,G)^2}} \exp( - d(F,G)^2 / 2t ) \|f\|_{L^2} \|g\|_{L^2}$

where ${d(F,G)}$ is the Euclidean distance between ${F}$ and ${G}$.

Proof: By Fourier inversion one has

$\displaystyle e^{-t\xi^2/2} = \frac{1}{\sqrt{2\pi t}} \int_{\bf R} e^{-s^2/2t} e^{is\xi}\ ds$

$\displaystyle = \sqrt{\frac{2}{\pi t}} \int_0^\infty e^{-s^2/2t} \cos(s \xi )\ ds$

for any real ${\xi}$, and thus

$\displaystyle \langle e^{t\Delta/2} f, g\rangle = \sqrt{\frac{2}{\pi}} \int_0^\infty e^{-s^2/2t} \langle \cos(s \sqrt{-\Delta} ) f, g \rangle\ ds.$

By finite speed of propagation, the inner product ${\langle \cos(s \sqrt{-\Delta} ) f, g \rangle\ ds}$ vanishes when ${s < d(F,G)}$; otherwise, we can use Cauchy-Schwarz and the contractive nature of ${\cos(s \sqrt{-\Delta} )}$ to bound this inner product by ${\|f\|_{L^2} \|g\|_{L^2}}$. Thus

$\displaystyle |\langle e^{t\Delta/2} f, g\rangle| \leq \sqrt{\frac{2}{\pi t}} \|f\|_{L^2} \|g\|_{L^2} \int_{d(F,G)}^\infty e^{-s^2/2t}\ ds.$

Bounding ${e^{-s^2/2t}}$ by ${e^{-d(F,G)^2/2t} e^{-d(F,G) (s-d(F,G))/t}}$, we obtain the claim. $\Box$

Observe that the argument is quite general and can be applied for instance to other Riemannian manifolds than ${{\bf R}^d}$.

Many fluid equations are expected to exhibit turbulence in their solutions, in which a significant portion of their energy ends up in high frequency modes. A typical example arises from the three-dimensional periodic Navier-Stokes equations

$\displaystyle \partial_t u + u \cdot \nabla u = \nu \Delta u + \nabla p + f$

$\displaystyle \nabla \cdot u = 0$

where ${u: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}^3}$ is the velocity field, ${f: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}^3}$ is a forcing term, ${p: {\bf R} \times {\bf R}^3/{\bf Z}^3 \rightarrow {\bf R}}$ is a pressure field, and ${\nu > 0}$ is the viscosity. To study the dynamics of energy for this system, we first pass to the Fourier transform

$\displaystyle \hat u(t,k) := \int_{{\bf R}^3/{\bf Z}^3} u(t,x) e^{-2\pi i k \cdot x}$

so that the system becomes

$\displaystyle \partial_t \hat u(t,k) + 2\pi \sum_{k = k_1 + k_2} (\hat u(t,k_1) \cdot ik_2) \hat u(t,k_2) =$

$\displaystyle - 4\pi^2 \nu |k|^2 \hat u(t,k) + 2\pi ik \hat p(t,k) + \hat f(t,k) \ \ \ \ \ (1)$

$\displaystyle k \cdot \hat u(t,k) = 0.$

We may normalise ${u}$ (and ${f}$) to have mean zero, so that ${\hat u(t,0)=0}$. Then we introduce the dyadic energies

$\displaystyle E_N(t) := \sum_{|k| \sim N} |\hat u(t,k)|^2$

where ${N \geq 1}$ ranges over the powers of two, and ${|k| \sim N}$ is shorthand for ${N \leq |k| < 2N}$. Taking the inner product of (1) with ${\hat u(t,k)}$, we obtain the energy flow equation

$\displaystyle \partial_t E_N = \sum_{N_1,N_2} \Pi_{N,N_1,N_2} - D_N + F_N \ \ \ \ \ (2)$

where ${N_1,N_2}$ range over powers of two, ${\Pi_{N,N_1,N_2}}$ is the energy flow rate

$\displaystyle \Pi_{N,N_1,N_2} := -2\pi \sum_{k=k_1+k_2: |k| \sim N, |k_1| \sim N_1, |k_2| \sim N_2}$

$\displaystyle (\hat u(t,k_1) \cdot ik_2) (\hat u(t,k) \cdot \hat u(t,k_2)),$

${D_N}$ is the energy dissipation rate

$\displaystyle D_N := 4\pi^2 \nu \sum_{|k| \sim N} |k|^2 |\hat u(t,k)|^2$

and ${F_N}$ is the energy injection rate

$\displaystyle F_N := \sum_{|k| \sim N} \hat u(t,k) \cdot \hat f(t,k).$

The Navier-Stokes equations are notoriously difficult to solve in general. Despite this, Kolmogorov in 1941 was able to give a convincing heuristic argument for what the distribution of the dyadic energies ${E_N}$ should become over long times, assuming that some sort of distributional steady state is reached. It is common to present this argument in the form of dimensional analysis, but one can also give a more “first principles” form Kolmogorov’s argument, which I will do here. Heuristically, one can divide the frequency scales ${N}$ into three regimes:

• The injection regime in which the energy injection rate ${F_N}$ dominates the right-hand side of (2);
• The energy flow regime in which the flow rates ${\Pi_{N,N_1,N_2}}$ dominate the right-hand side of (2); and
• The dissipation regime in which the dissipation ${D_N}$ dominates the right-hand side of (2).

If we assume a fairly steady and smooth forcing term ${f}$, then ${\hat f}$ will be supported on the low frequency modes ${k=O(1)}$, and so we heuristically expect the injection regime to consist of the low scales ${N=O(1)}$. Conversely, if we take the viscosity ${\nu}$ to be small, we expect the dissipation regime to only occur for very large frequencies ${N}$, with the energy flow regime occupying the intermediate frequencies.

We can heuristically predict the dividing line between the energy flow regime. Of all the flow rates ${\Pi_{N,N_1,N_2}}$, it turns out in practice that the terms in which ${N_1,N_2 = N+O(1)}$ (i.e., interactions between comparable scales, rather than widely separated scales) will dominate the other flow rates, so we will focus just on these terms. It is convenient to return back to physical space, decomposing the velocity field ${u}$ into Littlewood-Paley components

$\displaystyle u_N(t,x) := \sum_{|k| \sim N} \hat u(t,k) e^{2\pi i k \cdot x}$

of the velocity field ${u(t,x)}$ at frequency ${N}$. By Plancherel’s theorem, this field will have an ${L^2}$ norm of ${E_N(t)^{1/2}}$, and as a naive model of turbulence we expect this field to be spread out more or less uniformly on the torus, so we have the heuristic

$\displaystyle |u_N(t,x)| = O( E_N(t)^{1/2} ),$

and a similar heuristic applied to ${\nabla u_N}$ gives

$\displaystyle |\nabla u_N(t,x)| = O( N E_N(t)^{1/2} ).$

(One can consider modifications of the Kolmogorov model in which ${u_N}$ is concentrated on a lower-dimensional subset of the three-dimensional torus, leading to some changes in the numerology below, but we will not consider such variants here.) Since

$\displaystyle \Pi_{N,N_1,N_2} = - \int_{{\bf R}^3/{\bf Z}^3} u_N \cdot ( (u_{N_1} \cdot \nabla) u_{N_2} )\ dx$

we thus arrive at the heuristic

$\displaystyle \Pi_{N,N_1,N_2} = O( N_2 E_N^{1/2} E_{N_1}^{1/2} E_{N_2}^{1/2} ).$

Of course, there is the possibility that due to significant cancellation, the energy flow is significantly less than ${O( N E_N(t)^{3/2} )}$, but we will assume that cancellation effects are not that significant, so that we typically have

$\displaystyle \Pi_{N,N_1,N_2} \sim N_2 E_N^{1/2} E_{N_1}^{1/2} E_{N_2}^{1/2} \ \ \ \ \ (3)$

or (assuming that ${E_N}$ does not oscillate too much in ${N}$, and ${N_1,N_2}$ are close to ${N}$)

$\displaystyle \Pi_{N,N_1,N_2} \sim N E_N^{3/2}.$

On the other hand, we clearly have

$\displaystyle D_N \sim \nu N^2 E_N.$

We thus expect to be in the dissipation regime when

$\displaystyle N \gtrsim \nu^{-1} E_N^{1/2} \ \ \ \ \ (4)$

and in the energy flow regime when

$\displaystyle 1 \lesssim N \lesssim \nu^{-1} E_N^{1/2}. \ \ \ \ \ (5)$

Now we study the energy flow regime further. We assume a “statistically scale-invariant” dynamics in this regime, in particular assuming a power law

$\displaystyle E_N \sim A N^{-\alpha} \ \ \ \ \ (6)$

for some ${A,\alpha > 0}$. From (3), we then expect an average asymptotic of the form

$\displaystyle \Pi_{N,N_1,N_2} \approx A^{3/2} c_{N,N_1,N_2} (N N_1 N_2)^{1/3 - \alpha/2} \ \ \ \ \ (7)$

for some structure constants ${c_{N,N_1,N_2} \sim 1}$ that depend on the exact nature of the turbulence; here we have replaced the factor ${N_2}$ by the comparable term ${(N N_1 N_2)^{1/3}}$ to make things more symmetric. In order to attain a steady state in the energy flow regime, we thus need a cancellation in the structure constants:

$\displaystyle \sum_{N_1,N_2} c_{N,N_1,N_2} (N N_1 N_2)^{1/3 - \alpha/2} \approx 0. \ \ \ \ \ (8)$

On the other hand, if one is assuming statistical scale invariance, we expect the structure constants to be scale-invariant (in the energy flow regime), in that

$\displaystyle c_{\lambda N, \lambda N_1, \lambda N_2} = c_{N,N_1,N_2} \ \ \ \ \ (9)$

for dyadic ${\lambda > 0}$. Also, since the Euler equations conserve energy, the energy flows ${\Pi_{N,N_1,N_2}}$ symmetrise to zero,

$\displaystyle \Pi_{N,N_1,N_2} + \Pi_{N,N_2,N_1} + \Pi_{N_1,N,N_2} + \Pi_{N_1,N_2,N} + \Pi_{N_2,N,N_1} + \Pi_{N_2,N_1,N} = 0,$

which from (7) suggests a similar cancellation among the structure constants

$\displaystyle c_{N,N_1,N_2} + c_{N,N_2,N_1} + c_{N_1,N,N_2} + c_{N_1,N_2,N} + c_{N_2,N,N_1} + c_{N_2,N_1,N} \approx 0.$

Combining this with the scale-invariance (9), we see that for fixed ${N}$, we may organise the structure constants ${c_{N,N_1,N_2}}$ for dyadic ${N_1,N_2}$ into sextuples which sum to zero (including some degenerate tuples of order less than six). This will automatically guarantee the cancellation (8) required for a steady state energy distribution, provided that

$\displaystyle \frac{1}{3} - \frac{\alpha}{2} = 0$

or in other words

$\displaystyle \alpha = \frac{2}{3};$

for any other value of ${\alpha}$, there is no particular reason to expect this cancellation (8) to hold. Thus we are led to the heuristic conclusion that the most stable power law distribution for the energies ${E_N}$ is the ${2/3}$ law

$\displaystyle E_N \sim A N^{-2/3} \ \ \ \ \ (10)$

or in terms of shell energies, we have the famous Kolmogorov 5/3 law

$\displaystyle \sum_{|k| = k_0 + O(1)} |\hat u(t,k)|^2 \sim A k_0^{-5/3}.$

Given that frequency interactions tend to cascade from low frequencies to high (if only because there are so many more high frequencies than low ones), the above analysis predicts a stablising effect around this power law: scales at which a law (6) holds for some ${\alpha > 2/3}$ are likely to lose energy in the near-term, while scales at which a law (6) hold for some ${\alpha< 2/3}$ are conversely expected to gain energy, thus nudging the exponent of power law towards ${2/3}$.

We can solve for ${A}$ in terms of energy dissipation as follows. If we let ${N_*}$ be the frequency scale demarcating the transition from the energy flow regime (5) to the dissipation regime (4), we have

$\displaystyle N_* \sim \nu^{-1} E_{N_*}$

and hence by (10)

$\displaystyle N_* \sim \nu^{-1} A N_*^{-2/3}.$

On the other hand, if we let ${\epsilon := D_{N_*}}$ be the energy dissipation at this scale ${N_*}$ (which we expect to be the dominant scale of energy dissipation), we have

$\displaystyle \epsilon \sim \nu N_*^2 E_N \sim \nu N_*^2 A N_*^{-2/3}.$

Some simple algebra then lets us solve for ${A}$ and ${N_*}$ as

$\displaystyle N_* \sim (\frac{\epsilon}{\nu^3})^{1/4}$

and

$\displaystyle A \sim \epsilon^{2/3}.$

Thus, we have the Kolmogorov prediction

$\displaystyle \sum_{|k| = k_0 + O(1)} |\hat u(t,k)|^2 \sim \epsilon^{2/3} k_0^{-5/3}$

for

$\displaystyle 1 \lesssim k_0 \lesssim (\frac{\epsilon}{\nu^3})^{1/4}$

with energy dissipation occuring at the high end ${k_0 \sim (\frac{\epsilon}{\nu^3})^{1/4}}$ of this scale, which is counterbalanced by the energy injection at the low end ${k_0 \sim 1}$ of the scale.

As in the previous post, all computations here are at the formal level only.

In the previous blog post, the Euler equations for inviscid incompressible fluid flow were interpreted in a Lagrangian fashion, and then Noether’s theorem invoked to derive the known conservation laws for these equations. In a bit more detail: starting with Lagrangian space ${{\cal L} = ({\bf R}^n, \hbox{vol})}$ and Eulerian space ${{\cal E} = ({\bf R}^n, \eta, \hbox{vol})}$, we let ${M}$ be the space of volume-preserving, orientation-preserving maps ${\Phi: {\cal L} \rightarrow {\cal E}}$ from Lagrangian space to Eulerian space. Given a curve ${\Phi: {\bf R} \rightarrow M}$, we can define the Lagrangian velocity field ${\dot \Phi: {\bf R} \times {\cal L} \rightarrow T{\cal E}}$ as the time derivative of ${\Phi}$, and the Eulerian velocity field ${u := \dot \Phi \circ \Phi^{-1}: {\bf R} \times {\cal E} \rightarrow T{\cal E}}$. The volume-preserving nature of ${\Phi}$ ensures that ${u}$ is a divergence-free vector field:

$\displaystyle \nabla \cdot u = 0. \ \ \ \ \ (1)$

If we formally define the functional

$\displaystyle J[\Phi] := \frac{1}{2} \int_{\bf R} \int_{{\cal E}} |u(t,x)|^2\ dx dt = \frac{1}{2} \int_R \int_{{\cal L}} |\dot \Phi(t,x)|^2\ dx dt$

then one can show that the critical points of this functional (with appropriate boundary conditions) obey the Euler equations

$\displaystyle [\partial_t + u \cdot \nabla] u = - \nabla p$

$\displaystyle \nabla \cdot u = 0$

for some pressure field ${p: {\bf R} \times {\cal E} \rightarrow {\bf R}}$. As discussed in the previous post, the time translation symmetry of this functional yields conservation of the Hamiltonian

$\displaystyle \frac{1}{2} \int_{{\cal E}} |u(t,x)|^2\ dx = \frac{1}{2} \int_{{\cal L}} |\dot \Phi(t,x)|^2\ dx;$

the rigid motion symmetries of Eulerian space give conservation of the total momentum

$\displaystyle \int_{{\cal E}} u(t,x)\ dx$

and total angular momentum

$\displaystyle \int_{{\cal E}} x \wedge u(t,x)\ dx;$

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

$\displaystyle \int_{\Phi(\gamma)} u^*$

for any closed loop ${\gamma}$ in ${{\cal L}}$, or equivalently pointwise conservation of the Lagrangian vorticity ${\Phi^* \omega = \Phi^* du^*}$, where ${u^*}$ is the ${1}$-form associated with the vector field ${u}$ using the Euclidean metric ${\eta}$ on ${{\cal E}}$, with ${\Phi^*}$ denoting pullback by ${\Phi}$.

It turns out that one can generalise the above calculations. Given any self-adjoint operator ${A}$ on divergence-free vector fields ${u: {\cal E} \rightarrow {\bf R}}$, we can define the functional

$\displaystyle J_A[\Phi] := \frac{1}{2} \int_{\bf R} \int_{{\cal E}} u(t,x) \cdot A u(t,x)\ dx dt;$

as we shall see below the fold, critical points of this functional (with appropriate boundary conditions) obey the generalised Euler equations

$\displaystyle [\partial_t + u \cdot \nabla] Au + (\nabla u) \cdot Au= - \nabla \tilde p \ \ \ \ \ (2)$

$\displaystyle \nabla \cdot u = 0$

for some pressure field ${\tilde p: {\bf R} \times {\cal E} \rightarrow {\bf R}}$, where ${(\nabla u) \cdot Au}$ in coordinates is ${\partial_i u_j Au_j}$ with the usual summation conventions. (When ${A=1}$, ${(\nabla u) \cdot Au = \nabla(\frac{1}{2} |u|^2)}$, and this term can be absorbed into the pressure ${\tilde p}$, and we recover the usual Euler equations.) Time translation symmetry then gives conservation of the Hamiltonian

$\displaystyle \frac{1}{2} \int_{{\cal E}} u(t,x) \cdot A u(t,x)\ dx.$

If the operator ${A}$ commutes with rigid motions on ${{\cal E}}$, then we have conservation of total momentum

$\displaystyle \int_{{\cal E}} Au(t,x)\ dx$

and total angular momentum

$\displaystyle \int_{{\cal E}} x \wedge Au(t,x)\ dx,$

and the diffeomorphism symmetries of Lagrangian space give conservation of circulation

$\displaystyle \int_{\Phi(\gamma)} (Au)^*$

or pointwise conservation of the Lagrangian vorticity ${\Phi^* \theta := \Phi^* d(Au)^*}$. These applications of Noether’s theorem proceed exactly as the previous post; we leave the details to the interested reader.

One particular special case of interest arises in two dimensions ${n=2}$, when ${A}$ is the inverse derivative ${A = |\nabla|^{-1} = (-\Delta)^{-1/2}}$. The vorticity ${\theta = d(Au)^*}$ is a ${2}$-form, which in the two-dimensional setting may be identified with a scalar. In coordinates, if we write ${u = (u_1,u_2)}$, then

$\displaystyle \theta = \partial_{x_1} |\nabla|^{-1} u_2 - \partial_{x_2} |\nabla|^{-1} u_1.$

Since ${u}$ is also divergence-free, we may therefore write

$\displaystyle u = (- \partial_{x_2} \psi, \partial_{x_1} \psi )$

where the stream function ${\psi}$ is given by the formula

$\displaystyle \psi = |\nabla|^{-1} \theta.$

If we take the curl of the generalised Euler equation (2), we obtain (after some computation) the surface quasi-geostrophic equation

$\displaystyle [\partial_t + u \cdot \nabla] \theta = 0 \ \ \ \ \ (3)$

$\displaystyle u = (-\partial_{x_2} |\nabla|^{-1} \theta, \partial_{x_1} |\nabla|^{-1} \theta).$

This equation has strong analogies with the three-dimensional incompressible Euler equations, and can be viewed as a simplified model for that system; see this paper of Constantin, Majda, and Tabak for details.

Now we can specialise the general conservation laws derived previously to this setting. The conserved Hamiltonian is

$\displaystyle \frac{1}{2} \int_{{\bf R}^2} u\cdot |\nabla|^{-1} u\ dx = \frac{1}{2} \int_{{\bf R}^2} \theta \psi\ dx = \frac{1}{2} \int_{{\bf R}^2} \theta |\nabla|^{-1} \theta\ dx$

(a law previously observed for this equation in the abovementioned paper of Constantin, Majda, and Tabak). As ${A}$ commutes with rigid motions, we also have (formally, at least) conservation of momentum

$\displaystyle \int_{{\bf R}^2} Au\ dx$

(which up to trivial transformations is also expressible in impulse form as ${\int_{{\bf R}^2} \theta x\ dx}$, after integration by parts), and conservation of angular momentum

$\displaystyle \int_{{\bf R}^2} x \wedge Au\ dx$

(which up to trivial transformations is ${\int_{{\bf R}^2} \theta |x|^2\ dx}$). Finally, diffeomorphism invariance gives pointwise conservation of Lagrangian vorticity ${\Phi^* \theta}$, thus ${\theta}$ is transported by the flow (which is also evident from (3). In particular, all integrals of the form ${\int F(\theta)\ dx}$ for a fixed function ${F}$ are conserved by the flow.