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

where is the velocity field, and 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 or 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

where is the active scalar, and 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 . Indeed, taking the curl of (1), we obtain the vorticity equation

while the velocity can be recovered from the vorticity via the Biot-Savart law

The system (4), (5) has some features in common with the system (2), (3); in (2) it is a scalar field that is being transported by a divergence-free vector field , which is a linear function of the scalar field as per (3), whereas in (4) it is a vector field that is being transported (in the Lie derivative sense) by a divergence-free vector field , 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 for SQG and a vector field 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 order operator in (5).

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

where 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 is the identity operator. As discussed in this previous blog post (which used to denote the inverse of the operator denoted here as ), this generalised Euler system has many of the same features as the original Euler equation, such as a conserved Hamiltonian

the Kelvin circulation theorem, and conservation of helicity

Also, if we require 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 do not depend on the vertical coordinate , thus

and

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

where is the two-dimensional material derivative

Also, divergence-free nature of then becomes

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

for some scalar field , so that (7) becomes

The first two components of (8) become

which rearranges using (9) to

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

Finally, the last component of (8) is

At this point, we make the following choice for :

where is a real constant and is the Leray projection onto divergence-free vector fields. One can verify that for large enough , 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

so that (12) simplifies to

This implies (formally at least) that if vanishes at time zero, then it vanishes for all time. Setting , we then have from (10) that

and from (11) we then recover the SQG system (2), (3). To put it another way, if and solve the SQG system, then by setting

then solve the modified Euler system (6), (7) with given by (13).

We have , so the Hamiltonian for the modified Euler system in this case is formally a scalar multiple of the conserved quantity . The momentum for the modified Euler system is formally a scalar multiple of the conserved quantity , 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 vanishes, and other conserved quantities for SQG (such as the Hamiltonian ) 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.

An extremely large portion of mathematics is concerned with locating solutions to equations such as

for in some suitable domain space (either finite-dimensional or infinite-dimensional), and various maps or . To solve the fixed point iteration equation (1), the simplest general method available is the fixed point iteration method: one starts with an initial *approximate solution* to (1), so that , and then recursively constructs the sequence by . If behaves enough like a “contraction”, and the domain is complete, then one can expect the to converge to a limit , which should then be a solution to (1). For instance, if is a map from a metric space to itself, which is a contraction in the sense that

for all and some , then with as above we have

for any , and so the distances between successive elements of the sequence decay at at least a geometric rate. This leads to the contraction mapping theorem, which has many important consequences, such as the inverse function theorem and the Picard existence theorem.

A slightly more complicated instance of this strategy arises when trying to *linearise* a complex map defined in a neighbourhood of a fixed point. For simplicity we normalise the fixed point to be the origin, thus and . When studying the complex dynamics , , of such a map, it can be useful to try to conjugate to another function , where is a holomorphic function defined and invertible near with , since the dynamics of will be conjguate to that of . Note that if and , then from the chain rule any conjugate of will also have and . Thus, the “simplest” function one can hope to conjugate to is the linear function . Let us say that is *linearisable* (around ) if it is conjugate to in some neighbourhood of . Equivalently, is linearisable if there is a solution to the Schröder equation

for some defined and invertible in a neighbourhood of with , and all sufficiently close to . (The Schröder equation is normalised somewhat differently in the literature, but this form is equivalent to the usual form, at least when is non-zero.) Note that if solves the above equation, then so does for any non-zero , so we may normalise in addition to , which also ensures local invertibility from the inverse function theorem. (Note from winding number considerations that cannot be invertible near zero if vanishes.)

We have the following basic result of Koenigs:

Theorem 1 (Koenig’s linearisation theorem)Let be a holomorphic function defined near with and . If (attracting case) or (repelling case), then is linearisable near zero.

*Proof:* Observe that if solve (2), then solve (2) also (in a sufficiently small neighbourhood of zero). Thus we may reduce to the attractive case .

Let be a sufficiently small radius, and let denote the space of holomorphic functions on the complex disk with and . We can view the Schröder equation (2) as a fixed point equation

where is the partially defined function on that maps a function to the function defined by

assuming that is well-defined on the range of (this is why is only partially defined).

We can solve this equation by the fixed point iteration method, if is small enough. Namely, we start with being the identity map, and set , etc. We equip with the uniform metric . Observe that if , and is small enough, then takes values in , and are well-defined and lie in . Also, since is smooth and has derivative at , we have

if , and is sufficiently small depending on . This is not yet enough to establish the required contraction (thanks to Mario Bonk for pointing this out); but observe that the function is holomorphic on and bounded by on the boundary of this ball (or slightly within this boundary), so by the maximum principle we see that

on all of , and in particular

on . Putting all this together, we see that

since , we thus obtain a contraction on the ball if is small enough (and sufficiently small depending on ). From this (and the completeness of , which follows from Morera’s theorem) we see that the iteration converges (exponentially fast) to a limit which is a fixed point of , and thus solves Schröder’s equation, as required.

Koenig’s linearisation theorem leaves open the *indifferent case* when . In the *rationally indifferent* case when for some natural number , there is an obvious obstruction to linearisability, namely that (in particular, linearisation is not possible in this case when is a non-trivial rational function). An obstruction is also present in some *irrationally indifferent* cases (where but for any natural number ), if is sufficiently close to various roots of unity; the first result of this form is due to Cremer, and the optimal result of this type for quadratic maps was established by Yoccoz. In the other direction, we have the following result of Siegel:

Theorem 2 (Siegel’s linearisation theorem)Let be a holomorphic function defined near with and . If and one has the Diophantine condition for all natural numbers and some constant , then is linearisable at .

The Diophantine condition can be relaxed to a more general condition involving the rational exponents of the phase of ; this was worked out by Brjuno, with the condition matching the one later obtained by Yoccoz. Amusingly, while the set of Diophantine numbers (and hence the set of linearisable ) has full measure on the unit circle, the set of non-linearisable is generic (the complement of countably many nowhere dense sets) due to the above-mentioned work of Cremer, leading to a striking disparity between the measure-theoretic and category notions of “largeness”.

Siegel’s theorem does not seem to be provable using a fixed point iteration method. However, it can be established by modifying another basic method to solve equations, namely Newton’s method. Let us first review how this method works to solve the equation for some smooth function defined on an interval . We suppose we have some initial approximant to this equation, with small but not necessarily zero. To make the analysis more quantitative, let us suppose that the interval lies in for some , and we have the estimates

for some and and all (the factors of are present to make “dimensionless”).

Lemma 3Under the above hypotheses, we can find with such thatIn particular, setting , , and , we have , and

for all .

The crucial point here is that the new error is roughly the square of the previous error . This leads to extremely fast (double-exponential) improvement in the error upon iteration, which is more than enough to absorb the exponential losses coming from the factor.

*Proof:* If for some absolute constants then we may simply take , so we may assume that for some small and large . Using the Newton approximation we are led to the choice

for . From the hypotheses on and the smallness hypothesis on we certainly have . From Taylor’s theorem with remainder we have

and the claim follows.

We can iterate this procedure; starting with as above, we obtain a sequence of nested intervals with , and with evolving by the recursive equations and estimates

If is sufficiently small depending on , we see that converges rapidly to zero (indeed, we can inductively obtain a bound of the form for some large absolute constant if is small enough), and converges to a limit which then solves the equation by the continuity of .

As I recently learned from Zhiqiang Li, a similar scheme works to prove Siegel’s theorem, as can be found for instance in this text of Carleson and Gamelin. The key is the following analogue of Lemma 3.

Lemma 4Let be a complex number with and for all natural numbers . Let , and let be a holomorphic function with , , andfor all and some . Let , and set . Then there exists an injective holomorphic function and a holomorphic function such that

and

for all and some .

*Proof:* By scaling we may normalise . If for some constants , then we can simply take to be the identity and , so we may assume that for some small and large .

To motivate the choice of , we write and , with and viewed as small. We would like to have , which expands as

As and are both small, we can heuristically approximate up to quadratic errors (compare with the Newton approximation ), and arrive at the equation

This equation can be solved by Taylor series; the function vanishes to second order at the origin and thus has a Taylor expansion

and then has a Taylor expansion

We take this as our definition of , define , and then define implicitly via (4).

Let us now justify that this choice works. By (3) and the generalised Cauchy integral formula, we have for all ; by the Diophantine assumption on , we thus have . In particular, converges on , and on the disk (say) we have the bounds

In particular, as is so small, we see that maps injectively to and to , and the inverse maps to . From (3) we see that maps to , and so if we set to be the function , then is a holomorphic function obeying (4). Expanding (4) in terms of and as before, and also writing , we have

for , which by (5) simplifies to

From (6), the fundamental theorem of calculus, and the smallness of we have

and thus

From (3) and the Cauchy integral formula we have on (say) , and so from (6) and the fundamental theorem of calculus we conclude that

on , and the claim follows.

If we set , , and to be sufficiently small, then (since vanishes to second order at the origin), the hypotheses of this lemma will be obeyed for some sufficiently small . Iterating the lemma (and halving repeatedly), we can then find sequences , injective holomorphic functions and holomorphic functions such that one has the recursive identities and estimates

for all and . By construction, decreases to a positive radius that is a constant multiple of , while (for small enough) converges double-exponentially to zero, so in particular converges uniformly to on . Also, is close enough to the identity, the compositions are uniformly convergent on with and . From this we have

on , and on taking limits using Morera’s theorem we obtain a holomorphic function defined near with , , and

obtaining the required linearisation.

Remark 5The idea of using a Newton-type method to obtain error terms that decay double-exponentially, and can therefore absorb exponential losses in the iteration, also occurs in KAM theory and in Nash-Moser iteration, presumably due to Siegel’s influence on Moser. (I discuss Nash-Moser iteration in this note that I wrote back in 2006.)

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space , and is a vector in that Hilbert space, then one has

in the strong topology, where is the -invariant subspace of , and is the orthogonal projection to . (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if is a countable amenable group acting on a Hilbert space by unitary transformations for , and is a vector in that Hilbert space, then one has

for any Folner sequence of , where is the -invariant subspace, and is the average of on . Thus one can interpret as a certain average of elements of the orbit of .

In a previous blog post, I noted a variant of this ergodic theorem (due to Alaoglu and Birkhoff) that holds even when the group is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem)Let be an arbitrary group acting unitarily on a Hilbert space , and let be a vector in . Then is the element in the closed convex hull of of minimal norm, and is also the unique element of in this closed convex hull.

I recently stumbled upon a different way to think about this theorem, in the additive case when is abelian, which has a closer resemblance to the classical mean ergodic theorem. Given an arbitrary additive group (not necessarily discrete, or countable), let denote the collection of finite non-empty multisets in – that is to say, unordered collections of elements of , not necessarily distinct, for some positive integer . Given two multisets , in , we can form the sum set . Note that the sum set can contain multiplicity even when do not; for instance, . Given a multiset in , and a function from to a vector space , we define the average as

Note that the multiplicity function of the set affects the average; for instance, we have , but .

We can define a directed set on as follows: given two multisets , we write if we have for some . Thus for instance we have . It is easy to verify that this operation is transitive and reflexive, and is directed because any two elements of have a common upper bound, namely . (This is where we need to be abelian.) The notion of convergence along a net, now allows us to define the notion of convergence along ; given a family of points in a topological space indexed by elements of , and a point in , we say that *converges* to along if, for every open neighbourhood of in , one has for sufficiently large , that is to say there exists such that for all . If the topological space is Hausdorff, then the limit is unique (if it exists), and we then write

When takes values in the reals, one can also define the limit superior or limit inferior along such nets in the obvious fashion.

We can then give an alternate formulation of the abstract ergodic theorem in the abelian case:

Theorem 2 (Abelian abstract ergodic theorem)Let be an arbitrary additive group acting unitarily on a Hilbert space , and let be a vector in . Then we havein the strong topology of .

*Proof:* Suppose that , so that for some , then

so by unitarity and the triangle inequality we have

thus is monotone non-increasing in . Since this quantity is bounded between and , we conclude that the limit exists. Thus, for any , we have for sufficiently large that

for all . In particular, for any , we have

We can write

and so from the parallelogram law and unitarity we have

for all , and hence by the triangle inequality (averaging over a finite multiset )

for any . This shows that is a Cauchy sequence in (in the strong topology), and hence (by the completeness of ) tends to a limit. Shifting by a group element , we have

and hence is invariant under shifts, and thus lies in . On the other hand, for any and , we have

and thus on taking strong limits

and so is orthogonal to . Combining these two facts we see that is equal to as claimed.

To relate this result to the classical ergodic theorem, we observe

Lemma 3Let be a countable additive group, with a F{\o}lner sequence , and let be a bounded sequence in a normed vector space indexed by . If exists, then exists, and the two limits are equal.

*Proof:* From the F{\o}lner property, we see that for any and any , the averages and differ by at most in norm if is sufficiently large depending on , (and the ). On the other hand, by the existence of the limit , the averages and differ by at most in norm if is sufficiently large depending on (regardless of how large is). The claim follows.

It turns out that this approach can also be used as an alternate way to construct the Gowers–Host-Kra seminorms in ergodic theory, which has the feature that it does not explicitly require any amenability on the group (or separability on the underlying measure space), though, as pointed out to me in comments, even uncountable abelian groups are amenable in the sense of possessing an invariant mean, even if they do not have a F{\o}lner sequence.

Given an arbitrary additive group , define a *-system* to be a probability space (not necessarily separable or standard Borel), together with a collection of invertible, measure-preserving maps, such that is the identity and (modulo null sets) for all . This then gives isomorphisms for by setting . From the above abstract ergodic theorem, we see that

in the strong topology of for any , where is the collection of measurable sets that are essentially -invariant in the sense that modulo null sets for all , and is the conditional expectation of with respect to .

In a similar spirit, we have

Theorem 4 (Convergence of Gowers-Host-Kra seminorms)Let be a -system for some additive group . Let be a natural number, and for every , let , which for simplicity we take to be real-valued. Then the expressionconverges, where we write , and we are using the product direct set on to define the convergence . In particular, for , the limit

converges.

We prove this theorem below the fold. It implies a number of other known descriptions of the Gowers-Host-Kra seminorms , for instance that

for , while from the ergodic theorem we have

This definition also manifestly demonstrates the cube symmetries of the Host-Kra measures on , defined via duality by requiring that

In a subsequent blog post I hope to present a more detailed study of the norm and its relationship with eigenfunctions and the Kronecker factor, without assuming any amenability on or any separability or topological structure on .

Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper “Random matrices: tail bounds for gaps between eigenvalues“. This is a followup paper to my recent paper with Van in which we showed that random matrices of Wigner type (such as the adjacency matrix of an Erdös-Renyi graph) asymptotically almost surely had simple spectrum. In the current paper, we push the method further to show that the eigenvalues are not only distinct, but are (with high probability) separated from each other by some negative power of . This follows the now standard technique of replacing any appearance of discrete Littlewood-Offord theory (a key ingredient in our previous paper) with its continuous analogue (inverse theorems for small ball probability). For general Wigner-type matrices (in which the matrix entries are not normalised to have mean zero), we can use the inverse Littlewood-Offord theorem of Nguyen and Vu to obtain (under mild conditions on ) a result of the form

for any and , if is sufficiently large depending on (in a linear fashion), and is sufficiently large depending on . The point here is that can be made arbitrarily large, and also that no continuity or smoothness hypothesis is made on the distribution of the entries. (In the continuous case, one can use the machinery of Wegner estimates to obtain results of this type, as was done in a paper of Erdös, Schlein, and Yau.)

In the mean zero case, it becomes more efficient to use an inverse Littlewood-Offord theorem of Rudelson and Vershynin to obtain (with the normalisation that the entries of have unit variance, so that the eigenvalues of are with high probability), giving the bound

for (one also has good results of this type for smaller values of ). This is only optimal in the regime ; we expect to establish some eigenvalue repulsion, improving the RHS to for real matrices and for complex matrices, but this appears to be a more difficult task (possibly requiring some *quadratic* inverse Littlewood-Offord theory, rather than just *linear* inverse Littlewood-Offord theory). However, we can get some repulsion if one works with larger gaps, getting a result roughly of the form

for any fixed and some absolute constant (which we can asymptotically make to be for large , though it ought to be as large as ), by using a higher-dimensional version of the Rudelson-Vershynin inverse Littlewood-Offord theorem.

In the case of Erdös-Renyi graphs, we don’t have mean zero and the Rudelson-Vershynin Littlewood-Offord theorem isn’t quite applicable, but by working carefully through the approach based on the Nguyen-Vu theorem we can almost recover (1), except for a loss of on the RHS.

As a sample applications of the eigenvalue separation results, we can now obtain some information about *eigenvectors*; for instance, we can show that the components of the eigenvectors all have magnitude at least for some with high probability. (Eigenvectors become much more stable, and able to be studied in isolation, once their associated eigenvalue is well separated from the other eigenvalues; see this previous blog post for more discussion.)

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