On the infinite dimension limit of invariant measures and solutions of Zeitlin's 2D Euler equations

In this work we consider a finite dimensional approximation for the 2D Euler equations on the sphere, proposed by V. Zeitlin, and show their convergence towards a solution to Euler equations with marginals distributed as the enstrophy measure. The method relies on nontrivial computations on the structure constants of $\mathbb{S}^2$, that appear to be new. In the last section we discuss the problem of extending our results to Gibbsian measures associated with higher Casimirs.


Introduction
The 2D Euler equations are a fundamental mathematical model for studying ideal fluids, i.e. incompressible, inviscid, deformable bodies in which the dependence on one spatial dimension can be neglected.In particular, for barotropic incompressible fluids on some surface S embedded in the Euclidean space R 3 , the Euler equations take the simple form where ω and ψ are respectively the vorticity and the stream function, ∇ and ∆ are respectively the Riemannian gradient and the Laplace-Beltrami operator on S.
One of the most intriguing aspects of these equations is the fact that they possess an infinite amount of conserved quantities, i.e. the integrals ˆS ψωdvol S , ˆS f (ω)dvol S , where the first one represents the total kinetic energy and the second one the Casimir functions, defined for any f ∈ C 1 (R).These conservation laws are crucial in understanding the long-time behaviour of the fluid.Indeed, as Kraichnan showed in [13], the conservation of both energy and enstrophy (i.e. the Casimir for f (x) = x 2 ) is responsible for the remarkable phenomenon of the formation and the persistence of large coherent vortices.
A first attempt to understand the statistical properties of 2D ideal fluids is due to Lars Onsager [18], who showed that in the simplified point-vortex model the equilibrium statistical mechanics predicts the concentration of vortices with the same sign.More recently, the theory of Miller, Robert and Sommeria [15,20] extended the ideas of Onsager and Kraichnan taking into account all the invariants.Via a mean field approach considering formally defined an invariant microcanonical measure for the Euler equations, they derive a functional relationship between the equilibrium average vorticity and the stream function [5].Even though the MRS theory has been quite recognized, several critical aspects and discrepancies with respect to experiments and numerical simulations have been found [7,17].From a mathematical point of view, the MRS theory is purely formal and does not give any precise definition of the invariant measures considered.
At the moment only energy and enstrophy invariant measures have been rigorously constructed, and extending the existing results to other Casmirs is still an open problem.Albeverio and Cruzeiro in [2] showed the existence of solutions to the 2D Euler equations as stochastic processes limit of Galerkin approximation of the Euler equations with vorticity in H −α , such that the enstrophy and the (renormalized) energy Gibbs measures are invariant for the flow.
In this paper, we consider a different finite dimensional approximation for the 2D Euler equations, valid on any orientable compact surface.This model was derived by V. Zeitlin [22,23], based on the theory of geometric quantization of compact Kähler manifolds [4].One of the main feature of Zeitlin's finite dimensional model is to posses a number of conserved quantities, which is proportional to the level of discretization and such that, for a sufficiently regular vorticity field, they approximate the original Casimirs of the 2D Euler equations.In particular, for any level of discretization, the Zeitlin's model admits energy and enstrophy analogue, which are simply a spectral truncation of the original ones.
The aim of this work is to set a new theoretical framework in which developing a rigorous statistical theory for the Euler equations.Indeed, one of the main open problems is defining Gibbsian invariant measures which takes into account other conserved quantities than energy and esntrophy.Since these measures have distributional support, it is not clear, even up to renormalization, how to deal with higher order Casimirs of the Euler equations.In this paper, we show that it is possible starting from the Zetlin's model to recover the results of Albeverio and Cruzeiro in [2], but also that the Zeitlin's model gives new insights in the problem, that in the future could allow to deal also with the other Casimirs.Furthermore, a main novelty of our work is that we perform explicit calculations on the structure constants for the 2-sphere S 2 (cfr.Appendix A), which are technically more involved than those on the flat 2-torus (recalled in Appendix B for completeness).
The paper is structured as follows.In section 2 we present the geometric background necessary to set up the quantized version of Euler equations on S 2 : we introduce isometries between subspaces of functions on S 2 and spaces of matrices in the Lie algebra su(N ), as well as suitable Sobolev norms on su(N ).In section 3 we rigorously define a sequence of Gaussian measure on su(N ) whose pull-back converges weakly towards the enstrophy measure, and prove useful bounds on stationary solutions of quantized Euler equations.In section 4 we show the existence of a subsequence of solutions of quantized Euler equations converging towards a limiting process ω taking values in a space of distributions: as a consequence of previous results, we are able to prove that ω is a stationary process with marginals distributed as the enstrophy measure, and that it solves a symmetrized version of Euler equations on S 2 .Finally, in section 5 we discuss open problems, in particular concerning the difficulties encountered in trying to solve Euler equations having as invariant measure a Gibbsian measure associated to higher-order Casimirs, and we point out a tentative approach involving the evaluation of line integrals and Kelvin Theorem.

Fundamental concepts and definitions
In this section, we introduce the fundamental concepts and notations that we employ throughout the paper.In particular, in order to introduce the Zeitlin's model, we observe that the right hand side in the first equation of ( The Laplace-Beltrami operator is replaced by a linear operator ∆ N defined on su(N ), with the same spectrum (up to truncation) of ∆.In this paper, we perform our calculations on the 2-sphere S 2 embedded in the Euclidean space R 3 (in Appendix B we show that the same results can be derived for the Zeitlin's model on the 2D flat torus).The Zeitlin's model relies on the theory of geometric quantization of the Poisson algebra (C ∞ (S 2 ), {•, •}), [4].Let Y ℓ,m ∈ C ∞ (S 2 ) and T N ℓ,m ∈ su(N ) denote respectively the standard spherical harmonics and spherical matrices defined in [11], for ℓ ∈ N, m ∈ Z, |m| ≤ ℓ.For this domain, the relationship between the functions and matrices is explicitly given in terms of spherical harmonics and spherical matrices.Let us define the linear projectors: In the following, we denote jN : su(N ) → L 2 N (S 2 ) the inverse of the restriction of Π N to L 2 N (S 2 ), and The discrete Laplacian ∆ N : su(N ) → su(N ) acts on the basis and thus we can define for ω that is a good Sobolev norm on su(N ), in the sense that Aubin-Lions and Simon compactness criterions hold [21].
The quantized Euler equations can be written as [17]: These equations have as conserved quantities the energy the linear momentum and the Casimirs

Gaussian measures
In this section we introduce the Gaussian measure on su(N ) that permits us to prove the existence of stationary solutions to quantized Euler equations (2.4).For this purpose, let Q N : su(N ) → su(N ) be the covariance operator defined as where Z = ´C e − 1 2 |x| 2 dx is a suitable renormalization constant, and d N = N 2 − 1.The covariance operator Q N is just a convenient rewriting of the identity operator on su(N ), which is the content of the following: Proof.For notational convenience, let us relabel the basis (T N ℓ,m ) ℓ=1,...,N −1,|m|≤ℓ as Let us rearrange the product inside the integral in the following way.Denote {k, h} the set with elements k and h, and let card{k, h} be its cardinality, so that card{k, h} = 1 if k = h and card{k, h} = 2 if k = h.Since in the previous expression the integration with respect to dc j produces only a factor Z for j = k, h, we can rewrite where we deduce the last line from Proof.It follows immediately from (2.2) and the identity , given by the previous lemma.
su(N ) dW the Gaussian measure on su(N ) with covariance Q N , and let ν N be its pull-back on C ∞ (S 2 ) given by ν ).The enstrophy measure is defined as the centered Gaussian measure ν on H −1− (S 2 ) := ∩ s>0 H −1−s (S 2 ) with covariance Q = Id, or equivalently with reproducing kernel L 2 (S 2 ).The previous corollary implies ν N ⇀ ν as measures on H −1− (S 2 ).
We can now state the main results of this section.
Lemma 3.For every ǫ > 0 and p ∈ [1, ∞) there exists a finite constant C ǫ,p such that ˆsu(N) For the measure ν N the desired bound is classical, see for instance [1,Section 3].
Then for every ǫ > 0, p ∈ [1, ∞) and κ sufficiently large there exists a finite constant C ǫ,p,κ such that Proof.First of all, notice that there exists a unique stationary solution to (2.4) by a suitable adaptation of non-explosion results in [6, Section 3].The dynamics of Ŵ N W0 is given by Ẇ , and therefore the dynamics of ω N ω0 is given by , by the previous formula we deduce Having said that, by (2.3) and change of variables Let us consider the two terms separately.The first one is easy to control, indeed as for the second one, since r N is stationary as well we get .
Where we have used the fact that m = m + m ′ .We have the following equality of the 3j−symbols (cfr.Appendix A) 1 : 1 This can be directly derived from the relation of the 3j−symbols with the Clebsch-Gordan coefficients and the definition of the latter.
Hence, the first term on the right hand side vanishes.Therefore, we have: (3.4) We split the sum in two parts.We say that ℓ ≫ ℓ if ℓ ≥ 2ℓ(log(ℓ) + 1), and where C denotes, from now, on a positive suitable constant.The numerator also satisfies: Hence, for ℓ ≫ ℓ: By the Proposition 13 in Appendix A, we get: which goes to 0 for N → ∞ for κ > 5/2.For ℓ ≈ ℓ, ℓ ′ can be as small as 1.Hence: By the Proposition 13 in Appendix A, we get: which goes to 0 for N → ∞ for κ > 7/2. 4. Identification of the limit Proposition 5. Fix ǫ > 0. There exist a subsequence (N m ) m∈N , a common probability space ( Ω, F , P) and random variables ωm , ω : for every m ∈ N and ωm → ω almost surely with respect to P.
Proof.Convergence in law up to a subsequence follows from Corollary 4, exploiting Simon compactness criterion [21, Corollary 9] and Prokhorov Theorem.Almost sure convergence in an auxiliary probability space is then a consequence of Skorokhod Theorem.
In the following we say that a random variable taking values in H −1−ǫ (S 2 ) is a white noise if distributed as ν.We recall the following result from [8], here adapted in order to consider functions defined on the sphere S 2 .Proposition 6. [8, Theorem 8].Let ω : Ω → H −1−ǫ (S 2 ) be a white noise, and for a fixed test function φ ∈ C ∞ (S 2 ) denote Assume to have a sequence of symmetric functions H N φ ∈ H 2+2ǫ (S 2 × S 2 ), N ∈ N that approximates H φ in the following sense: Then the sequence of random variables ω ⊗ ω, H N φ , N ∈ N is a Cauchy sequence in L 2 (Ω).Moreover, the limit is independent of the sequence H N φ , that is: if HN φ , N ∈ N is another sequence satisfying (4.1) and (4.2), then Remark 7.There exists a sequence H N φ satisfying (4.1) and (4.2).It can be constructed by mollification of the Biot-Savart kernel: see [8,Remark 9] for details.
Proof.Let ωm , ω be given by Proposition 5, and fix φ ∈ C ∞ (S 2 ).Recalling (3.1), it is easy to check for every m ∈ N and P-a.s.
for every t ∈ [0, T ], where rNm is distributed as j Nm r Nm .Since ωm → ω as m → ∞ almost surely with respect to the C([0, T ], H −1−ǫ ) topology, we have ωm t , φ − ωm 0 , φ → ωt , φ − ω0 , φ as m → ∞, with probability one.Concerning the second summand on the right-hand-side, we notice that H jN m ΠN m φ − H φ converges to zero in L 2 (S 2 × S 2 ) and therefore as m → ∞, which implies the almost sure convergence up to a subsequence (that we still denote m with a little abuse of notation): ) → 0 as m → ∞, which implies the almost sure convergence up to a subsequence (that we still denote m with a little abuse of notation): Finally, let us focus on the first term on the right-hand-side.Let H M φ , M ∈ N, be a sequence of H 2+2ǫ (S 2 × S 2 ) functions that approximates H φ in the sense of Proposition 6 above, and exists by Remark 7. We can decompose, for fixed M ∈ N: Now, by condition (4.1) for every δ > 0 there exists M ∈ N such that ≤ δ; moreover, since it is easy to check that ω is a white noise by Corollary 2, by Proposition 6 and Definition 8 for every δ > 0 there exists M ∈ N such that Having fixed such M , we have and the proof is complete.

Open problems
5.1.Gibbs measure associated to Casimirs.The 2D Euler equations on a compact surface S have inifinitely many conservation laws.The following integrals, when defined, are invariants for the dynamics: where f : R → R can be any C 1 function.In particular, for f (x) = x 2 , we have the enstrophy E (ω) = ´ω2 dvol S .The presence of these conservation laws comes from the fact that 2D Euler equations are an infinite dimensional Lie-Poisson system on the dual of the Lie algebra of smooth divergence-free vector fields on S [3].This space can be identified with the space of smooth functions on S. Therefore, because of the Hamiltonian nature of the Euler equations, we formally have that the "flat measure" on C ∞ (S) is an invariant measure.Hence, heuristically we can define the following family of invariant measures for α, β, γ p ≥ 0: where [dω] is the formal "flat measure" on C ∞ (S) and Z is the partition function.In order to make this more rigorous, we cannot use the formal "flat measure" [dω].Instead, we take the enstrophy measure ν as reference measure on H −1− (S) = ∩ ǫ>o H −1−ǫ (S) (cfr.Section section 3).We could then define µ as: where We notice that the measure µ for γ p = 0 can be defined using the theory of Gaussian measures on H −1− (S).However, for instance, taking β = 0 and γ p = 0 only for p = 4, the measure it is not well defined, since we do not have a precise meaning of a power of an element in H −1− (S).In order to make sense of this operation, one would like to use the renormalization theory, that allows to define the renormalized power of a suitable Gaussian measure ω as the mean square limit of the renormalized power where ω ε is a mollification of ω and C ε → ∞ is a suitable renormalization constant.Unfortunately, the current renormalization theory does not cover Gaussian measures associated with Casimirs higher than the enstrophy.The quantized Euler equations (2.4) in su(N ) have the following invariants: for p = 2, . . ., N .It is known that for smooth ω, we get [16]: for N → ∞.In section 3, we have seen that ν N ⇀ ν as measures on H −1− (S 2 ).Let, for instance p = 4. Defining we would like to show that j * N η N has a weak limit in H −1− (S 2 ).5.2.Line integrals and Kelvin theorem.Developing the machinery needed to prove invariance theorems based on line integrals is also an appealing question, having in mind especially Kelvin theorem, see [14].In the generalized setting of the enstrophy measure, where all fields are distributional, this looks a formidable task, still open.However, we would like to emphasize that line integrals on deterministic curves are well defined, in spite of an apparent difficulty.It is the generalization to random curves which is open and, unfortunately, necessary to develop invariance properties, since one should consider curves moving with the fluid, hence random.
Let us thus show that line integrals are well defined on deterministic closed curves.We follow the approach developed for the definition of line integrals of the Gaussian Free Field, see for instance [12].Let us restrict ourselves for simplicity to curves which are boundaries of bounded open connected sets A ⊂ S 2 .Assume that ∂A is a Lipschitz boundary and assume that γ : [a, b] → S 2 is a Lipschitz continuous curve parametrizing ∂A.Assume that the parametrization is regular, namely that the derivative γ ′ (t), which exists a.s., has the property |γ ′ (t)| ≥ c > 0 a.s., for some positive constant c.It is known that the map originally defined on W s,2 S 2 ∩ C S 2 , for some s > 1  2 , extends to a bounded linear map from W s,2 S 2 to L 2 (∂A) (in fact it takes values in W s− 1 2 ,2 (∂A)).Thanks to regularity of γ, we can say that the function Moreover, for every s > 1 2 there is a constant C s > 0 such that (5.2) Associated to the rectifiable curve γ we may define, for every s > 1 2 , the rectifiable current Indeed notice that, by (5.1) the integral is finite and by (5.2) the map Γ is bounded.Thus Γ is a bounded linear functional on W s,2 S 2 , R 2 , namely it is an element of the dual of W −s,2 S 2 , R 2 , and this holds for every s > 1 2 : Let now µ be the enstrophy measure on S 2 defined in Section section 3, namely the centered Gaussian measure, supported on W −1−ǫ,2 S 2 , R with identity covariance for all ϕ, ψ ∈ W 1+ǫ,2 S 2 , R , where •, • inside the integral is the dual pairing, outside the scalar product in L 2 S 2 , R .Let K be the Biot-Savart map from W −1−ǫ,2 S 2 , R to W −ǫ,2 S 2 , R 2 and let (we use the notation K * interpreting K as a kernel) ξ = K * µ be the centered Gaussian velocity field associated to the enstrophy measure, namely a centered Gaussian measure, supported on W −ǫ,2 S 2 , R 2 , such that Formally we aim to define The key remark is that the covariance property above of the measure ξ (dv) allows to extend the definition of the Gaussian random variable v, w , v selected by ξ (dv), from vector fields w of class W ǫ,2 S 2 , R 2 to vector fields of class W −1,2 S 2 , R 2 , which includes the space where Γ lives, see (5.3).
Since the rectifiable current Γ, associated to a regular Lipschitz curve γ : [a, b] → S 2 as done above, is of class (5.3), the r.v.v, Γ is well defined and we take it as the definition of ´b a ξ (γ (t)) • γ ′ (t) dt.
Let us explain why the Gaussian random variable v, w is well defined also for which implies (by the convergence properties of θ ǫ * in W 1,2 S 2 , R 2 ) that the family v, θ ǫ * w is Cauchy in L 2 with respect to the measure ξ (dv).We call v, w its limit, which is a centered Gaussian r.v. with variance equal to (−∆) −1 w, w .
These properties are based on the fact that Γ is deterministic.As said at the beginning, the extension to random curves is an open problem.
Within the quantized Euler equations (2.4) in su(N ), it is possible to identify the discrete analogue of the line integrals of the velocity field.Alternatively to the usual choice for the Casimirs C N n (W ) = T r(W n ), for n = 2, . . ., N , one can equivalently consider the eigenvalues λ i of W . Indeed it holds The first choice of the Casimirs corresponds to a discrete version of the momenta of the continuous vorticity C n (ω) = ´S2 ω n dS, for n > 1, whereas the second one corresponds to the conserved quantities given by the Kelvin circulation theorem.Indeed, we have that the Kelvin circulation theorem implies that for any material domain A(t) ∈ S 2 , i.e. a domain A = A(t) evolving accordingly to the fluid motion, the integral ´A(t) ωdS = ´∂A(t) u • ds is invariant in time, where ∇ × u = ωn, for n normal vector on S 2 .We now want to show the heuristic analogy among the eigenvalues of W and the integrals ´A(t) ωdS.Let us consider then spectral decomposition of W : for E unitary and Λ purely imaginary diagonal.Let e i , for i = 1, . . ., N be the columns of E and the λ i the eigenvalues of W . Then we can write: The matrices e i e * i are pairwise orthogonal with respect to the Frobenius inner product.Hence, T r(W * e i e * i ) = λ i .
The heuristic analogy with the Kelvin's theorem reads as: iT r(W * e i e * i ) ≈

ˆA(t) ωdS,
for some domain A(t) which corresponds to the support of j N (ie i e * i ) ∈ C ∞ (S 2 ), for N → ∞.
Analogously for the other choice of Casimirs, we can define the invariant measure on su(N ) as we would like to show that j * N η N has a weak limit in H −1− (S 2 ). to the Y ℓ,m basis and the Poisson bracket (2.1) (see [19]).Then we have that2 : where where, for odd values of L = ℓ + ℓ ′ + ℓ, Note that for even values of L = ℓ + ℓ ′ + ℓ, P may be arbitrarily defined.Developing C (N )ℓm ℓm,ℓ ′ m ′ with respect to µ = 1 N +1 , one finds that the even powers of the series vanish.In fact, one can check the following identities: These imply, relabelling k with L − k, that: and so for even powers of µ only even L terms survive but because of the coefficient (1 − (−1) l+ℓ ′ +ℓ ) in C (N )ℓm ℓm,ℓ ′ m ′ , these can be ignored.Finally, since the term the calculations above imply that the linear convergence proved in [19] is actually quadratic for ℓ, ℓ ′ , ℓ ≪ N , i.e. for ℓ, ℓ ′ , ℓ fixed while N → Lemma 11 (C ℓm ℓm,ℓ ′ m ′ bounds).There exists a constant C > 0 such that the structure constants of the spherical harmonics in the usual basis satisfy the following bound: ℓℓ, ℓ ′ ℓ}, for any ℓ, ℓ ′ , ℓ = 1, 2, ..., satisfying the triangular inequality.
Proof.We have seen that the structure constants C ℓ ℓmℓ ′ m ′ can be written in the following way Step 1. Let's first focus on Using the Stirling approximation of the factorial we get: where we have repeatedly used the equality: L 1 + L 2 + L 3 = L. From this, using the definition of the L i and the fact that the ℓ, ℓ ′ , ℓ satisfy the triangular inequality, it is straightforward to check that: Step 2. For any ℓ * ∈ {ℓ, ℓ ′ , ℓ}, we have (see [19]): Step 3. Finally, using the results in Step 1 and Step 2, we get: ℓm,ℓ ′ m ′ satisfy the following bounds.There exists some constant C > 1 such that: Proof.
By the Isserlis-Wick formula: which goes to 0 for N → ∞ for s > 9/2 .