Energy conditional measures and 2D turbulence
Franco Flandoli Dejun Luo





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Energy conditional measures and 2D turbulence
    Franco Flandoli[Email: franco.flandoli@sns.it. Scuola Normale Superiore of Pisa, Italy.]  and
Dejun Luo[Email: luodj@amss.ac.cn. RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing 100049, China. ]
    today
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We show that the invariant measure of point vortices, when conditioning the Hamiltonian to a finite interval, converges weakly to the enstrophy measure by conditioning the renormalized energy to the same interval. We also prove the existence of solutions to 2D Euler equations having the energy conditional measure as invariant measure. Some heuristic discussions and numerical simulations are presented in the last section.


Keywords: point vortices, Hamiltonian, white noise, renormalized energy, microcanonical ensemble






§ INTRODUCTION


For the dynamics of point vortices associated to the 2D Euler equations on the torus ^2 = ^2/^2, the Lebesgue  measure is invariant. If we also randomize the intensities, a measure like
  
    λ_N(x̣_1, …, x̣_N, ξ̣_1,…, ξ̣_N)= x̣_1 …x̣_N 𝒩(ξ̣_1) …𝒩(ξ̣_N)

is invariant too, where 𝒩 denotes the standard Gaussian distribution on . Define the Hamiltonian
  
    ℋ_N((ξ_1, x_1),…, (ξ_N, x_N))= -1/2N∑_1≤ i≠ j≤ Nξ_i ξ_j G(x_i-x_j),

where G is the Green function on ^2. The coefficient -1/2 here is chosen so that ℋ_N converge weakly to the renormalized energy of the white noise, see Proposition <ref> for details. We can rewrite ℋ_N as a functional of the point vortices
  
    ω_N= 1/√(N)∑_i=1^N ξ_i δ_x_i.

Indeed, if we regard G as a function on ^2×^2 by setting G(x,y)= G(x-y) and G(x,x)=0 for all x,y∈^2, then
  
    ℋ_N= -1/2ω_N⊗ω_N, G .

Let μ_N be the distribution of ω_N on H^-1-(^2) under the measure λ_N, where H^-1-(^2) is the intersection of all the Sobolev spaces H^s(^2) of order less than -1. It is proved in <cit.> that μ_N is weakly convergent to the enstrophy measure μ, which is supported by H^-1-(^2).

Recall that μ is the distribution of the white noise on ^2. Let =̋ :ℋ(ω): be the renormalized energy of a white noise ω (see Section 2.2 for its definition); we can regard $̋ as a random variable defined on the probability space(H^-1-(^2), ℬ(H^-1-(^2) ), μ), whereℬ(H^-1-(^2) )is the collection of Borel measurable sets. Fixa,b∈witha<b; from Proposition <ref> below, we always haveμ({∈̋[a,b]}) >0.Therefore, the conditional measure
  
    μ^a,b(A) = μ(A ∩{∈̋[a,b]})/μ({∈̋[a,b]}),    A∈ℬ(H^-1-(^2) )

is well defined. On the other hand, we shall prove in Proposition <ref> thatlim_N→∞ μ_N({ℋ_N ∈[a,b]}) = μ({∈̋[a,b]}) >0, hence we can define in the same way the energy conditional measures for the point vortices:
  
    μ_N^a,b(A) = μ_N(A ∩{ℋ_N ∈ [a,b]})/μ_N({ℋ_N ∈ [a,b]}),    A∈ℬ(H^-1-(^2) ).

The main result in the present paper is


The family {μ_N^a,b}_N≥ 1 of energy conditional measures converge weakly to μ^a,b.


This result shows the convergence of a class of microcanonical measures. We mention that, when the intensities{ξ_i }_i≥1are i.i.d. centered Bernoulli random variables, Benfatto et al. <cit.> proved that the canonical Gibbs measures of the point vortices, with appropriately regularized Green functions, converge to the Gaussian measureμ_β, γ(ω̣) = e^-βH -γE ω̣(β, γ>0,Hand Eare the energy and enstrophy functionals), which are invariant for the 2D Euler flow. In the recent work <cit.>, analogous result was proved without smoothing the Green function; see <cit.> for related result concerning the generalised inviscid surface quasi-geostropic equations.

It is worth pointing out, since the interaction in the Euler dynamics is of long range, that the general principle of equivalence of ensembles is not necessarily valid, see <cit.> for more discussions on nonequivalence of ensembles. In the mean field regime of Onsager theory it holds true, see <cit.>, but in the regime studied here the infinite particle limit is not the Gibbs measure associated to the renormalized energy. It seems that the relevant statistical ensemble is the microcanonical measure, see <cit.> for detailed explanations.

The results presented here are meant to be fragments of a more general investigation on invariant measures of 2D Euler equations, in the attempt to capture some features of inverse cascade turbulence. Onsager theory, extremely relevant for the explanation of large scale coherent vortex structures, does not provide a description of inverse stationary turbulence; but unfortunately also the regime studied here is not the correct description. In a sense, Onsager theory and the regime considered here are two extremes, both with relevant features, but turbulence is somewhat in between. In the last section we discuss this issue.

The above theorem will be proved in Section 3. To this end, we first make some necessary preparations in Section 2, including the definitions ofω⊗ω, Gand of the renormalized energy=̋  :ℋ(ω):for a white noiseω; the relation between them will be clarified in Section 2.3. We study in Section 4 the limit behaviors of the correlation functions of the energy conditional measures on the “flat space”(×^2)^N, following some arguments in <cit.> (see also <cit.> for related results). Based on the results in <cit.>, we prove in Section 5 the existence of solutions to the 2D Euler equations having the energy conditional measureμ^a,bas invariant measure. Finally, we present in the last section some heuristical discussions together with numerical simulations of the spectrum functions for point vortices, illustrating the relevance of our results with 2D turbulence.



§ PRELIMINARY RESULTS ON THE RENORMALIZED ENERGY


In this section we make some preparations regarding the renormalized energy of a white noiseωon^2. Firstly, we follow the idea in <cit.> to define the quantityω⊗ω, G, whereGis the Green function on^2. Secondly, we recall the definition of the renormalized energy$̋ using the Galerkin approximation; based on the series expansion of $̋, we are able to show that its distribution has full support on the real line. Finally, we study the relation betweenω⊗ω, Gand$̋, see Theorem <ref>.



 §.§ Definition of ω⊗ω, G for a white noise ω
 

In this part we follow the approach in <cit.> (see also <cit.>) to define the quantity ω⊗ω, G when ω is a white noise on ^2. The results below are proved in <cit.> and we omit the proofs here.




  (i) If ω: Θ→ C^∞(^2)' is a white noise and f∈ H^2+(^2×^2), then for every p≥ 1 there is constant C_p>0 such that
  
    [|ω⊗ω, f^p|] ≤ C_p f_∞^p.



  (ii) We have ω⊗ω, f = ∫_^2 f(x,x) x̣.


  (iii) If f is symmetric, then
  
    [ |ω⊗ω, f - ω⊗ω, f|^2]= 2 ∫_^2×^2 f(x,y)^2 x̣ỵ.




Based on these facts we can give a definition of ω⊗ω, G when ω is a white noise on ^2.


Let ω: Θ→ C^∞(^2)' be a white noise. Assume that G_n∈ H^2+(^2×^2) are symmetric and approximate G in the following sense:
  
    lim_n→∞∫_^2×^2 (G_n-G)^2(x,y) x̣ỵ =0,   lim_n→∞∫_^2 G_n(x,x) x̣ =0.

Then the sequence of random variables ω⊗ω, G_n is a Cauchy sequence in mean square. We denote by ω⊗ω, G its limit.

Moreover, the limit is the same if G_n is replaced by G̃_n with the same properties and such that lim_n→∞∫_^2×^2 (G_n- G̃_n)^2(x,y) x̣ỵ =0.



The proofs are the same as those of <cit.>; we recall them here for completeness. Since lim_n→∞∫_^2 G_n(x,x) x̣ =0, it is equivalent to show that ω⊗ω, G_n- ∫_^2 G_n(x,x) x̣ is a Cauchy sequence in mean square. We have
  
    [|ω⊗ω, G_n- ∫_^2 G_n(x,x) x̣- ω⊗ω, G_m+ ∫_^2 G_m(x,x) x̣|^2 ]
    
      =    [|ω⊗ω, G_n-G_m- ∫_^2 (G_n-G_m)(x,x) x̣|^2 ]
    
      =    2 ∫_^2×^2 (G_n-G_m)^2(x,y) x̣ỵ,

where the last equality follows from (ii) and (iii) of Lemma <ref>. This implies the Cauchy property, and thus ω⊗ω, G is well defined. The invariance property is proved similarly.


Here is an example of the approximating functions G_n. Let χ: ^2=[-1/2, 1/2]^2 → [0,1] be a smooth and symmetric function with support in a small ball B(0,r), and equal to 1 in B(0,r/2). For any n≥ 1, set χ_n(x) = χ(nx),  x∈^2. Define
  
    G_n(x) = 
      G(x)(1-χ_n(x)),     x≠ 0;
    
      0,     x=0.

We regard G_n as a function on ^2×^2 by setting G_n(x,y) = G_n(x-y). Since G_n(x,x) ≡ 0, we have the following estimate (cf. Lemma <ref>(iii)):
  
    [(ω⊗ω, G_n- ω⊗ω, G)^2] ≤ 2 ∫_^2×^2 (G_n-G)^2(x,y) x̣ỵ.




 §.§ Definition of the renormalized energy $̋

In this subsection we recall the definition of the renormalized energy$̋ via the Galerkin approximation. To this end, let {e_k}_k∈_0^2 be defined as
  
    e_k(x) = √(2)cos(2π k· x),     k∈^2_+ , 
    sin(2π k· x),     k∈^2_-,

where _0^2= ^2 ∖{0} and ^2_+ = {k∈^2_0: (k_1 >0)  (k_1=0,  k_2>0) } and ^2_- = -^2_+. This family of functions is an orthonormal basis of square integrable functions on ^2 with vanishing mean. Let ω be a white noise on ^2, then the random series
  
    ω = ∑_k∈_0^2ω, e_k e_k

converge in mean square in H^-1-δ(^2) for any δ>0. For N≥ 1, define Λ_N = {k∈_0^2: |k|≤ N} and
  
    ω̅_N = ∑_k∈Λ_Nω, e_k e_k,    u_N = K∗ω̅_N,

where K is the Biot–Savart kernel:
  
    K(x)= ∇^⊥ G(x) = - i/2π∑_k∈_0^2k^⊥/|k|^2 e^2π i k· x,

with ∇^⊥ = (∂_2, -∂_1) and k^⊥ = (k_2, -k_1). Set
  
    ℰ_N = 1/2∫_^2 |u_N(x)|^2 x̣,   ℰ̃_N = ℰ_N - ℰ_N.

The following result is well known (see e.g. <cit.> or <cit.>).


The sequence {ℰ̃_N }_N≥ 1 is Cauchy in L^2(Θ, ). Denote its limit by $̋ and call it the renormalized energy; one has=̋1/8π^2∑_k∈_0^21/|k|^2(ω, e_k^2 -1)and( ||̋^2 ) = 1/32π^4∑_k∈_0^21/|k|^4.We give the detailed computations since we need the exact coefficients. Note that
  K(x) =  1/2π∑_l∈_0^2l^⊥/|l|^2sin(2π l· x).
Therefore, if k∈^2_+, then
   (K∗ e_k)(x)   = 1/2π∑_l∈_0^2l^⊥/|l|^2∫_^2sin(2π l· (x-y)) √(2)cos(2π k· y) ỵ
   = √(2)/2πk^⊥/|k|^2sin(2π k· x) = - 1/2πk^⊥/|k|^2 e_-k(x),
  
where the second equality is due to the fact that the integral vanishes unless l= ± k. Similarly, if k∈^2_-, then
  (K∗ e_k)(x) = 1/2π∑_l∈_0^2l^⊥/|l|^2∫_^2sin(2π l· (x-y)) √(2)sin(2π k· y) ỵ = - 1/2πk^⊥/|k|^2 e_-k(x).
Thus,
  u_N = K∗ω̅_N = - 1/2π∑_k∈Λ_Nk^⊥/|k|^2ω, e_k e_-k.
As a result,
  ℰ_N = 1/2∫_^2 |u_N(x)|^2 x̣ = 1/8π^2∑_k,l∈Λ_Nk· l/|k|^2 |l|^2ω, e_kω, e_le_-k, e_-l = 1/8π^2∑_k∈Λ_N1/|k|^2ω, e_k^2. 
Consequently,
  
    ℰ̃_N = 1/8π^2∑_k∈Λ_N1/|k|^2 (ω, e_k^2 -1).


Next,
  [(ℰ̃_N)^2 ]    = 1/64π^4∑_k,l∈Λ_N1/|k|^2 |l|^2[(ω, e_k^2 -1) (ω, e_l^2 -1) ] 
   = 1/64π^4∑_k∈Λ_N1/|k|^4[ω, e_k^4 -1 ] = 1/32π^4∑_k∈Λ_N1/|k|^4,
  
since ω, e_k^4 =3. The same calculations imply that {ℰ̃_N }_N≥ 1 is a Cauchy sequence in L^2(Θ, ) and the two desired equalities.


As an application of the expression for the renormalized energy, we can prove


The distribution of $̋ is supported on the whole real line.

For any a,b∈,  a<b, it suffices to show that Z^a,b:= ({∈̋[a,b]})>0. Without loss of generality, assume b-a≤ 1.

We define δ_0:= (b-a)/5 and the remainder
  ℛ_N:= 1/8π^2∑_|k|>N1/|k|^2 (ω, e_k^2 -1).
Then =̋ℰ̃_N + ℛ_N and, for all N≥ 1, the two r.v.'s ℰ̃_N and ℛ_N are independent of one another. Moreover,
  {∈̋[a,b]}⊃{ℰ̃_N ∈ [a+δ_0, b-δ_0] }∩{|ℛ_N| ≤δ_0},
therefore,
  ({∈̋[a,b]}) ≥( {ℰ̃_N ∈ [a+δ_0, b-δ_0] }) ({|ℛ_N| ≤δ_0}).
Since ℛ_N tends to 0 in the norm L^2(Θ,) as N→∞, we can find N_0∈_+ such that ({|ℛ_N| ≤δ_0}) ≥ 1/2 for all N≥ N_0. Thus, it is enough to show that
  
    ( {ℰ̃_N_0∈ [a+δ_0, b-δ_0] })>0.


We define
  L= 1/8π^2∑_k∈Λ_N_01/|k|^2
and consider three different cases according to the location of the origin 0 w.r.t. the middle subinterval [a+2δ_0, a+3δ_0].

(i) a+2δ_0 >0. Since [a+3δ_0, a+4δ_0] ⊂ [a+δ_0, b-δ_0], it is sufficient to prove that
  
    ( {ℰ̃_N_0∈ [a+3δ_0, a+4δ_0] })>0.

Set c_1:= (a+3δ_0)/L and c_2:= (a+4δ_0)/L which are positive constants. Recall that {ω,e_k}_k∈_0^2 is a family of i.i.d. standard Gaussian r.v.'s; we have
  p_1 := ( {ω,e_k ^2∈ [1+c_1, 1+c_2] })>0.
The desired property (<ref>) follows from the next inclusion between events:
  {ℰ̃_N_0∈ [a+3δ_0, a+4δ_0] }⊃⋂_k∈Λ_N_0{ω,e_k ^2∈ [1+c_1, 1+c_2] }.

(ii) a+2δ_0 ≤ 0 ≤ a+3δ_0. In this case, we have [-δ_0, δ_0] ⊂ [a+δ_0, b-δ_0]. Similar to case (i), we deduce the desired result from the two facts below:
  {ℰ̃_N_0∈ [-δ_0, δ_0] }⊃⋂_k∈Λ_N_0{ω,e_k ^2∈ [1-δ_0/L, 1+δ_0/L] }
and
  p_2:= ( {ω,e_k ^2∈ [1-δ_0/L, 1+δ_0/L] })>0.

(iii) a+3δ_0< 0. In this case, it suffices to show that
  
    ( {ℰ̃_N_0∈ [a+δ_0, a+2δ_0] })>0.

We assume N_0 is big enough such that the constant L> -a; then
  -1< c_3:= (a+δ_0)/L< c_4:= (a+2δ_0)/L<0.
We can get the inequality (<ref>) from the facts that
  {ℰ̃_N_0∈ [a+δ_0, a+2δ_0] }⊃⋂_k∈Λ_N_0{ω,e_k ^2∈ [1+c_3, 1+c_4] }
and
  p_3:= ( {ω,e_k ^2∈ [1+c_3, 1+c_4] })>0.

Summarizing the above three cases, we complete the proof of (<ref>).


 §.§ The relation between ω⊗ω, G and $̋

In this part, for a white noiseω, we follow the idea in <cit.> to show the relation betweenω⊗ω, Gand$̋ . Although we mainly work with real valued functions, we shall make use of the canonical complex orthonormal basis of L^2(^2, ℂ): ẽ_k(x)=  e^2π i k· x,  k∈^2, x∈^2. Note that {ẽ_k⊗ẽ_l }_k,l∈^2 is an orthonormal basis of L^2(^2×^2, ℂ).


Let ω be a white noise on ^2. Assume f∈ C^∞(^2×^2,) is symmetric and ∫_^2 f(x,x) x̣=0. Then
  ω⊗ω, f = ∑_k,l∈^2 f_k,lω, ẽ_kω, ẽ_l   L^2(Θ, ),
where
  f_k,l= f, ẽ_k⊗ẽ_l= ∫_^2×^2 f(x,y)ẽ_k(x) ẽ_l(y) x̣ỵ.
Denote by
  
    Λ̂_N = {k∈^2: |k|≤ N} = Λ_N∪{0}.

Since f∈ C^∞(^2×^2), the partial sum of the Fourier series
  f_N(x,y):= ∑_k,l∈Λ̂_N f_k,l ẽ_k(x) ẽ_l(y) 
converges to f, uniformly on ^2×^2 and in L^2(^2×^2). In particular,
  
    lim_N→∞∫_^2 f_N(x,x) x̣ = ∫_^2 f(x,x) x̣ =0.

It is obvious that f_N(x,y) is smooth and symmetric. By (ii) and (iii) in Lemma <ref>,
  
    [ (ω⊗ω, f- f_N + ∫_^2 f_N(x,x) x̣)^2 ] = 2 ∫_^2×^2 (f- f_N)^2(x,y)  x̣ỵ.

As a result,
  
    [ω⊗ω, f- f_N^2 ] ≤ 4∫_^2×^2 (f- f_N)^2(x,y)  x̣ỵ + 2 (∫_^2 f_N(x,x) x̣)^2.


Next, note that
  ω⊗ω, f_N= ∑_k,l∈Λ̂_N f_k,lω, ẽ_kω, ẽ_l.
Therefore, by (<ref>),
     [(ω⊗ω, f - ∑_k,l∈Λ̂_N f_k,lω, ẽ_kω, ẽ_l)^2 ] 
≤     4∫_^2×^2 (f- f_N)^2(x,y)  x̣ỵ + 2 (∫_^2 f_N(x,x) x̣)^2.
  
Thanks to (<ref>), the desired result follows by letting N→∞.


We need the following simple equality.


Let {a_k,l}_k,l∈Λ̂_N⊂ℂ be satisfying a_k,l= a_l,k, a_k,l= a_-k,-l. Then
  
    [ |∑_k,l∈Λ̂_N a_k,lω, ẽ_kω, ẽ_l - ∑_k∈Λ̂_N a_k,-k|^2] = 2∑_k,l∈Λ̂_N |a_k,l|^2.

Since ω, ẽ_k= ω, ẽ_-k, it is clear that ∑_k,l∈Λ̂_N a_k,lω, ẽ_kω, ẽ_l is real and
  
    ∑_k∈Λ̂_N a_k,-k = (∑_k,l∈Λ̂_N a_k,lω, ẽ_kω, ẽ_l).

We have
  (∑_k,l∈Λ̂_N a_k,lω, ẽ_kω, ẽ_l)^2 = ∑_k,l,m,n ∈Λ̂_N a_k,l a_m,nω, ẽ_kω, ẽ_lω, ẽ_mω, ẽ_n, 
and by the Isserlis–Wick theorem,
  (ω, ẽ_kω, ẽ_lω, ẽ_mω, ẽ_n) =     (ω, ẽ_kω, ẽ_l) ( ω, ẽ_mω, ẽ_n)
    + (ω, ẽ_kω, ẽ_m) ( ω, ẽ_lω, ẽ_n) 
    + (ω, ẽ_kω, ẽ_n) ( ω, ẽ_lω, ẽ_m) 

  =    δ_k,-lδ_m,-n + δ_k,-mδ_l,-n + δ_k,-nδ_l,-m.
  
Therefore,
  (∑_k,l∈Λ̂_N a_k,lω, ẽ_kω, ẽ_l)^2    = ∑_k,m ∈Λ̂_N a_k,-k a_m,-m + ∑_k,l ∈Λ̂_N a_k,l a_-k,-l  + ∑_k,l ∈Λ̂_N a_k,l a_-l,-k
   = (∑_k ∈Λ̂_N a_k,-k)^2 + 2 ∑_k,l ∈Λ̂_N |a_k,l|^2,
  
where we have used the facts a_-l,-k = a_-k,-l = a_k,l. Combining this equality with (<ref>) finishes the proof.


Recall the definition of Λ̂_N in (<ref>). To simplify the notations, we introduce
  ω̂_N = Π̂_N ω = ∑_k∈Λ̂_Nω, ẽ_kẽ_k.
Then
  
    ω̂_N⊗ω̂_N, G = ∑_k,l∈Λ̂_NG, ẽ_k⊗ẽ_lω, ẽ_kω, ẽ_l

is the partial sum of the series.


We have
  ω̂_N⊗ω̂_N, G = -1/4π^2∑_k∈Λ_N1/|k|^2ω, e_k^2.
Recall that
  G(x) = -1/4π^2∑_k∈_0^21/|k|^2 e^2π i k· x = -1/4π^2∑_k∈_0^21/|k|^2ẽ_k(x). 
Therefore, for l≠ 0,
   (G∗ẽ_l)(x) = -1/4π^21/|l|^2ẽ_l(x),
which implies that
  G, ẽ_k⊗ẽ_l = ∫_^2ẽ_k(x) (G∗ẽ_l)(x) x̣= -1/4π^21/|l|^2δ_k,-l. 
Hence,
  ω̂_N⊗ω̂_N, G = -1/4π^2∑_k∈Λ_N1/|k|^2ω, ẽ_kω, ẽ_-k = -1/4π^2∑_k∈Λ_N1/|k|^2 |ω, ẽ_k|^2.
The desired identity follows from |ω, ẽ_k|^2 = 1/2 (ω, e_k^2 + ω, e_-k^2) for all k∈Λ_N.


Now we can prove the main result of this section.


Let ω be a white noise on ^2. Almost surely, it holds
  ω⊗ω, G =-2  .
Let G_n be the smooth functions defined at the end of Section <ref>.  We have
  
    [(ω⊗ω, G + 2 )^2 ] 
    ≤     4 [ω⊗ω, G - G_n^2 ] + 4 [(ω⊗ω, G_n - ω̂_N⊗ω̂_N, G_n)^2 ] 
        + 4 [ ( ω̂_N⊗ω̂_N, G_n + 2 ℰ̃_N )^2 ] + 16 [( ℰ̃_N - )^2 ].


We deal with these terms one-by-one. By (<ref>),
  
    [ω⊗ω, G-G_n^2 ] ≤ 2 ∫_^2×^2 ( G-G_n)^2(x,y)  x̣ỵ.

Next, for any fixed n≥ 1, Lemma <ref> implies
  
    [ (ω̂_N⊗ω̂_N, G_n - ω⊗ω, G_n)^2]→ 0    N→∞.

Moreover, by Proposition <ref>, the last term in (<ref>) vanishes as N→∞.

It remains to treat the third term on the r.h.s. of (<ref>). By (<ref>) and Lemma <ref>,
  
    -2 ℰ̃_N= ω̂_N⊗ω̂_N, G - ω̂_N⊗ω̂_N, G.

Therefore,
      [ ( ω̂_N⊗ω̂_N, G_n + 2 ℰ̃_N )^2 ] 

  =     [ ( ω̂_N⊗ω̂_N, G_n-G - ω̂_N⊗ω̂_N, G_n-G + ω̂_N⊗ω̂_N, G_n)^2 ] 
≤     2 [ ( ω̂_N⊗ω̂_N, G_n-G - ω̂_N⊗ω̂_N, G_n-G)^2 ] + 2 [ω̂_N⊗ω̂_N, G_n]^2.
  
By (<ref>) and Lemma <ref>,
      [(ω̂_N⊗ω̂_N, G_n- G  - ω̂_N⊗ω̂_N, G_n- G )^2 ]

  =      2 ∑_k,l∈Λ̂_N| G_n- G, ẽ_k⊗ẽ_l |^2 ≤ 2 ∫_^2×^2 (G_n- G)^2(x,y) x̣ỵ .
  
Hence,
  [ ( ω̂_N⊗ω̂_N, G_n + 2ℰ̃_N )^2 ] ≤     4 ∫_^2×^2 (G_n- G)^2(x,y) x̣ỵ + 2 [ ω̂_N⊗ω̂_N, G_n ]^2.
  
As a result of (<ref>),
  
    lim_N→∞ω̂_N⊗ω̂_N, G_n =  ω⊗ω, G_n = ∫_^2 G_n(x,x) x̣=0,

where the second step is due to Lemma <ref>(ii). Thus,
  lim sup_N→∞[ ( ω̂_N⊗ω̂_N, G_n + 2 ℰ̃_N )^2 ] ≤ 4 ∫_^2×^2 (G_n- G)^2(x,y) x̣ỵ.
Combining the above inequality with (<ref>)–(<ref>), letting N→∞ in (<ref>) yield
  lim sup_N→∞[(ω⊗ω, G + 2 )^2 ] ≤ 24 ∫_^2×^2 (G_n- G)^2(x,y) x̣ỵ.
We finish the proof by sending n→∞.


§ PROOF OF THE MAIN RESULT


In this section, we first show that the Hamiltonian ℋ_N converge weakly to the renormalized energy $̋, by making use of the weak convergence of the random point vortices to the white noise. Thanks to the fact that the distribution of$̋ has a density, finally we are able to prove Theorem <ref>.

First of all, we prove the following intermediate result.


The Hamiltonian ℋ_N defined in (<ref>) converge weakly to the renormalized energy $̋.


By (<ref>) and Theorem <ref>, it suffices to prove thatω_N ⊗ω_N, Gconverge weakly toω⊗ω, G, whereω_Nis the random point vortices andωis a white noise on^2. This result seems to be obvious, thanks to the weak convergence ofω_Ntoω, see <cit.>. However, in view of the proof of our main result, we give the details here.

The following equality will be very useful in the sequel (cf. <cit.>): iff∈ L^2(^2×^2,)is symmetric andf(x,x)≡ 0, then
  
    [ ω_N ⊗ω_N, f^2 ] = 2 ∫_^2×^2 f(x,y)^2 x̣ỵ.

Note that, we do not need the boundedness offfor proving the above equality and, by convention,G(x,x)=0for allx∈^2.

Recall thatμ_Nis the law ofω_NonH^-1-(^2)and that the sequence{μ_N}_N≥ 1converge weakly to the enstrophy measureμ. Thus by the Skorokhod theorem, there exists a new probability space( Θ̃, ℱ̃, )and a sequence of random variablesω̃_N: Θ̃→ H^-1-(^2), andω̃: Θ̃→ H^-1-(^2)such that

  (a)ω̃_N Ł∼μ_N and ω̃Ł∼μ;

  (b)-a.s., ω̃_N converge  to ω̃ as N→∞.

In particular,ω̃is a white noise on^2.

Letf: →be any bounded and uniformly continuous function. We need to showlim_N→∞ f(ω_N ⊗ω_N, G) =  f(ω⊗ω, G).We havef(ω_N ⊗ω_N, G) -  f(ω⊗ω, G) = f(ω̃_N ⊗ω̃_N, G) - f(ω̃⊗ω̃, G).From the next result we deduce that the above quantity vanishes asN→∞.


We have
  lim_N→∞|ω̃_N ⊗ω̃_N, G- ω̃⊗ω̃, G| =0.
Denote the expectation by I_N. Let G_n be the approximating functions given at the end of Section <ref>. By the triangle inequality,
  
    I_N ≤    |ω̃_N ⊗ω̃_N, G - ω̃_N ⊗ω̃_N, G_n| + | ω̃_N ⊗ω̃_N, G_n - ω̃⊗ω̃, G_n| 
       + | ω̃⊗ω̃, G_n - ω̃⊗ω̃, G|.

We denote the three terms by I_N,i,  i=1,2,3. Cauchy's inequality yields
  I_N,1≤(|ω̃_N ⊗ω̃_N, G - ω̃_N ⊗ω̃_N, G_n|^2 )^1/2 = (2∫_^2×^2 (G-G_n)^2(x,y)  x̣ỵ)^1/2,
where in the second step we have used (<ref>). Similarly, by (<ref>),
  I_N,3≤(2∫_^2×^2 (G-G_n)^2(x,y)  x̣ỵ)^1/2.
Next, for any fixed n≥ 1, the family {ω̃_N ⊗ω̃_N, G_n}_N≥ 1 is bounded in L^2() by (<ref>), hence it is uniformly integrable. Moreover, -a.s.,
  ω̃_N ⊗ω̃_N, G_n→ω̃⊗ω̃, G_n   N→∞, 
due to the a.s. convergence of ω̃_N to ω̃. Therefore,
  lim_N→∞ I_N,2 =0.
Summarizing the above discussions, we first let N→∞ and then n→∞ in (<ref>) to deduce that lim_N→∞ I_N =0.


As a consequence, we can prove


For any nontrivial interval [a,b], one has
  
    lim_N→∞({ℋ_N ∈ [a,b]}) = ({∈̋[a,b]}).

By <cit.> (see also <cit.>), the renormalized energy $̋ is infinitely differentiable in the sense of Malliavin and it is non-degenerate, which implies that, as a real valued random variable, the distributionνof$̋ has a density w.r.t. the Lebesgue measure on . Thus any interval [a,b] is a ν-continuous set, that is, the boundary of [a,b] (i.e. {a,b}) is ν-negligible. On the other hand, Proposition <ref> tells us that the distributions on  of ℋ_N converge weakly to ν as N→∞. Therefore, the desired limit holds true.


Finally we are ready to prove the main result.


Taking into account Proposition <ref> and Corollary <ref>, it is sufficient to show that, for any bounded and uniformly continuous function F:H^-1-(^2) →, one has
  lim_N→∞[ F(ω_N)  1_[a,b](ℋ_N) ]= [ F(ω)  1_[a,b]()̋],
where ω_N and ω denote the random point vortices and the white noise respectively.

We follow the idea of the last subsection and use the Skorokhod theorem. Then, adopting the notations given there,
  [ F(ω_N)  1_[a,b](ℋ_N) ] - [ F(ω)  1_[a,b]()̋] = [ F(ω̃_N)  1_[a,b](ℋ̃_N) ] - [ F(ω̃)  1_[a,b]() ], 
where the notations with a tilde denote quantities on the new probability space ( Θ̃, ℱ̃, ). Denote by J_N the difference on the right hand side; then,
  |J_N| ≤|F(ω̃_N) - F(ω̃)| + F_∞| 1_[a,b](ℋ̃_N) -  1_[a,b]() |.
The first term tends to zero by the dominated convergence theorem and the -a.s. convergence of ω̃_N to ω̃. To show that the second one also vanishes as N→∞, we take a sequence of bounded continuous functions such that:
   f_n(t) = 
  1,     t∈ [a, b]; 

  0,     t∈ (-∞, a-1/n ] ∪ [b+ 1/n, +∞ ); 
,     t∈ [a-1/n, a ] ∪ [b, b+ 1/n].
  
Then,
  
    | 1_[a,b](ℋ̃_N) -  1_[a,b]() | 
    ≤    | 1_[a,b](ℋ̃_N) - f_n(ℋ̃_N)| + |f_n(ℋ̃_N) - f_n() | + | f_n() -  1_[a,b]() |.

By Lemma <ref>, we know that ℋ̃_N = -1/2ω̃_N ⊗ω̃_N, G converge in L^1() to = -1/2ω̃⊗ω̃, G. For fixed n∈, f_n is Lipschitz continuous with f_n_ Lip =n; therefore,
  lim_N→∞|f_n(ℋ̃_N) - f_n() |=0.
Next, let ν_N be the distribution of ℋ_N, and thus also of ℋ̃_N. We have
  | 1_[a,b](ℋ̃_N) - f_n(ℋ̃_N)| ≤ν_N([a-1/n, a ] ∪ [b, b+ 1/n ] ),
hence, by Proposition <ref>,
  lim sup_N→∞| 1_[a,b](ℋ̃_N) - f_n(ℋ̃_N)| ≤ν([a-1/n, a ] ∪ [b, b+ 1/n ] ),
where ν is the distribution of $̋ which is the same as that of. Finally,|f_n() -  1_[a,b]() | ≤ν([a-1/n, a ] ∪ [b, b+ 1/n ] ).Recall thatνis absolutely continuous w.r.t. the Lebesgue measure. Therefore, first lettingN→∞and thenn→∞in (<ref>), we complete the proof.


§ TRIVIALITY OF CLUSTER POINTS


In this part, following the discussions at the end of <cit.> (see also <cit.>), we study the limit behavior of the correlation functions (i.e. marginal distributions) of the energy conditional measuresλ_N^a,bon the “flat space”(×^2)^N. Here, fora,b∈,  a<b,
  
    λ_N^a,b = 1/Z_N^a,b 1_{ℋ_N((ξ_1,x_1), …, (ξ_N,x_N))∈ [a,b]}λ_N,

whereλ_Nis defined in (<ref>) andZ_N^a,bis the normalizing constant:Z_N^a,b= ∫_(×^2)^N 1_{ℋ_N((ξ_1,x_1), …, (ξ_N,x_N))∈ [a,b] } λ̣_N =({ℋ_N ∈ [a,b]}).Note that the measureμ_N^a,bdefined in the introduction is the image measure ofλ_N^a,bunder the map𝒯_N: (×^2)^N → H^-1-(^2) defined as
  
    ((ξ_1, x_1), …, (ξ_N, x_N)) 𝒯_N⟶1/√(N)∑_i=1^N ξ_i δ_x_i.


We introduce the notationsx̃_i=(ξ_i, x_i)∈×^2 andX_N= (x̃_1, …, x̃_N). Denote bỵ̃x_i = x̣_i 𝒩(ξ̣_i)andX̣_N= ̣̃x_1 …̣̃x_N. Letρ^N(X_N)=  1_{ℋ_N(X_N)∈ [a,b] } /Z_N^a,bbe the density function of the conditional probability measureλ_N^a,bon(×^2)^N. The correlation functionsρ^N_j  (1≤ j≤ N)are defined as follows:ρ^N_N= ρ^Nand for1≤ j≤ N-1,ρ^N_j(x̃_1, …, x̃_j)= ∫_(×^2)^N-jρ^N(X_N)  ̣̃x_j+1…̣̃x_N.Each function is a probability density (for the firstjpoint vortices) and is symmetric in(x̃_1, …, x̃_j), thanks to the symmetry ofρ^Nin(x̃_1, …, x̃_N). To simplify the notations, we introduceX_j= (x̃_1, …, x̃_j),    X^N-j= (x̃_j+1, …, x̃_N),    1≤ j≤ N-1.First of all, we have the following simple result (see <cit.> or <cit.>).


For any 1≤ j≤ N-1,
  ∫_(×^2)^Nρ^N logρ^N X̣_N ≥∫_(×^2)^jρ^N_j logρ^N_j X̣_j + ∫_(×^2)^N-jρ^N_N-jlogρ^N_N-j X̣^N-j.
We include the proof for the reader's convenience. It is well known that tlog t ≥ t -1 for all t≥ 0. Therefore,
  ρ^N(X_N) log(ρ^N(X_N)/ρ^N_j(X_j) ρ^N_N-j(X^N-j)) + ρ^N_j(X_j) ρ^N_N-j(X^N-j) -ρ^N(X_N) ≥ 0,
which implies
  ∫_(×^2)^Nρ^N(X_N) log(ρ^N(X_N)/ρ^N_j(X_j) ρ^N_N-j(X^N-j))  X̣_N ≥ 0.
This is equivalent to the desired inequality.


Next, we prove


For any fixed j≥ 1, {ρ^N_j }_N≥ j is weakly compact in L^1( (×^2)^j, X̣_j ).

For any N≥ j, there exist m= m(j,N)∈ℕ and r=r(j,N)∈{0,1,…, j-1} such that N= mj+r. By Lemma <ref>,
  ∫_(×^2)^Nρ^N logρ^N X̣_N ≥ m ∫_(×^2)^jρ^N_j logρ^N_j X̣_j + r ∫_×^2ρ^N_1 logρ^N_1 X̣_1.
Using the inequality tlog t ≥ t -1 (t≥ 0), it is clear that ∫_×^2ρ^N_1 logρ^N_1 X̣_1≥ 0. Thus
  ∫_(×^2)^jρ^N_j logρ^N_j X̣_j ≤1/m∫_(×^2)^Nρ^N logρ^N X̣_N = 1/mlog1/Z_N^a,b,
where the last step follows from the definition of ρ^N. Noticing that 1/m = O(1/N), thus by (<ref>), the right hand side vanishes as N→∞. In particular, we conclude that {ρ^N_j logρ^N_j }_N≥ j is bounded in L^1( (×^2)^j, X̣_j ). The proof is complete.


We say that a family{ρ_j}_j≥ 1of probability densities is a weak cluster point of{ρ^N_j }_j≥ 1if there exists a subsequence{N_k}_k≥ 1of integers, such that, for anyj≥ 1,ρ^N_k_jconverge weakly toρ_jinL^1( (×^2)^j, X̣_j ). Now we prove the main result of this section.


Any weak cluster point {ρ_j}_j≥ 1 of {ρ^N_j }_j≥ 1 is trivial, that is, for any j≥ 1, ρ_j = 1 almost surely on (×^2)^j. Consequently, for any j≥ 1, the whole sequence {ρ^N_j }_N≥ j converge weakly to 1.

Fix any >0 and j≥ 1; let
  𝒞_ = { u∈ L^1( (×^2)^j, X̣_j ): u ≥ 0, ∫_(×^2)^j ulog u X̣_j ≤}.
Let {N_k}_k≥ 1 be the subsequence such that ρ^N_k_j converge weakly to ρ_j in L^1( (×^2)^j, X̣_j ). By the proof of Proposition <ref>, we have ρ^N_k_j ∈𝒞_ for all k big enough. Therefore, ρ_j is a weak cluster point of 𝒞_, which is a convex subset of L^1( (×^2)^j, X̣_j ). Since the weak closure of 𝒞_ coincides with the strong one, there exists a sequence of functions {u_n }⊂𝒞_ which converge strongly to ρ_j in L^1( (×^2)^j, X̣_j ). Along a subsequence, u_n converge to ρ_j almost everywhere. Therefore, by Fatou's lemma, we have
  ∫_(×^2)^jρ_jlogρ_j X̣_j ≤lim inf_n→∞∫_(×^2)^j u_nlog u_n X̣_j ≤.
The arbitrariness of >0 leads to ∫_(×^2)^jρ_jlogρ_j X̣_j =0, which implies ρ_j = 1 almost surely. The last assertion follows from the weak compactness of {ρ^N_j }_N≥ j and the uniqueness of the weak limit.


Theorem <ref> is a propagation-of-chaos type result, which means, in the limit, the joint distributionρ_jofjpoint vortices is thej-th power of that of one point vortex. The weak cluster point obtained above gives a trivial solution to the mean field equationρ(ξ,x)= 1/Z_β e^-βξ U_ρ (x),  β∈,whereZ_βis the normalizing constant andU_ρis the averaged stream function:U_ρ (x)= ∫_×^2ξ G(x,y) ρ(ξ, y) 𝒩(ξ̣) ỵ,   x∈^2.In our case,ρ=1a.s. andU_ρ =0a.s. The corresponding free energyF(1)= S(1) + β E(1)=0, where the entropyS(ρ)= ∫_×^2ρ(x̃)logρ(x̃)  ̣̃xand the energyE(ρ)= ∫_(×^2)^2ℋ_2(x̃_1, x̃_2) ρ(x̃_1)ρ(x̃_2) ̣̃x_1̣̃x_2.We conclude this section by showing that, under the measureλ_N^a,b= ρ^N(X_N) X̣_N, the empirical measure1/N∑_i=1^N δ_x̃_iconverges weakly to the trivial measurẹ̃x = 𝒩(ξ̣) x̣on×^2.


For any ϕ∈ C_b(×^2),
  lim_N→∞∫_(×^2)^N|1/N∑_i=1^N ϕ(x̃_i) - ∫_×^2ϕ(x̃)  ̣̃x|^2  λ̣_N^a,b=0.
We denote by I_N the integral. Expanding the square in the integral leads to
  
  I_N =     1/N^2∑_i,j=1^N ∫_(×^2)^Nϕ(x̃_i) ϕ(x̃_j)  λ̣_N^a,b +(∫_×^2ϕ(x̃)  ̣̃x )^2 
    - 2/N(∫_×^2ϕ(x̃)  ̣̃x ) ∑_i=1^N ∫_(×^2)^Nϕ(x̃_i)  λ̣_N^a,b.
  
Note that λ_N^a,b= ρ^N(X_N) X̣_N. Using the marginal densities ρ^N_j,  j=1,2, we have
  
  I_N =    N-1/N∫_(×^2)^2ϕ(x̃_1) ϕ(x̃_2) ρ^N_2(x̃_1, x̃_2) ̣̃x_1̣̃x_2 + 1/N∫_×^2ϕ(x̃_1)^2 ρ^N_1(x̃_1) ̣̃x_1 
     + (∫_×^2ϕ(x̃)  ̣̃x )^2 - 2 (∫_×^2ϕ(x̃)  ̣̃x ) ∫_×^2ϕ(x̃_1) ρ^N_1(x̃_1) ̣̃x_1 .
  
Now we finish the proof by letting N→∞ and using Theorem <ref>.


§ ENERGY CONDITIONAL SOLUTIONS TO 2D EULER EQUATIONS


In this part we show the existence of solutions to 2D Euler equations whose renormalized energy is confined in an interval[a,b]. First we give the precise meaning of the solution.


Let a,b∈, a<b be fixed. A stochastic process ω_· defined on some probability space (Θ, ℱ, ) with trajectories in C( [0,T], H^-1-(^2)) is called an energy conditional solution of the 2D Euler equations if for any t∈ [0,T], ω_t has the distribution μ^a,b defined in (<ref>), and for any ϕ∈ C^∞(^2), -a.s.,
   
    ω_t, ϕ= ω_0, ϕ + ∫_0^t ω_s⊗ω_s, H_ϕ ṣ   t∈ [0,T].


The above equation is called the weak vorticity formulation of the 2D Euler equation (see <cit.>). Here, for anyϕ∈ C^∞(^2),H_ϕ(x,y)= 1/2 K(x-y)· (∇ϕ(x)- ∇ϕ(y)),   x,y∈^2, x≠ y,in whichKis the Biot–Savart kernel on^2. We shall setH_ϕ(x,x)=0for allx ∈^2. Note thatμ^a,bis absolutely continuous with respect to the enstrophy measureμ, with a density function bounded by1/Z^a,b, whereZ^a,b= μ({∈̋[a,b]}) >0. Thus by <cit.>, the nonlinear termω_s⊗ω_s, H_ϕis well defined.


We recall that Cipriano showed in <cit.> the existence of solutions to 2D Euler equations with given energy a∈, as long as the density function of $̋ is positive ata. It is interesting to prove the same result by lettingb→ ain the above definition. The key ingredient is to show uniform estimates (independent ofaandb) of the type in Lemma <ref> (without the parameterN). For the moment we do not know how to do this.


Now we state our main result in this part.


There exists a probability space (Θ,ℱ, ) with the following properties.

   (i) There exists a stochastic process ω:[0,T]×Θ→ H^-1-(^2) such that it is an energy conditional solution of the 2D Euler equations in the sense of Definition <ref>.

   (ii) On (Θ,ℱ, ) one can define a subsequence of random point vortex system which converges -a.s. in C( [0,T], H^-1-(^2)) to the solution of point  (i).

   (iii) On (Θ,ℱ, ) one can define a subsequence of functions ω^(n)(θ,t,x),  (θ,t,x)∈Θ× [0,T]×^2, such that for -a.s. θ∈Θ, the functions (t,x)↦ω^(n)(θ,t,x) are L^∞-solutions of 2D Euler equations, and converge to ω_·(θ) in C( [0,T], H^-1-(^2)).


Recall thatμ^a,b= (Z^a,b)^-1 1_{:ℋ: ∈ [a,b]}μ, it may seem that the above result follows from <cit.>. However, the initial density function in the present case is not continuous, thus our result is not a direct consequence of <cit.>. A careful investigation of the proof in <cit.> reveals that the continuity of density function was only used there to show that the normalizing constantsC_Ntend to 1 asN→∞(see the arguments below <cit.>). Since we have already shown in (<ref>) the convergenceZ_N^a,b→ Z^a,b, we can follow the ideas in <cit.> to prove Theorem <ref>.  In the sequel we present the main preparations which are needed in the proof.

LetN≥ 2be fixed, we consider the point vortex dynamics on^2:
  
    X̣^i,N_t/ṭ= 1/√(N)∑_j=1^N a_j K(X^i,N_t -X^j,N_t ),    i=1,…, N

with the vortex intensities(a_1,…, a_N)∈ (∖{0})^Nand initial positions(X^1,N_0, …, X^N,N_0 )∈ (^2)^N ∖Δ_N, whereΔ_N= { (x_1, …, x_N)∈ (^2)^N: ∃  i≠ j  x_i= x_j}is the generalized diagonal. It is well known that, for Leb_^2^⊗ N-a.e. initial condition(X^1,N_0, …, X^N,N_0 )∈ (^2)^N ∖Δ_N, the above system of equations has a global solution, that is, the vortex points do not collapse, cf. <cit.>. Therefore, we can define the vorticity fieldω^N_t = 1/√(N)∑_i=1^N a_i δ_X^i,N_t,   t≥ 0,which satisfies, for anyϕ∈ C^∞(^2),
  
    ω^N_t, ϕ= ω^N_0, ϕ + ∫_0^t ω^N_s⊗ω^N_s, H_ϕ ṣ,    t≥ 0.

We mention that an interesting model involving the creation and damping of point vortices is studied in the recent work <cit.>.

We shall consider the point vortex dynamics with random intensities and random initial conditions. Thus, on a probability space(Θ, ℱ, ), let{ξ_i}_i≥ 1be an i.i.d. sequence of random variables with the standard Gaussian distributionN(0,1), and{X^i_0}_i≥ 1an i.i.d. sequence of random variables with uniform distribution on^2; assume the two families are independent. Note that the measuresλ_Nandμ_Ndefined in Section 1 are the laws of((ξ_1, X^1_0), …, (ξ_N, X^N_0))and ofω^N_0= 1/√(N)∑_i=1^N ξ_i δ_X^i_0, respectively.

For the reader's convenience, we recall the following result which is taken from <cit.> (the last property is not mentioned there, but it is obvious since the Hamiltonian is invariant for point vortices).


Consider the point vortex dynamics (<ref>) with random intensities (ξ_1, …, ξ_N) and random initial positions ( X^1_0, …, X^N_0) distributed as λ_N. -almost surely, the dynamics ( X^1,N_t, …, X^N,N_t) is well defined in (^2)^N ∖Δ_N for all t≥ 0, and ( (ξ_1, X^1,N_t), …, (ξ_N, X^N,N_t) ) has the invariant law λ_N. The associated measure-valued vorticity ω^N_t is a stochastic process with the invariant marginal law μ_N; moreover, -a.s., ω^N_t satisfies (<ref>) and
  
    ℋ_N (ω^N_t )= ℋ_N( (ξ_1, X^1,N_t), …, (ξ_N, X^N,N_t) ) = ℋ_N (ω^N_0 ),    t≥ 0.


Next we confine the point vortex dynamics to those initial configurations with energy belonging to the interval[a,b]. To this end, we introduce the conditional probability measures on(Θ, ℱ):_N^a,b=  1_{ℋ_N(ω^N_0)∈ [a,b]}/({ℋ_N(ω^N_0)∈ [a,b] }) =  1_{ℋ_N(ω^N_0) ∈ [a,b]}/Z_N^a,b.The measureλ_N^a,bdefined in (<ref>) is the distribution on(×^2)^Nof( (ξ_1, X^1_0), …, (ξ_N, X^N_0) )under_N^a,b, and we haveμ_N^a,b = (𝒯_N)_#λ_N^a,b, where𝒯_Nis defined in (<ref>). From Proposition <ref> we deduce the following result.


Consider the point vortex dynamics (<ref>) with random intensities (ξ_1, …, ξ_N) and random initial positions ( X^1_0, …, X^N_0) distributed as λ_N^a,b. Then, _N^a,b-a.s., for all t≥ 0, the dynamics ( X^1,N_t, …, X^N,N_t) is well defined in (^2)^N ∖Δ_N, and ( (ξ_1, X^1,N_t), …, (ξ_N, X^N,N_t) ) has the invariant distribution λ_N^a,b. The associated measure-valued vorticity ω^N_t is a stochastic process with the invariant marginal law μ_N^a,b; moreover, _N^a,b-a.s., ω^N_t satisfies (<ref>) and
  ℋ_N (ω^N_t ) = ℋ_N (ω^N_0 )∈ [a,b],    t≥ 0.
Since the conditional probability measure _N^a,b is absolutely continuous with respect to , the properties that hold -a.s. also hold almost surely with respect to _N^a,b. It remains to show that λ_N^a,b is the invariant distribution of ( (ξ_1, X^1,N_t), …, (ξ_N, X^N,N_t) ). Once this is proved, we deduce that ω^N_t has the invariant marginal law μ_N^a,b since
   law(ω^N_t ) = (𝒯_N)_# law((ξ_1, X^1,N_t), …, (ξ_N, X^N,N_t) ) = (𝒯_N)_#λ_N^a,b =μ_N^a,b.

To simplify the notations, we write ξ= (ξ_1, …, ξ_N) and X^N_t= (X^1,N_t, …, X^N,N_t ). For any bounded measurable function F: (×^2)^N→, by the definition of _N^a,b and (<ref>),
  ∫_Θ F(ξ, X^N_t) _N^a,b   = 1/Z_N^a,b∫_Θ F(ξ, X^N_t)  1_{ℋ_N(ω^N_0) ∈ [a,b]} 
   = 1/Z_N^a,b∫_Θ F(ξ, X^N_t)  1_{ℋ_N (ξ, X^N_t) ∈ [a,b]} 
   = 1/Z_N^a,b∫_Θ F (ξ, X^N_0)  1_{ℋ_N (ξ, X^N_0) ∈ [a,b]} ,
  
where the last step is due to the fact that, under , (ξ, X^N_t) has the invariant distribution λ_N. Therefore,
  
    ∫_Θ F(ξ, X^N_t) _N^a,b = ∫_Θ F (ξ, X^N_0)  _N^a,b,

which implies that, under the conditional probability measure _N^a,b, (ξ, X^N_t) has the same law as (ξ, X^N_0), i.e. λ_N^a,b.


To emphasize the dependence on the parametersa,b, we denote byω^N_a,b(·)the associated measure-valued vorticity field obtained in Proposition <ref>. The next lemma gives useful estimates onω^N_a,b(·).


Let N_0 be large enough such that Z_0:= inf_N≥ N_0 Z_N^a,b>0 and f:^2×^2→ be symmetric, bounded and measurable. Then for all p≥ 1 and δ>0, there are constants C_p, C_p,δ>0 such that for all N≥ N_0,
  __N^a,b[ω^N_a,b(t)⊗ω^N_a,b(t), f ^p] ≤ C_p f_∞^p / Z_0,   __N^a,b[ω^N_a,b(t)_H^-1-δ^p ] ≤ C_p,δ/ Z_0.
Moreover, if f(x,x)=0 for all x∈^2, then
  __N^a,b[ω^N_a,b(t)⊗ω^N_a,b(t), f ^2] ≤2/Z_0∫_^2×^2 f(x,y)^2 x̣ỵ.
Note that
  __N^a,b[ω^N_a,b(t)⊗ω^N_a,b(t), f ^p] = 1/Z_N^a,b_[ω^N_t⊗ω^N_t, f ^p  1_{ℋ_N(ω^N_0) ∈ [a,b]}], 
the first estimate follows from <cit.>. The proofs of the others are similar.


With these preparations in hand, we can complete the proof of Theorem <ref>. More precisely, letQ^Nbe the law of the process{ω^N_a,b(t) }_t∈ [0,T]on𝒳= C([0,T], H^-1-(^2) ). Using the equation (<ref>) and the estimates in Lemma <ref>, we can show that the family{Q^N}_N≥ N_0is tight on𝒳, see the beginning part of <cit.> for details. By Prohorov's theorem, there is a subsequence{N_k}_k≥ 1such thatQ^N_kconverges weakly to some probability measureQon𝒳. Skorokhod's theorem implies that there exist a probability space( Θ̃, ℱ̃,)and processesω̃^N_kandω̃, with trajectories in𝒳, such that their laws areQ^N_kandQ, respectively; and-a.s.,ω̃^N_kconverges toω̃in the topology of𝒳. Moreover, the processesω̃^N_kcan be represented as a sum of Dirac deltas:ω̃^N_k_t= 1/√(N_k)∑_i=1^N_kξ̃_i δ_X̃^i,N_k_t,    t∈ [0,T],where((ξ̃_1, X̃^i,N_k_t),…, (ξ̃_N_k, X̃^N_k,N_k_t))is a random vector with the invariant lawλ_N_k^a,bfor allt∈[0, T], and it solves the point vortex dynamics (<ref>), see <cit.> for the detailed proof.

Next we prove the law ofω̃_tis the energy conditional measureμ^a,bfor allt∈ [0,T]. For anyF∈ C_b( H^-1-(^2) ), sinceω̃^N_kconverges-a.s. toω̃in the topology of𝒳,∫_Θ̃ F(ω̃_t) ̣̃   = lim_k→∞∫_Θ̃ F(ω̃^N_k_t )  ̣̃ = lim_k→∞∫_H^-1-(^2) F(ω) μ̣_N_k^a,b(ω) = ∫_H^-1-(^2) F(ω) μ̣^a,b(ω),where in the last step we have used the weak convergence ofμ_N^a,btoμ^a,bproved in Section 3.

Finally, using again the estimates in Lemma <ref> and repeating the arguments below <cit.>, we can show thatω̃_·satisfies the weak vorticity formulation (<ref>) of the 2D Euler equation. Summarizing the above discussions, we have proved the first two assertions of Theorem <ref>. The last assertion is proved in the same way as the end of <cit.>.




§ STRUCTURES AND INTERMEDIATE REGIMES



In classical Onsager theory the microcanonical measure is defined as the
uniform measure on configurations(  x_1,… ,x_N)  such
that
  
    ∑_i≠ jξ_iξ_jlog1/| x_i-x_j|∼ N^2 a

for some value ofa>0(in this section we heuristically writeainstead of[a,b]since, fora>0, it is the value ofawhich plays a practical role, independently ofb). For typical configurations(  x_1
,… ,x_N)  , whenNis large, the empirical measure

    1/N∑_i=1^Nξ_iδ_x_i

is close to the solutions of a certain mean field equation (Onsager theory).
There is a natural explanation, fora≫ 0: in order to have
(<ref>) we need roughlyN^2terms in the sum∑_i≠
jwith valueξ_iξ_jlog1/| x_i
-x_j|close toa(this argument is very rough). The “only”
way to reach such result is to group positive vortices together, all very
close to each other, and similarly for the negative ones, with the two clusters
not so close to each other: roughly(  N/2)  ^2terms will be
positive and close toa(those corresponding to positive pairs), other(  N/2)  ^2terms will be positive as well and close toa(those corresponding to negative pairs), and the remaining pairs, composed of
vortices of opposite signs, have small value ofξ_iξ_jlog1/| x_i-x_j|because the two points belong to
clusters which are relatively far from each other.

In our “white noise” model, the microcanonical measure corresponds to the constraint
  
    ∑_i≠ jξ_iξ_jlog1/| x_i-x_j|∼ N a.

The typical configurations(  x_1,… ,x_N)  , for largeN, have the renormalized empirical measure
  
    1/√(N)∑_i=1^Nξ_iδ_x_i

close to white noise conditioned to renormalized energy equal toa. In Figure 1 below, we show the histogram of the interaction energy of 200 point vortices (its features do not change by increasing the number of vortices). It shows the typical values of “a” in formula (<ref>). They are very small and the corresponding configurations are quite disordered, opposite to the structures of Onsager theory and coherently with the white noise limit. The theoretical energy spectrum of free white noise ensemble (not constrained by the energy) decays ask^-1, opposite to the predicted decayk^-5/3of inverse stationary 2D turbulence. The question then is the decay of the spectrum for the microcanonical ensemble, especially for large values ofa, when we expect some degree of clustering of the vortices and then, potentially, the emergence of a more interesting spectrum.


    < g r a p h i c s >
Left: the histogram of 10,000 samples of the interaction energies of 200 uniformly distributed point vortices. Right: the curve is the spectrum function computed from the 11 samples with largest interaction energies, while the straight line has the slope -5/3.

It is very difficult to compute theoretically the spectrum function of the microcanonical measure (<ref>), thus we do some numerical simulations. We generate 10,000 samples of uniformly distributed point vortices; each sample consists of 200 vortices, in which half of them have intensity1/√(200)and the rest-1/√(200). We single out the 11 samples which have the largest interaction energies (a= 0.51in this case), and compute their average spectrum function. The results are shown in Figure <ref>. It shows that the slope of the spectrum is still very close to-1like in the case of the free ensemble, far from-5/3.

Deviations of the spectrum slope from the flat value-1are due to clusterization of point vortex configuration. To prove numerically this claim we proceed as follows: we produce artificially an initial condition with small clusters and then let it evolve by point vortex dynamics. We do not have a theorem of convergence to equilibrium but hope that after some time the configuration is more typical for the microcanonical ensemble. Precisely, we generate a point vortex configuration which, apart from some uniformly distributed point vortices, contains small clusters with 2, 4 and 8 vortices (these numbers are chosen for convenience). The clusters have uniformly distributed centers and their diameters are of the order 0.01. To get a smoother spectrum function, we produce 10 such samples (with average energy 1.364966) and compute the averaged spectrum function, which is shown by the thin line on the right of Figure <ref>. We see that it is close to the line with slope-5/3in a certain range oflog(k). We take these special configurations as initial conditions and run the dynamics (use the Heun algorithm), with a small time steph=0.0001. In Figure <ref>, we show the vortex distribution of one of the samples after 120,000 steps of evolution:+and∘represent vortices of positive and negative intensity, respectively. The graph of the final spectrum function is shown by the dashed line on the right of Figure <ref>, which, on the rangelog(k)∈ [1,3], has the approximative slope-1.775. Opposite to the cases considered in Figure <ref>, here we find a slope considerably different from-1and in the direction of-5/3.


    < g r a p h i c s >
Left: the configuration of (major part of) vortices after 120,000 steps of evolution. Right: spectrum functions before and after running dynamics.

The question then is how to obtain spontaneously some degree of local clusterization, from an invariant measure and in particular from a microcanonical ensemble. Compared to turbulence, it seems that the two regimes of Onsager and conditional white noise are two “extremes”. Turbulence is in the middle: typical configurations are not so uniformly distributed as in the white noise case, they have locally a great degree of clustering. But only locally, at small scale, not globally as the two big clusters of the Onsager case. Thus in the turbulence regime we expect that each vortex interacts neither with all those of the same sign (as in (<ref>)) nor only with very few of the same sign (as in (<ref>)), but with an intermediate amount.

A natural microcanonical condition is therefore
  
    ∑_i≠ jξ_iξ_jlog1/| x_i-x_j|∼ c(N) e

for some
  
    N≪ c(N)≪ N^2.

The mathematical question then is whether it is possible to study the limit asN→∞of this intermediate regime. For finiteNthe microcanonical measure with normalizing constantc(  N)  is invariant for Euler dynamics, but we do not know a corresponding invariant measure obtained asN→∞.  We leave this question open but hope the clarifications of this work help to address the question.


Acknowledgements. The second author is grateful to the financial supports of the grant “Stochastic models with spatial structure” at the Scuola Normale Superiore di Pisa, the National Natural Science Foundation of China (Nos. 11571347, 11688101), and the Youth Innovation Promotion Association, CAS (2017003).


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