Existence for stochastic 2D Euler equations with positive $H^{-1}$ vorticity
Zdzisław Brzeźniak Mario Maurelli


Improved Distributed Approximations for Maximum-Weight Independent Set
    Ken-ichi KawarabayashiNII, Japan, 
		Seri KhouryUniversity of California, Berkeley,  seri_khoury@berkeley.eduAaron SchildUniversity of California, Berkeley, Gregory SchwartzmanNII, Japan, 
    
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We prove the existence of non-negative measure and H^-1-valued vorticity solutions to the stochastic 2D Euler equations in the vorticity form  with the transport type noise, starting from any non-negative vortex sheet. This extends the result by Delort <cit.> to the stochastic case.







§ INTRODUCTION



In this paper we consider the two-dimensional  stochastic Euler equations, in the vorticity form, with the transport type gaussian noise with the periodic boundary conditions. In other words we study the following stochastic  evolution equation on the two-dimensional torus 𝕋^2:

    ∂_t ξ + u·∇ξ + ∑_k σ_k·∇ξ∘  Ẇ^k =0,
       u = K ∗ξ,

see below for explanation of the notation used here.

This equation represents the motion of an incompressible fluid in a periodic domain, perturbed by a noise of the transport-type. In our main result, Theorem <ref>, we prove the weak existence,  in both the probabilistic and the analytic sense,  of a certain measure-valued solution for this equation.  More precisely, we  prove that for any initial datum ξ_0 ∈ H^-1 which is also a  non-negative measure, there exists a weak, measure and H^-1-valued solution to (<ref>). This result extends the existence result by Delort <cit.> from the deterministic to the stochastic case.

The deterministic incompressible, d-dimensional Euler equations written in the following form

    ∂_t u +(u·∇) u +∇ p = 0,
        u =0,

describe the evolution of the velocity field  u(t,x)∈^d and the scalar pressure p(t,x)∈ of an incompressible fluid.  For the simplicity sake we assume that x belongs to d-dimensional torus  ^d. The local (in time) well-posedness of (<ref>) withing a class of  smooth solutions has been known    since at l;east of the seminal work  <cit.>.  A general description of the Euler equations can be found in many excellent works, for example in  <cit.>, <cit.>, <cit.>. In the two-dimensional case, i.e. when d=2, the Euler equations (<ref>) are known to be, at a heuristic level,  equivalent to the following  non-linear transport type equation, called  the vorticity equation,

    ∂_t ξ + u·∇ξ =0,
       u(t) = K ∗ξ(t),   .

Here ξ(t,x):=  u(t,x) = ∂_x_1u^2(t,x) -∂_x_2u^1(t,x) is the scalar vorticity of the fluid, which describes  how fast the fluid rotates around a point x.  The kernel K:^2→^2 is given by K=∇^⊥ G, where G:^2→ is the Green function of the Laplacian on L^2(^2), restricted to   mean zero functions, see <cit.>. Alternatively, u(t) =  ∇^⊥Δ_0^-1ξ(t), t≥ 0.

 The global well-posedness within the class of  essentially bounded solutions  has been proved in <cit.> and <cit.>. See also <cit.> and <cit.> for an extension to almost bounded functions and <cit.> for a different proof using flows.  In the case of deterministic 2D Euler Equations,  there is a huge number of papers using the vorticity formulation
(<ref>)  of these equations with the initial data belonging to various functional, e.g. Besov,  spaces, see for instance Serfati <cit.>,  Vishik <cit.>, Bernicot et al <cit.>, Bahouri and Chemin <cit.>.


In larger classes than of  bounded solutions, the global well-posedness  has been established
by Majda and Di Perna in <cit.> for the initial  vorticity belonging to the spaces  L^p(^2), with 1<p<∞.  To be more precise, the result in <cit.>  is given in  the full euclidean space ℝ^2, but for the simplicity sake, in this introduction we will not distinguish between the cases of the full space and the torus. The case  we are interested in is of the initial  the vorticity being a  measure and the solution is measure-valued. One of the main results in this context is the global existence result by Delort <cit.>, where the vorticity is assumed to have  a distinguished sign and to be an element of the Sobolev space H^-1(^2). More  precisely, Delort proved that   for any  initial datum ξ_0 ∈ H^-1(^2) which is also a non-negative measure, there exists a non-negative measure- and H^-1(^2)-valued solution ξ to the vorticity equation (<ref>)  such that ξ(0)=ξ_0. Delort's result  includes the case of an initial vortex sheet, that is when ξ_0 is concentrated on a C^1 curve. Later papers by Schochet <cit.> and Poupaud <cit.> gave a somehow clearer argument which we will  mostly use  in the rest of the paper. A more general existence result, where the vorticity is a sum of a non-negative measure H^-1(^2) and an L^p(^2) function, for some p≥ 1, has been studied  in <cit.>; <cit.> show the short-time existence when the initial vortex sheet consists of two parts with opposite sign and disjoint supports, and establish non-trivial bounds on propagation of the support of solutions. Studying  of irregular vorticity solutions has several motivations.  Firstly, such solutions  represent physically relevant situations, where the vorticity is concentrated on sets of   Lebesgue measure zero, see e.g. <cit.>.  Secondly, such solutions are   related to the possible anomalous energy dissipation; to the best of our knowledge, the energy conservation or dissipation remains an open problem for class of the Delort solutions, see <cit.> for energy conservation for unbounded vorticity. Thirdly, but not lastly, they are also related  to the boundary layers phenomena, see e.g. <cit.>. However we do not pursue  these topics here. Let us only briefly discuss the two other classes of solutions. The authors of <cit.> introduce another (and different) concept of a “measure-valued solution”.  This concept refers to a different, weaker type of a solution, i.e. in the sense of the Young measures, which we do not consider here in this paper. An interested reader can consult the paper <cit.> by Majda for more details about comparison between the framework of <cit.> and that of <cit.>. Recently a variety of types of weak solutions to (<ref>), where ξ does not need to be a function or measure, have been constructed using methods related to convex integration, showing the non-uniqueness in certain functional spaces, as for instance u∈ L^∞, and energy dissipation, see e.g. <cit.>. In particular, <cit.> exhibits infinitely many weak solutions to (<ref>) with an initial condition ξ_0 being a non-negative H^-1 measure. However, the author claims that, for those weak solutions, the vorticity ξ =  u is in general not measure-valued. We are not aware whether one can construct wild solutions to (<ref>) by convex integration methods whose vorticity is measure-valued, see also the discussion in <cit.>.



Before moving on  to the stochastic case, let us present  a short idea of the proof of the Delort result. The strategy passes through an usual compactness and convergence argument.  Take a sequence of approximants for which we can  show uniform a'priori  bounds, prove  the compactness and  show the stability of the equation in the limit. The main problem comes from the stability step, in particular from the proof of the stability of the non-linear term. The main idea used  by Delort, and later also by Schochet <cit.>,  is that the non-linearity  is continuous with respect to  non-negative, or more-generally bounded from below, H^-1(^2) measures, in the weak topologies. The precise description of the topologies involved will be given later. Hence it suffices to show the conservation  of the non-negativity, or at least a bound from below, and the uniform a'priori bounds on the total variation  and the H^-1(^2) norms of the approximations of the  solution. The transport and divergence-free nature of the vorticity equation (<ref>) implies easily the non-negativity and the a priori bound on the total variation norm. Since the  H^-1(^2) norm of ξ is equivalent to the L^2(^2) norm of u=K ∗ξ, i.e.  the energy, so the a'priori H^-1(^2) bound is equivalent to an a priori bound on the energy of u, which is also classical and available. This allows to have compactness and stability as required.


The transport and divergence-free nature of the vorticity equation (<ref>) implies easily the formal conservation of mass of ξ and hence a uniform bound on the total variation norm of ξ, the main issue is the stability under the limit, in particular the stability of the non-linear term. The main point by Schochet is that the drift u·∇ξ (with u=K ∗ξ) reads formally as a distribution N(ξ) defined by

    N(ξ),φ = ∫_^2∫_^2 F_φ(x,y) ξ( x)ξ( y),   φ∈ C^∞(^2),

where, for any test function φ, F_φ is bounded everywhere and continuous outside the diagonal {x=y}. As a consequence, the function ξ↦ N(ξ) is continuous among those ξ which are non-negative and non-atomic (the precise topologies will be given later). Hence it suffices to show that the approximants ξ^ and the limit measure ξ are non-negative and non-atomic.


The stochastic  Euler Equations perturbed   by gaussian noise have been investigated in a large number  of publications.  Here we only give a review of a small selection of them. Many articles study such equations with either   additive or multiplicative noise but dependent on u and not on its gradient. The first works are <cit.>, which shows  the global well-posedness of strong solutions in the case of   additive noise, for bounded vorticity in a 2D domain (though adaptedness is not proven there), <cit.> and <cit.>, which show the global existence of martingale solutions, with multiplicative noise, for L^2(^2) and, in the second work, L^p(^2) vorticity also in the  2D domains. Let us  also mention the paper <cit.> for a geometric approach in the case of finite-dimensional additive noise and <cit.> for an approach via nonstandard analysis. The works <cit.>, <cit.> contain proofs of a  local strong well-posedness of  smooth solutions, with additive and, in the second work, multiplicative noise, in general dimensions. The latter paper  contains also a proof of  a  global well-posedness  of  smooth solutions in 2D for additive or   multiplicative linear noise. The work <cit.> deals with  low regularity solutions to the stochastic Euler equations driven by an additive noise, in any space dimension. More precisely,
the authour shows the existence of the s measure-valued solutions, in the sense of Young measures as in <cit.>, see the discussion above,  for the initial vorticity in the H^-1 space.



This is different from our results as we prove an existence of solutions for the initial vorticity being a non-negative measure from the H^-1 space and we prove that this space is preserved in time. Moreover, the our noise is multiplicative (of transport type) while Kim considers an additive noise and hence the is no intersection between our results.
In fact, one cannot expect the non-negativity of the vorticity to be preserved for Euler Equations driven by an additive noise.
It is important to stress that we work here in a framework which corresponds to, in   the deterministic case, the framework  of Delort <cit.>, Schochet <cit.>, Poupaud <cit.> and Majda <cit.> of weak solutions and not in the framework of DiPerna and Lions <cit.> of the so called measure valued solutions. An interesting reader can consult the paper <cit.> by Majda for more details about comparison between these two approaches.



The transport type noise for the stochastic Navier-Stokes Equations has been first introduced in early papers <cit.> and later in, e.g. <cit.> and <cit.>.
In the case of the stochastic Euler Equations, the transport type noise has been first
 studied in <cit.>, <cit.>, <cit.>, where the transport term acts  on the velocity and not on the vorticity.  These works show the global existence of martingale solutions for L^2(^2) vorticity in 2D domains. The model (<ref>) has been introduced  in <cit.>, where the authours  prove the  global strong well-posedness in the class of  bounded vorticity solutions on the two-dimensional torus. In the recent papers <cit.> and <cit.>, the authors consider  the same model, but with very irregular vorticity, not in H^-1(^2) as in the present study, and  show the  existence of solutions for random  initial data.
 which is invariant for the dynamics or almost invariant (that is, it is absolutely continuous with respect to the invariant measure).
   
   An analogue of the model (<ref>) in the 3D case, where a stochastic advection term also appears, has been recently  considered in <cit.>, where the authors  show the local strong well-posedness and a  Beale-Kato-Majda type criterion for the non-explosion. Let us  also point out the work <cit.>, in which the authours consider equation  (<ref>) with the initial mass concentrated in a finite number of point vortices and shows a regularization by noise phenomenon, precisely that, with full probability, no collapse of vortices happens with a certain noise, while it does happen without noise.

There are several reasons why we use transport noise. One of them, maybe the most important one,  is that the equations driven by   a transport type noise preserve the transport structure of the vorticity equation.  At a formal level, a solution ξ(t,x,ω) follows the characteristics of the associated SDEs, i.e.  ξ(t,X(t,x,ω),ω)=ξ(0,x), where X(t,x,ω) is the stochastic flow solution to the following SDEs

    dX(t) = u(t,X(t))dt +∑_kσ_k(X(t)) ∘ dW^k(t).

This fact can be proven by   the Itô formula.   The Stratonovich form of the noise is essential, because the Itô formula for this noise works as the chain rule, without second-order corrections. Furthermore, there is a derivation of models with stochastic transport term in <cit.>, <cit.>, there are applications using the transport noise to model uncertainties, e.g. <cit.>, also the linear stochastic transport equation has been used as a toy model for turbulence, see e.g. <cit.>, though we will not go into these directions here. The main feature of interest here is the fact that the transport noise preserves,  at least the heuristic level,  the L^∞(^2) norm of the solution and also, when the coefficients σ_k are divergence-free as here, the L^p(^2) norm for any 1≤ p≤∞ (this can be seen for example via a priori bounds for the L^p norm). As a consequence, the transport noise preserves the total mass of the vorticity and its positivity[ i.e. if ξ_0≥0, then also ξ_t≥0,] two properties that are crucial to apply the Schochet's argument <cit.>. This is not the case for a generic additive or multiplicative noise as considered in the above mentioned papers. Indeed the additive noise does not guarantee positivity of the solution, while linear multiplicative noise does not guarantee mass conservation. We should admit though that additive  or certain multiplicative noise could be used for our purposes. Indeed what we really need to have the  Delort's result in the stochastic case  are  some uniform M(^2) bounds of the vorticity and some uniform L^∞ bounds, or even L^p bounds in the line of the more general result in <cit.>, on the negative part of the vorticity. Now a linear multiplicative noise acting on  on the vorticity of the form σ_kξ∘  Ẇ^k preserves the positivity and may also give some uniform bound on the mass; for additive noise, a bound from below of the vorticity may be proved via a transformation à la Doss-Sussmann from the SPDE to a random PDE.


Let us also  remark here that the stochastic vorticity equation (<ref>) preserves  the enstrophy, that is the L^2(^2) norm of the vorticity solution ξ, but not the energy, that is the L^2(^2) norm of the velocity u=K ∗ξ. Indeed, the equation for the velocity field u(t,x,ω) can be written, in a purely  formal way,  as

    ∂_t u +(u·∇) u +∑_k(σ_k·∇ +Dσ_k)u ∘  Ẇ^k = -∇ p -γ,
        u=0,

with unknown   pressure p(t,x,ω) and a time dependent constant γ(t,ω). The constant  γ
 is needed here in order  to keep u with zero spatial mean.[Thus, as the pressure is often viewed as the Lagrange multiplier corresponding to the divergence zero constraint, the constant γ could be seen as the
  the Lagrange multiplier corresponding to the  mean  zero constraint.]

 In the above  equation (<ref>), a new  zero order term (Dσ_k) u ∘  Ẇ^k  causes the L^2-norm of the velocity not to be preserved anymore.

Let us also mention that the transport noise has been used to show the so called regularization by noise phenomena for the stochastic  transport equations, see <cit.> and <cit.> among many others.  Such results have been so far mostly limited to the linear cases, with a few positive results for other nonlinear hyperbolic cases, see e.g. <cit.>.
A  generalisation of the results from the paper   <cit.> to the case of a more general, measure-valued vorticity provides new and challenging  difficulties.



Before embarking to describing contributions of our paper, let us  point out two peculiarities of the strategy used, in the stochastic case, in order to prove the existence of a  martingale solution, see for example <cit.>. Analogously to the deterministic case, the strategy passes through the usual tightness and convergence argument. One  takes a sequence of approximants, shows a'priori uniform bounds for suitable moments and deduces  the tightness of the laws, then one  shows the stability of the equation in the limit. We want to emphasize  two facts, which we will use in our proof as well. Firstly, to prove the  tightness of the laws, say, in C([0,T];X), where X is a suitable  functional space on the torus ^2, one needs a uniform bound on the marginals and also a uniform bound in C^α([0,T];Y), where Y is another functional space, typically a negative-order Sobolev space, containing X. The latter is the stochastic counterpart of the Aubin-Lions Lemma, used  since at least the paper  <cit.>. Secondly, for the stability, one option is to exploit the Skorokhod representation theorem which allows one to pass from convergence in law to a.s. convergence in the aforementioned space C([0,T];X).

Now we are ready to describe in detail  our paper. The main result, i.e.  Theorem <ref> extends the result by Delort <cit.> to  the stochastic setting with the transport type noise. To the best of our knowledge, no such extension has been studied before. Our proof combines the argument by Schochet <cit.> to deal with the nonlinear term,   with the argument by e.g. <cit.> to deal with the stochasticity, as explained before.  More details are given in Seciton <ref>.

Two comments are in place. Firstly, as mentioned earlier, the transport type noise is responsible for the conservation  of mass and of positivity and thus allows to extend easily the Schochet argument concerning these aspects. On the contrary, a generic, additive or multiplicative, noise would fail at this point, though uniform bounds could work also for certain choices of noise.


Secondly, the Schochet argument does not immediately goes through with respect  the uniform L^2(^2) bounds on the velocity u. Indeed,  as we have seen above,  the equation  (<ref>) for the velocity filed u does not preserve the energy. To overcome this difficulty, we assume a certain structure of
the coefficients σ_k so that  uniform L^2(^2) bounds on u can be still deduced.
These assumptions are not restrictive as they allow  to deal with the relevant cases, as for example the "locally isotropic" convariance matrices, see the discussion in Section <ref>.

The  last comment we make in this Introduction concerns  an alternative approach  to  the proof. In <cit.>, we suggested an alternative proof of the well-posedness within the class of  bounded solutions, which is based on the Doss-Sussmann transformation.  To be precise,  ψ is the stochastic flow solutions corresponding to the following SDEs on the torus 𝕋^2:

    dψ_t = ∑_kσ_k(ψ_t) ∘ dW^k_t,

then the process  ξ̃(t,x,ω):= ξ(t,ψ(t,x,ω),ω) satisfies a random Euler-type PDE of the following type

    ∂_t ξ̃ +ũ·∇ξ̃ =0,
       ũ(t,x,ω) = Dψ^-1(t,x,ω) ∫_^2 K(ψ(t,x,ω)-ψ(t,y,ω)) ξ̃(t,y,ω)dy,

where ψ^-1 is the inverse flow. Equation (<ref>) is a nonlinear transport type equation, where the kernel K has been replaced by the above random kernel. In the case that σ_k=e_k 1_k∈{1,2}, where e_k is the canonical basis on ^2, the PDE (<ref>) is exactly the Euler vorticity equation. Indeed in this case the stochastic Euler equations correspond simply to a random shift in Lagrangian coordinates. In the general case however, this is not the Euler vorticity equation, yet it enjoys similar properties since the random kernel has similar regularity properties, hence the well-posedness among bounded solutions can be derived applying the deterministic arguments to the random PDE (<ref>). One may then wonder if a similar argument is possible here, namely using Schochet's arguments for the random PDE (<ref>). In this context however, there are two problems at least. Firstly, even if we could just apply the Delort's and/or Schochet's result to equationm (<ref>), at ω fixed, this would only give the existence of a measure-valued and H^-1-valued solution to (<ref>) at ω fixed, with no adaptedness property; so a compactness argument would probably be needed anyway. Secondly, from (<ref>) it is not immediately clear how to get a uniform H^-1(^2) bound on ξ̃. Moreover this type of arguments à la Doss-Sussmann does not work intrinsically on the stochastic PDE. Let us remark anyway that the argument may still give some hints: for example, it would be interesting to see if, using the equation for ũ instead of the equation for u, one can get a uniform H^-1(^2) bound without the additional assumptions on σ that are needed here. We leave this point for future research.












§ THE SETTING




 §.§ Notation


We recall some notation frequently used in the paper. We use the letters t, x, ω for a generic element in [0,T], ^2, Ω respectively. The coordinates of x are denoted by  (x^1,x^2), while the partial derivatives are denoted by ∂_x_j=D_j=∂/∂ x_j, j=1,2. Unless differently specified, the derivatives ∇, D and  Δ are meant to be  with respect to the space x. The Sobolev, not the Besov,  spaces are denoted by ^s,p, for s ∈ℝ and p∈ [1,∞).  For p=2, we put H^s = ^s,2.

By L_0^2(^2,^2) we denote the closed Hilbert subspace of L^2(^2,^2) consisting of of  mean zero vector fields
and  by  we denote the closed Hilbert subspace of L_0^2(^2,^2) consisting of  divergence free  vector fields. Alternatively,
  is equal to the closed Hilbert subspace of L^2(^2,^2) consisting of  mean zero and divergence free  vector fields. We endow  with the inner product from the space
L^2(^2,^2) and introduce the Leray-Helmholtz projection Π which by definition is the orthogonal projection from L^2(^2,^2) onto .

For a differentiable map f=(f^1,⋯,f^m):^2→^m, m∈ℕ,
we denote by Df(a) the m × 2-matrix of the Frèchet derivative d_af ∈ℒ(ℝ^2,ℝ^m) of f at a=(a_1,a_2)∈^2,  in the standard bases of ℝ^2 and ℝ^m,  i.e.  Df(a) =(D_jf^i)=[ D_1 f^1 D_2 f^1;       …       …; D_1 f^m D_2 f^m ].  If b: ^2 →^2 then we put 
    b ·∇ f:={^2 ∋ x ↦∑_j=1^2b_j(x) D_jf(x) ∈^m}.

If m=1, then by ∇ f(a) we will understand the unique vector in ℝ^2 such that ∇ f(a),x=d_af(x), x∈ℝ^2.
 By a ⊗ b we denote the tensor product of two vectors  a,b∈ℝ^2, i.e. a linear operator in ℝ^2 defined by ^2 ∋ x ↦ a  b,x∈^2. The matrix of
 in the standard bases of ^2 is [a_ib_j]_i,j=1^2=a_1b_1a_1b_2a_2b_1a_2b_2.
 Identifying a with a 2×1-matrix, we see that the last  matrix  is equal to ab^t.

For function  spaces, we often put the input variables (t, x, ω) as subscripts. For example, the spaces L^2(^2), C([0,T]), is denoted in short by L^2_x, C_t respectively.  As another example, the space C([0,T];(H^-4(^2),w)) of continuous functions on [0,T], which take values in H^-4(^2) endowed with  the weak topology, is denoted in short by C_t(H^-4_x,w). The symbol f∗ g stands for the convolution in the space variable  using the group structure of  ^2,  between two functions or distributions f and g on ^2.

The space M_x=M(^2) is the space of finite ℝ-valued  measures on ^2. It is a Banach space, endowed with the total variation norm ·_M_x and is the dual of the space C_x= C(^2) of the space of continuous functions on ^2. The notation μ,φ will be used to denote the duality pairing  between a measure μ∈M_x and a function φ∈ C_x. The closed ball of center 0 and radius M on M_x is denoted by M_x,M; the radius here refers to the  norm ·_M_x. Following the above notation, the space C([0,T];(M_x,M,w^∗ )) of continuous functions on [0,T] taking values in the closed ball of radius M in  M_x endowed with the weak-^∗ topology, is denoted in short by  C_t(M_x,M,w^∗ ). We  also use notation M_x,+ to denote  the set of non-negative finite measures on ^2. Finally, by  M_x,M,+,no-atom we denote  the set of non-negative non-atomic measures with total mass ≤ M, M_x,y for the space of finite measures on 𝕋^2× 2.


Given a probability space (Ω,A,ℙ) endowed with a filtration 𝔽=(F_t)_t, by the  symbol ℬ𝔽 we  denote  the progressively measurable σ-algebra associated with 𝔽. The letter C will be used for constants which may change from one line to another.





 §.§ Assumptions on the noise











Here we give the assumptions on the noise coefficients σ_k.



We assume the following assumptions.


  (A.1) For every k∈ℕ,  σ_k ∈ C^1(^2,^2)∩,  and

    ∑_kσ_k_C^1_x^2<∞.

In particular, we assume that there exists a continuous function a:^2×^2→ℒ(^2,^2), called the infinitesimal covariance matrix, such that




    a(x,y) =∑_k 
    σ_k(x)⊗σ_k(y)
    ,     (x,y)∈^2,

with the convergence in the space C_x,y. Moreover, function  a is differentiable separately w.t.t.  x and  y and

    ∂_x_ia(x,y) =∑_k ∂_x_iσ_k(x) ⊗σ_k(y) ,   i=1,2,

with the convergence in C_x,y. Analogously for the derivatives  w.r.t. y.

  (A.2) There exists c≥ 0 such that the function a defined above satisfies,

    a(x,x)=cI_2,   x∈,

where I_2 is the identity map in ^2. 

  (A.3) The function a defined above satisfies

    ∂_y_ia(x,y)|_y=x =0,   i=1,2,    x∈^2.







The equality (<ref>) in Assumption  (A.2) reads as follows:

    ∑_kσ_k^i(x)σ_k^j(x) = cδ_ij,    x,    i,j=1,2.


The equality (<ref>) in Assumption  (A.3) reads as follows:

    ∑_kσ_k^i(x) ∂_x_lσ_k^j(x) = ∂_y_la^i,j(x,y)|_y=x = 0,    x,    i,j,l=1,2.



Assumption (A.2) allows us  to simplify the Itô-Stratonovich correction term  
    1/2∑_k σ_k·∇[σ_k·∇ξ]
  in equation (<ref>). It cancels the first order term, i.e.

    1/2∑_k (σ_k·∇) σ_k ·∇ξ=
    1/2∑_k ∑_i,jσ_k^i D_i σ_k^jD_j ξ=0
  and  makes the second order term equal to a constant multiplied by the Laplacian, i.e.

    1/2∑_k ∑_i,jσ_k^iσ_k^jD_iD_j ξ= 1/2∑_k  [σ_k⊗σ_k
    D^2ξ]=c/2Δξ.
 The latter equality is obvious and   former   follows from equality (<ref>) with l=i. Indeed,

    ∑_k ∑_i,jσ_k^i D_i σ_k^jD_j ξ=∑_i,j[   ∑_k  σ_k^i D_i σ_k^j] D_j ξ = ∑_i,j 0 ×  D_jξ=0.

Hence the Stratonovich integral  in equation (<ref>) takes the following form

    ∫_0^t ∑_k σ_k·∇ξ∘ dW^k(s) =∑_k σ_k·∇ξ dW^k  - c/2∫^t_0 Δξ(x)   dr.

Finally, we infer that  the vorticity equation (<ref>)  can be written as the following  Itô equation:

    ∂_t ξ + u·∇ξ + ∑_k σ_k·∇ξ  Ẇ^k = 1/2 cΔξ,
       u(t) = K ∗ξ(t),     t ≥ 0.


The assumption <ref> for the for well-posedness of bounded vorticity solutions  in  the present paper are   stronger than those  used  in <cit.>.
Indeed, in <cit.> only the first two assumptions are needed;  the second in a slightly weaker form. Moreover,  the second assumption is not essential  because the  first-order Itô correction can be treated via lengthly computations. On the other hand, in the current paper,   it seems unclear from the proof whether Assumptions (A.2) and (A.3)  can be removed. Indeed these two assumptions guarantee that the equation for the velocity u has no other  first order terms but   the transport one. The last property   in turn implies a key energy bound of solution, i.e.  the L^2_x norm of the solution u(t).



The following Example provides a non-trivial example, i.e.  non constant, of vector fields   σ_k satisfying Assumption <ref>.



Let β>3 and define, with ℤ_∗^2 :={ k∈ℤ^2 }∖{0} and,  k^⊥=(-k_2,k_1) for k=(k_1,k_2)∈ℤ^2,

    σ_k(x) = (cos(k· x)+sin(k· x)) k^⊥/|k|^β,    x∈^2,    k ∈ℤ_∗^2∖{0}.

Since β>3, we infer that ∑_kσ_k_C^1_x^2 <∞ and we can calculate that the function a defined in (<ref>) satisfies

    a(x,y)    = ∑_k∈ℤ_+^2cos(k· (x-y)) k^⊥⊗ k^⊥/|k|^2β,

and (<ref>) with c=∑_k∈ℤ_+^22/|k|^2β-2,
 where

    ℤ_+^2:={ k∈ℤ^2: k_1≥0, k_2>0 }.




Note that, in this example, the infinitesimal covariance matrix a is translation-invariant, i.e  a(x,y) = a(x-y),  and even, i.e a(x)=a(-x). Moreover, if the function a is translation invariant and even, then it satisfies  Assumption (A.3). 
Indeed

    ∂_y_ia(x,y)|_y=x = -∂_z_ia(z)|_z=0 =0.


We could essentially include the class of isotropic infinitesimal covariance matrices in our setting. On ^2, an infinitesimal covariance matrix a is called isotropic if it is translation- and rotation-invariant, that is a(x,y)=a(x-y) for all (x,y) and R^ta(Rx)R =a(x) for every rotation matrix R, and a(0)=cI for some c>0. Rotation invariance implies that a is even, take for instance Rx=-x. Hence a sufficiently regular isotropic matrix a satisfies the Assumption <ref>. More precisely, if  a is regular, then  one can find vector fields σ_k  safisfying Assumption <ref>.

The isotropy condition on a means essentially that the noise ∑_k σ_k(x)∘Ẇ_t is Gaussian, white in time, coloured and isotropic in space. Such a class has been considered in the mathematical and physical literature, see e.g. <cit.>, <cit.>, even without the nonlinear term: for example, the same type of noise, but irregular in space, provides a simplified model for the study of passive scalars in a turbulent motion (see <cit.>, <cit.>, <cit.>). Strictly speaking we are not allowed here to take an isotropic matrix as infinitesimal covariance matrix a here, for the simple reason that the torus itself (considered as [-1,1]^2 with periodic boundary conditions) is not rotation-invariant. However, one may still take a translation-invariant and rotation-invariant on a neighborhood of the diagonal {x=y}. Moreover the torus setting here is taken to avoid technicalities at infinity, but we believe a similar construction, including isotropic vector fields, would go through also in the full space case.







 §.§ The nonlinear term


We focus now on the definition and properties of the nonlinear term ξ,u^ξ·∇φ. For this, we introduce some important notation. By  M_x=M(^2) we denote  the space of ℝ-valued measures on ^2. It is a Banach space, endowed with the total variation norm ·_M_x.  The space  M_x is the dual of the space C_x= C(^2) of  all continuous ℝ-valued functions on ^2,  see e.g. <cit.>.  The space M_x is endowed with the Borel σ-field generated by the weak-^∗ topology. The notation f, g denotes either the L^2 product between two functions
in L^2 or a duality between a  distribution and a test function  on ^2.  Similarly,  μ,φ will be also used to denote the duality  between a measure μ∈M_x and a function φ∈ C_x.

It follows from  Lemma <ref> that if  ξ∈ L^4/3_x, then u^ξ∈^1,4/3_x and thus  u^ξ∈ L^4_x by the Sobolev Embedding Theorem, see <cit.>.  Hence the product ξ u^ξ∈ L^1_x and the nonlinear term makes sense for a smooth function  φ. However, if ξ does not belong  to  L^4/3_x, the product ξ u^ξ is, in general,  not defined.
To overcome this difficulty, we use the following Lemma, due to Schochet <cit.>. For φ∈ C^2_x, we define the function F_φ:^2×^2→ by

    F_φ(x,y) := 1/2 K(x-y) · (∇_xφ(x)-∇_yφ(y))1_x≠ y,   (x,y)∈^2×^2.




The following properties are satisfied.


  * If φ∈ C^2_x, the the function F_φ defined in (<ref>) is bounded  by Cφ_C^2_x and continuous outside the diagonal {(x,y): x=y}. Moreover, the averages of F_φ with respect to both x and y are equal to zero, i.e.

    ∫ F_φ(x,y)   x =0   ∀ y∈ ,  ∫ F_φ(x,y)   y =0  ∀ x∈^2.


  * For any measure ξ∈M_x, the formula

    C^∞_x ∋φ↦ N(ξ), φ := ∫_^2∫_^2 F_φ(x,y) ξ( x)ξ( y) ∈ℝ,

has a unique extension to a bounded liner functional on H^4_x, i.e.
N(ξ) ∈ H^-4_x> Moreover, there exists C>0 such that

    N(ξ)_H^-4_x≤ Cξ_M_x^2,   ξ∈M_x.


  * The map N:M_x→ H^-4_x defined above is Borel measurable, where  M_x is endowed with the Borel σ-algebra generated by the weak-^∗ topology and
H^-4_x=(H_x^4)^∗ is endowed with the Borel σ-field generated by the weak-^∗ topology as well.

  * The map N coincides with the nonlinear term in equation (<ref>) on  L^4/3_x, i.e.

    N(ξ) =ξ,u^ξ·∇φ,   ξ∈ L^4/3_x, φ∈ C^∞_x.


  * For every M>0, the map N, restricted to the set M_x,M,+,no-atom of non-negative non-atomic measures on ^2 with total mass ≤ M and with values in H^-4_x, is continuous, i.e.  ξ↦ N(ξ),φ is continuous for every φ∈ H^4.  Moreover,  N is the unique  continuous extension of the nonlinear term L^4/3_x ∋φ↦ξ,u^ξ·∇·∈ H^-4_x    to  M_x,M,+,no-atom.

  * If ξ∈M_x and α∈ℝ, then N(ξ+α)=N(ξ).



Thanks to this Lemma, we will use the right-hand side of identity (<ref>) as the definition of the nonlinear term ξ∇ u^ξ   in equation (<ref>). Note that  by Lemma <ref> stated later, the set M_x,M,+,no-atom contains the set  ξ∈M_x,+∩ H^-1_x we are interested in here. The last assertion of Lemma <ref> is a consequence of Lemma <ref>, while the other assertions are proved in  Appendix <ref>.


 
 One important point is how to make sense of the term ξ,u^ξ·∇φ. Recall that, for 1< p<∞, ξ is in L^p if and only if u^ξ is in ^1,p (see Lemma <ref>). For p>2, by Sobolev embedding u is continuous bounded and therefore ξ,u^ξ·∇φ makes sense easily. For 1<p≤ 2, u is not bounded in general and, for p close to 1, u is expected (again by Sobolev embedding) to be slightly better than L^2, in particular we cannot expect the nonlinear term to make sense just by Hölder inequality. Obviosly the same problem appears when ξ is a measure.

 However, we can use the following trick by Schochet ...: Recall that K is odd (see again Lemma <ref>), then we can write formally
 
    ξ,u^ξ·∇φ = ∫_^2∫_^2 F_φ(x,y) ξ(x)ξ(y)  x y1/2∫_^2∫_^2[ N^⊥(x-y) (φ(x)-φ(y))(∇φ(x)- ∇φ(y))^t/|x-y|^2] ξ(x)ξ(y)  x y

 where the function F_φ:^2×^2→^2× 2 is defined as
 N^⊥:^2×^2→^2× 2 is defined as
 
    F_φ(x,y) := 1/2 K(x-y) · (∇φ(x)-∇φ(y))1_x≠ y.

 In particular F_φ is bounded everywhere and continuous outside the diagonal {(x,y): x=y}. Therefore, for general ξ∈M_x, we define the nonlinear term using the RHS in (<ref>). For ξ∈ L^p_x with p>2, the equality above holds true and so this definition is equivalent to the classical term ξ,u^ξ·∇φ.
 in particular N^⊥ is bounded everywhere and continuous outside the diagonal {(x,y): x=y}. For regular φ, the fraction term is estimated by D^2φ_L^∞_x and therefore the RHS above makes sense for any ξ∈M_x.
 

 
 The above computation tells that the expression (<ref>) extends the nonlinear term ξ, u^ξ·∇φ for regular ξ. One may wonder in which sense the equation with this extended nonlinearity is the original Euler equations, or more precisely whether the extension is continuous or unique. Unfortunately the answer is negative in the general case. Indeed, first we are allowed to choose any value for F_φ on the diagonal {x=y}: this choice does not affect the nonlinear term for regular ξ but it does affect it when ξ⊗ξ gives positive mass to the diagonal, which corresponds to the case of ξ with non-zero atomic component. Moreover, even for non-atomic measures ξ, the term (<ref>) is in general non-continuous with respect to the weak-^∗ topology on measures: we may have a sequence of non-atomic ℝ-valued  measures, converging weakly-∗ to 0, but whose total variation measures converge to a Dirac delta.

 Nevertheless Lemma <ref>, in the proof of convergence of approximants, shows that the term (<ref>) is continuous, with respect to the weakly-∗ topology on measures, when restricted to the space of non-negative non-atomic measures, which, by Lemma <ref>, includes the case of ξ∈M_x,+∩ H^-1_x we are interested in here. In particular, (<ref>) is the unique continuous extension of the classical nonlinear term ξ, u^ξ·∇φ.
 

 
 Note that, for any measure ξ, (K ∗ξ)·∇ξ, extended as in Remark <ref>, makes sense when tested against C^2_x functions, in particular as distribution: precisely, for any measure ξ, the formula
 
    N(ξ), φ := ξ⊗ξ, F_φ(x,y)

 defines a distribution on ^2 and we have
 
    | N(ξ),φ|≤ξ_M_x^2sup_x,y|F_φ|≤ Cξ_M_x^2φ_C^2_x.

 Since H^4_x is embedded in C^2_x, N(ξ) is actually in H^-4_x and we have
 
    N(ξ)_H^-4_x≤ Cξ_M_x^2.

 Finally, the map N:M_x→ H^-4_x is Borel. Indeed, the map ξ↦ξ⊗ξ is Borel (it is actually continuous in the strong topologies) and so, for every φ∈ C^2_x, the map ξ⊗ξ, F_φ(x,y) is Borel by Remark <ref>. Hence N is weakly-∗ Borel as an H^-4_x-valued map, hence it is Borel by Lemma <ref>.
 






 §.§ A definition of a measure-valued solution



Now we can give the definition of a measure-valued solution to the stochastic vorticity equation. Again we give some notation. For any fixed M>0, M_x,M is the set of all finite ℝ-valued  measures μ on ^2 with total variation μ_M_x≤ M. This will be preceded by presenting some additional notation.  Let us first observe that by the Banach-Alaoglu Theorem and  <cit.>, the set  M_x,M endowed with the weak-^∗ topology induced from  M_x,  is a compact Polish space. This set is endowed  with the Borel σ-field  generated by this topology. Given a filtration 𝔽=(F_t)_t, we denote by  ℬ𝔽 the associated progressive σ-field.  By an 𝔽-cylindrical Wiener process  we understand  a sequence W=(W^k)_k of independent 𝔽-Brownian motions.

The following  definition provides a rigorous   formulation of equation (<ref>) in the Itô form (<ref>).



Assume that  M>0 and T>0. Assume that the vector fields σ_k, k∈ℕ,  satisfy  Assumption <ref>. A weak distributional M_x,M-valued solution to the vorticity equation (<ref>) is an object 
    (Ω,A,𝔽,ℙ,W,ξ),

 where 𝔽=(F_t)_t∈ [0,T] is a filtration,
(Ω,A,𝔽,ℙ) is a filtered probability space satisfying the usual assumptions,  W=(W^k)_k is an 𝔽-cylindrical Wiener processon time interval [0,T],
ξ:[0,T]×Ω→M_x,M is ℬ𝔽 Borel measurable function and, ℙ-a.s.[The ℙ-exceptional set being independent of t.] the following equalities are satisfied
  in H^-4_x,

    ξ_t = ξ_0 -∫^t_0 N(ξ_r)  r -∑_k ∫^t_0 σ_k·∇ξ_r  W^k_r +c/2∫^t_0 Δξ_r  r,    t≥ 0.



The above definition as well as Theorem <ref> are formulated on an arbitrary (but fixed) time interval [0,T]. One can change these to an infinite interval [0,∞), see for instance <cit.>.




Let us observe that by Lemmata <ref> and <ref>  the map ξ :[0,T]×Ω→ H_x^-4  as well as  the integrands in (<ref>) are B𝔽 Borel measurable H^-4_x-valued maps.  Moreover, by inequality (<ref>) we infer that

    ∑_k ∫^T_0 σ_k·∇ξ_r^2_H^-4_x r ≤ M^2 ∑_kσ_k_C_x^2.

Hence the deterministic Bochner integrals and the stochastic Itô integral exists in the Hilbert space  H^-4_x.

Moreover, Lemma <ref> shows that (<ref>) is equivalent to the following weak formulation, i.e. that  for every φ∈ C^∞_x,

    ξ_t,φ   = ξ_0,φ +∫^t_0  N(ξ_r),φ r +∑_k ∫^t_0 ξ_r,σ_k·∇φ W^k_r 
          +1/2∫^t_0 ξ_r, cΔφ r,  for every t,  ℙ-a.s.,

with the ℙ-exceptional set being independent of t and test functions φ.

We will sometimes say that a process ξ is an L^p_x-valued solution of problem (<ref>) iff it  has finite L^m_t,ω(L^p_x) norm for some 1≤ m≤∞, where we identify a measure with its density if the latter  exists. Similarly we can define am   H^-1_x-valued solutions.



Note that, if ξ is in L^m_t,ω(L^p_x), then there exists a version of ξ which is jointly measurable in (t,x,ω), we will take such version when possible.

We make sense of the Stratonovich integral in the following way: We impose that, for every function ψ in C_t,x, ξ_t,ψ_t has a modification ξ(ψ) that is a continuous semi-martingale and that, for every φ∈ C^∞_t,x, equation (<ref>) is satisfied for ξ(φ)_t, namely

    ξ(φ)_t    = ξ_0,φ_0 +∫^t_0 ξ_r,(∂_t+ u_r^ξ·∇) φ_r  r
          +∑_k ∫^t_0 ξ(σ_k·∇φ)_r ∘ W^k_r

Using the above equation, we obtain that the quadratic covariation between ξ(σ_k·∇φ) and W^k is (TODO: summability in k)

    [ξ(σ_k·∇φ),W^k]_t = ξ(σ_k·∇(σ_k·∇φ))_t

Now the local translation-invariance property in Assumption <ref> implies that ∑_k σ_k(x)σ_k^t(x)=C(0) and that ∑_k∇σ_k(x)σ_k(x)^t = ∇ C(0) = 0. Therefore equation (<ref>) reads in Itô form, for a.e. (t,ω),

    ξ_t,φ_t    = ξ_0,φ_0 +∫^t_0 ξ_r,(∂_t+ u_r^ξ·∇) φ_r  r
          +∑_k ∫^t_0 ξ_r,σ_k·∇φ_r  W^k_r +1/2∫^t_0 ξ_r,[C(0)D^2φ_r]  r.



Assume on σ_k ... (needed assumption on Dσ_k). A weak distributional L^2_x-valued solution of the velocity equation is the object (Ω,A,(F_t)_t,P,(W^k)_k,u), where u:[0,T]×Ω→ L^2(^2)^2 is B𝔽-weakly measurable, divergence-free in x and in L^2_t,ω(L^2_x) and, for every divergenge-free φ∈ C^∞_c([0,T]×^2;^2), it holds for a.e. (t,ω),

    u_t,φ_t    =  u_0,φ_0 +∫^t_0  u_r,(∂_t+ u_r^ξ·∇) φ_r  r
          +∑_k ∫^t_0  u_r,σ_k·∇φ_r +(Dσ_k)^∗φ_r ∘ W^k.

We say that the solution is ^1,p_x-valued, for 1<p<∞, if it is in L^m_t,ω(^1,p_x) for some 1≤ m≤∞.


We make sense of the Stratonovich integral analogously as before: We impose that, for every test function φ, u_t,φ_t has a modification u(φ) that is a continuous martingale and that equation (<ref>) is satisfied with u(φ)_t in the LHS ....

By Sobolev embedding, u is in L^∞_t,ω(L^p^∗_x)) for some p^∗>2, in particular the nonlinear term u,(u·∇)φ makes sense.


Itô form of the equation:

    2∂_t u +2(u·∇) u +2(σ_k·∇) u   Ẇ^k +2Dσ_k· u  Ẇ^k 
       = [σ^j_k∂_jσ^l_k∂_lu^i +σ^j_kσ^l_k∂_j∂_ku^i
          +σ^j_k∂_i∂_jσ^l_ku^l +σ^i_k∂_jσ^l_k∂_iu^l
          +∂_iσ^j_k∂_jσ^l_ku^l +∂_iσ^j_kσ^l_k∂_lu^j]_i

to be understood in the sense of distibution, with suitable assumptions on σ_k; the summation in the RHS runs over all indices bu i. If it can be of help, the RHS can be written as

    [D C]· u + C ·(∇ u +(∇ u)^∗) +[∂_iσ^j_kσ^l_k∂_lu^j]_i.

When the covariance matrix C(x,y) = ∑_k σ_k(x)σ_k(y)^∗ is isotropic, all the terms but the second order term in u cancel and we obtain

    ∂_t u +(u·∇) u +(σ_k·∇) u  Ẇ^k +Dσ_k· u  Ẇ^k = 1/2[C(0)D^2 u]



The following lemma states the equivalence of the two concepts of solution. Note that the map u↦ξ^u=  u, from the subspace of divergence free, centered ^1,p functions to L^p_0 (the space of L^p functions with 0 mean) is bijective, with inverse given by ξ↦ u^ξ=K ∗ξ.


Assume on σ_k ... (needed assumption on Dσ_k). Then the vorticity equation and the velocity equation are equivalent, in the following sense:


  * If ξ is a M_x-valued solution which is also in L^2_t,ω(H^-1_x), then u^ξ is an L^2_x-valued solution.

  * If u is an L^2_x-valued solution such that u is in L^2_t,ω(M_x), then, for every ξ̅ in , ξ=  u +ξ̅ is a M_x-valued solution (which is also in L^2_t,ω(H^-1_x)).

  * For any 1<p<∞, ξ is in L^m_t,ω(L^p_x) if and only if u is in L^m_t,ω(^1,p_x).




From ξ to u: use ∑_i Ǩ^i *φ^i (where Ǩ(z) = K(-z)) as test function in (<ref>) to get (<ref>). From u to ξ: use ∇^⊥φ as test function in (<ref>) to get (<ref>).






§ THE GLOBAL EXISTENCE FOR INITIAL VORTICITY IN THE SPACE  M_X,+∩ H^-1_X


The objective  of this section is to present the main result of this paper.


The corresponding deterministic result has been proved by Delort 1991, we follow the approach by Poupaud 2002, combined with the strategy of Brzezniak+Peszat 2001. Note that a nonnegative distribution is automatically a measure.

Here is the main result in the stochastic case:




Assume the Assumption <ref> on the coefficients σ_k.  Then for all    M>0 and  T>0,  and every  ξ_0 ∈M_x,M,+∩ H^-1_x, there exists a weak distributional M_x,M-valued solution (Ω,A,𝔽,ℙ,(W^k)_k,ξ), where 𝔽=(F_t)_t∈ [0,T],  to the vorticity equation (<ref>) such that   ξ∈ C_t(ℳ_x,M,+,w^∗ )∩ L^2_t,ω(H^-1_x) ℙ-a.s..



Note that, if ξ is a solution to the vorticity equation (<ref>) and α∈ℝ is a real constant, (in space, time and Ω),
then ξ+α is also a solution, see Lemma <ref>. Hence the Theorem <ref> can be generalized as follows. 

 For every ξ_0 ∈M_x,M∩ H^-1_x with negative part bounded by a constant α, then there exists a weak distributional M_x,M-valued solution ξ to (<ref>),  whose paths belong to  C_t(ℳ_x,M,+,w^∗ )∩ L^2_t,ω(H^-1_x), with negative part bounded by α,  for all t, ℙ-a.s..

This generalization is important because,  in particular,   if our departure point is the initial   velocity u_0, then, unless (u_0) =0,  (u_0) cannot be non-negative, but it can be bounded from below.


The strategy of the proof of Theorem <ref> is  as follows. We begin by fixing M>0 and T>0.

Compactness argument. We take  a family ξ^ϵ, >0,  of solutions with regular bounded initial datum ξ^ϵ_0. We will show that the family of laws of these solutions is  tight on a suitable
topological space,  via appropriate  a'priori bounds. This  will be accomplished in the following  steps.


  * Prove the non-negativity and a uniform L^∞_t,ω(ℳ_x,+) bound on ξ^.  This will follow from the conservation of non-negativity and from the conservation of mass for the vorticity equation (<ref>).

  * Prove a uniform L^2_t,ω(H^-1_x) bound on ξ^.  Since the L^2_t,ω(H^-1_x) norm of ξ is equivalent to the L^2_t,ω(L^2_x) norm of the corresponding velocity u^:=u^ξ^,  we will  prove a uniform energy bound on u^   by rewriting the vorticity equation (<ref>)  in an equivalent form (<ref>) for the velocity u^. Note that, contrary  to the deterministic case, the energy, i.e. the L^2 norm of the velocity u^,  is not preserved. This is  due to the presence of an additional term (∇σ_k)· u ∘ Ẇ^k in the equation. To prove the energy bound, the assumptions on σ_k play a crucial role.

  * Prove, for α<1/2,  a uniform L^2_ω(C^α_t(H^-4_x)) bound on ξ^. This will  follow from the Lipschitz bounds w.r.t the  time variable  of the Bochner  integrals in the vorticity equation (<ref>) and the Hölder bound w.r.t the  time variable of  the H^-4-valued stochastic integrals.

  * Show tightness  of laws of u^ on  C_t(ℳ_x,M,w^∗ )∩ (L^2_t(H^-1_x),w), where by w^∗, respectively w,   we denote  the weak-^∗ topology on ℳ_x,M, respectively  the weak topology on L^2_t(H^-1_x)).  This will follow from the previous uniform bounds.  In fact, we could   prove tightness simply in C_t(ℳ_x,M,w^∗ ), without using the L^2_t(H^-1_x) bound, but  our present proves to be useful in the convergence part.


Convergence argument. We will show that any limit point ξ of ξ^ϵ, as → 0,  solves the vorticity equation (<ref>). This  will be accomplished in the following  steps.


  * Pass to an a.s. convergence.  By the Skorokhod-Jakubowski Theorem, see <cit.> and <cit.>, we have, up to a  subsequence and a choice of a new probability space, an a.s. convergence of  ξ^ to some ξ with trajectories  in C_t(ℳ_x,M,w^∗ )∩ (L^2_t(H^-1_x),w).

  * Show that   ξ  satisfies (<ref>).  For the nonlinear term, we will use the Schochet approach and in particular  the continuity of the nonlinear term among non-negative non-atomic measures as in Lemma <ref> and the fact that H^-1_x measures are non-atomic. For the stochastic term, we use the approximation of the stochastic integral via the Riemann sums as in <cit.>.


The main result will follow from Lemma <ref>, which will show that the limit ξ of any subsequence satisfies (<ref>).





















§ PROOF OF THE MAIN THEOREM <REF> 


We fix M>0 and T>0 and the initial condition ξ_0 ∈M_x,M,+∩ H^-1_x. We also assume, for the whole section,  that the coefficients σ_k satisfy
 Assumption <ref>.




 §.§ A priori bounds



For >0, we take ξ^ to be the L^∞_t,x,ω solution to the stochastic vorticity equation (<ref>), with initial condition ξ_0^ = ξ_0*ρ_, where (ρ_)_ >0 is the standard approximation of identity  on ^2. The convolution is understood w.r.t. the group structure of ^2. Note that by inequality (4) in <cit.>,   ξ_0^_M_x≤ M.

The existence and the uniqueness of such  solution follows from our joint paper  <cit.> with Flandoli. To be more precise,    <cit.>  implies the existence and the  uniqueness of the stochastic flow Φ^: [0,∞)×^2×Ω→^2
such that


  (i)
for almost every ω∈Ω, the map [0,∞)×^2 ∋ (t,x) ↦Φ^(t,x,ω) ∈^2 is continuous,

  (ii)
for every x ∈^2, the process
Φ^(·,·,ω): [0,∞)×^2  →^2 is progressively measurable,

  (iii)
for almost every ω∈Ω,  Φ^(t,·,ω):  ^2  →^2 preserves the Lebuesgue
measure for every t≥ 0,

  (iv)
and for every x∈^2, the process Φ^(t,x) is solution to the following  SDE

    Φ^(t,x) = ∫_^2 K(Φ^(t,x)-Φ^(t,y)) ξ^_0(y)  y  t +∑_k σ_k(Φ^(t,x))  W^k_t.



Moreover,  if we define  ξ^_t(ω) to be  the image (i.e. the push-forward) of  measure ξ_0^, see <cit.>,  by the map Φ(t,·,ω)), i.e.

    ξ^_t(ω)=(Φ(t,·,ω))_#ξ^_0,    t≥ 0,

 then,  by  <cit.> we infer that  for a.e. ω, for every t ≥ 0, the random variable  ξ^_t(ω) admits a density with respect to the Lebesgue measure, which belongs to  L^∞_t,x,ω and, for every φ∈ C^∞_x, the process ξ^_t,φ is progressively measurable and satisfies (<ref>). It follows, see Remark <ref>, that, up to taking an indistinguishable version, such ξ^ is a weak distributional M_x,M-valued solution in the sense of Definition <ref>.






















  §.§.§ Bound in L^∞_t,ω(M_x)


We start with a uniform L^∞_t,ω(M_x) bound:


For every >0 fixed, for a.e. ω, we have

    sup_t∈[0,T]ξ^_t_L^∞_x≤sup_t∈[0,T]ξ^_t_M_x≤ξ^_0_M_xξ_0_M_x≤ M.

In particular, up to taking an indistinguishable version, ξ^ is a M_x,M-valued solution (in the sense of Definition <ref>). Moreover, for a.e. ω, we have: for every t, ξ^_t≥ 0 (that is, it is a non-negative measure).



The non-negativity follows directly from the assumption  that ξ_0 is a non-negative measure  and so ξ_0^ are non-negative functions and from the representation formula (<ref>). Concerning the bound, this also follows from the representation formula (<ref>), but we prefer giving a  short,  PDEs in spirit,    proof. Using φ≡ 1 as test function in (<ref>), we get, for a.e. ω: for every t,

    ∫_^2ξ^_t  x = ∫_^2ξ^_0  x.

Since ξ^_t≥ 0, we get that ξ^_t_L^∞_t,ω(M_x)≤ξ_0_M_x for every t, for a.e. ω. Defining ξ^=0 outside the exceptional set where the bound is not satisfied, we get that ξ^ is in C_t(M_x,M) and so is a M_x,M-valued solution.



In the proof of the previous Lemma, we used the fact that ξ^ is positive (and so ∫_^2 ξ^ is the M_x norm of ξ^), but this is not essential. Indeed, the vorticity equation is a transport equation with divergence-free velocity field, therefore the mass is conserved at least under suitable regularity assumption on the velocity, which are satisfied here.































  §.§.§ Equation for the velocity and for its energy



In view of a uniform H^-1_x bound on ξ^, we will:

 
   1) get an equation for the velocity u^=u^ξ^=K ∗ξ^,
 
   2) get an equation for the energy of u^, that is the L^2_x norm of u^,
 
   3) conclude a uniform L^2_x bound on u^ by Lemma <ref>, this bound is equivalent to a uniform H^-1_x bound on ξ^, up to the  average of ξ^ on ^2.


Here we consider a solution ξ to the vorticity equation (<ref>), with sufficiently integrability to include the (bounded) approximants ξ^ϵ. As we have seen in the introduction, see formula (<ref>), in this case the corresponding velocity process u is a solution to the following

    ∂_t u +(u·∇) u +∑_k(σ_k·∇ +(Dσ_k)^t)u ∘ Ẇ^k = -∇ p -γ,
        u=0,

where processes p:[0,T]×^2×Ω→ and γ:[0,T]×Ω→^2 are also unknown. A  rigorous formulation of this result can be stated  as follows.

  Assume that  2<p<∞.  Assume  that a process    ξ belongs to L^p_t,ω(L^p_x)  and that
  ξ is  an M_x,M-valued distributional solution to the stochastic vorticity equation (<ref>).   Define a process u by
  
    u(t,·,ω) =K ∗ξ (t,·,ω),    (t,ω) ∈ [0,∞)×Ω.

   Then u belongs to  L^p_t,ω(^1,p_x) and it is a distributional solution to the stochastic Euler equation, i.e.  for every t≥ 0, ℙ-a.s.

    u_t    = u_0 -∫^t_0 Π[(u_r·∇) u_r]  r
          -∑_k ∫^t_0 Π[σ_k·∇ u_r +(Dσ_k)^T u_r]  W^k(r) 
          +c/2∫^t_0  Δ u_r  r,   H_x^-1 .

 Here Π: H_x^-1→ H_x^-1 is the unique bounded extension of the Leray-Helmholtz  projection Π:L^2(^2,^2) →.

Remark Let us observe that the range Π(H_x^-1) is contained in  a closed subspace of   H^-1_x consisting of  divergence-free zero-mean elements.

The proof is essentially based on the following heuristic identity, which rigorous proof is given in the Appendix.
If  the vector fields v:^2→^2 and w:^2→^2  are of sufficient regularity and v is divergence-free, then

    [v·∇ w +(Dv)^t w] = v·∇ [w],

where Dv= [  D_1v_1  D_1v_2; D_2 v_1 D_2 v_2 ]
 and ^t is the transpose operation. Using the above, one can  pass from the velocity equation (<ref>) to the vorticity equation (<ref>).
From the equation for  the velocity  (<ref>) we can deduce  the equation for the expected valued of the energy:


Under the assumptions of Lemma <ref>,

    u_t_L^2_x^2    =
    u_0_L^2_x^2 -2∫^t_0 ∫_^2 u·Π[(u ·∇) u]  x  r 
        -c ∫^t_0 ∫_^2 |∇ u|^2  x  r 
        +∫^t_0 ∑_k ∫_^2 |Π[(σ_k·∇ +(Dσ_k)^t) u]|^2  x  r,   t≥ 0.



One can get this equation formally from (<ref>) by applying the Itô formula to u_L^2_x^2. However this is not possible rigorously, because the rigorous equation (<ref>) holds in H^-1_x and the square of the L^2_x norm is not continuous on H^-1_x. The rigorous proof of the formula (<ref>) is based on a regularization argument and is postponed to the Appendix <ref>.



  §.§.§ Bound in L^2_t,ω(H^-1_x)


Now we give a uniform L^2_t,ω(H^-1_x) bound on the approximants ξ^. This bound is not essential for the compactness argument, but it is essential for the convergence argument, as we will see.




There exist C>0 such that for all ∈ (0,1] the following inequality  holds

    ξ^_t_H^-1_x^2≤ C (ξ_0_H^-1_x^2 +ξ_0_M_x^2),  for every t.



In the proof of this lemma, we use crucially the assumptions <ref> on σ_k. We recall that u^=u^ξ^=K ∗ξ^.


By Remark <ref>, the H^-1_x norm is a Borel function on (M,w^∗ ), therefore ξ^_H^-1_x and u^_L^2_x are progressively measurable and the expectations of their moments make sense. By Lemma <ref>, applied to ξ^_t-∫ξ^_t(y) dy, we have, for every t,

    ξ^_t_H^-1_x≤ Cu^_t_L^2_x + |∫_^2ξ^_t dx|,
       u^_t_L^2_x≤ Cξ^_t_H^-1_x.

By Lemma <ref>, the L^1 norm of ξ^ is uniformly bounded by ξ_0_M_x. Hence it is enough to show, for every t,

    u^_t_L^2_x^2≤ Cu_0_L^2_x^2.

We will show the above bounds by  using the velocity equation (<ref>).

We start with equation (<ref>) applied to u^. Since u^ is divergence-free, the nonlinear term in (<ref>) vanishes. Indeed, we have the following train of equalities,

    ∫_^2 u^·Π[(u^·∇) u^]  x = ∫_^2Π[u^]· (u^·∇) u^ x
       = ∫_^2 u^· (u^·∇) u^ x = 0.

For the term with σ_k, since the Leray-Helmholtz projection Π is a contraction  in the space L^2_x, we infer that

    ∑_k ∫_^2 |Π[(σ_k·∇ +(Dσ_k)^t) u^]|^2  x ≤∑_k ∫_^2 |(σ_k·∇ +(Dσ_k)^t) u^|^2  x
       = ∫_^2 [∑_k|σ_k ·∇ u^|^2 +∑_k|(Dσ_k)^t u^|^2 +2∑_i,j,h∑_kσ_k^i ∂_x_jσ_k^h (u^)^h ∂_x_i (u^)^j ]  x.

Now we use the assumptions <ref>, precisely that ∑_kσ_k^iσ_k^j = cδ_ij and that ∑_kσ_k^i ∂_x^jσ_k^h = 0 for all i,j,h, with uniform (with respect to x) convergence in the series over k: we get

    ∑_k ∫_^2 |Π[(σ_k·∇ +(Dσ_k)^t) u^]|^2  x ≤∫_^2 c|∇ u^|^2  x +∑_kσ_k_C^1_x∫_^2 |u^|^2  x .

Putting all together, we obtain for every t

    u^_t_L^2_x^2 ≤u^_0_L^2_x^2 +∑_kσ_k_C^1_x∫^t_0 u^_r_L^2_x^2  r.

By applying the  Gronwall Lemma to the function u^_t_L^2_x^2,  we conclude, that

    u^_t_L^2_x^2 ≤u^_0_L^2_x^2 exp[t∑_kσ_k_C^1_x],     t≥ 0.

This  implies the desired bound since u^_0 is deterministic and u^_0_L^2_x≤ Cu_0_L^2_x. The proof of Lemma <ref> is complete.




  §.§.§ Bound in L^m_ω(C_t^α(H^-4_x))


Now we prove a uniform L^m_ω(C_t^α(H^-4_x)) bound, for m≥ 2:


Assume that T>0,  m ∈ [2,∞) and  0<α<1/2. Then   there  exists a constant C=C(m,α,T,M) such that for all ∈ (0,1],


    ξ^_C^α_t(H^-4_x)^m ≤ C (ξ_0_M_x^2m +ξ_0_M_x^m).



 Let us  choose and fix α∈ (0,1/2).  Choose then m such that α<1/2-1/m.

Note that, by Remark <ref>, ω↦ξ^_C_t^α(H^-4_x) is measurable. Using the equation (<ref>), we get, for all s, t ∈ [0,T] such that s≤ t,

    ξ^_t - ξ^_s _H^-4_x^m
       ≤ C∫^t_s N(ξ^_r)  r _H^-4_x^m
          +C∑_k ∫^t_s σ_k·∇ξ^_r  W^k _H^-4_x^m
          +Cc^m∫^t_s Δξ^_r  r _H^-4_x^m.

By Lemma <ref>, we get the following inequality for the   for the nonlinear term

    ∫^t_s N(ξ^_r)  r _H^-4_x^m ≤ C(t-s)^m ξ^_L^∞_t,ω(M_x)^2m.

What concerns  the stochastic integral, by the Burkholder-Davis-Gundy inequality and Lemma <ref>  we infer that

    ∑_k ∫^t_s σ_k·∇ξ^_r ∘ W^k _H^-4_x^m
       ≤ C ( ∑_k ∫^t_s σ_k·∇ξ^_r _H^-4_x^m  r )^m/2
       ≤ C(t-s)^m/2(∑_kσ_k_C_x^2)^m/2ξ^_L^∞_t,ω(M_x)^m

Finally,  for the second order term,  again by Lemma <ref>  we deduce that

    ∫^t_s Δξ^_r  r _H^-4_x^m≤ C ξ^_L^∞_t,ω(M_x)^m (t-s)^m

We put all together and we recall the a priori bound on ξ^_L^∞_t,ω(M_x) in Lemma <ref>: we obtain

    ξ^_t - ξ^_s _H^-4_x^m ≤ C (t-s)^m/2 (ξ_0_M_x^m +ξ_0_M_x^2m),   0≤ s≤ t≤ T,

where the constant C depends on ∑_kσ_k_B_x^2 and on c. By the Kolmogorov criterion, or the Sobolev embedding  as in <cit.>,  recalling that ξ^ is already continuous as H^-4_x-valued process, we infer that

    ξ^_C^α_t(H^-4_x)^m ≤ C(ξ_0_M_x^m +ξ_0_M_x^2m).

The proof is complete.





 §.§ Tightness

Let us recall  that M>0 is fixed such that ξ_0_M_x≤ M. We also fix T>0, see Remark <ref>. Whenever we use subscript _t we mean that the corresponding functions are defined on the closed time interval [0,T].

In this section we prove the tightness of laws on C_t(M_x,M,w^∗ ) ∩ (L^2_t(H^-1_x),w) of the family of processes  (ξ^)_∈ (0,1].

We recall that (M_x,M,w^∗ ) is metrizable with the distance d_M_x,M(μ,ν) = ∑_j 2^-j |μ-ν, φ_j |, therefore C_t(M_x,M,w^∗ ) is metrizable as well, see Remark <ref> with X=C_x. The intersection space C_t(M_x,w^∗ )∩ (L^2_t(H^-1_x),w) is defined as the set consisting of those elements  of C_t(M_x,w^∗ ) which have  finite L^2_t(H^-1_x) norm.
On this set, which is a subspace of C_t(M_x,w^∗ ), the topology is induced by the C_t(M_x,w^∗ )  and the (L^2_t(H^-1_x),w) topologies, i.e.
it is the weakest topology such that the natural
embeddings to  C_t(M_x,w^∗ )  and  (L^2_t(H^-1_x),w) are continuous. It follows that a function from another topological space with values in the intersection space is continuous if and only if it is composition with those two natural embeddings are continuous. Moreover, one can show that a function from another measurable  space with values in the intersection space is Borel measurable if and only if
it is composition with those two natural embeddings are Borel measurable.


Let us finish by pointing out that by Lemma <ref>,  the Borel σ-algebra generated by (L^2_t(H^-1_x),w) coincides with the Borel σ-algebra generated by the strong topology on L^2_t(H^-1_x),

Note that, by Lemmas <ref> and <ref>, for any >0, ξ^_t takes values in M_x,M∩ H^-1_x for every t, ℙ-a.s. and so for every ω up to taking an indistinguishable version. Hence, by Lemma <ref>, ξ is Borel measurable as C_t(M_x,M,w^∗ )-map and as (L^2_t(H^-1_x),w)-valued maps, i.e. both spaces being endowed with their Borel σ-algebras. Hence, by the above we infer that   ξ^ is Borel measurable as a  C_t(M_x,M,w^∗ ) ∩ (L^2_t(H^-1_x),w)-valued map, i.e.
ξ^ is a  C_t(M_x,M,w^∗ ) ∩ (L^2_t(H^-1_x),w)-valued random variable.


 The  family of laws of (ξ^)_∈ (0,1] is tight on C_t(M_x,M,w^∗ ) ∩ (L^2_t(H^-1_x),w).


We start with a generalization of <cit.>. The latter is a refined version of the compactness argument in <cit.>, which can be seen as a stochastic version of the Aubin-Lions lemma. Given a Banach space X, we call B_M^X the closed ball in X of radius M.



Let X, Y be separable Banach spaces with Y densely embedded in X. Then, for every M≥0, α>0, a≥0, the set

    A_a = { z ∈ C_t(B_M^X^∗,w^∗ ) : z_C^α_t(Y^∗)≤ a }

is compact in C_t(B_M^X^∗,w^∗ ).



For the proof, we recall the following facts. First, the ball B_M^X^∗ endowed with the weak-^∗ topology is metrizable with the distance d_B_M^X^∗(w,w') = ∑_j 2^-j | w-w', φ_j |, where (φ_j)_j is a dense sequence in B^X_1, see <cit.>. Hence the set C_t(B_M^X^∗,w^∗ ) is metrizable with the distance

    d(z,z') = sup_t∈[0,T]∑_j 2^-j | z-z', φ_j |,   z,z'∈ C_t(B_M^X^∗,w^∗ ).

Moreover, given an C_t(B_M^X^∗,w^∗ )-valued  sequence (z^n)_n, an element  z ∈ C_t(B_M^X^∗,w^∗ ) and a dense subset D ⊂ X, the following three conditions are equivalent.


  ∙ the sequence z^n converges to z in C_t(B_M^X^∗,w^∗ );

  ∙ for every φ∈ X, the sequence z^n,φ  in C_t to z,φ;

  ∙ for every φ∈ D, the sequence z^n,φ converges  in C_t to z,φ.

Equivalence between the first two conditions can be seen using the distance defined in (<ref>). Equivalence  between the last two conditions can be seen by approximating a generic element  φ∈ X by an D-valued sequence  and using the uniform bound sup_n sup_t∈ [0,T]z_n_X^∗≤ M.



Since the topological space  space C_t(B_M^X^∗,w^∗ ) is metrizable, the compactness is equivalent to the sequential compactness. Let (z^n)_n be a sequence in A_a. We have to find a subsequence (z^n_k)_k which converges in C_t(B_M^X^∗,w^∗ ) to an element of the set  A_a.

For fixed t in ℚ∩ [0,T], (z^n_t)_n is an B^X^∗_M-valued sequence.  Hence, by the Banach-Alaoglu Theorem, there exists a subsequence (z^n_k_t)_k which is convergent weakly-∗ to an element z̃_t ∈ B^X^∗_M. By a diagonal procedure, we can find  a sequence (n_k)_k independent of t in ℚ∩ [0,T].

On the other side, let D be a countable dense set in Y, and so in X. The fact that z^n are equicontinuous and equibounded in Y^∗ implies that, for every φ∈ D, the functions t↦ z^n_k_t,φ are equicontinuous and equibounded, their C^α norm being bounded by aφ_Y. Hence, by the Ascoli-Arzelà Theorem, there exists a subsequence converging  in C_t to some element f^φ={[0,T] ∋ t↦ f^φ_t}, which also satisfies f^φ_C^α_t≤ aφ_Y. By a diagonal procedure, we can choose the subsequence independent of φ∈ D. With a small abuse of notation, we continue using n_k for this subsequence. Then, for all t in ℚ∩ [0,T], for all φ∈ D, z̃_t,φ = f^φ_t.

Fix t in [0,T] and let (t_j)_j be a sequence in ℚ∩ [0,T] converging to t. Since the sequence (z̃_t_j)_j takes values in the ball  B^X^∗_M,  up to a subsequence, it converges weakly-∗ to an element z_t in B^X^∗_M. On the other hand, for every φ∈ D, by the continuity of the function [0,T]∋ t↦ f^φ_t, we infer that  z_t,φ = f^φ_t.


Since the map [0,T]∋ t↦ z_t,φ = f^φ_t is continuous for every φ∈ D, and in fact for every φ∈ X, by an approximation argument, we infer that  z=[0,T]∋ t ↦ z_t  belongs to  C_t(B^X*_M,w^∗ ). Moreover

    z_t-z_s_Y^∗ = sup_φ∈ D,φ_Y≤ 1 | z_t-z_s,φ|
       = sup_φ∈ D,φ_Y≤ 1 |f^φ_t-f^φ_s| ≤ a|t-s|^α,

and similarly for z_t_Y^∗ alone. Hence z_C^α_t(Y^∗)≤ a and so z is in A_a.

Finally, for every φ∈ D, z^n_k,φ converges uniformly to f^φ =  z,φ. Therefore, by Remark <ref>, we infer that z^n_k converges to z in C_t((B_M^X^∗,w^∗ ). The proof is complete.
















As a consequence of the previous Lemma and the Banach-Alaoglu Theorem, we get the following result.


For  all  M>0, α>0 and  a,b≥0, the set

    A_a,b=A_a,b(M,α) = {μ∈ C_t(M_x,M,w^∗ ) ∩ L^m_t(H^-1_x) : μ_C^α_t(H^-4_x)≤ a,  μ_L^m_t(H^-1_x)≤ b }

is metrizable and compact subset of  C_t(M_x,M,w^∗ ) ∩ (L^m_t(H^-1_x),w).



Since the topologies on C_t(M_x,M,w^∗ ) and on the closed ball of radius b in (L^m_t(H^-1_x),w^∗ ) are metrizable, A_a,b is metrizable as well and the compactness is equivalent to the sequential compactness.

Let (μ^n)_n be a sequence in A_a,b. By the previous Lemma, applied to X=C_x and Y=H^4_x, there exists a sub-subsequence (μ^n_k)_k converging to some μ in C_t(M_x,M,w^∗ ) with μ_C^α_t(H^-4_x)≤ a. On the other hand, by the Banach-Alaoglu theorem, there exists a subsequence, which we can assume (μ^n_k)_k up to relabelling, converging to some ν in (L^m_t(H^-1_x),w) with ν_L^m_t(H^-1_x)≤ b. Using these two limits, for every g in C_t and every φ∈ C^1_x, we have

    ∫^T_0 g(t) μ_t, φ t = ∫^T_0 g(t) ν_t, φ t.

Hence μ=ν and so μ is the limit in A_a,b of the subsequence (μ^n_k)_k. The proof is complete.


We are ready to prove tightness of ξ^.


As we have seen at the beginning of this section, by Lemmas <ref> and <ref>, for any >0, ξ^ is, up to an indistinguishable version, a C_t(M_x,w^∗ )∩ (L^2_t(H^-1_x),w)-valued random variable. Lemma <ref> ensures that the set A_a,b defined in that Lemma is metrizable and compact in C_t(M_x,w^∗ )∩ (L^2_t(H^-1_x),w). The Markov inequality gives

    P{ξ^∉ A_a,b}≤ P{ξ^_C_t^α(H^-4_x)>a } + P{ξ^_L^m_t(H^-1_x)>b }
       ≤ a^-mξ^_C_t^α(H^-4_x)^m + b^-mξ^_L^m_t(H^-1_x)^m.

By Lemmas <ref> and <ref>, the right-hand side above can be made arbitrarily small, uniformly in , taking a and b large enough. The tightness is proved.


As a consequence, we have actually:


The family (ξ^,W)_∈ (0,1], where W=(W^k)_k is tight on the space χ := [C_t(M_x,M,w^∗ )∩(L^m_t(H^-1_x),w)] × C_t^ℕ.



The tightness of (ξ^,W) follows easily from the tightness of the marginals.





 §.§ Convergence


Let us begin this subsection with an observation that we can apply the Skorohod-Jakubowski representation theorem, see <cit.> and <cit.> to the family (ξ^,W)_ and the space

    Ξ = [C_t(M_x,M,w^∗ ) ∩ (L^m_t(H^-1_x),w)] × C_t^ℕ.

 Indeed the family (ξ^,W)_ is tight by Corollary <ref> and the space Ξ satisfies the assumption (10) in <cit.>: for given sequences (t_i)_i dense in [0,T], (φ_j)_j dense in C_x, the maps f_i,j and g_i,k, defined on Ξ by f_i,j(μ,γ)=μ_t_i,φ_j and g_i,k(μ,γ)= arctan(γ^k_t_i), form a sequence of continuous, uniformly bounded maps separating points in Ξ.

Hence, by the Skorohod-Jakubowski Theorem there exist an infinitesimal sequence (_j)_j, a probability space (Ω,A,ℙ), a Ξ-valued sequence (ξ^j, W^(j))_j and a Ξ-valued random variable (ξ,W) such that (ξ^j,W^(j)) has the same law of (ξ^_j,W) and (ξ_j,W^(j)) converges to (ξ,W) a.s. in C_t(M_x,M,w^∗ ) ∩ (L^m_t(H^-1_x),w). For notation, we use W^(j),k and W^k for the k-th component of the C_t^ℕ-valued random variables W^(j) and W.

Let us denote  𝔽^0=(F_t^0)_t∈ [0,T] the filtration generated by the processes ξ, W and the ℙ-null sets on (Ω,A,ℙ). We also put 𝔽=(F_t)_t∈ [0,T], where  F_t = ∩_s>tF^0_s. Similarly, we denote by  𝔽^0,j=( F_t^0,j)_t∈ [0,T] the filtration generated by the processes ξ^j, W^(j) and the ℙ-null sets on (Ω,A,ℙ) and finally we put 𝔽^j=( F_t^j)_t∈ [0,T], where  F^j_t = ∩_s>tF^0,j_s.


The filtration 𝔽 is complete and right-continuous and W is a cylindrical 𝔽-Wiener process. Moreover, the process  ξ is an (M_x,M,w^∗ )-valued (F_t)_t∈ [0,T]-progressively measurable. Similar statements hold for  𝔽^j, W^(j) and ξ^j, for every j ∈ℕ.


The proof of this Lemma is simple but technical and postponed to  Appendix <ref>.

Now we will prove that each copy ξ^j of the approximating process  ξ^_j is a solution to the stochastic vorticity equation.


For every j∈ℕ, the object 
    (Ω,A,𝔽^j,ℙ,W^(j),ξ^j)

is a M_x,M-valued solution to the vorticity equation (<ref>) with the initial condition ξ^j_0=ξ_0^_j ℙ-a.s.. Moreover Lemmata <ref> and <ref> hold for ξ^j in place of ξ^ and, ℙ-a.s., ξ_t is non-negative for every t∈ [0,T].


Also the proof of this Lemma is technical and postponed to Appendix <ref>.



 §.§ Limiting equation


Now we show that ξ satisfies the vorticity equation with W as Brownian motion. With this Lemma, Theorem <ref> is proved.


The object (Ω,A,(F_t)_t∈ [0,T],ℙ,W,ξ) is a M_x,M-valued solution to the vorticity equation (<ref>), which is also in C_t(M_x,M,w^∗ ) ∩ (L^2_t(H^-1_x),w).




 The proof of Lemma <ref> will occupy the remaining parts of the current subsection.

To prove Lemma <ref> we will show (<ref>) for (ξ,W) for every test function φ∈ C^∞_x. By Lemma <ref>, for each j, (ξ^j,W^(j)) satisfies (<ref>) for every φ∈ C^∞_x. Hence it is enough to pass to the ℙ-a.s. limit, as j→∞, in each term of (<ref>) for (ξ^j,W^(j)), possibly choosing a subsequence, for every t and every φ∈ C^∞_x. We fix t in [0,T] and φ∈ C^∞_x.

We start with the deterministic linear terms: ξ^j_t, φ, ξ^j_0, φ and

    ∫^t_0 ξ^j_r, cΔφ r

converge ℙ-a.s. to the corresponding terms without the superscript j, thanks to the convergence of ξ^j to ξ in C_t(M_x,M,w^∗ ).



  §.§.§ The nonlinear term


Concerning the nonlinear term, we recall Lemma <ref>) and we follow the Schochet argument, see Schochet <cit.> and Poupaud <cit.>. The first main ingredient for the convergence is the following:


Fix M>0. For every φ∈ C^2_x, the map μ↦ N(μ),φ is continuous on the subset M_x,M,+,no-atom of M_x,M of non-negative non-atomic measures with total mass bounded by M, endowed with the weak-^∗ topology.


We use the following result, a version of the classical Portmanteau theorem (which deals with probability measures rather than non-negative measures):


Let X be a compact metric space. Assume that (ν^k)_k is a sequence of non-negative bounded measures and converges to ν in (M(X),w^∗ ). Let F be a closed set in X with ν(F)=0 and let ψ:X→ be a bounded Borel function, continuous on X∖ F. Then the sequence (ν^k,ψ)_k converges to ν,ψ.



Let >0, we have to prove that |ν^k-ν, ψ| <C for k large enough, for some constant C. The fact that ν(F)=0 implies the existence of δ>0 such that ν(B̅(F,δ))<, where B̅(F,δ):={x∈ X: d(x,F)≤δ}. As the function 1_B̅(F,δ) is upper semi-continuous and (ν^k)_k converges weakly-∗ to ν, there exists k̅ such that ν^k(B̅(F,δ)) < for all k≥k̅. By Urysohn lemma, there exists a continuous function ρ with 0≤ρ≤ 1, ρ=1 on F and ρ=0 on B̅(F,δ)^c; it is easy to see that ψ(1-ρ) is then continuous on all X. Now we split

    |ν^k-ν, ψ| ≤ |ν^k, ψρ| +|ν^k-ν, ψ(1-ρ) | +|ν, ψρ|.

For the first term in the right-hand side, we have

    |ν^k, ψρ|   ≤sup_k |ν^k(B̅(F,δ))| sup_X|ψ|
    
       = sup_k ν^k(B̅(F,δ)) sup_X|ψ| ≤sup_X|ψ|.

The same inequality holds for the third term in the right-hand side of (<ref>). Finally, the second term in (<ref>) is bounded by  provided k is large enough, by weak-^∗ convergence of (ν^k)_k. The proof is complete.




It is only in (<ref>) in the proof of Lemma <ref> that we had  to use that the fact that the process ξ takes values in non-negative measures.




We have to show that, for every sequence (μ^n)_n converging to ξ∈M_x,M,+,no-atom, the sequence N(μ^n),φ converges to N(μ),φ. By Lemma <ref> in the Appendix, (μ^n⊗μ^n)_n converges weakly-∗ to ξ⊗ξ. Moreover, since μ has no atoms, then μ⊗μ gives no mass to the diagonal D={(x,y): x=y}. Indeed,  by the Fubini theorem,

    (μ⊗μ)(D) = ∫_^2μ( x) ∫_{x}μ( y) = 0.

We are now in a position to apply Lemma <ref> to the sequence (μ^n⊗μ^n)_n, with the state space X=^2 ×^2, with F=D and with ψ=F_φ, which is continuous outside the diagonal D: we get that

    μ^n⊗μ^n , F_φ→μ⊗μ , F_φ,

which is exactly the desired convergence. The proof is complete.


The second ingredient for the convergence of the nonlinear term is the following:


Let μ∈M_x ∩ H^-1_x. Then the measure μ has no atoms.




Fix x_0 in ^2, we have to prove that μ({x_0})=0. Let ρ:^2→ be a smooth function with 0≤ρ≤ 1, supported on B_1(0) (the ball centered at 0 with radius 1) and with ρ(x)=1 if and only if x=0. For n positive integer, call ρ_n(x)=ρ(n(x-x_0)) and take its periodic version on ^2, which, with small abuse of notation, we continue calling ρ_n. Now (ρ_n)_n is a nonincreasing sequence which converges pointwise to 1_{x_0}, so (μ,ρ_n )_n converges to μ({x_0}).

On the other hand |μ,ρ_n |≤μ_H^-1_xρ_n_H^1_x. For the H^1_x norm of ρ_n, we have

    ∇ρ_n_L^2_x^2  = ∫_^2 n^2 |∇ρ(nx)|^2  x = ∫_B_1/n(0) |∇ρ(y)|^2  y

and so ∇ρ_n_L^2 converges to 0 as n→∞. In a similar and easier way one sees that ρ_n_L^2 converges to 0. So ρ_n_H^1 tends to 0. Hence (μ,ρ_n )_n tends also to 0 and therefore μ({x_0})=0.




It is only for the previous Lemma that we need to use that the process ξ is H^-1_x-valued.


We are now able to conclude the convergence of the nonlinear term in (<ref>). Fix ω in a full measure set such that (ξ^j)_j converges to ξ in C_t(M_x,M,w^∗ ). Since ξ belongs to L^2_t(H^-1_x), ξ_r belongs to H^-1_x for all r in a full measure set S of [0,T]. In particular, by Lemma <ref>, ξ_r has no atoms. Hence, for all r in S, Lemma <ref> implies the convergence of

    N(ξ^j),φ = ξ^j⊗ξ^j, F_φ

towards the same term without j. By the dominated convergence theorem (in r), its time integral converges as well. This proves convergence for the nonlinear term.



  §.§.§ The stochastic integral


It remains to prove convergence of the stochastic term. We follow the strategy in <cit.>. We use the notation t^l_i = 2^-li, and define, for t∈ [0,T],

    Y_j,k(t) = ξ^j_t, σ_k·∇φ_t ,
       Y_j,k^K,l(t) = 1_{k≤ K}∑_i Y_j,k(t^l_i)1_[t^l_i,t^l_i+1[(t)

and similarly without j. Finally we call ρ_j,k the modulus of continuity of Y_j,k, namely

    ρ_j,k(a) = sup{  |Y_j,k(t)-Y_j,k(s)|: t,s, ∈ [-,T]  |t-s|≤ a}

and similarly without j. Note that ρ_j,k(a) and ρ_k(a) are F̃_T-measurable on Ω̃, since the above supremum can be restricted to rational times t,s. We split

    |∑_k ∫^t_0 ξ^j, σ_k·∇φW^(j),k - ∫^t_0 ξ, σ_k·∇φW^k|
       ≤| ∑_k ∫^t_0 (Y_j,k-Y^K,l_j,k) W^(j),k|
          + | ∑_k ∫^t_0 Y^K,l_j,kW^(j),k - ∫^t_0 Y^K,l_kW^k|
          + | ∑_k ∫^t_0 (Y_k-Y^K,l_k) W^k|
       =:T_1+T_2+T_3.

Concerning the first term  T_1, we have

    |  T_1 |^2 = ∑_k ∫^t_0 |Y_j,k-Y^K,l_j,k|^2  r.

In order to have uniform estimates with respect to j, we want to use the convergence in C_t of Y^j,k to Y^k. For this, we split again the right-hand side:

    ∑_k ∫^t_0 |Y_j,k-Y^K,l_j,k|^2  r
       ≤ C∑_k ∫^t_0 |Y^K,l_j,k-Y^K,l_k|^2  r + C ∑_k ∫^t_0 |Y_k-Y^K,l_k|^2  r + C ∑_k ∫^t_0 |Y_j,k-Y_k|^2  r
       =:T_11+T_12+T_13.

For T_11, we have

    T_11= C ∑_k≤ K∫^t_0 |Y^K,l_j,k-Y^K,l_k|^2  r
       ≤ C ∑_k≤ Ksup_r |Y_j,k(r)-Y_k(r)|^2

For T_13, we have similarly

    T_13= C ∑_k≤ K∫^t_0 |Y_j,k-Y_k|^2  r + C ∑_k>K∫^t_0 |Y_j,k-Y_k|^2  r
       ≤ C ∑_k≤ Ksup_r |Y_j,k(r)-Y_k(r)|^2 + C ∑_k>Kσ_k_C_x^2

where we have used that sup_r|Y_j,k(r)|≤ Cσ_k_C_x^2. The constant C here  depends on M, the upper bound of ξ^j_M_x and φ. For T_12, we have

    T_12 = C ∑_k≤ K∫^t_0 |Y_k-Y^K,l_k|^2  r + C ∑_k>K∫^t_0 |Y_k|^2  r
       ≤ C ∑_k≤ Kρ_k(2^-l)^2 + C ∑_k>Kσ_k_C_x^2.

This complete the bound for T1. Concerning the term T_3, we have

    | T_3|^2 = ∑_k ∫^t_0 |Y_k-Y^K,l_k|^2  r,

which is the term T_12 up to a multiplicative constant and can therefore bounded as T_12. Finally, we note that the term T_2 can be written as

    T_2 = | ∑_k≤ K∑_i [Y_j,k(t^l_i) (W^(j),k_t^l_i+1-W^(j),k_t^l_i) - Y_k(t^l_i) (W^k_t^l_i+1-W^k_t^l_i)] |

Putting all together, we find

    |∑_k ∫^t_0 ξ^j, σ_k·∇φW^(j),k - ∫^t_0 ξ, σ_k·∇φW^k|^2
       ≤ C ∑_k≤ Kρ_k(2^-l)^2 + C ∑_k>Kσ_k_C_x^2
     + C ∑_k≤ Ksup_r |Y_j,k(r)-Y_k(r)|^2
          + C| ∑_k≤ K∑_i [Y_j,k(t^l_i) (W^(j),k_t^l_i+1-W^(j),k_t^l_i) - Y_k(t^l_i) (W^k_t^l_i+1-W^k_t^l_i)] |^2.

We first choose a natural number  K such that ∑_k>Kσ_k_C_x^2<. For each k, since Y_k is a continuous function, ρ_k(2^-l) converges to 0 as l→∞ ℙ-a.s.. Moreover Y_k is also essentially bounded, therefore, by dominated convergence theorem, ρ_k(2^-l)^2 also converges to 0. Hence, for K fixed as before, we can choose l such that

    ∑_k≤ Kρ_k(2^-l)^2<.

Again for each k, due to the convergence of ξ^j in C_t(M_x,M,w^∗ ), we infer that sup_r |Y_j,k(r)-Y_k(r)|^2 converges to 0 as j→∞ ℙ-a.s.. Moreover Y_j,k are bounded uniformly in j, therefore, by dominated convergence theorem, sup_r |Y_j,k(r)-Y_k(r)|^2 also converges to 0. Hence, for K fixed as before, we can choose j̅∈ℕ such that, for every j≥j̅,

    ∑_k≤ Ksup_r |Y_j,k(r)-Y_k(r)|^2<.

Finally, for K, l fixed as before, the term

    ∑_k≤ K∑_i [Y_j,k(t^l_i) (W^(j),k_t^l_i+1-W^(j),k_t^l_i) - Y_k(t^l_i) (W^k_t^l_i+1-W^k_t^l_i)]

converges to 0 as j→∞ ℙ-a.s.. Therefore, by the Lebesgue Dominated Convergence Theorem, also its second moment converges to 0. Hence, for K, l fixed as before, we can choose  j̅∈ℕ such that, for every j≥j̅,

    | ∑_k≤ K∑_i [Y_j,k(t^l_i) (W^(j),k_t^l_i+1-W^(j),k_t^l_i) - Y_k(t^l_i) (W^k_t^l_i+1-W^k_t^l_i)] |^2 <.

This proves that the stochastic term in (<ref>) converges in L^2_ω norm, and so ℙ-a.s. up to subsequences.

We have proved that all the terms in (<ref>) passes to the ℙ-a.s. limit, up to subsequences, and therefore ξ is a solution to (<ref>), so to (<ref>) with W as a Brownian Motion. The proof of Lemma <ref> is thus complete.






§ THE NONLINEAR TERM IN THE EULER EQUATIONS




A proof of this  Lemma is essentially due to Schochet <cit.>. We use here a version due to  Poupaud <cit.>.


  * Since K is a smooth function outside the diagonal {x=y}, also the function F_φ is smooth outside the diagonal. Recall that by Lemma <ref> we have |K(x-y)|≤ C|x-y|^-1,  (x,y)∈^2. Therefore, since  ∇φ is Lipschitz, we infer that

    |F_φ(x,y)|≤1/2 |K(x-y)|D^2φ_C_x |x-y|≤ Cφ_C^2_x,

which gives the desired bound on F_φ. The zero-average property (<ref>) of F_φ is a consequence of the fact that K is divergence-free.

  * For every ξ∈M_x, for every φ∈ C^2_x, F_φ is bounded and so N(ξ),φ is well-defined and

    | N(ξ),φ| = |∫∫ F_φ(x,y) ξ( x)ξ( y)|
       ≤ Cφ_C^2_xξ^⊗ 2_M_x,y≤ Cφ_H^4_xξ_M_x^2,

where we used the Sobolev embedding in the last inequality. In particular N(ξ) is a well-defined linear bounded functional on H^4.

  * By Lemma <ref>  the map M_x∋ξ↦ξ⊗ξ∈M_x,y is Borel (with respect to the Borel σ-algebras generated by the weak-^∗ topologies on M_x and M_x,y). Moreover, by Lemma <ref>,   for every φ∈ C^2_x, the map

    M_x,y∋μ↦μ, F_φ∈

is Borel. Therefore, for every φ∈ C^2_x, the function

    M_x∋ξ↦ N(ξ),φ∈

is Borel. Hence we deduce that  N: M_x →  H^-4_x is weakly-∗ Borel. Since H^-4_x is a separable reflexive Banach space, by Lemma <ref> we infer that N is also Borel.

  * Recall that K is odd by Lemma <ref>, therefore with u=K∗ξ we have

    ∫ξ(x) u(x)·∇φ(x)  x = ∫∫ξ(x)ξ(y) K(x-y) ·∇φ(x)  x y
       = - ∫∫ξ(x)ξ(y) K(y-x) ·∇φ(x)  x y
       = - ∫∫ξ(x)ξ(y) K(x-y) ·∇φ(y)  x y,

where the  last equality follows by  swapping  x with y. Hence

    ∫ξ(x) u(x)·∇φ(x)  x    =∫ξ(x) u(x)·∇φ(x)  x
       = 1/2∫∫ξ(x)ξ(y) K(x-y) · (∇_xφ(x) -∇_yφ(y))  x y.


  * Continuity of N on M_x,M,+,no-atom follows from Lemma <ref>.

  * The fact that N(ξ+α)=N(ξ) follows from the zero-average property (<ref>) of F_φ.





§ SOME TECHNICAL LEMMATA




Assume that the vector fields σ_k, k∈ℕ,  satisfy Assumption <ref>.  Then the following three  maps

    M_x ∋μ↦μ∈ H^-4_x,
       M_x ∋μ↦σ_k·∇μ∈ H^-4_x,   k∈ℕ,
       M_x ∋μ↦Δμ∈ H^-4_x

are well defined and  linear and continuous in both the norm-to-norm and the weak-^∗-to-weak sense and there exists  a number C>0 such that for every μ∈M_x,

    μ_H^-4_x +Δμ_H^-4_x≤ Cμ_M_x,
    σ_k·∇μ_H^-4_x≤ Cσ_k_C_xμ_M_x.

Moreover,  if  M_x is endowed with the weak-^∗ topology,  these three maps are  Borel.



The maps in (<ref>), tested against a test function φ, read formally

    μ↦μ,φ,
       μ↦ -μ,σ_k·∇φ,
       μ↦μ, Δφ.

Now, by Sobolev embedding, for any φ∈ H^4_x, the functions φ, σ_k·∇φ, Δφ are continuous with

    φ_C_x≤ Cφ_H^4_x,
       σ_k·∇φ_C_x≤σ_k_C_x∇φ_C_x≤ Cσ_k_C_x∇φ_H^4_x,
       Δφ_C_x≤ Cφ_H^4_x.

Hence the maps in (<ref>) are weak-^∗-to-weak continuous. Taking the supremum over φ in the unit ball of H^4_x, we get also the norm-to-norm continuity and the desired bounds.



Assume Assumption <ref> on σ_k and  fix M>0. Then for any object (Ω,A,𝔽,ℙ,W,ξ), where 𝔽=(F_t)_t and W=(W^k)_k is a filtration and, respectively an 𝔽-cylindrical Wiener process on a filtered probability space (Ω,A, 𝔽,ℙ) satisfying  the usual assumptions,  and ξ:[0,T]×Ω→M_x,M is B𝔽 Borel measurable function, equality  (<ref>) holds if and only if equality (<ref>) holds for every φ∈ C^∞_x.



Assume that (<ref>) holds, fix φ∈ C^∞_x. Then (<ref>) follows by applying the linear continuous functional ·,φ to (<ref>) and exchanging the functional with the integrals.

Conversely, assume that (<ref>) holds for every φ∈ C^∞_x. Lemma <ref> implies that ξ and all the integrands in (<ref>) are progressively measurable as H^-4_x processes and that the deterministic and stochastic integrals are well-defined (see (<ref>)). Now we have for every test function φ∈ C^∞_x, by (<ref>) exchanging ·,φ and the integrals,

    ξ_t,φ   = ξ_0,φ +∫^t_0 N(ξ_r)  r, φ
          -∑_k ∫^t_0 σ_k·∇ξ_r  W^k_r , φ
          +c/2∫^t_0 Δξ_r  r, φ,   for every t,  ℙ-a.s.,

that is (<ref>) tested against φ, where the ℙ-exceptional set can depend on φ. Taking φ in a countable set of C^∞_x, dense in H^4_x, we deduce (<ref>) in H^-4_x. The proof is complete.





Let ξ^_t(ω)=(Φ^(t,·,ω))_#ξ^_0 be defined as at the beginning of Section <ref>. For a.e. ω, for every t, ξ^_t ∈M_x,M.  Moreover, the process ξ^_t,φ is progressively measurable and, because it satisfies (<ref>),  continuous  for every φ∈ C^∞_x, and hence, by the density of C^∞_x in C_x, for every φ∈ C_x. Hence, by Lemma <ref> and Remark <ref>, up to redefining ξ^=0 on the ℙ-exceptional set where ξ^_t is not in M_x,M^ for some t, ξ^ is B𝔽 Borel as M_x-valued map and satisfies (<ref>) for every φ∈ C^∞_x. By Lemma <ref>, ξ^ is a M_x,M-valued solution in the sense of Definition <ref>.


Before we can embark on the proof of Lemma <ref> we formulate the following axially result.


  If the C^2-class vector fields  v,w:^2→^2  are such that v=0, then

    [v·∇ w +(Dv)^t w] = v·∇ (  w).

Here Dv= [  D_1v^1  D_2v^1; D_1 v^2 D_2 v^2 ]
 and ^t is the transpose operation.

 
 The expression (Dv)^t w is understood here as the 2× 1 matrix  that is equal to the  multiplication of a 2× 1 matrix w^1w^2   by a 2× 2 matrix D_1v^1 D_1v^2 D_2 v^1D_2 v^2, i.e.

    (Dv)^t w:= [  D_1v^1  D_1v^2; D_2 v^1 D_2 v^2 ]w^1w^2
                                                =w^1 D_1v^1   +  w^2 D_1v^2 w^1 D_2 v^1 + w^2 D_2 v^2

Therefore, since u:= D_1u^2-D_2u^1 and D_1D_2-D_2D_1=0 we get

    [ (Dv)^t w]   = D_1( w^1 D_2 v^1 + w^2 D_2 v^2 ) -D_2( w^1 D_1v^1   +  w^2 D_1v^2)
    
       = ( D_1D_2 v^1-   D_2D_1v^1 )w^1+ ( D_1D_2 v^2- D_2D_1v^2 )w^2
    
       + D_1 w^1 D_2 v^1 + D_1 w^2 D_2 v^2  -D_2 w^1 D_1v^1   -D_2 w^2 D_1v^2
    
       = D_1 w^1 D_2 v^1 + D_1  w^2 D_2 v^2  -D_2  w^1 D_1v^1    -D_2    w^2 D_1v^2.

Now we will prove another auxiliary identity:

    [ v·∇ w]=  v·∇[ w]+ ∑_i (D_1 v^iD_iw^2  -D_2 v^iD_iw^1 )

In the above formula, by v·∇ w we mean the following vector field ∑_i v^iD_iw and for ℝ-valued function f, we put
v·∇ f:=∑_i v^iD_if.

Let us now prove identity  (<ref>).

    [ v·∇ w]   =
    [ ∑_i v^iD_iw]= D_1[ ∑_i v^iD_iw^2 ] -D_2[ ∑_i v^iD_iw^1 ]
    
       =  ∑_i v^i D_1 D_iw^2  + ∑_i D_1 v^iD_iw^2
    
       -  ∑_i v^i D_2 D_iw^1  - ∑_i D_2 v^iD_iw^1
     
       =  ∑_i v^i ( D_1 D_iw^2 -D_2 D_iw^1)+ ∑_i (D_1 v^iD_iw^2  -D_2 v^iD_iw^1 )
    
       =  ∑_i v^i D_i( D_1 w^2 -D_2 w^1)+ ∑_i (D_1 v^iD_iw^2  -D_2 v^iD_iw^1 )
    
       =  ∑_i v^i D_i(  w)+ ∑_i (D_1 v^iD_iw^2  -D_2 v^iD_iw^1 ).



 By (<ref>) and (<ref>) we have

    [v·∇ w +(∇ v)^t· w] - v·∇ [w]
    
       =  D_1 v^1D_1w^2   -D_2 v^1D_1w^1  +  D_1 v^2D_2w^2  -D_2 v^2D_2w^1
    
       +  D_1 w^1 D_2 v^1 + D_1  w^2 D_2 v^2  -D_2  w^1 D_1v^1    -D_2    w^2 D_1v^2 .
    
       =  D_1 v^1D_1w^2 -D_2 v^2D_2w^1 + D_1  w^2 D_2 v^2  -D_2  w^1 D_1v^1
    
       = D_1 v^1 (D_1w^2  -D_2  w^1)+ D_2 v^2( D_1  w^2 -D_2w^1)
    
       =  [v] [w].

Note that the above identity is valid for all C^2-class vector fields. But if additionally v=0, it implies the identity
 (<ref>). The prof is complete.
 



Assume that p ∈ (2,∞). Then for all  ξ∈ L^p_x and  v ∈^1,p_x ∩, the following equality holds in  H^-1_x:
 
    Π[v·∇ K ∗ξ +(Dv)^t K ∗ξ] = K*[v·∇ξ].


 Let us choose and  fix p ∈ (2,∞).
 We begin by observing  that by part 5 of Lemma <ref>, identity (<ref>) implies that

    Π[v·∇ w +(D v)^t w] = K*[v·∇ w],

where Π the Leray-Helmholtz  projection.
 Take w=K ∗ξ, for a regular scalar function ξ:^2→. By  Lemma <ref>, [w] = ξ -γ, where γ = ∫_^2ξ∈ℝ.
 If we take  a regular and divergence-free vector field  v:^2→^2,  then by applying  the above formula  we infer that
 (<ref>)  holds.
 We claim that   equality (<ref>)  in the H^-1_x sense  is satisfied for every  ξ∈ L^p_x and  v ∈^1,p_x ∩.
 Since
   by  Lemma <ref>, both sides of (<ref>) are continuous bilinear maps from L^p_x ×^1,p_x ∩ to H^-1_x, the result follows.


After having proved the last Lemma we are ready to embark on our main task.

 Let us choose and  fix p ∈ (2,∞).
Assume  that a process    ξ belongs to L^p_t,ω(L^p_x)  and that
  ξ is  an M_x,M-valued distributional solution to the stochastic vorticity equation (<ref>) (according to Definition <ref>).
  Thus equality  (<ref>) is satisfied in H_x^-4.
   Define a process u by
  
    u(t,·,ω) =K ∗ξ (t,·,ω),    (t,ω) ∈ [0,∞)×Ω.


Since ξ belongs to  L^p_t,ω(L^p_x), by by Lemma <ref> we infer that  u belongs  L^p_t,ω(^1,p_x). We  also observe by Lemma <ref>,  that since  ξ∈ L^p_t,ω(L^p_x), the nonlinear term can be written as u·∇ξ, and the equation (<ref>) holds actually in H^-2_x.  Indeed, ξ and all the integrands of (<ref>) takes values in H^-2_x and are progressively measurable as H^-2_x-valued processes (and their deterministic and stochastic H^-2_x-valued integrals coincide with the H^-4_x-valued integrals). Now we apply to (<ref>) the convolution operator

    K∗: H^-2_x ∋ξ↦ u=: K∗ξ∈  H^-1_x,

which by Lemma <ref> is linear and bounded. By the first part of this proof, we get, for a.e. ω, as equality in H^-1_x: for every t,

    u_t    = u_0 - ∫^t_0 Π[u_r·∇ u_r +(Du_r)^t u_r]  r
       - ∑_k ∫^t_0 Π[ σ_k·∇ u_r +(Dσ_k)^t u_r ]  W^k_r
       +c/2∫^t_0 Δ u_r  r

Now we note that (Du_r)^t u_r = ∇ [|u_r|^2]/2 and hence π( (Du_r)^t u_r)=0. Thus we deduce  (<ref>) and so
the proof of Lemma <ref> is complete.



Leu us fix p∈ (2,∞) and the processes ξ and u as in the proof  of Lemma <ref>.

For any δ∈ (0,1], we consider  an operator

    R^δ:  H^-1_x u ↦ρ_δ∗ u ∈  L^2_x

where (ρ_δ)_δ is a standard family of mollifiers on ^2.
Let us observe that the operators  R^δ  are uniformly  bounded maps  on the spaces L^p_x and ^1,p_x. In particular, the processes   R^δ u  belong uniformly to the same spaces as the process u. Hence, by the Sobolev embedding, the L^p_t,ω(L^∞_x) norm of  R^δ u is uniformly bounded.



Note that R^δ f → f in L^2_x, resp. H^-1_x for every f ∈ L^2_x, resp. f ∈ H^-1_x. By applying the  map  R^δ to equality (<ref>) we get

    R^δ u_t = R^δ u_0 -∫^t_0 R^δΠ[u_r·∇ u_r] dr
       -∫^t_0 R^δΠ[ σ_k·∇ u_r +(Dσ_k)^t u_r ]  W^k_r
       +c/2∫^t_0 R^δΔ u_r  r,    L^2_x

 for every t ≥ 0. Next we can apply the Itô formula  <cit.> to the square of the L^2_x norm, which is obviously of C^2 class on the space L^2_x,  with uniformly continuous derivatives on bounded subsets of L^2_x. We infer that, for every t,

    R^δ u_t_L^2_x^2 = R^δ u_0_L^2_x^2 -2∫^t_0  R^δ u_r, R^δΠ[u_r·∇ u_r]  dr
       -2∫^t_0  R^δ u_r, R^δΠ[ σ_k·∇ u_r +(Dσ_k)^t u_r ]  W^k_r
       +∫^t_0  R^δ, cR^δΔ u_r  r + ∫^t_0 ∑_k R^δΠ[ σ_k·∇ u_r +(Dσ_k)^t u_r_L^2_x^2  r.

Since

    ∑_k ∫^T_0 | R^δ u_r, R^δΠ[ σ_k·∇ u_r +(Dσ_k)^t u_r ] |^2_L^2_x r
       ≤ C u_L^2_t,ω(L^∞_x)^2 (∑_k σ_k_C^1_x) u_L^2_t,ω(^1,2_x)^2.

we infer that  the stochastic integral is an L_x^2-martingale with zero mean. Similarly the integrands in the deterministic integrals have finite L^1_t,ω norm and we can take expectation: we get

    R^δ u_t_L^2_x^2 = R^δ u_0_L^2_x^2 -2∫^t_0  R^δ u_r, R^δΠ[u_r·∇ u_r]  dr
       -∫^t_0 cR^δ∇ u_r_L^2_x^2  r + ∫^t_0 ∑_k R^δΠ[ σ_k·∇ u_r +(Dσ_k)^t u_r_L^2_x^2  r,

where we have used integration by parts and that R^δ commutes with Δ and ∇. Finally, we note that R^δ f → f in L^2_x for every f in L^2_x. We exploit this fact for f=u_r, f=Π[u_r·∇ u_r], f=σ_k·∇ u_r +(Dσ_k)^t u_r and f=∇ u_r, and use the dominated convergence theorem in r and ω and k, to pass δ→ 0 and obtain (<ref>). The proof is complete.




































































Let X be a closed convex subset of a topological vector space, endowed with its Borel σ-algebra. Assume that X is also a Polish space. Let ζ:[0,T]×Ω→ X be a B([0,T])×A Borel measurable map.


  * The set C_t(X) is a Polish space and, if, for every ω, t↦ζ_t is in C_t(X), then ω↦ζ(·,ω) is A Borel measurable as C_t(X)-valued map.

  * If X is a separable reflexive Banach space and, for every ω, t↦ζ_t is in L^2_t(X) (more precisely, has finite L^2_t(X) norm), then ω↦ζ(·,ω) (more precisely, its equivalence class) is A Borel measurable as L^2_t(X)-valued map.




For the first point, the fact that C_t(X) is a Polish space is well known. Moreover the Borel σ-algebra B(C_t(X)) on C_t(X) is generated by the evaluation maps π_t(γ)=γ_t. Indeed,  B(C_t(X)) is generated by the maps

    γ↦ d(γ(t),g(t)),   t∈ [0,T]∩ℚ,   g∈ C_t(X),

with d distance on X, and these maps are measurable in the σ-algebra generated by the evaluation maps (because they are composition of the evaluation maps and a Borel function on X). Now, for every t, the map π_t(ζ)=ζ_t is A Borel, by the Fubini theorem, hence, if ζ is C_t(X)-valued, then it is A Borel measurable as C_t(X)-valued map.



For the second point, we note that, by Lemma <ref>, it is enough to show that ζ is weakly progressively measurable. Since the dual of L^2_t(X) is L^2_t(X^∗) (see <cit.>), it is enough to show that, for every φ∈ L^2_t(X^∗),

    ω↦∫^T_0 ζ(t,ω), φ(t)_X,X^∗ t

is measurable. But this follows from Fubini theorem. The proof is complete.



Let us denote by 𝔽^00=(F^00_t)_t, where F^00_t = σ{ξ_s,W_s : 0≤ s≤ t},   the filtration generated by ξ and W.
Clearly the processes W and ξ are adapted to F^00. We claim that W is a cylindrical F^00-Wiener process. Indeed, W is a cylindrical Brownian motion with respect to its natural filtration, as a.s. limit of cylindrical Brownian motions. Moreover, for every 0≤ s_1≤…≤ s_h≤ s < t, W^(j)_t-W^(j)_s is independent of (ξ^j_s_1,W^(j)_s_1,…ξ^j_s_h,W^(j)_s_h), therefore W_t-W_s is independent of (ξ_s_1,W_s_1,…ξ_s_h,W_s_h). This proves our claim.

Recall that 𝔽^0=(F_t^0)_t is the filtration generated by 𝔽^00 and the ℙ-null sets on (Ω,A,ℙ) and that F_t = ∩_s>tF^0_s. We argue as in the proof of <cit.> (note that the proof is valid for any filtration making W^k Brownian motions) and we get that the filtration (F_t)_t is complete and right-continuous and W is still a cylindrical Brownian motion with respect to it. Finally ξ is an (M_x,M,w^∗ )-valued (F_t)_t-adapted and continuous process, hence also progressively measurable.

In a similar (and easier) way, one gets the result for (F^j_t)_t, W^(j) and ξ^j, for each j.



Let us fix j. We have to verify equation (<ref>) for (ξ^j,W^(j)) for every φ∈ C^∞_t,x. The idea is taken by <cit.>: it is enough to verify that, for every φ∈ C^∞_x, for every t, the random variables

    Z_t   := ξ^_j_t,φ_t  - ξ^_j_0,φ -∫^t_0  N(ξ^_j_r),φ r
        -∑_k ∫^t_0 ξ^_j_r,σ_k·∇φ W^k_r -1/2∫^t_0 ξ^_j_r, cΔφ r

and Z_t, obtained as in (<ref>) replacing (ξ^_j,W) with (ξ^j,W^(j)), have the same law. We fix t and φ∈ C^∞_x. By Lemma <ref> and Lemma <ref>, all the terms in (<ref>) but the nonlinear term and the stochastic integral are Borel functions of ξ^_j with respect to the C_t(M_x,M,w^∗ ) topology. Concerning the stochastic integral we use an approximation argument. For every positive integers K and N, calling t^N_i =2^-Ni for i integer, the map

    C_t(M_x,M,w^∗ )× C_t^ℕ∋ (ξ,W)↦∑_k=1^K ∑_i,t^N_i+1≤ tξ_t^N_i, σ_k·∇φt^N_i (W_t^N_i+1-W_t^N_i)

is a continuous, in particular Borel function. By the continuity of t↦ξ_t, σ_k·∇φ_t for every k, for a.e. ω, and by the square-summability of σ_k_C_x, we get via the dominated convergence theorem that, as (N,K) tends to ∞,

    ∑_k ∫^T_0 |ξ_t, σ_k·∇φ_t  - 1_k≤ K∑_i ξ^_j_t^N_i, σ_k·∇φt^N_i 1_[t^N_i+1,t^N_i)(t)|^2  r → 0,

so by the Itô isometry we obtain that, as (N,K)→∞,

    ∑_k=1^K ∑_i,t^N_i+1≤ tξ^_j_t^N_i, σ_k·∇φt^N_i (W_t^N_i+1-W_t^N_i) →∑_k ∫^t_0 ξ^_j_r,σ_k·∇φ_r  W^k_r   in L^2_ω.

Similarly for (ξ^j,W^(j)) (with convergence in L^2_ω̃). We conclude that

    Z_t = F_t(ξ^_j) +L^2_ω-lim_N,K G_N,K,t(ξ^_j,W),
       Z̃_t = F_t(ξ̃^j) +L^2_ω̃-lim_N,K G_N,K,t(ξ̃^j,W̃^(j))

for some Borel maps F_t and G_N,K,t. Since (ξ^_j,W) and (ξ^j,W^(j)) have the same law, also Z_t and Z_t have the same law. Since, ℙ-a.s., Z_t=0 for every t, also, ℙ-a.s., Z_t=0 for every t, and so, by Lemma <ref>, (Ω,A,(F^j_t)_t,ℙ,W^(j),ξ^j) solves (<ref>).














Concerning Lemmas <ref> and <ref>, for any integer h, as a consequence of Remark <ref>, the maps

    C_t(M_x,M,w^∗ )∋ξ↦ (ξ_t_H^h_x)_t ∈ C_t,
       C_t(M_x,M,w^∗ )∋ξ↦ξ_C^α_t(H^h_x)∈

are Borel. Hence ξ^_j_t_H^h_x and ξ^j_t_H^h_x have the same laws (as C_t-valued random variables) and so Lemma <ref> holds for ξ^j. Similarly ξ^_j_t_H^h_x and ξ^j_t_H^h_x have the same laws and so Lemma <ref> holds for ξ^j.

Finally, concerning non-negativity, we note that the set {ξ_t≥ 0, ∀ t} is Borel in C_t(M_x,w^∗ ), because it can be written as ξ_t,φ≥ 0 for all rational t and all φ in a countable dense set in C_x. Since ξ^_j is concentrated on {ξ_t≥ 0, ∀ t}, also ξ^j is concentrated on this set. The proof is complete.





§ THE TORUS AND THE CORRESPONDING  GREEN FUNCTION



We consider the torus ^2 as the two-dimensional manifold obtained from [-1,1]^2 identifying the opposite sides; we call π: ℝ^2 →𝕋^2 the quotient map. A continuous (C_x) function is understood here as a continuous periodic function on ^2, with period 2 on both x_1 and x_2 directions, and can be identified with a continuous function on the torus ^2. For s positive integer, a C^s_x function on ^2 is a C^s periodic function on ^2 (with period 2). Similarly, for s positive integer and 1≤ p ≤∞, a ^s,p_x function on ^2 is a ^s,p_loc periodic function on ^2 (with period 2). One can also define a Riemannian structure on the torus via the quotient map π so that π is a local isometry; the local isometry implies that the gradient, the covariant derivatives etc transform naturally, moreover the C^s and ^s,p spaces defined via the Riemannian structure coincide with the corresponding spaces of periodic functions as defined above.

The space of distribution 𝒟^'_x on ^2 is understood as the dual space of C^∞ periodic functions on ^2. The spaces of functions can be identified with subspaces of distribution via the L^2 scalar product f,g =∫_[-1,1[^2 f(x) g(x)  x. The space of measures M_x is the space of distributions on ^2 which are continuous (precisely, can be extended continuously) on C_x; the space M_x can be identified with the space of finite ℝ-valued  measures on ^2 and with the quotient space of finite ℝ-valued  measures on [-1,1]^2 under the map π, via the L^2_x scalar product:

    f, μ = ∫_[-1,1[^2 f(x) μ( x),  ∀ f∈ C_x.

For s>0  and 1<p<∞, denoting by  p^∗ the conjugate exponent of p, the space ^-s,p^∗ is the space of distributions on ^2 which can be continuously extended to ^s,p.

The convolution on the torus is understood as

    f∗ g(x) = ∫_[-1,1[^2f(y)g(x-y)  y

for f, g periodic functions on ^2.

We recall here some standard facts on the Green function G of the Laplacian on the zero-mean functions, that is

    Δ G(·, y) = δ_y,  ∀ y∈𝕋^2.



The following facts hold true.


  * The Green function G is translation invariant, that is 
    G(x,y)=G(x-y), ,
 even, regular outside 0, with

    -C^-1log|x|≤ G(x)≤ -Clog|x| .


  * The kernel K=∇^⊥ G is divergence-free (in the distributional sense), odd, regular outside 0, with C^-1|x|^-1≤ |K(x)|≤ C|x|^-1 in a neighborhood of 0.

  * Let ξ be a distribution on ^2 with zero mean, define u=K ∗ξ. Then u =0 and ξ =   u.

  * Let u be a vector-valued distribution on ^2 with zero mean and with u =0, define ξ=   u. Then u=K ∗ξ.

  * Let u be a vector-valued distribution on ^2, define ξ=   u. Then Π u = K ∗ξ, where Π is the Leray projector on zero-mean divergence-free distributions.

  * Let ξ be a distribution on ^2 with zero mean, define u=K ∗ξ. For any 1<p<∞, for any  n ∈ℤ, ξ∈^n,p_x if and only if u ∈^n+1,p_x and  there exist a constant C=C(n) independent of ξ such that

    C^-1ξ_^n,p_x≤u_^n+1,p_x≤ Cξ_^n,p_x.




For the proof, we recall the following facts:


  * Any distribution f on ^2 can be written in Fourier series as f= ∑_k a_k e^ik· x (the convergence being when tested against a smooth periodic function), see <cit.> and <cit.>.

  * For any integer s and any 1<p<∞, the Sobolev space ^s,p can also be written in terms of Fourier series, that is

    ^s,p_x = {f ∈𝒟^'_x : f̃^s := ∑_k a_k (1+|k|^2)^s/2 e^ik· x∈ L^p_x },

with f̃^s_L^p_x as equivalent norm, see <cit.>. This fact is well-known for s≥ 0. We give a sketch of the proof for s<0 for completeness. We have to show that the above right-hand side is the dual space of ^-s,p_x. Indeed,  for every distributions f continuous on ^-s,p^∗_x, it holds

    |φ̃^-s, f̃^s | = |φ , f | ≤ Cφ_^-s,p^∗_x≤ C'φ̃^-s_L^p^∗_x,   ∀φ∈^-s,p^∗_x,

hence f̃^s belongs to L^p_x, so f belongs to the right-hand side of (<ref>).

  * Regularity theory: The Laplacian operator Δ, indended in the sense of distribution, acts multiplying each Fourier coefficient a_k by |k|^2. In particular, it is invertible on the subspace of zero-mean distributions and its inverse acts multiplying each Fourier coefficient a_k by |k|^-21_k≠ 0. It follows that the inverse Δ^-1 of the Laplacian (on zero-mean distributions) maps ^s,p into ^s+2,p, for any integer s and any 1<p<∞.

  * Hodge decomposition: if f is a ^2-valued distributions with f=0 and f=0, then f is a constant. Indeed, if a_k are the Fourier coefficients of f, we have a_k · k=0 and a_k· k^⊥ =0 for every k, therefore a_k=0 for every k≠ 0.





  * The fact that G is translation-invariant is due to the translation invariant property of the torus: if φ is periodic and zero-mean and solves Δφ =δ_0 in the distributional sense, then φ_y(x):=φ(x-y) is still periodic and zero-mean and solves Δφ= δ_y. For the even and regularity property and the bounds, see e.g. <cit.> and references therein.

  * The fact that K is divergence-free, odd and regular outside 0 is a consequence of the definition of K and the properties of G. For the bounds, see again <cit.>.

  * The fact that u is divergence-free follows from the same property of K. Call ψ=(-Δ)^-1ξ = -G*ξ. Then u=-∇^⊥ψ and so

    u = ∂_x_1u^2 - ∂_x_2u^1 = -Δψ = ξ,

where all the computations are intended using test functions.

  * Call ũ=K ∗ξ. We deduce from the previous points that (u-ũ) = 0 and that (u-ũ) = 0. From this we conclude that u-ũ is a constant, therefore is =0 as both functions have zero mean.

  * This follows from the previous point, applied to Π u in place of u.

  * If u is in ^s+1,p then ξ=  u is in ^s,p. Conversely, if ξ is in ^s,p, then ψ = (-Δ)^-1ξ is in ^s+2,p and so u = -∇^-1ψ is in ^s+1,p.

























































§ MEASURABILITY



We include here various standard concepts and results about measurability.

We recall the definition of strong, weak, weak-^∗ and Borel measurability for a Banach-space valued map. We are given a σ-finite measure space (E,ℰ,μ), a Banach space V and a function f:E→ V:


  * we say that f is strongly measurable if it is the pointwise
(everywhere) limit of a sequence of V-valued simple measurable
functions (i.e. of the form ∑_i=1^Nv_i1_A_i for A_i
in ℰ and v_i in V);

  * we say that f is weakly measurable if, for every φ in
V^∗, x↦⟨ f(x),φ⟩_V,V^∗ is measurable;

  * if V=U^∗ is the dual space of a Banach space U, we say that
f is weakly-∗ measurable iff, for every φ∈ U, x↦⟨ f(x),φ⟩_V,U
is measurable;

  * we say that f is resp. strongly Borel, weakly Borel, weakly-∗ Borel measurable if, for every open set A in
V resp. in the strong, weak, weak-^∗ topology, f^-1(A) is in ℰ. We omit strongly/weakly/weakly-∗ when clear.


The following result is essentially the  Pettis Measurability Theorem. The present version is a consequence of <cit.>.


Assume that V is a separable Banach space. Then the notions of the strong measurability, the weak measurability, the strongly Borel measurability and the weakly Borel measurability coincide. They also coincide with the weak-^∗ measurability and weakly-∗ Borel measurability if in addition V is reflexive.








We prove here a statement concerning weak-^∗ and weakly-∗ Borel measurability, which applies in particular to M_x=(C_x)^∗. We call B̅_R the closed centered ball in V of radius R (in the strong topology).


Assume that V=U^∗ is the dual space of a separable Banach space U. Then the notions of weak-^∗ measurability and of weakly-∗ Borel measurability coincide. Moreover, for any sequence (φ_k)_k dense in the unit centered ball of U, the Borel σ-algebra associated to the weak-^∗ topology is generated by

    ·,φ_k.




We recall that, if (E,E) is a measurable space, I generates the σ-algebra E and F is a subset of E, then the σ-algebra E|_F={A∩ F: A∈E} on F is the σ-algebra generated on F by I|_F={I∩ F: I∈I}. In particular, the Borel σ-algebra restricted to a subset F is the Borel σ-algebra on F (with the topology restricted on F) and the previous statement can be extended to subsets of U.



We fix the sequence (φ_k)_k. We call B the Borel σ-algebra associated to the weak-^∗ topology and C the σ-algebra generated by the maps ·,φ for φ∈ U. Since φ_k are dense in the unit centered ball of U, C is generated by the maps ·,φ_k. We will show that B=C, what implies both statements in the Lemma. Since the maps ·,φ, for φ∈ U, are continuous in the weak-^∗ topology, C⊆B. For the converse inclusion, it is enough to show that, for any R>0, for any open set A in the weak-^∗ topology, the sets B̅_R and A∩B̅_R are in C, where B̅_R is the closed centered ball in V of radius R (in the strong topology). By separability of U, we can fix a sequence (φ_k)_k which is dense in the unit centered ball of U. For any R>0, the ball B̅_R is in C because the strong norm on V is C-measurable. Indeed,  it can be written as

    v = sup_k | v,φ_k|.

We recall that the weak-^∗ topology, restricted on B̅_R is separable and metrizable, see <cit.> for the metrizability and  separability follows from the  compactness, with the distance

    d(v,v') = ∑_k 2^-k| v-v',φ_k|.

Now, for every v in B̅_R, d(v,·) is C-measurable, hence any open ball with respect to d is in C. Moreover, for any open set A in the weak-^∗ topology, A∩B̅_R can be written as countable union of open balls with respect to d, hence A∩B̅_R is in C. The proof is complete.


Now we give a measurability property of the testing against bounded, but not necessarily continuous maps. Here, given a Polish space X, M(X) is the set of finite measures on X.


Let F:X→ be a bounded Borel function on a compact metric space X (in particular X=^2). Then the map

    Ψ_F: M(X) ∋μ↦∫_X F(x) μ( x) ∈

is Borel with respect to the weak-^∗ topology on M(X).



If F is continuous, then also Ψ_F is continuous in the weak-^∗ topology, in particular weakly-∗ Borel. If F=1_A is the indicator of an open set A in ^2, then 1_A is the pointwise (everywhere) non-decreasing limit on ^2 of continuous functions F_n; so, by the dominated convergence theorem, Ψ_1_A is the pointwise limit of Ψ_F_n and so it is also weakly-∗ Borel. For the case of general F, we use the monotone class theorem. We consider the set W of Borel functions F on X such that Ψ_F is weakly-∗ Borel. Then W contains the indicators of all the open sets, it is a vector space and it is stable under monotone non-decreasing convergence. Indeed, if (F_n)_n is a non-decreasing sequence in W converging pointwise to F, then, by the dominated convergence theorem, Ψ_F is the pointwise limit of Ψ_F_n, in particular weakly-∗ Borel, and so F belongs also to W. Then, by the Monotone Class Theorem, W contains all bounded Borel functions F on ^2, which gives the result.












We recall a classical fact for the product of measures. For a compact metric space X, we call M(X) the set of finite measures on X, dual to the space C(X) of continuous function on X, and, for M>0, M_M(X) the closed centered ball on M(X) of radius M.


For any compact metric space X, the map G:M(X)∋μ↦μ⊗μ∈M(X× X) is Borel with respect to the weak-^∗ topologies. Moreover, for any M>0, the map G, restricted on M_M(X) with values in M_M^2(X× X), is continuous with respect to the weak-^∗ topologies.



For M>0, we call G_M:M_M(X) →M_M^2(X× X) the map G restricted on M_M(X) with values in M_M^2(X× X). We start showing the continuity of G_M. By metrizability of M_M(X) and M_M^2(X× X), it is enough to show that, if (μ^n)_n is a sequence in M_M(X) converging weakly-∗ to μ, then (μ^n⊗μ^n)_n converges weakly-∗ to μ⊗μ. For every two continuous functions φ, ψ on X, we have

    φ⊗ψ, μ^n⊗μ^n  = φ,μ^nψ,μ^n→φ⊗ψ, μ⊗μ.

Now the set of all linear combinations of φ⊗ψ for all continuous functions φ, ψ is a subalgebra of C(X× X) which separates point, hence, by the Stone-Weierstrass theorem, it is dense in C(X× X). Then, for any ϕ continuous function on X× X, by a standard approximation argument on ϕ we get that (ϕ,μ^n⊗μ^n )_n converges to ϕ,μ⊗μ. This shows continuity of the map G restricted to M_M(X).

For Borel measurability on the full space, take any open set A in M(X× X), then G^-1(A) is the non-decreasing union of G_M^-1(A∩M_M^2(X× X)) for M in ℕ. By continuity of G_M, G_M^-1(A∩M_M^2(X× X)) is open, hence Borel, in M_M(X). Moreover M_M(X) is itself a Borel set in M(X). Indeed the closed centered ball B̅_R in a dual space V=U^∗ is Borel, as shown in the proof of Lemma <ref>. So G_M^-1(A∩M_M^2(X× X)) is Borel in M(X). Therefore A is Borel in M(X). The proof is complete.


We conclude on measurability of the H^h norms:


For any fixed integer h, the H^h_x norm can be written as supremum of |·,φ| over a set D of φ∈ C_x, with D countable and dense in H^h_x. Therefore the H^h_x norm is a lower semi-continuous function and Borel function on (M_x,w^∗ ).

The C_t^α(H^h_x) norm can be written as

    f_C^α_t(H^h_x)= sup_t∈ℚ∩[0,T]f_t_H^h_x +sup_s,t∈ℚ∩[0,T],s<tf_t-f_s_H^h_x/|t-s|^α

(note the supremum over a countable set of times). Therefore, for any fixed M>0, the C_t(H^h_x) norm is a lower semi-continuous function, in particular a Borel function, on C_t(M_x,M,w^∗ ).




Acknowledgement. We would like to thank to James-Michael Leahy for pointing out the presence of the constant γ in the equation (<ref>) for the velocity, and to  Jasper Hoeksema and Oliver Tse for pointing out a mistake in Lemma <ref>, in a previous draft of this paper, and a suggestion  how to correct it. This work was undertaken mostly when M.M. was at the University of York, supported by the Royal Society via the Newton International Fellowship NF170448 “Stochastic Euler Equations and the Kraichnan model”. Finally, we would like to thank Philippe Serfati for pointing out several references on deterministic Euler equations.

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