# Conservation of Total Vorticity for a 2D Stochastic Navier Stokes Equation

Authors: Peter M. Kotelenez; Bradley T. Seadler
Journal: Advances in Mathematical Physics
 (2011)

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## Abstract

We considerN point vortices whose positions satisfy a stochastic ordinary differential equation on ℝ2N perturbed by spatially correlated Brownian noise. The associated signed point measure-valued empirical process turns out to be a weak solution to a stochastic Navier-Stokes equation (SNSE) with a state-dependent stochastic term. As the number of vortices tends to infinity, we obtain a smooth solution to the SNSE, and we prove the conservation of total vorticity in this continuum limit.


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## Body

## 1. Introduction

Our aim is to show that for a two-dimensional incompressible fluid, the total vorticity of the fluid is a conserved quantity (where the vorticity for a rigid body is twice the angular velocity). Following Kotelenez [1] and Marchioro and Pulvirenti [2] (cf. also Amirdjanova, [3, 4], Amirdjanova and Xiong [5]), the distribution of the vorticity satisfies the following:
(1.1)∂∂tX(r,t)=νΔX(r,t)-∇•(U(r,t)X(r,t)),X(r,t)=curl⁡U(r,t)=∂U2∂r1-∂U1∂r2,∇⋅U≡0,
where U(r,t) is the velocity field, r∈ℝ2, ν≥0 is the kinematic viscosity, Δ is the Laplacian, ∇ is the gradient, and • denotes the scalar product on ℝ2. If ν>0, we obtain the Navier-Stokes equation for the vorticity. If the fluid is inviscid (or ideal), that is, ν=0, we obtain the Euler equation. Note that by the incompressibility condition ∇•U≡0, we obtain
(1.2)U(r,t)=∫(∇⊥g)(r-q)X(q,t)dq,
where g(r):=(1/2π)ln(|r|) with |r|2=r12+r22 and ∇⊥=(-(∂/∂r2),(∂/∂r1))T with T denoting the transpose; ∫()dq denotes integration over ℝ2 with respect to the Lebesgue measure. Consequently, we can obtain the velocity field, U, from the vorticity distribution.Let0<δ≤1. Let gδ(s) be at least twice continuously differentiable with bounded derivatives up to order 2 with |gδ′(s)|≤|g′(s)| and |gδ′′(s)|≤|g′′(s)|, for s>0 such that gδ(r)≡g(r), for δ≤|r|≤1/δ. Set
(1.3)Kδ(r):=∇⊥gδ(|r|).We may assume without loss of generality thatgδ′(0)=0, which implies Kδ(0)=0. Thus, we have the smoothed Navier-Stokes equation (NSE)
(1.4)∂∂tX(r,t)=νΔX(r,t)-∇•(Uδ(r,t)X(r,t)),Uδ(r,t):=∫Kδ(r-q)X(q,t)dq.ConsiderN point vortices with intensities ai∈ℝ, and let ri be the position of the ith vortex in ℝ2. Abbreviate rN:=(r1,…,rN)∈ℝ2N. Assume that the positions satisfy the stochastic ordinary differential equation (SODE)
(1.5)dri(t)=∑j=1NajKδ(ri-rj)dt+2νdmi(rN,t),i=1,…,N.Themi(rN,t) are ℝ2-valued square-integrable continuous martingales (i=1,…,N), which may depend on the positions of the vortices. Let us for the moment assume that for suitably adapted square-integrable initial conditions, (1.5) has a unique (Itô) solution rN(t)=(r1(t),…,rN(t)). Set
(1.6)XN(t):=∑i=1Naiδri(t),
where ri(t) are the solutions of (1.5) and δr is the point measure concentrated in r. We will call 𝒳N(t) the empirical process associated with the SODE (1.5). Let Lp(ℝ2,dr) be the standard Lp-spaces of real-valued functions on ℝ2 with p∈[1,∞], where dr is the Lebesgue measure. Set H0:=L2(ℝ2,dr) and denote by 〈·,·〉0 and ∥·∥0 the standard scalar product and its associated norm on H0. Further, let 〈·,·〉 be the extension of 〈·,·〉0 to a duality between distributions and smooth functions. For m∈ℕ, we define C0m(ℝ2,ℝ) to be the functions from ℝ2 into ℝ which are m times continuously differentiable in all variables such that all derivatives vanish at infinity.(i)
Ifν=0, X(t) is a solution to the Euler equation (1.1) and the initial condition satisfies X(0)∈L1(ℝ2,dr)∩L∞(ℝ2,dr), then there is a sequence Kδ(N)(r)→K(r):=∇⊥g(r) as N→∞ such that for ϕ∈C02(ℝ2,ℝ) (cf. Marchioro and Pulvirenti [2]),
(1.7)〈XN(t),ϕ〉⟶〈X(t),ϕ〉asN⟶∞.(ii)
Supposeν>0 and X(t) is a solution to the Navier-Stokes equation (1.1). Choose mi(rN,t):=βi(t), where βi(t) are i.i.d. ℝ2-valued standard Wiener processes and half of the intensities ai equal to (a+)(2/N) for a+>0 and the other half to (-a-)(2/N) with a->0. Let φ∈C02(ℝ2,ℝ). Apply Itô's formula and compute the quadratic variation
(1.8)d[〈φ,XN(t)〉]=2ν∑±[∑j=1N/22a±N{(∇φ)(rj(t))•βj(dt)}]=2ν∑±∑j=1N/2(2a±)2N2{∑k=12(∂kφ)2(rj(t))}dt,
where we used the independence of βi(·). Hence, for t≤T,
(1.9)[〈φ,XN(t)〉]=O(1N,φ,T).In other words, the empirical vorticity distribution𝒳N(t) becomes macroscopic, as N→∞. Choosing a sequence Kδ(N)(r)→K(r) and assuming a suitable convergence of the initial conditions 𝒳N,0 towards the initial condition in (1.1), we may expect that
(1.10)〈φ,XN(t)〉⟶〈φ,X(t)〉,
where X(t) is the solution to (1.1). As N→∞ describes a continuum limit (in this limit, the discrete point particle distribution may become a smooth particle distribution with densities, etc.), (1.10) implies that the macroscopic limit and the continuum limit coincide. Marchioro and Pulvirenti [2] (cf. also Chorin [6] and the references therein) prove a somewhat weaker result: for the case mi(rN,t):=βi(t), assuming in addition to the previous conditions, 〈E𝒳N(0),φ〉→〈X(0),φ〉 as N→∞, they prove that for all t>0,
(1.11)〈EXN(t),φ〉⟶〈X(t),φ〉asN⟶∞,
where E denotes the mathematical expectation with respect to the underlying probability space (Ω,ℱ,ℱt,P). (All our stochastic processes are assumed to live on Ω and to be ℱt-adapted (including all initial conditions in SODE’s and SPDE’s), where the filtration ℱt is assumed to be right continuous. Moreover, the processes are assumed to be (dP⊗dt)-measurable, where dt is the Lebesgue measure on [0,∞).)In order to separate the macroscopic and continuum limits and to derive a mesoscopic vorticity distribution, Kotelenez [1] introduces spatial correlations of the Brownian noise as follows through correlation functionals convolved with space-time Gaussian white noise as follows: for i=1,2, wi(dp,ds) are i.i.d. space-time Gaussian white noise fields, (this is the multiparameter generalization of the increments of a real valued Brownian motion β(ds). cf. Walsh [7] and Kotelenez [8]) ɛ>0 is the spatial correlation length and define a 2×2-matrix-valued correlation kernel by
(1.12)Γ̂ϵ=(Γ̃ϵ,11Γ̃ϵ,12Γ̃ϵ,12Γ̃ϵ,22).Γ̃ɛ,ij:ℝ2→ℝ+ are symmetric, bounded, Borel-measurable functions such that the following integrability conditions are satisfied, where i∈{1,2} (the integration domain for ∫ in what follows is always ℝ2, unless it is specified to be different)
(1.13)∫Γ̃ϵ,ii2(r-p)dp=1,
and there is a finite positive constant c such that for any r,q∈ℝ2 and i,j∈{1,2}(1.14)∫(Γ̃ϵ,ij(r-p)-Γ̃ϵ,ij(q-p))2dp≤cϱ(r-q),
where ϱ(r-q):=|r-q|∧1 and ∧ denotes the minimum of two numbers. In this case, the following choice for the square-integrable martingales is made:
(1.15)mi(rN,t):=∫0t∫Γ̂ϵ(ri(s)-p)w(dp,ds).The two-particle and one-particle diffusion matrices are given by (cf. Kotelenez and Kurtz [9]) (1.16)D̃ϵ,kl(ri-rj):=∫∑m=12Γ̃ϵ,km(ri-q)Γ̃ϵ,ml(rj-q)dq∀i,j=1,…,N,Dϵ,kl:=D̃ϵ,kl(0),
where the spatial shift invariance of the two-particle diffusion matrix and the state independence of the one-particle diffusion matrix follows from the shift invariance of the kernels. Let r1(t),r2(t) be continuous ℝ2-valued processes, which describe the positions of point particles or point vortices. Levy's theorem then implies that the marginal processes
(1.17)∫0t∫Γ̂ϵ(r1(s)-p)w(dp,ds),∫0t∫Γ̂ϵ(r2(s)-p)w(dp,ds),
are ℝ2-valued Brownian motions, whereas by (1.16), the joint ℝ4-valued motion is not Brownian. If we now assume in addition that for all δ>0(1.18)lim⁡ϵ↓0sup⁡|ri-rj|>δ|D̃ϵ,kl(ri-rj)|=0k,l∈{1,2},
then the joint ℝ4-valued motion is approximately Brownian, if the separation between the point vortices is sufficiently large (cf. Kotelenez and Kurtz [9, (2.6)]. Further, in colloids it is an empirical fact that at close range Brownian particles are attracted to one another as a result of the depletion phenomenon. cf. Asakura and Oosawa [10] as well as Kotelenez et al. [11], and the references therein).Example 1.1.
ChooseΓ̃ϵ,kk(r):=-cϵ(∂/∂rk)(1/2πϵ)e-|r|2/4ϵ, where cϵ>0, k=1,2, and for the off-diagonal elements (k≠ℓ) set Γ̃ϵ,kl(r):=0.
Another class of examples can be obtained by taking, for example, the square root of a standard normal distribution with varianceϵ as Γϵ,ii, i=1,2, and 0 for the off-diagonal elements of Γ̃ϵ. The Chapman-Kolmogorov equation yields
(1.19)|D̃ϵ,kl(ri-rj)|≈O(ϵ).The method of perturbing the motion of the point vortices has the benefit that the empirical process𝒳N(t) associated with (1.5) satisfies a smoothed stochastic Navier-Stokes equation (SNSE) by Ito's formula (cf. (3.8) in Section 3). From this, Kotelenez [1] derives a priori estimates used to generalize the solution to arbitrary adapted initial conditions. However, the metric used in Kotelenez [1] does not define a metric on the signed measures and, therefore, cannot be used to prove the conservation of total vorticity in the continuum limit. Consequently, we proceed as follows.(i)
(Section2) We introduce metrics on the space of signed measures. In view of the method of correlation functions, we desire a metric that is complete. Instead, we derive a metric that satisfies a useful “partial-completeness” result which aides in the conservation of vorticity argument.(ii)
(Section3) We analyze the SNSE equation and derive the existence of solutions. Furthermore, using the results of Section 2, we show that the total vorticity is conserved.(iii)
(Section4) Based on the recent work of Kotelenez and Kurtz [9], we conjecture the macroscopic limit for the stochastic Navier-Stokes equation.


## 2. Metrics on the Space of Signed Measures

Letℝ2 be equipped with the complete, separable metric ϱ(r-q)=|r-q|∧1. We define Mf to be the set of finite, Borel measures on ℝ2 and Mf,s the set of finite, Borel signed measures on ℝ2. For a finite, signed Borel measure, μ, we let μ+,μ- denote the Hahn-Jordan decomposition of μ. We also let CL,∞(ℝ2,ℝ) be the set of Lipschitz functions from ℝ2 to ℝ which are also bounded. We endow CL,∞(ℝ2,ℝ) with the norm ∥·∥L,∞, where ∥f∥L,∞:=∥f∥∞∨∥f∥L for(2.1)‖f‖∞:=sup⁡q|f(q)|,‖f‖L:=sup⁡r,q∈R2,r≠q{|f(r)-f(q)|ϱ(r-q)},
and ∨ denotes maximum.Further, we define forμ,μ̃∈Mf,s(2.2)γf(μ,μ̃):=sup⁡‖f‖L,∞≤1|〈μ-μ̃,f〉|.By Kotelenez [8] and Dudley [12], it follows that (Mf,s,γf) is a metric space on which γf is actually a norm. Furthermore, restricting γf to Mf implies that (Mf,γf) is a complete and separable metric space (where the linear combination of point measures forms a dense set) (cf., e.g., de Acosta [13]). However, by Kotelenez and Seadler [14], (Mf,s,γf) is not a complete space. Hence, we introduce the product metric γ̂f on M̂f,s:=Mf,s×Mf,s. For μ̂=(μ1,μ2), ν̂=(ν1,ν2)∈M̂f,s, we define(2.3)γ̂f(μ̂,ν̂):=γf(μ1,ν1)+γf(μ2,ν2).It follows that(M̂f,s,γ̂f) is a metric space, restricted to M̂f:=Mf×Mf, the metric is complete and separable. We can identify a signed measure μ with the measure pair formed by the Hahn-Jordan decomposition, μ̂±=(μ+,μ-). However, under this identification, (Mf,s,γ̂f) is still not a complete space. We finally introduce the following quotient-type metric on Mf,s (cf. Kotelenez and Seadler [14]):(2.4)γf,s(μ,ν):=inf⁡η̂∈D̂+(γ̂f(μ̂±-ν̂±+η̂)∨γ̂f(μ̂±-ν̂±-η̂)),
where μ,ν∈Mf,s and
(2.5)D̂+:={ν̂=(ν1,ν2)∈M̂f:ν1=ν2}.(Mf,s,γf,s) is not a complete space, but it satisfies the following useful partial completeness result (cf. Kotelenez and Seadler [14]).Theorem 2.1.
(i)γf,s is a metric on  Mf,s.
(ii) A Cauchy sequence{μn}n≥1 for  γf,s converges if and only if there exists a subsequence  {μnk}k≥1 such that  μ̂nk±→μ̂± in  γ̂f (i.e., the limit is in Hahn-Jordan form).Proof.
(Sketch) For the forward implication, if given a Cauchy sequence for{μn}n≥1 for γf,s, it is routine to show that we can extract a subsequence {μ̂nk±}k≥1 in M̂f that is Cauchy for γ̂f. Consequently, as (M̂f,γ̂f) is complete, there is a limit μ̂∈M̂f. One can show by standard estimates that
(2.6)μ̂nk±⟶μ̂=μ̂±+ϕ̂whereϕ̂∈D̂+.
Thus, to prove the forward implication, it suffices to establish the following lemma.
Lemma 2.2.
Letψ:M̂f→Mf,s by ψ((ν1,ν2)):=ν1-ν2, then
(2.7)γf,s(ψ(μ̂nk±),ψ(μ̂±+ϕ̂))⟶0iffϕ̂=0̂.Proof.
(i) “⇐” follows from γf,s(ψ(μ̂nk±),ψ(μ̂±))≤γ̂f(μ̂nk±-μ̂±).
(ii) Supposeγ̂f(ϕ̂)>0. Then,
(2.8)γf,s(ψ(μ̂nk±),ψ(μ̂±+ϕ̂))≥inf⁡η̂∈D̂+γ̂f(μ̂nk±-μ̂±-η̂)∨inf⁡η̂∈D̂+γ̂f(μ̂±-μ̂nk±-η̂).
Let ϵ>0. We compute the second term
(2.9)inf⁡η̂∈D̂+γ̂f(μ̂±-μ̂nk±-η̂)=inf⁡η̂∈D̂+γ̂f(μ̂±+ϕ̂-μ̂nk±-ϕ̂-η̂)≥inf⁡η̂∈D̂+γ̂f(-ϕ̂-η̂)-γ̂f(μ̂±+ϕ̂-μ̂nk±)≥inf⁡η̂∈D̂+γ̂f(ϕ̂+η̂)-ϵ,
for sufficiently large n, since, by assumption, γ̂f(μ̂±+ϕ̂-μ̂nk±)→0. Hence,
(2.10)inf⁡η̂∈D̂+γ̂f(μ̂±-μ̂nk±-η̂)≥inf⁡η̂∈D̂+γ̂f(ϕ̂+η̂)=γ̂f(ϕ̂)>0.
Therefore, we also have
(2.11)γf,s(ψ(μ̂nk),ψ(μ̂+ϕ̂))≥γ̂f(ϕ̂)>0.
Part (i) of the proof of Lemma2.2 also implies the reverse implication for Theorem 2.1.Although the above metrics provide an understanding of the difficulty of completeness for the signed measures, we will also need the Wasserstein metrics on the set of signed measures. Further, for nonnegative numbersb+,b-,A, an arbitrary Borel set in ℝ2, and μ+,μ-, we write
(2.12)μ±(A)=b±iffμ+(A)=b+,μ-(A)=b-.Ifμ,μ̃∈Mf,s, we will call positive Borel measures Q± on ℝ4 joint representations of (μ+,μ̃+) [(μ-,μ̃-), resp.] if Q±(A×ℝ2)=μ±(A)μ̃±(ℝ2) and Q±(ℝ2×B)=μ̃±(B)±μ±(ℝ2) for arbitrary Borel A,B⊂ℝ2. In this paper, we will assume unless stated otherwise that μ,μ̃∈Mf,s with μ±(ℝ2)=μ̃±(ℝ2)=a±, where a+,a->0. The set of all joint representations of (μ+,μ̃+) [(μ-,μ̃-), resp.] will be denoted by C(μ+,μ̃+) [C(μ-,μ̃-), resp.]. For μ,μ̃∈Mf,s and p=1,2, set
(2.13)γW,p(μ,μ̃):=[inf⁡Q+∈C(μ+,μ̃+)∬Q+(dr,dq)ϱp(r,q)+inf⁡Q-∈C(μ-,μ̃-)∫∫Q-(dr,dq)ϱp(r,q)]1/p.γW,p is a metric on M̂f, and its restriction to Mf is a metric, but it is not a metric on Mf,s. As we will work with the various types of metrics, we note some basic inequalities relating γW,1 and γW,2. It follows from the Cauchy-Schwarz inequality and the assumption that ϱ is bounded by one that
(2.14)γW,1(μ,μ̃)≥γW,22(μ,μ̃)≥12(a+∨a-)γW,12(μ,μ̃).Furthermore, when restricted to probability measures,γW,1 and γf define the same metric by the Kantorovich-Rubinstein theorem and Kotelenez [8].


## 3. Existence and Uniqueness of the SNSE and Conservation of Vorticity

Let us return to the SODE (1.5) and note that if we define mi(rN,t) by (1.15), then (1.5) becomes
(3.1)dri(t):=∑j=1NajKδ(ri-rj)dt+2ν∫Γ̂ϵ(ri-p)w(dp,dt).For metric spaces𝕄1,𝕄2, and C(𝕄1,𝕄2) is the space of continuous functions from 𝕄1 into 𝕄2. If μ is a finite (signed) Borel measure on ℝ2, we set
(3.2)μ∫Γ̂ϵ(⋅-p)w(dp,t):=∫Γ̂ϵ(⋅-p)w(dp,t)μ(d⋅),
that is, ∫Γ̂ϵ(·-p)w(dp,t) is treated as a density with respect to μ.Ifμ itself has a density with respect to the Lebesgue measure, we will denote this density also by μ (i.e., 〈ϕ,μ〉=〈ϕ,μ〉0), and the above expression reduces to pointwise multiplication between μ and the stochastic integral.Lemma 3.1.
To eachℱ0-adapted initial condition  qN(0)∈ℝ2N, (3.1) has a unique  ℱt-adapted solution  qN(·)∈C([0,∞);ℝ2N) a.s.; which is an  ℝ2N-valued Markov process.Proof.
Compare with Kotelenez [1, 8].For the square-integrable martingales in (1.5), we denote by 〈〈mki(rN,t),mlj(rN,t)〉〉 the mutual quadratic variation process of the one-dimensional components where k,l∈{1,2},i,j∈{1,…,N} (cf. Métivier and Pellaumail [15]). For ϕ∈C02(ℝ2;ℝ), it follows from Ito's formula that the empirical process associated with (1.5), 𝒳N (defined by (1.6)) satisfies the following:(3.3)d〈XN(t),ϕ〉=〈XN(t),(Uδ,N•∇)ϕ〉dt+ν∑i=1Nai∑k,l=12∂2∂rk∂rlϕ(ri(t))d〈〈mki(rN,t),mli(rN,t)〉〉+2ν∑i=1Nai(∇ϕ(ri(t)))•dmi(rN,t),
where r=(r1,r2) and
(3.4)Uδ,N(r,t):=∫Kδ(r-q)XN(dq,t).Recalling that the marginals in (1.15) are standard ℝ2-valued Brownian motions, we obtain
(3.5)ν∑i=1Nai∑k,l=12∂2∂rk∂rlϕ(ri(t))d〈〈mki(rN,t),mli(rN,t)〉〉=ν∑i=1NaiΔϕ(ri(t))dt=ν〈(Δϕ)(⋅),XN(t)〉,
and (in what follows, we use the duality between ℝ2-valued functions and ℝ2-valued generalized functions by first applying the scalar product • and then for each component computing the duality)(3.6)2ν∑i=1Nai(∇ϕ(ri(t)))•∫Γ̂ϵ(ri(t)-p)w(dp,dt)=〈(∇ϕ(⋅),∫2νΓ̂ϵ(ri(t)-p)w(dp,dt)XN(t)〉.Therefore, (3.3) yields
(3.7)d〈XN(t),ϕ〉=〈XN(t),(Uδ,N•∇)ϕ〉dt+ν〈(Δϕ)(⋅),XN(t)〉dt+〈(∇ϕ(⋅),∫2νΓ̂ϵ(ri(t)-p)w(dp,dt)XN(t)〉.Integrating in (3.7) by parts in the sense of generalized funtions, we obtain that the empirical process is the weak solution of the following stochastic Navier-Stokes equation (SNSE) on Mf,s:
(3.8)dX(t)=[νΔX-∇•(ŨδX)]dt-2ν∇•(XΓ̂ϵ(⋅-p))w(dp,dt),Ũδ(r,t):=∫Kδ(r-q)X(dq,t).Lemma 3.2 (conservation of vorticity for discrete initial conditions).
Consider(3.9)XN±(R2,t)=XN±(R2,0)=a±a.s.Proof.
By Kotelenez and Seadler [14], it suffices to verify that the coefficients of (3.1) satisfy Lipschitz conditions. For the stochastic component, we note that if r,q∈ℝ2, we obtain that if {ϕ̃n}n∈ℕ is a complete orthonormal system in L2(ℝ2,dr) and
(3.10)ϕn:=(ϕ̃n00ϕ̃n),
then it follows that
(3.11)∫Γ̂ϵ(r-p)w(dp,t)=∑n=1∞∫Γ̂ϵ(r-p)ϕn(p)dpβn(t),
where βn(t):=∫ϕn(p)w(dp,t) are i.i.d. ℝ2-valued standard Wiener processes (cf. Kotelenez [8]). It follows from the definition of the correlation function that for i,j∈{1,2}(3.12)∑n=1∞[∫(Γ̃ϵ,ij(r-p)-Γ̃ϵ,ij(q-p))ϕ̃n(p)dp]2=∫(Γ̃ϵ,ij(r-p)-Γ̃ϵ,ij(q-p))2dp≤cϱ2(r-q).
Now, we note that the drift coefficient can be represented byF(𝒳N(t),r):Mf,s×ℝ2→ℝ2, where F(μ,r):=Kδ*μ(r), * denotes convolution, and μ∈Mf,s. For μ1,μ2∈Mf,s,r1,r2∈ℝ2, we have the following (cf. Kotelenez [8, page 81]):
(3.13)|F(μ1,r1)-F(μ2,r2)|≤cK,δ(aϱ(r1,r2)+γ̂f(μ̂1±,μ̂2±)),
where μ̂±=(μ+,μ-) and μ+,μ- is the Hahn-Jordan decomposition of μ and a:=a++a- is the total vorticity.We wish to extend the result of Lemma3.2 to arbitrary adapted initial conditions and not just discrete adapted initial conditions. To accomplish this, we must derive a priori estimates on the empirical distribution. We first introduce the following notation.If(Y,λ) is some metric space and p≥1, Lp(Ω;Y) is the metric space of Y-valued p-integrable random variables with metric (Eλp(ξ,η)1/p) for ξ,η∈Lp(Ω;Y). Set(3.14)Mf,s,d:={μ∈Mf,s:μisafinitelinearcombinationofpointmeasureson⁡R2},M̃0:=L2(Ω;Mf,s,d),M0:=L2(Ω;Mf,s),M[0,T]:=L2(Ω;C([0,T];Mf,s)).LetrN(t):=rN(t,Z1(t),𝒳N,0) and qN(t):=qN(t,Z2(t),𝒴N,0) be the solutions of the SODE (3.1) with ℱ0-measurable initial empirical distributions 𝒳N,0,𝒴N,0∈Mf,s. Denote the empirical processes associated with rN(t,Z1(t),𝒳N,0) and qN(t,Z2(t),𝒴N,0) by 𝒳N(t,Z1,𝒳N,0) and 𝒴N(t,Z2,𝒴N,0), respectively, where Z1(·),Z2(·) are also adapted Mf,s-valued processes, replacing in (3.1) the empirical processes. Set
(3.15)ϱN(rN,qN):=max⁡i=1,…,Nϱ(rNi,qNi).Theorem 3.3.
For anyT>0,
(3.16)Esup⁡0≤t≤T∧τϱN2(rN(t),qN(t))≤cT,δ,ϵ(EϱN2(rN(0),qN(0))+E∫0tγ̂f2(Z1(s),Z2(s))ds).Proof.
This follows from Theorem  2.1 in Kotelenez and Seadler [14].We can now derive the necessary a priori estimates to extend from discrete initial conditions to arbitrary adapted initial conditions.Lemma 3.4.
For anyT>0, there is a  c:=cδ,ϵ,T>0 such that for all  N∈ℕ(3.17)Esup⁡0≤t≤Tγ̂f2(X̂N(t),ŶN(t))≤cEγ̂f2(X̂N(0),ŶN(0)).Proof.
(i) Define𝒳N+(t,Zk):=∑ai≥0aiδri(t,Zk,r0i) and 𝒳N-(t,Zk)=∑ai<0aiδxi(t,Zk,r0i). Similarly, we decompose 𝒴N(t,Zk), where k=1,2. 							(3.18)Esup⁡0≤t≤Tγ̂f2(X̂N(t,Z1),ŶN(t,Z2))≤4(Esup⁡0≤t≤Tγf2(XN+(t,Z1),YN+(t,Z2))+Esup⁡0≤t≤Tγf2(XN-(t,Z1),YN-(t,Z2))).
We show the estimate forEsup0≤t≤Tγf2(𝒳N+(t,Z1),𝒴N+(t,Z2)) as a similar estimate will hold for the second term. Note that
(3.19)Esup⁡0≤t≤Tγf2(XN+(t,Z1),YN+(t,Z2))≤4(Esup⁡0≤t≤Tγf2(XN+(t,Z1),XN+(t,Z2))+Esup⁡0≤t≤Tγf2(XN+(t,Z2),YN+(t,Z2))).
Recall that𝒳N,0 and 𝒴N,0 are the initial measures of 𝒳(t) and 𝒴(t), respectively. Let Q0∈C(𝒳N+(0),𝒴N+(0)), then, by Cauchy-Schwarz and Theorem 3.3, the right hand side of the last inequality equals
(3.20)4Esup⁡0≤t≤T(sup⁡‖f‖L,∞≤1∫(f(ri(t,Z1,q))-f(ri(t,Z2,q)))XN,0(dq))2+4Esup⁡0≤t≤T(sup⁡‖f‖L,∞≤1∬(f(ri(t,Z2,q))-f(qi(t,Z2,q̃)))Q0(dq,dq̃))2≤Esup⁡0≤t≤Tc∫ϱ2(ri(t,Z1,q)-ri(t,Z2,q))XN,0(dq)+Esup⁡0≤t≤Tc∬ϱ2(ri(t,Z2,q)-qi(t,Z2,q̃))Q0(dq,dq̃).≤c̃T,a(Eγf2(XN,0+,YN,0+)+∫0TEγ̂f2(Ẑ1(s),Ẑ2(s))ds),
by Theorem 3.3, (2.14), and the Kantorovich-Rubinste in theorem.
Combining the terms for the positive and negative parts, choosingZ1≡𝒳N and Z2≡𝒴N and applying Gronwall's Inequality yields the claim.The following theorem asserts that the total vorticity of the fluid is conserved.Theorem 3.5 (conservation of vorticity in the continuum limit).
(1) The map𝒳N(0)↦𝒳ϵ(·,𝒳N(0)) from  ℳ̃0 into  ℳ[0,T] extends uniquely to a map  𝒳0↦𝒳ϵ(·,𝒳0) from  ℳ0 into  ℳ[0,T], and this extension is a weak solution of (3.8). Moreover, for any  𝒳0,𝒴0∈ℳ0, there exists a constant  c̃:=c̃T,δ,a<∞(3.21)Esup⁡0≤t≤Tγ̂f2(X̂ϵ(t,X0),X̂ϵ(t,Y0))≤c̃Eγ̂f2(X̂0,Ŷ0),
(2)(3.22)Xϵ±(R2,t,X0)=Xϵ±(R2,0,X0)=a±a.s.Proof.
(i) By (3.17), 𝒳̂N(0)↦𝒳̂ϵ(·,𝒳̂N(0)) is uniformly continuous. Hence, we can extend our solutions of (3.8) by continuity to all 𝒳0∈ℳ0 by the density of ℳ̃0 in ℳ0. We can see that (3.21) follows from (3.17).
Sinceϕ∈C02(ℝ2;ℝ), ∥Δϕ∥L.∞<∞ and ∥(∂/∂rl)ϕ∥L,∞<∞,l=1,2. So, the right-hand side of (3.7) is defined if we replace 𝒳N(t) by 𝒳ϵ(t,𝒳0).
(ii) SetfN(t):=𝒳ϵ(t,𝒳0)-𝒳(t,𝒳N(0)). Then,
(*)E(∫0t〈fN(s),∫Γ̂ϵ(⋅-p)w(dp,ds)⋅∇ϕ〉)2=∑l,k,l̃,k̃=12E∫0t∬fN(s,dr)fN(s,dq)×∫Γ̃ϵ,lk(r,p)Γ̃ϵ,l̃k̃(q,p)dp∂∂rlϕ(r)∂∂ql̃ϕ(q)ds.
Since for anyq,
(3.23)|∫[Γ̃ϵ,lk(r-p)-Γ̃ϵ,lk(r̃-p)]Γ̃ϵ,l̃k̃(q-p)dp|≤c(∫(Γ̃ϵ,lk(r-p)-Γ̃ϵ,lk(r̃-p))2)1/2≤c̃ϱ(q,q̃),
we obtain that ∫Γ̃ϵ,lk(r,p)Γ̃ϵ,l̃k̃(q,p)dp(∂/∂rl)ϕ(r) is a bounded Lipschitz function in r uniformly in q. Similarly, if the roles of r and q are reversed, the right-hand side of * tends to zero as N→∞ as a consequence of Lemma 3.4.
(iii) Becausesupp∥Kδ(·-p)∥L,∞≤c<∞, the analogue to step (ii) also holds for the deterministic integrals on the right-hand side of (3.8).
Next, we must show conservation of vorticity for all𝒳0∈ℳ0. For 𝒳0∈ℳ0, choose a sequence {𝒳N,0}N≥1⊂ℳ̃0 such that 𝒳̂N,0→𝒳̂0 in γ̂f. We first solve (3.8) on the product space by Corollary  3.3 in Kotelenez and Seadler [14]. By Lemma 3.4 and the fact that γ̂f dominates γ̂f,s, we have that 𝒳ϵ(·,𝒳N,0) is a converging Cauchy sequence for γf,s. By Theorem 2.1, we have that the limit, 𝒳ϵ(·,𝒳0) must be in Hahn-Jordan form. Since positive and negative vorticities are conserved, we have conservation of total vorticity (3.22).


## 4. The Macroscopic Limit

Set(4.1)ΛN:={rN∈R2N:∃(i,j),1≤i<j≤N,suchthatri=rj},
and denote by “⇒” weak convergence. Based on the recent work of Kotelenez and Kurtz [9], we conjecture the following.Conjecture 4.1.
For eachN∈ℕ suppose  rN(0)∉ΛN a.s. and suppose that the two-dimensional coordinates of  rN(0) are exchangeable. Let  ϕ∈C02(ℝ2,ℝ) and suppose that  〈𝒳N,0,ϕ〉⇒〈X(0),ϕ〉0, as  N→∞. Then, there is a sequence  δ(N)→0, as  N→∞ such that for any  t>0,
(4.2)〈Xϵ,δ(N)(t),ϕ〉⟹〈X(t),ϕ〉0asϵ⟶0,N⟶∞,
where X(·) is the solution of (1.1).



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*Source: 862186-2011-11-09.xml*