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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 862186, 14 pages
Conservation of Total Vorticity for a 2D Stochastic Navier Stokes Equation
Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA
Received 19 July 2011; Revised 15 September 2011; Accepted 17 September 2011
Academic Editor: Rémi Léandre
Copyright © 2011 Peter M. Kotelenez and Bradley T. Seadler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider point vortices whose positions satisfy a stochastic ordinary differential equation on 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.
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  and Marchioro and Pulvirenti  (cf. also Amirdjanova, [3, 4], Amirdjanova and Xiong ), the distribution of the vorticity satisfies the following: where is the velocity field, , is the kinematic viscosity, is the Laplacian, is the gradient, and denotes the scalar product on . If , we obtain the Navier-Stokes equation for the vorticity. If the fluid is inviscid (or ideal), that is, , we obtain the Euler equation. Note that by the incompressibility condition , we obtain where with and with denoting the transpose; denotes integration over with respect to the Lebesgue measure. Consequently, we can obtain the velocity field, , from the vorticity distribution.
Let . Let be at least twice continuously differentiable with bounded derivatives up to order 2 with and , for such that , for . Set
We may assume without loss of generality that , which implies . Thus, we have the smoothed Navier-Stokes equation (NSE)
Consider point vortices with intensities , and let be the position of the th vortex in . Abbreviate . Assume that the positions satisfy the stochastic ordinary differential equation (SODE)
The are -valued square-integrable continuous martingales (), 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 . Set where are the solutions of (1.5) and is the point measure concentrated in . We will call the empirical process associated with the SODE (1.5). Let be the standard -spaces of real-valued functions on with , where is the Lebesgue measure. Set and denote by and the standard scalar product and its associated norm on . Further, let be the extension of to a duality between distributions and smooth functions. For , we define to be the functions from into which are times continuously differentiable in all variables such that all derivatives vanish at infinity.(i)If , is a solution to the Euler equation (1.1) and the initial condition satisfies , then there is a sequence as such that for (cf. Marchioro and Pulvirenti ), (ii)Suppose and is a solution to the Navier-Stokes equation (1.1). Choose , where are i.i.d. -valued standard Wiener processes and half of the intensities equal to for and the other half to with . Let . Apply Itô's formula and compute the quadratic variation where we used the independence of . Hence, for ,
In other words, the empirical vorticity distribution becomes macroscopic, as . Choosing a sequence and assuming a suitable convergence of the initial conditions towards the initial condition in (1.1), we may expect that where is the solution to (1.1). As 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  (cf. also Chorin  and the references therein) prove a somewhat weaker result: for the case , assuming in addition to the previous conditions, as , they prove that for all , where denotes the mathematical expectation with respect to the underlying probability space . (All our stochastic processes are assumed to live on and to be -adapted (including all initial conditions in SODE’s and SPDE’s), where the filtration is assumed to be right continuous. Moreover, the processes are assumed to be -measurable, where is the Lebesgue measure on .)
In order to separate the macroscopic and continuum limits and to derive a mesoscopic vorticity distribution, Kotelenez  introduces spatial correlations of the Brownian noise as follows through correlation functionals convolved with space-time Gaussian white noise as follows: for , are i.i.d. space-time Gaussian white noise fields, (this is the multiparameter generalization of the increments of a real valued Brownian motion . cf. Walsh  and Kotelenez ) is the spatial correlation length and define a -matrix-valued correlation kernel by
are symmetric, bounded, Borel-measurable functions such that the following integrability conditions are satisfied, where (the integration domain for in what follows is always , unless it is specified to be different) and there is a finite positive constant such that for any and where and denotes the minimum of two numbers. In this case, the following choice for the square-integrable martingales is made:
The two-particle and one-particle diffusion matrices are given by (cf. Kotelenez and Kurtz ) 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 be continuous -valued processes, which describe the positions of point particles or point vortices. Levy's theorem then implies that the marginal processes are -valued Brownian motions, whereas by (1.16), the joint -valued motion is not Brownian. If we now assume in addition that for all then the joint -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  as well as Kotelenez et al. , and the references therein).
Example 1.1. Choose , where , , and for the off-diagonal elements set .
Another class of examples can be obtained by taking, for example, the square root of a standard normal distribution with variance as , , and 0 for the off-diagonal elements of . The Chapman-Kolmogorov equation yields
The method of perturbing the motion of the point vortices has the benefit that the empirical process 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  derives a priori estimates used to generalize the solution to arbitrary adapted initial conditions. However, the metric used in Kotelenez  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)(Section 2) 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)(Section 3) 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)(Section 4) Based on the recent work of Kotelenez and Kurtz , we conjecture the macroscopic limit for the stochastic Navier-Stokes equation.
2. Metrics on the Space of Signed Measures
Let be equipped with the complete, separable metric . We define to be the set of finite, Borel measures on and the set of finite, Borel signed measures on . For a finite, signed Borel measure, , we let denote the Hahn-Jordan decomposition of . We also let be the set of Lipschitz functions from to which are also bounded. We endow with the norm , where for and denotes maximum.
Further, we define for
By Kotelenez  and Dudley , it follows that is a metric space on which is actually a norm. Furthermore, restricting to implies that is a complete and separable metric space (where the linear combination of point measures forms a dense set) (cf., e.g., de Acosta ). However, by Kotelenez and Seadler , is not a complete space. Hence, we introduce the product metric on . For , , we define
It follows that is a metric space, restricted to , the metric is complete and separable. We can identify a signed measure with the measure pair formed by the Hahn-Jordan decomposition, . However, under this identification, is still not a complete space. We finally introduce the following quotient-type metric on (cf. Kotelenez and Seadler ): where and
is not a complete space, but it satisfies the following useful partial completeness result (cf. Kotelenez and Seadler ).
Theorem 2.1. (i) is a metric on .
(ii) A Cauchy sequence for converges if and only if there exists a subsequence such that in (i.e., the limit is in Hahn-Jordan form).
Proof. (Sketch) For the forward implication, if given a Cauchy sequence for for , it is routine to show that we can extract a subsequence in that is Cauchy for . Consequently, as is complete, there is a limit . One can show by standard estimates that
Thus, to prove the forward implication, it suffices to establish the following lemma.
Lemma 2.2. Let by , then Proof. (i) “” follows from .
(ii) Suppose . Then, Let . We compute the second term for sufficiently large , since, by assumption, . Hence, Therefore, we also have
Part (i) of the proof of Lemma 2.2 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 numbers ,, an arbitrary Borel set in , and , we write
If , we will call positive Borel measures on joint representations of [, resp.] if and for arbitrary Borel . In this paper, we will assume unless stated otherwise that with , where . The set of all joint representations of [, resp.] will be denoted by [, resp.]. For and , set
is a metric on , and its restriction to is a metric, but it is not a metric on . As we will work with the various types of metrics, we note some basic inequalities relating and . It follows from the Cauchy-Schwarz inequality and the assumption that is bounded by one that
Furthermore, when restricted to probability measures, and define the same metric by the Kantorovich-Rubinstein theorem and Kotelenez .
3. Existence and Uniqueness of the SNSE and Conservation of Vorticity
For metric spaces , and is the space of continuous functions from into . If is a finite (signed) Borel measure on , we set that is, is treated as a density with respect to .
If itself has a density with respect to the Lebesgue measure, we will denote this density also by (i.e., ), and the above expression reduces to pointwise multiplication between and the stochastic integral.
Lemma 3.1. To each -adapted initial condition , (3.1) has a unique -adapted solution a.s.; which is an -valued Markov process.
For the square-integrable martingales in (1.5), we denote by the mutual quadratic variation process of the one-dimensional components where (cf. Métivier and Pellaumail ). For , it follows from Ito's formula that the empirical process associated with (1.5), (defined by (1.6)) satisfies the following: where and
Recalling that the marginals in (1.15) are standard -valued Brownian motions, we obtain and (in what follows, we use the duality between -valued functions and -valued generalized functions by first applying the scalar product and then for each component computing the duality)
Therefore, (3.3) yields
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 :
Lemma 3.2 (conservation of vorticity for discrete initial conditions). Consider
Proof. By Kotelenez and Seadler , it suffices to verify that the coefficients of (3.1) satisfy Lipschitz conditions. For the stochastic component, we note that if , we obtain that if is a complete orthonormal system in and
then it follows that
where are i.i.d. -valued standard Wiener processes (cf. Kotelenez ). It follows from the definition of the correlation function that for
Now, we note that the drift coefficient can be represented by , where , denotes convolution, and . For , we have the following (cf. Kotelenez [8, page 81]): where and is the Hahn-Jordan decomposition of and is the total vorticity.
We wish to extend the result of Lemma 3.2 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 is some metric space and , is the metric space of -valued -integrable random variables with metric for . Set
Let and be the solutions of the SODE (3.1) with -measurable initial empirical distributions . Denote the empirical processes associated with and by and , respectively, where are also adapted -valued processes, replacing in (3.1) the empirical processes. Set
Theorem 3.3. For any ,
Proof. This follows from Theorem 2.1 in Kotelenez and Seadler .
We can now derive the necessary a priori estimates to extend from discrete initial conditions to arbitrary adapted initial conditions.
Lemma 3.4. For any , there is a such that for all
Proof. (i) Define and . Similarly, we decompose , where .
We show the estimate for as a similar estimate will hold for the second term. Note that
Recall that and are the initial measures of and , respectively. Let , then, by Cauchy-Schwarz and Theorem 3.3, the right hand side of the last inequality equals by Theorem 3.3, (2.14), and the Kantorovich-Rubinste in theorem.
Combining the terms for the positive and negative parts, choosing and 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 map from into extends uniquely to a map from into , and this extension is a weak solution of (3.8). Moreover, for any , there exists a constant
Proof. (i) By (3.17), is uniformly continuous. Hence, we can extend our solutions of (3.8) by continuity to all by the density of in . We can see that (3.21) follows from (3.17).
Since , and . So, the right-hand side of (3.7) is defined if we replace by .
(ii) Set . Then,
Since for any , we obtain that is a bounded Lipschitz function in uniformly in . Similarly, if the roles of and are reversed, the right-hand side of tends to zero as as a consequence of Lemma 3.4.
(iii) Because , 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 all . For , choose a sequence such that in . We first solve (3.8) on the product space by Corollary 3.3 in Kotelenez and Seadler . By Lemma 3.4 and the fact that dominates , we have that is a converging Cauchy sequence for . By Theorem 2.1, we have that the limit, 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 and denote by “” weak convergence. Based on the recent work of Kotelenez and Kurtz , we conjecture the following.
Conjecture 4.1. For each suppose a.s. and suppose that the two-dimensional coordinates of are exchangeable. Let and suppose that , as . Then, there is a sequence , as such that for any , where is the solution of (1.1).
The final version of the paper has profited from careful refereeing. P. M. Kotelenez and B. T. Seadler are supported by NSA Grant no. H98230-10-1-0206.
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