A Reflection Type Problem for the Stochastic 2-D Navier-Stokes Equations with Periodic Conditions

We prove the existence of a solution for the Kolmogorov equation associated with a reflection problem for 2-D stochastic Navier-Stokes equations with periodic spatial conditions and the corresponding stream flow in a closed ball of a Sobolev space of the torus $T^2$.


Introduction
We consider here the 2-D stochastic Navier-Stokes equation for an incompressible non-viscous fluid d X − ν∆X d t + (X · ∇)X d t = ∇p d t + dW t ∇ · X = 0 (1) This equation is considered on a 2-D torus, that we identify with the square T 2 = [0, 2π] × [0, 2π] and with periodic boundary conditions. Here ν is the viscosity of the fluid, X is the velocity field, p is the pressure and W is a cylindrical Wiener process.
If we denote by φ : T 2 → R the corresponding stream function, that is where ∇ ⊥ = (−D 2 , D 1 ), curl X = D 2 X 1 − D 1 X 2 , X = (X 1 , X 2 ) we may rewrite (1) in terms of the stream function φ (see [1], [2]) and formulate for (1) the corresponding reflection problem on the set where H 1−α is the Sobolev space of order 1 − α with α > 3 2 , with respect to the natural Gibbs measure µ given by enstrophy (see Section 2 below.) More precisely, we shall prove that the Kolmogorov equation associated with (1), (2) and (4) has at least one solution ϕ : T 2 → R. In terms of coordinates u j = 1 2π T 2 e i j·ξ φ(ξ) dξ this equation has the form where L is the Kolmogorov operator defined on a space C 2 b of cylindrical smooth functions. (The function B k is defined in (10).) The main result of this work, Theorem 1 below, amounts to saying that the Neumann problem (5) has at least one weak solution ϕ, but the uniqueness of this solution remains open. It should be said that the uniqueness is still an open problem in the case K = H 1−α and it is equivalent in the later case with the unique extension of operator L from C 2 b to an m-dissipative operator in L 2 (µ) see [3]. We mention, however, that L is essentially m-dissipative in L 1 (µ) when the viscosity ν is sufficiently large (Stannat [11]). It should mention also that in this way the study of stochastic process X = X t reduces to a linear infinite dimensional equation in the space H 1−α associated to the operator L.
There is a large number of works devoted to infinite dimensional stochastic reflection problems but most of them are, except a few notable works, concerned with Wiener processes W with finite covariance. So the existence theory for (13) is still open. Here following the way developped in [5], [6], we will treat instead of (1) its associated Kolmogorov equation which as noted in Introduction will lead to an infinite dimensional Neumann problem on the convex K. (The Kolmogorov equation [6] in the special case K = H 1−α was previously studied by Flandoli and Gozzi [9].) Previous results on infinite dimensional reflection problems, starting from [10] are essentially concerned with reversible systems. We believe that the present paper is the first attempt to study non symmetric infinite dimensional Kolmogorov operators with Neumann boundary conditions.

The functional setting
Consider the Sobolev space of order p ∈ R defined by u j e i j·ξ : The space H p is a complex Hilbert space with the scalar product Consider the Gibbs measure µ = µ ν given by the enstrophy, that is We recall (see [1], [3]) that for α > 0 we have and so the probability measure µ is supported by Given a closed convex subset K ⊂ H δ with smooth boundary we denote by H 1,2 There is a standard way (see [1], [2]) to reduce equation (1) to a differential equation in H 1−α we briefly present below. Namely applying the curl operator into (3) we get for ψ = curl X the equation and take W to be the cylindrical Wiener process where {W j } j∈Z 2 0 are independent Brownian motions in a probability space {Ω, , P, { t } t≥0 }. We note that By (7) we have Then (1) reduces to Here we have used the notation and It turns out that if p < −1 then the vector field B = {B j } j∈Z 2 0 is L q -integrable in the norm | · | p with respect to the Gibbs measure µ for all q ≥ 1. One also has (see [7]) Moreover, the measure µ is infinitesimally invariant for B (see [1], [7].) Equation (9) can be written in H 1−α as where We recall (see [1]) that A is a Hilbert-Schmidt operator on H 2 and |Au| 2 = |u| 1−α . Now, we associate with (12) the stochastic variational inequality where Rv = {k −2α v k } k∈Z 2 0 , K is a smooth closed and convex subset of H = H 1−α and ∂ I K : K → 2 H is the normal cone to K. Formally (13) can be written as where λ(t) ≥ 0 and n K (u) is the unit exterior normal to ∂ K.
Coming back to equation (1) and taking into account (2) the variational inequality (13) can be rewritten in terms of the velocity field X under the form where N (X ) is the normal cone to the closed convex set of {X ∈ L 2 (0, 2π) 2 ; ∇ · X = 0, X (0) = X (2π)} defined by, This is the reflection problem to the boundary of on the oblique normal direction N (x). In the special case of K given by (4) its meaning is that the stream value φ of the fluid is constrained to the set φ 1−α ≤ and when φ reaches the boundary ∂ K in the dynamic of fluid arises a convective acceleration oriented toward interior of K along an oblique direction. Indeed we have by definition of the normal cone N (X ), Recalling that by (2), (7), On the other hand, the normal cone N K (u) to K in H 1−α is given by and taking into account (13) and definition of this yields (14) as claimed.
We recall (see e.g., [1], [2], [3]) that the measure µ is invariant for operator L. As noticed earlier the essential m-dissipativity of L in the space L 2 (µ) is still an open problem.
Our aim here is to study the Neumann problem considered in some generalized sense to be precised below. and It is readily seen by (11) that (14) makes sense for all ψ ∈ C 2 b . Theorem 1 below is the main result. Theorem 1. Assume that α > 3 2 and then for each real valued f ∈ L 2 (K, µ) problem (5) has at least one weak solution ϕ ∈ H 1,2 −1 (K, µ) and the following estimates hold In (17) as well as in (16),(19) by D j ϕ we mean of course the distributional derivative D j of function ϕ which belongs to L 2 (µ).

Remark 1.
If ϕ is a smooth solution to elliptic problem (15) then it is easily seen via integration by parts that ϕ is also weak solution in the sense of Definition 1.

Proof of Theorem 1
To prove Theorem 1 we consider the approximating equation where L is given by (6) and (Here Π K is the projection on K.) We introduce also the measure and note that .
It should be mentioned that equation (21) in spite of its apparent simplicity is still unsolvable for all f ∈ L 2 (µ) and the reason is that as mentioned earlier we dont know whether the operator L is essentially m-dissipative. In order to circumvent this we shall define just a weak solution concept for (21) and prove the existence of such a solution.
and λ ϕ ψ dµ + for all real valued cylindrical functions ψ ∈ C 2 b . We note that Definition 2 is in the spirit of Definition 1 and that if ϕ is a smooth solution to (21) then we see by (21) via integration by parts that ϕ satisfies also (23). We note that because by enstrophy invariance we have (see e.g., [1], [2]) and has at least one weak solution ϕ which satisfies the estimates Proof. We shall use the Galerkin scheme for equation (21). Namely, we introduce the finite dimensional approximation B n k of B k (see [1]) and I n = {m ∈ Z 2 0 : 0 < |m| ≤ n}. Then B n = {B n k (u)} k∈I n , like B, has the properties (25) and the operator defined on the space of smooth functions ϕ = ϕ(u 1 , u 2 , . . . , u n ) has the invariant measure µ n = | j|≤n µ j . Then we consider the equation where (β n k ) = 1 1 − |u| H n u k and H n = {u j : j ∈ I n }. By standard existence theory for Kolmogorov equations associated with stochastic differential equations, the equation (29) has a unique solution ϕ n which is precisely the function and X n = {u n j : j ∈ I n } is the solution to stochastic equation (see [3]) We may assume therefore that ϕ is smooth and so multiplying (29) by ϕ n and integrating with respect to the measure On the other hand, taking into account that by (25) and letting n tend to infinity into the weak form of (29), that is (34) and recalling that {B n k } is strongly convergent to {B k } in L 2 (µ) (see Lemma 1.3.2 in [7]) we infer that ϕ is solution to (21) as claimed. Estimates (27), (28) follows by (31), (32), (33). This complete the proof of Proposition 1.

Remark 2.
Letting tend to zero into (29) it follows via integration by parts formula by a similar argument as in [5] that ϕ n → ϕ n , D j ϕ n → D j ϕ n in L 2 (H n , µ) where ϕ n is the solution to Neumann boundary value problem λ ϕ n − ν ∆ϕ n + B n (u n ) · Dϕ n = f in K n ∂ ϕ n ∂ n K n = 0 on ∂ K n .
where K n = K ∪ H n . Moreover, by elliptic regularity, ϕ n ∈ H 2 ( K n ). On the other hand, it is clear by the above energetic estimates in H 1−α that for n → ∞ {ϕ n } is convergent to a weak solution ϕ to (15). However, this solution is not necessarily that given by approximating process ϕ .