Stochastic Navier-Stokes Equations in Unbounded Channel Domains

In this paper we prove the existence and uniqueness of path-wise strong solution to stochastic viscous flow in unbounded channels with multiple outlets using local monotonicity arguments. We devise a construction for solvability using a stochastic basic vector field.


Introduction
This paper concerns with stochastic fluid dynamics in unbounded channel domains with noncompact boundaries generalizing the deterministic results in Sritharan [62]. Mathematical theory of viscous incompressible flow through unbounded channel has many applications such as hydraulics in water resources, hydraulic machinery, oil transport networks, flow in engines etc. Well-posedness theorem is an essential step for applications in optimal control theory (Sritharan [64]), convergence of numerical algorithms and nonlinear filtering (Sritharan [63], Fernando and Sritharan [23]). Solvability theory of generalized solutions to Navier-Stokes equations was pioneered by Leray [41], Hopf [31] and Ladyzhenskaya [35], [36]. Steady state flow through channels of various kinds has been studied by a number of authors including Amick [4], [5], Amick and Franenkel [6], Ladyzhenskaya and Solonnikov [37], [38], [39]. In [4], Amick discussed the steady flow of viscous incompressible fluid in channels and pipes in two and three dimensions which are cylindrical outside some compact set. The paper by Heywood [28] highlighted the question of uniqueness of the solution of the Navier-Stokes equations for certain unbounded domains modeling channels, tubes, or conduits of some kind and the importance of prescribing flux or the overall pressure difference. In [6], Amick and Fraenkel studied steady state solutions of the Navier-Stokes equations in various types of two dimensional channel domains. In [29], Heywood constructed classical solutions of the Navier-Stokes equations for both stationary and non-stationary boundary value problems in arbitrary three-dimensional domains with smooth boundaries. The time dependent flow through the threedimensional channels with outlets diverging at infinity has been studied by Ladyzhenskaya and Solonnikov [37] and, Solonnikov [59]. The paper by Solonnikov [59] presents solvability of boundary value problems for the Stokes and Navier-Stokes equations in noncompact domains with several oulets to infinity. Babin [7] considered the Navier-Stokes system in an unbounded planar channel-like domain and proved that when the external force decays at infinity, the semigroup generated by the system has a global attractor and its Hausdroff dimension is finite using weighted Sobolev estimates. The paper by Sritharan [62] addressed the following two important cases which were not considered in the earlier works: (i) time-dependent flow through two and three dimensional channels with finite cross section; (ii) time-dependent flow through two-dimensional channels with outlets diverge at infinity and provided a unique solvability theorem for the two-dimensional case of the problem type (i). Solvability of stochastic Navier-Stokes equations in unbounded channel-like domains with non-zero flux condition have remained as an open problem in both two and three-dimensions. To the best of the authors knowledge, this work appears to be the first systematic treatment of stochastic two-dimensional Navier-Stokes equations in such domains. In this paper we consider a stochastic version of the problem of type (i) in multichannel domains in two dimensions and prove a unique solvability theorem with a possible future extension to three dimensions (up to a stopping time determined by the size of the flux and the Reynolds number). The problem of type (ii) may possibly be resolved by suitably choosing a conformal mapping Figure 1. multi-channel domain (see Amick and Franenkel [6] for similar ideas in the case of steady flows) to straighten the diverging outlets. Let us consider the unbounded multi-channel domains, with several outlets as shown in the figure (see Figure 1). Let the outlets of the multi-channel domain be named as O 1 , O 2 , · · · , O N and outside a compact region let the outlets be of constant widths d 1 , d 2 , · · · , d N . Our first step is to construct a basic vector field through each of these outlets with the stochastic flux F i (t, ω) such that N i=1 F i (t, ω) = 0, P -a. s. The methodology of proof can be understood by considering a channel with two outlets having a unit width connected in a smooth way Θ = O 1 ∪ O 2 ∪ Θ 0 (see Figure 2).

Figure 2. channel with two outlets having unit width
Let us now discuss the problems of type (i) and examine the difficulties that arise in proving solvability. The time-dependent Navier-Stokes problem is usually treated using the method of Hopf [31] in the deterministic setting which relies on L 2 -energy estimates. The traditional methods of solvability fail in the absence of an energy inequality. For channels of finite cross section, as pointed out in Sritharan [62], in order for the net flux to be nontrivial, the velocity field should not decay to zero at infinity (upstream and downstream) and hence such velocity fields would then have infinite energy.
Below we give a heuristic argument regarding the infinite energy of such velocity fields. We also point out that in the absence of a rigorous decay theory, only a heuristic argument could be made in this regard.
Let the velocity field be stochastic and modeled on a complete probability space (Ω, F, F t , P). If the stochastic net flux of the velocity field u is F (t, ω), ω ∈ Ω, then across any cross section Γ, we have where n is the normal to the curve Γ and dS is the length element.
In this case, the velocity |u(x, t, ω)| 0 as |x| → ∞, (where x = (x, y) with y is of constant width) P-a. s. To see this let us take the 2-D case with the outlet O 2 = {(x, y) ∈ (0, ∞) × (0, 1)}. The flux across any cross section is same throughout the channel, due to divergence free condition. That is, if |u(x, t, ω)| → 0 as x → ∞, then the flux at the far field is zero. Hence the flux across any cross section is zero throughout the channel giving the net flux is zero. Thus for the flux to be non-zero, we need the condition that |u(x, t, ω)| 0 as x → ∞.
The plan of the paper is as follows. In section 2, the main result of this paper and the functional setting have been given. A divergence free vector field of infinite energy carrying a nontrivial net flux through the channel is constructed in section 3 using the solution of the heat equation. In section 4, we characterize the properties of the linear and bilinear operators that are associated with the Navier-Stokes problem. A perturbed vector field is constructed in section 5 using a suitable transformation involving the constructed basic vector field. A-priori estimates for the solutions of the perturbed vector field are obtained in section 6. In section 7, we prove the local monotonicity condition for the sum of the Stokes and the inertia operators as well as the existence and uniqueness of strong solutions to the perturbed vector field by exploiting this local monotonicity condition. In section 8, we mathematically characterize the perturbation pressure field using a generalization of the de Rham's Theorem to processes. section 9, completes the proof of the main result.

Basic Definitions and the Main Theorem
In this section, following Sritharan [62], we define the class of channel domains that will be analyzed. Definition 2.1. (Admissible channel domain) A simply connected open set Θ ⊂ R 2 with C ∞ boundary ∂Θ consisting of two disconnected components ∂Θ 1 and ∂Θ 2 is called an admissible channel domain (see Figure 3), if it is the union of three disjoint sets Θ 0 ∪ O 1 ∪ O 2 defined in the following way. Let O 1 and O 2 be two semi-infinite strips of width d 1 and d 2 respectively. These two straight channels are smoothly (not necessarily coaxially) joined by a bounded domain Θ 0 such that ∂Θ 1 ∪ ∂Θ 2 = ∂Θ ∈ C ∞ . Now let us consider the problem of accelerating a viscous incompressible fluid from rest to a given stochastic flux rate through an admissible channel domain. Let (Ω, F, F t , P) be a complete probability space. The mathematical problem is to find the velocity field u and pressure field p such that the momentum equation the incompressibility condition the non-slip boundary condition on the channel walls (2.5) and the flux condition are satisfied. The properties of the stochastic flux will be discussed in the later sections. Here ν > 0 is the coefficient of kinematic viscosity and Γ is any cross-sectional curve cutting the channel. In this formulation the stochasticity of fluid flow is due to an external random forcing and the random flux. Also we will assume that the external random forcingĠ (x, t, ω) and the random flux F (t, ω) are mutually independent processes. Further details about the noise have been provided in the subsequent sections. Now let us state the main result of this paper.
Theorem 2.2. Suppose that the flux rate satisfies the moment bound: for some prescribed T . Then for each such F (·, ω) there exits a unique strong solution u(x, t, ω) with the following estimates: for some divergence free vector field w(x, t, ω) that vanishes on the boundary ∂Θ and carries the prescribed flux We prove the above theorem in the subsequent sections. The following functional frame work is used in this paper.
C ∞ 0 (Θ) = the space of all infinitely differentiable vector fields with compact support in Θ, Let us denote the norm in H by | · | and the norm in V by · . If we identify H with its dual H = L (H; R) using the Riesz representation theorem, we obtain the continuous and dense embedding Also let us denote the duality pairing between V and V by (·, ·). Note, however, that (unlike in bounded domains) the embedding V ⊂ H is not compact since Θ is unbounded. Poincaré lemma holds for admissible channel domains (since they have finite cross section): φ L 2 (Θ) ≤ C ∇φ L 2 (Θ) , ∀φ ∈ H 1 0 (Θ) and hence, in V the norm of H 1 (Θ) is equivalent to that obtained by the Dirichlet integral ∇φ L 2 (Θ) .

Construction of the Basic Vector Field
Following the ideas from Sritharan [62], we will construct a divergence free basic vector field w(x, t, ω) in Θ which vanishes on ∂Θ and carries the prescribed random flux F (t, ω) through the channel. Note that this "constructed" vector field need not satisfy the Navier-Stokes equations in Θ although it does in O 1 and O 2 due to the nature of the construction used. The method can be described as follows. Using the solution of the one-dimensional heat equation with random flux F (t, ω), a vector field is constructed in each of the straight channel outlets O 1 and O 2 . We then smoothly join these two vector fields by constructing a smooth extension in Θ 0 . In O 1 and O 2 the vector field will have only one component, namely in the direction of the axes of the outlets.
In Θ 0 , however, in general both components of w(x, t, ω) will be nonzero. Let us now consider one of the outlets, say O 2 and assume for simplicity that the width of the channel is unity (see Figure 4). Let us define the outlet O 2 = {(x, y) ∈ (0, ∞) × (0, 1)} . Let us seek a divergence-free vector field in the form w(x, y, t, ω) = (w 1 (y, t, ω), 0) and a scalar field P (x, t, ω) in O 2 such that Here the function f (t, ω) needs to be determined from the prescribed flux F (t, ω). To resolve this problem we first write down the solution of the system (3.1)- (3.3) in terms of f (t, ω) and then use the condition (3.4) to evaluate f (t, ω) in terms of F (t, ω). The solution of (3.1)-(3.3) can be obtained by the method of separation of variables (see Cannon [17]). The existence and uniqueness of solution of the boundary value problem for the heat equation with stochastic boundary conditions has been proved in Cahlon [16].
From (3.12), using the uniform convergence of the above series in time t > 0 for a fixed y ∈ (0, 1), we have ∂ ∂t (w 1 (y, t, ω)) = −4νπ Next, let us calculate ∂ 2 ∂y 2 w 1 (y, t, ω). We have (3.14) Since K(·, ·) is uniformly convergent in y and ∂K ∂t (·, ·) is also uniformly convergent in y for a fixed t > 0, by a simple calculation, we have The above series is uniformly convergent in time t > 0 for a fixed y ∈ (0, 1) and hence from (3.14), we get The above integral is well defined and from equations (3.13) and (3.15), we have part (iv) of the theorem. Now we prove part (i) of the theorem. Since K is continuous in time t, for any given ε > 0, there exists a η > 0 such that 2 and by using Young's inequality and continuity of K(y, t) in t, we have since ε is arbitrary and as h → 0, t → s and K(y, t + h − s) → K(y, 0) = 1 for all y ∈ (0, 1). Similarly, E(w 1 (y + h, t, ω) − w 1 (y, t, ω)) 2 → 0 as h → 0.
Let us now prove part (ii) of the theorem. Since for a fixed y ∈ (0, 1), K(y, t) is uniformly convergent in time t > 0 and its derivative ∂K ∂t exists and also is uniformly convergent for t > 0, we have for a given ε > 0, there exists an η > 0 such that To prove (3.7), let us use the differentiability of K(y, t) in time t. For |h| < η, we have Note that in the time interval [t, t+h] as h → 0, s → t and hence the function Thus by the Lebesgue's differentiation theorem (Theorem 6, Appendix E.4 of Evans [21]), the last term of the right hand side of the above inequality goes to 0 as h → 0. Finally, since ε > 0 is arbitrary, we have the desired result (3.7). Similarly one can prove that there exists a stochastic function w 1yy (y, t, ω) ∈ L 2 (Ω;

16)
where h(t) = 8 Proof. From (3.4), we have where Since h is differentiable in time t > 0, for any given ε > 0, there exists an η > 0 such that For proving the existence of ∂ ∂t F (t, ω), let us use f ∈ L 2 (Ω; C[0, T ]), the differentiability of h and choose |h| < η to get Note that in the time interval [t, t+h] as h → 0, s → t and hence the function Thus by the Lebesgue's differentiation theorem (Theorem 6, Appendix E.4 of Evans [21]), the last term of the right hand side of the above inequality goes to 0 as h → 0. The arbitrariness of ε gives the required result.
Let us now obtain a stochastic version of the Theorem 2 (section 3) from Sritharan [62]. and Proof. We only need to verify the regularity. Let us denote by A the Friedrich's extension of the operator −∂ 2 y to H 1 0 (0, 1)∩H 2 (0, 1). Then from (3.9), by using the uniform convergence of the series solution w 1 (y, t, ω), Fubini's theorem and Young's inequality for convolutions, we have (3.33) Stochastic Navier-Stokes Equations in Unbounded Channel Domains 17 Thus noting that ∞ n=0 (2n + 1) −2+2ε < ∞ for ε < 1 2 , we obtain Then one can deduce that, A 1+ε/2 w 1 ∈ L 2 (Ω; L 2 (0, T ; L 2 (0, 1))) which implies w 1 ∈ L 2 (Ω; L 2 (0, T ; H 2+ε (0, 1))). Since A is the Friedrich's extension of the operator −∂ 2 y to H 1 0 (0, 1) ∩ H 2 (0, 1) and from (3.34), we get Let us now describe a method to extend the constructed flux F (·, ·) carried in O i into the domain Θ 0 in a smooth manner. LetΘ 0 be an open subset of Θ with compact closure such that Θ 0 ⊂Θ 0 Θ. Moreover, ∂Θ 0 is of class C ∞ (see Figure 5). Proposition 3.5. There exists a vector fieldŵ : for almost all ω ∈ Ω, in the sense of trace. Here w Proposition 3.5 can be deduced from the well-known method of extending the boundary data (with zero net flux) into the domain as a divergencefree vector field (Ladyzhenskaya [36]). Also we note that the regularity of the boundary data corresponds to that obtained in the solution of the heat equation and hence we have Let us useŵ(x, t, ω) inΘ 0 and w Sritharan [62] for more details about the construction).
Let ψ (i) (y, t, ω) be the stream function corresponding to the vector field (w Also due to the flux condition, we havê Let us construct a C ∞ function λ(·) : R → (0, 1) by mollifying a step function such that In this way we obtain a stream function which takesψ(x, y, t, ω) in Θ 0 and smoothly become ψ (i) (y, t, ω) in O i . On the lower wall ∂Θ 1 , we have ψ(x, t, ω) =ψ(x, t, ω) = ψ (i) (y, t, ω) = 0 and on the upper wall ∂Θ 2 , we have Hence we obtain the desired divergence free basic vector field as A similar extension can be constructed for the scalar field P (x, t, ω) so that in Θ 0 , P (x, t, ω) ∈ L 2 (Ω; L 2 (0, T ; H 1 (Θ 0 ))) and becomes −xf (t, ω) smoothly in O i .

Hence the inequality (3.41) becomes
The series ∞ n=0 (2n+1) 4ς−6 is convergent for 0 ≤ ς < 5 4 . Hence for 0 ≤ ς < 5 4 and for ρ < 1, we get By using the estimates (3.37) and (3.38), by taking ς = 0, 1 2 in (3.42), one can easily see that in O i . Since Θ 0 is bounded, by using property (iii) of the basic vector field and we extended the constructed flux F (t, ω) = Γ w · ndS carried in O i into the domain Θ 0 in a smooth manner, it can be easily shown that E β 2 20 (ω) T ≤ C F 2 L 2 (Ω;H 1 (0,T )) and E  For i = 1, 2, we can always get an upper bound for β 2i (t, ω) for all t ∈ [0, T ] and almost all ω ∈ Ω. By using (3.9) and Young's inequality for convolution of two functions, we obtain

The Linear and Multilinear Operators
In this section we define the Stokes operator, the inertia operator and the other operators relevant to our analysis. Most of the results obtained in this section has been taken from [62] given here for completeness.

The Stokes Operator
Let us first define the Stokes operator for the two-dimensional admissible channel domain and analyze its properties. Let us denote by a(·, ·) the symmetric bilinear form Let us now define the Stokes operator A and its domain D(A) in the following way. Given u ∈ V, if there exists an element g ∈ H such that then we say u ∈ D(A) and Au = g. By Poincaré inequality u L 2 (Θ) ≤ C ∇u L 2 (Θ) holds for the admissible channel domains, we obtain the coerciveness (V-elliptic) property as, a(u, u) = ∇u 2 L 2 (Θ) ≥ C u 2 , ∀ u ∈ V. Since the form a(·, ·) is symmetric, continuous, and positive definite, we have the following standard results obtained in [66].
with D(Ã) ⊂ V dense. Moreover, it is possible to extend this operator as an isomorphic onto mapÂ ∈ L (V, V ) such that The operatorsÃ andÂ are different Friedrich's extensions of the classical Stokes operator. In the remainder of the paper, both of these operators will be denoted by A. It follows from a general theorem of Lions [42] that V = D(A 1/2 ). Let the orthogonal Helmhotz-Hodge projection P H : L 2 (Θ, R 2 ) → H and Au = −P H ∆u ∀ u ∈ D(A).
Let us now state a regularity theorem. This give us an explicit representation of D(A).

The Bilinear Operator (Inertia Term)
Let us now define the trilinear form and the bilinear operator and their properties. The trilinear form is given by, It is well known that trilinear form b(·, ·, ·) : The following lemma is standard. Moreover, and Now we will provide a general setting for estimating the terms b(w, ·, ·) and b(·, w, ·), where w is the basic vector field constructed in section 3. Moreover, Proof. For proof see Lemma 6, section 4 of [62].

Lemma 4.7. There exists a continuous linear operators
Moreover, Proof. For proof see Lemma 7, section 4 of [62].
By taking the Helmhotz-Hodge projection P H , we get
Our problem is to find From the construction of the basic vector field it is clear that the supp{f w } Θ. Also one can notice that f w ≡ 0 for O i \Θ 0 . Moreover, form the estimate we have f w ∈ L 2 Ω; L 2 (0, T ; V ) . A-priori estimates for the solution can be obtained by assuming smoothness of v and sufficient decay at infinity. Let us now take the Helmhotz-Hodge projection on the equation (5.1) and using the fact that (∇q, v) = 0, ∀ v ∈ H, one can reduce the equation See Appendix of Mikulevicius and Rozovskii [49] for more details about the properties of the projection operator. Let the noise process be represented as a series dG k = k g k (x, t)dW k (t, ω), where g = (g 1 , g 2 , · · · ) is and 2 -valued function and W k are mutually independent standard one dimensional Brownian motions. The stochastic term gdW is thus an H -valued Wiener process with a trace-class covariance operator denoted by g * g = g * g(t) given by This means that the mapping is a continuous linear functional on H with probability 1 and the noise is the formal time-derivative of the process P H G (t) = t 0 g(t)dW (t). A multiplicative noise of the form g(u(x, t))dW (t), where g(u(x, t)) is a continuous operator from V into L 2 (0, T ; 2 (H)), can also be considered as the random forcing. HereĠ (·) is formally written and it is the time-derivative of · 0 g(s)dW (s). We have · 0 g(s)dW (s) ∈ L 2 (Ω; C([0, T ]; H)) implies its time-derivativeĠ (·) satisfiesĠ (·) ∈ L 2 (Ω; W −1,∞ (0, T ; H)), since ∂ t is linear continuous from C([0, T ]; H) into W −1,∞ (0, T ; H).

A-Priori Estimates for the Perturbed Vector Field
Let H n = span{e 1 , e 2 , · · · , e n }, where {e j } is any fixed orthonormal basis in H with e j ∈ D(A). Let P n denote the orthogonal projection of H into H n . Define v n = P n v. Let us define v n as the solution of the following stochastic differential equation in the variational form such that for each v ∈ H n , It can be shown that for all n ≥ 1, there exists an adapted process v n ∈ C([0, T ]; H n ) a. s. such that v n satisfies (6.1). For proof see corollary 2.1 of Albeverio, Brzeźniak, Wu [1], page 128 of Brzeźniak, Hausenblas, Zhu [13], Lemma 3.1 of Capiński and Peszat [15], Proposition 3.2 of Manna, Menaldi and Sritharan [45], Theorem 3.1.1 of Prévôt and Röckner [54]. Theorem 6.1. Let Θ be an admissible channel domain. Under the above mathematical setting, let v n (t) be an adapted process in C([0, T ]; H n ) which solves the stochastic ODE (6.1). Then we have the following a-priori estimates: For Proof. Let us consider the function e −δt |x| 2 and apply Itô's lemma (see Theorem 1, page 155 Gyöngy and Krylov [26], Metivier [48]) to the process v n (t), for any δ > 0 By applying the properties of the trilinear form, (∆v n , v n ) = − v n 2 and 2|(P H f w , v n )| ≤ δ|v n | 2 + 1 δ |f w | 2 in (6.4), we get Let G be the σ-algebra generated by v n in the probability space (Ω; F, F t , P).
Since we have assumed that the external random forcing and the random flux are mutually independent processes, the constructed vector field w is independent of G. Let us define the stopping time τ N by Note that τ N is adapted to G and w is independent of G. Now let us integrate the inequality (6.5) in t from 0 to t ∧ τ N to get |v n (t ∧ τ N )| 2 e −δt + 2ν By using the above estimate and using Young's inequality, we have for all Hence (6.6) becomes Let us take conditional expectation on both sides of (6.7) with respect to the σ-algebra G to get In (6.8), let us use the G-measurability of v n and independence of w with respect to G to get Tr(g * g)(s)e −δs ds Let us take expectation on both sides of the inequality (6.9) to obtain Note that the last term in the inequality (6.10) is a martingale with zero expectation, one obtains Tr(g * g)(s)e −δs ds.
Hence we get the required result.
Theorem 6.2. Let Θ be an admissible channel domain. Under the above mathematical setting, let v n (t) be an adapted process in C([0, T ]; H n ) which solves the stochastic ODE (6.1). Then we have the following a-priori estimates: Proof. Apply Itô's lemma (see Theorem 1, page 155 Gyöngy and Krylov [26], Metivier [48]) to the function |x| 2 and to the process v n (t). Proceeding similarly as in Theorem 6.1 and using the inequality , we get the required result.

Existence and Uniqueness of Strong Solutions for the Perturbed Vector Field
Monotonicity arguments were first used by Krylov and Rozovskii [34] to prove the existence and uniqueness of the strong solutions for a wide class of stochastic evolution equations (under certain assumptions on the drift and diffusion coefficients), which in fact is the refinement of the previous results by Pardoux [50,51] (also see Metivier [48]) and also the generalization of the results by Bensoussan and Temam [10]. Menaldi and Sritharan [47] further developed this theory for the case when the sum of the linear and nonlinear operators are locally monotone.
In this section, we will prove the local monotonicity of the sum of the Stokes operator and the inertia term of the perturbed vector field, and following Menaldi and Sritharan [47] we use a generalization of Minty-Browder technique to prove the existence and uniqueness result that avoids compactness method and hence applicable for unbounded domains directly.
We will start this section with a lemma known as the Gagliardo-Nirenberg inequality (section 1.2, Theorem 2.1 in DiBenedetto [20]).
where n is the dimension of the space. For every fixed number p, s ≥ 1, there exists a constant C depending only upon n, p and s such that , where α ∈ [0, 1], p, q ≥ 1 and s satisfies the following relation: The following lemma is a special case of Lemma 7.1 in two dimension which will be useful in our context.
where Θ is the admissible channel domain, we have the following estimate: The general proof in R 2 of this lemma can be obtained from Ladyzhenskaya [36] (Chapter 1, Lemma 1).
Remark 7.3. The above lemma suggests that V ∩ H ⊂ L 4 (Θ) and Definition 7.4. Let X be a Banach space and let X be its topological dual. An operator F : D → X , D ⊂ X is said to be monotone if F is said to be Λ-monotone if F + ΛI is monotone, where Λ ∈ R and I is the identity operator.
Next we prove the local monotonicity property.
Then the operator F + ΛI, for Λ ∈ R and sufficiently large |Λ|, is monotone in B , i.e., for any v ∈ V and x ∈ B , . β 20 is defined in (3.36) and η 2i 's are defined in Remark 3.7.
By a simple application of the property of the trilinear form, we have Similarly,

Now one can get
Also we have (ν∆z, z) = −ν z 2 and Similarly, one can prove that This gives Using Hölder's inequality and Sobolev embedding theorem, one gets Also, we have (B 1 (w)z, z) = b(w, z, z) = 0. By using the inequality |b(u, w, v)| ≤ β 20 |u| 1/2 u 1/2 |v| 1/2 v 1/2 + 2 k=1 β 2k (t)|u||v|, Young's inequality and Remark 3.7, we get Applying all these estimates in (7.3), one can deduce that Since Θ is an admissible channel domain and Poincaré inequality still holds, we get |z| ≤ C z and hence we have Definition 7.6 (Strong Solution). A strong solution v is defined on a given probability space (Ω, F, F t , P) as a L 2 (Ω; L 2 (0, T, V)∩C(0, T ; H)) valued adapted process which satisfies the stochastic channel flow model v(0) = 0, (7.5) in the weak sense and also the energy inequalities Definition 7.7. Let X be a Banach space and let X be its topological dual. An operator F : D → X , D ⊂ X is said to be hemicontinuous at x ∈ D, if y ∈ X, t n > 0, n = 1, 2, · · · , t n → 0 and x+t n y ∈ D imply F (x+t n y) → F (x) weakly. satisfying the stochastic model (7.5) and the a-priori bounds in (7.6) and (7.7).
Proof. Part I: Existence of Strong Solution.
We prove the existence of Strong solutions of the stochastic channel flow model (7.5) in the following four steps.

But 2
T 0 e −r(t) (g(t)dW (t), v n (t)) is a martingale having expectation zero. Then from the above equation, we have , v n (t) dt + T 0 e −r(t) Tr(g * g)(t)dt .
Since y(t) = y(ω, x, t) ∈ L 2 (Ω; L ∞ (0, T ; H m ) belongs to the closed L 4ball in V, from the Local Monotonicity Lemma 7.5, by setting On rearranging the terms in the inequality (7.21), we get Taking the limit as n → ∞ in (7.22), one obtains By using (7.19) and (7.23), we get From the above inequality, we get Stochastic Navier-Stokes Equations in Unbounded Channel Domains 41 On rearranging the terms in the inequality (7.25), we obtain This estimate holds for any y ∈ L 2 (Ω; L ∞ (0, T ; H m )) for any m ∈ N. It is clear by a density argument that the above inequality remains the same for any y ∈ L 2 (Ω; C(0, T ; H) ∩ L 2 (0, T ; V)). Indeed, for any y ∈ L 2 (Ω; C(0, T ; H) ∩ L 2 (0, T ; V)), there exits a strongly convergent sequence y m ∈ L 2 (Ω; C(0, T ; H)∩ L 2 (0, T ; V)) that satisfies inequality (7.26).
Let us now take for any λ > 0, y(t) = v(t) − λz(t) in (7.26). Note that v ∈ L 2 (Ω; C(0, T ; H)∩L 2 (0, T ; V)) satisfies the Itô differential equation given by (7.14) and z is an adapted process in L 2 (Ω; C(0, T ; H)∩L 2 (0, T ; V)). Then from (7.26), we have Dividing by λ on both sides of the above inequality and letting λ go to 0 and using the hemicontinuity property of F , one obtains (For more details see Fernando, Sritharan and Xu [22].) Since z is arbitrary, we conclude thatF 0 (t) =F (v(t)). Thus the existence of the strong solutions of the stochastic system (7.5) has been proved.
Part II: Pathwise Uniqueness of Strong Solution.
Hence the uniqueness of v(x, t, ω) satisfying the system of equations (5.1) -(5.7) has been proved.

Characterization of the Perturbation Pressure
In this section, we characterize the perturbation pressure, q(·, ·, ·) = p(·, ·, ·)− P (·, ·, ·) by using a generalization of de Rham's theorem ( [56]) to processes (see [40] for more details). We have constructed the basic vector field in such a way that the random basic pressure field P (x, t, ω) tends to infinity in each of the outlets O i as |x| → ∞. The proof for deterministic case can be found in Lemma 1.4.2, Chapter 2, page 202 of Sohr [58] for any general unbounded domain in R n , n ≥ 2. Also see the Remark 4.3 of Langa, Real, and Simon [40] for stochastic case.
We will now prove the existence and uniqueness of the perturbation pressure using Lemma 8.2. Theorem 8.3. There exists a unique scalar distribution q ∈ L 1 Ω; W −1,∞ (0, T ; L 2 loc (Θ)) such that (5.1)-(5.7) are satisfied in the sense of distributions.
Finally by applying Lemma 8.2, we get a unique q ∈ L 1 Ω; W −1,∞ (0, T ; L 2 loc (Θ)) such that χ = ∇q and the Navier Stokes Equations are satisfied in the distributional sense.
Proof. The existence of the solution u(x, t, ω), satisfying the prescribed flux condition F (t, ω) is given by the existence of the basic vector field w(x, t, ω), which carries the flux F (t, ω) and the divergence free, zero flux vector field v(x, t, ω).
For the pathwise uniqueness of the solutions, let us assume that there exists two solutions u 1 (x, t, ω) and u 2 (x, t, ω) satisfying the prescribed flux condition F (t, ω). Hence the difference field w(t) = w(x, t, ω) = u 1 (x, t, ω) − u 2 (x, t, ω) satisfies the equation dw(t) = −νAw(t)dt − [B(w(t), w(t)) + B(u 1 (t), w(t)) + B(w(t), u 2 (t))]dt, w(0) = 0, Γ w · ndS = 0, ∀ ω ∈ Ω, (9.1) with zero flux condition and which is similar to the system of equations in (5.8) without external noise term . The uniqueness of the solutions follows from the uniqueness of this system, the properties of linear and bilinear operators and by using the Poincaré inequality for the admissible channel domain. The uniqueness of the pressure field p(x, t, ω) follows from the uniqueness of the constructed basic scalar field P (x, t, ω) in section 3 and the uniqueness of the perturbed pressure q(x, t, ω).
Remark 9.2. Theorem 9.1 can be extended to the case, where the original problem (2.2)-(2.6) has a multiplicative Gaussian noise or a multiplicative Lévy noise as external forcing. For this case the basic vector field can be constructed in the same way as discussed in this paper. The method of proof of the existence and uniqueness of the perturbed vector field, under the suitable assumptions (growth property and Lipschitz's condition) on the noise coefficients, can be obtained from [44].