Stochastic Navier-Stokes Equations with Artificial Compressibility in Random Durations

Abstract

The existence and uniqueness of adapted solutions to the backward stochastic Navier-Stokes equation with artificial compressibility in two-dimensional bounded domains are shown by Minty-Browder monotonicity argument, finite-dimensional projections, and truncations. Continuity of the solutions with respect to terminal conditions is given, and the convergence of the system to an incompressible flow is also established.

1. Introduction

The Navier-Stokes equation (NSE for short), named in honor of Navier and Stokes, who were responsible for its formulation, is an acknowledged model for equation of motion for Newtonian fluid. It is closely connected to the theory of hydrodynamic turbulence, the time dependent chaotic behavior seen in many fluid flows.

The well-posedness of the Navier-Stokes equation has been studied extensively by Ladyzhenskaya [1], Constantin and Foias [2], and Temam [3], among others. Although some ingenious approaches have been made, the problem has not been fully understood. The nonlinearity, part of the cause of turbulence, made the problem extraordinarily difficult. In hope of taking advantage of the noise, randomness has been introduced into the system and some pioneer work has been done by Flandoli and Gatarek [4], Mikulevicius and Rozovsky [5], Menaldi and Sritharan [6], and others. Although the introduction of randomness is not very successful in overcoming the difficulty, it provides a more realistic model than deterministic Navier-Stokes equations and is interesting in itself.

The vast majority of work on the Navier-Stokes equations is done for viscous incompressible Newtonian fluids. In a suitable Hilbert space and under the incompressibility assumption ∇⋅𝐮=0, the two-dimensional stochastic Navier-Stokes equation in a bounded domain 𝐺⊂ℝ² with no-slip condition reads 𝜕𝐮+(𝐮⋅∇)𝐮𝑑𝑡−𝜈Δ𝐮𝑑𝑡=−∇𝑝𝑑𝑡+𝐟(𝑡)𝑑𝑡+𝜎(𝑡,𝐮)𝑑𝕎(𝑡), (1.1) where 𝜈 is the constant viscosity, 𝐮 is the velocity, 𝑝 is the pressure, 𝐟 is the external body force and 𝕎 is the infinite-dimensional Wiener process. The assumption of incompressibility works well even for compressible fluids such as air at room temperature. But there are extreme phenomena, such as the diffusion of sound, that are closely related to fluid compressibility. Also the constraint caused by the incompressibility creates computational difficulties for numerical approximation of the Navier-Stokes equations. The method of artificial compressibility was first introduced by Temam [3] to surmount this obstacle. It also describes the slight compressibility existed in most fluids. The model has its own interest, and is given below with the parameter 𝜀: 𝜕_𝑡𝐮_𝜀−𝜈Δ𝐮_𝜀+(𝐮_𝜀⋅∇)𝐮_𝜀+12(∇⋅𝐮_𝜀)𝐮_𝜀+∇𝑝_𝜀=𝐟,𝜀𝜕_𝑡𝑝_𝜀+∇⋅𝐮_𝜀=0. (1.2) Backward stochastic Navier-Stokes equations (BSNSEs for short) arise as an inverse problem wherein the velocity profile at a time 𝑇 is observed and given, and the noise coefficient has to be ascertained from the given terminal data. Such a motivation arises naturally when one understands the importance of inverse problems in partial differential equations (see Lions [7, 8]). Linear backward stochastic differential equations were introduced by Bismut in 1973 [9], and the systematic study of general backward stochastic differential equations (BSDEs for short) were put forward first by Pardoux and Peng [10], Ma, Protter, Yong, Zhou, and several other authors in a finite-dimensional setting. Ma and Yong [11] have studied linear degenerate backward stochastic differential equations motivated by stochastic control theory. Later, Hu et al. [12] considered the semilinear equations as well. Backward stochastic partial differential equations were shown to arise naturally in stochastic versions of the Black-Scholes formula by Ma and Yong [13]. A nice introduction to backward stochastic differential equations is presented in the book by Yong and Zhou [14], with various applications.

The usual method of proving existence and uniqueness of solutions by fixed point arguments does not apply to the stochastic system on hand since the drift coefficient in the backward stochastic Navier-Stokes equation is nonlinear, non-Lipschitz and unbounded. The drift coefficient is monotone on bounded 𝐿⁴(𝐺) balls in 𝑉, which was first observed by Menaldi and Sritharan [6]. The method of monotonicity is used in this paper to prove the existence of solutions to BSNSEs. The proof of the uniqueness and continuity of solutions also relies on the monotonicity assumption of the coefficients. Existence and uniqueness of solutions are shown to hold under the ℍ¹₀ boundedness on the terminal values.

The structure of the paper is as follows. The functional setup of the paper is introduced and several frequently used inequalities are listed in Section 2. The a priori estimates for the solutions of projected BSNSEs are given under different assumptions of the terminal conditions and external body force in Section 3. The existence and uniqueness of solutions of projected BSNSEs are shown in Section 4. Also the existence of solutions of BSNSEs under suitable assumptions is shown by Minty-Browder monotonicity argument. The uniqueness of the solution under the assumption that terminal condition is uniformly bounded in 𝐻¹ sense is given in Section 5. The continuity of solutions and the convergence as 𝜀 approaches zero are also studied.

2. Preliminaries

Suppose that 𝐺 is a domain bounded in ℝ² with smooth boundary conditions. Let 𝜀 be a positive parameter which vanishes to 0. The artificial state equation for a slightly compressible medium is defined as 𝜌=𝜌₀+𝜀𝑝, (2.1) where 𝜌 is the density, 𝑝 is the pressure, and 𝜌₀ is the first approximation of the density. By adjusting the equations of motion according to the state equation, we obtain the following family of perturbed systems associated with the parameter 𝜀: 𝜕_𝑡𝐮_𝜀−𝜈Δ𝐮_𝜀+(𝐮_𝜀⋅∇)𝐮_𝜀+12(∇⋅𝐮_𝜀)𝐮_𝜀+∇𝑝_𝜀=𝐟,𝜀𝜕_𝑡𝑝_𝜀+∇⋅𝐮_𝜀=0, (2.2) where 𝐮_𝜀∈𝕃²=𝕃²(𝐺) is the velocity, 𝑝_𝜀∈𝐿²=𝐿²(𝐺) is the pressure, 𝐟∈𝕃² is the external body force, and 𝜈 is the kinematic viscosity. Readers may refer to Temam [3] for details.

Denote by (⋅,⋅) the inner product of 𝕃², (⋅,⋅)_ℍ¹₀ the inner product of ℍ¹₀=ℍ¹₀(𝐺), ℍ⁻¹ the dual space of ℍ¹₀, and ⟨⋅,⋅⟩ the duality pairing between ℍ¹₀ and ℍ⁻¹. Let |⋅| be the norm of 𝕃² and let ‖⋅‖ be the norm of ℍ¹₀. Without causing any confusion, we also use the same notations to denote the norms of 𝐿² and 𝐻¹₀=𝐻¹₀(𝐺). For any 𝐱∈𝕃² and 𝐲∈ℍ¹₀, there exists 𝐱^′∈ℍ⁻¹, such that (𝐱,𝐲) = ⟨𝐱^′,𝐲⟩. Then the mapping 𝐱↦𝐱^′ is linear, injective, compact and continuous. A similar result holds for 𝐻⁻¹ and 𝐿².

Suppose that (Ω,ℱ,𝑃) is a complete probability space. Let 𝕎(𝑡) be an 𝕃²-valued ℚ-Wiener process, where ℚ is a trace class operator on 𝕃². Let {𝐞_𝑗}^∞_𝑗=1∈𝕃²∩ℍ¹₀∩𝕃⁴ be a complete orthonormal system in 𝕃² such that there exists a nondecreasing sequence of positive numbers {𝜆_𝑗}^∞_𝑗=1, lim_𝑗→∞𝜆_𝑗=∞ and −Δ𝐞_𝑗=𝜆_𝑗𝐞_𝑗 for all 𝑗. Let ℚ𝐞_𝑘=𝑞_𝑘𝐞_𝑘 with ∑^∞_𝑘=1𝑞_𝑘<∞, and {𝑏^𝑘(𝑡)} be a sequence of independent standard Brownian motions in ℝ. Then Wiener process 𝕎(𝑡) is taken as 𝕎(𝑡)=∑^∞_𝑘=1√𝑞_𝑘𝑏^𝑘(𝑡)𝐞_𝑘.

Let 𝑄 be a trace class operator on 𝐿². Similarly, we can define a complete orthonormal system {𝑒_𝑗}^∞_𝑗=1, a nondecreasing sequence of positive numbers {𝜅_𝑗}^∞_𝑗=1 such that −Δ𝑒_𝑗=𝜅_𝑗𝑒_𝑗, and positive numbers 𝑞^′_𝑗 such that 𝑄𝑒_𝑗=𝑞^′_𝑗𝑒_𝑗 and ∑^∞_𝑗=1𝑞^′_𝑗<∞. Let 𝑊(𝑡)=∑^∞_𝑗=1𝑞^′_𝑗𝑏^𝑗(𝑡)𝑒_𝑗. Then 𝑊(𝑡) is an 𝐿²-valued 𝑄-Wiener process. From now on, let {ℱ_𝑡} be the natural filtration of {𝕎(𝑡)} and {𝑊(𝑡)}, augmented by all the 𝑃-null sets of ℱ. A complete definition of Hilbert space-valued Wiener processes can be found in [15].

With inner product ⟨𝐅,𝐆⟩_𝐿ℚ=tr(𝐅ℚ𝐆^∗)=tr(𝐆ℚ𝐅^∗) (2.3) for all 𝐅 and 𝐆∈𝐿_ℚ, let 𝐿_ℚ denote the space of linear operators 𝐄 such that 𝐄ℚ^1/2 is a Hilbert-Schmidt operator from 𝕃² to 𝕃². Similarly, we define 𝐿_𝑄 for 𝑄, the trace class operator on 𝐿².

To be realistic in nature, let us introduce randomness into the system to obtain 𝜕𝐮_𝜀(𝑡)𝜕𝑡−𝜈Δ𝐮_𝜀(𝑡)+(𝐮_𝜀(𝑡)⋅∇)𝐮_𝜀1(𝑡)+2(∇⋅𝐮_𝜀(𝑡))𝐮_𝜀(𝑡)+∇𝑝_𝜀(𝑡)=𝐟(𝑡)+𝜎(𝑡)𝑑𝕎(𝑡),𝑑𝑡𝜀𝜕𝑝_𝜀(𝑡)+∇⋅𝐮_𝜀𝐮(𝑡)𝑑𝑡=0,_𝜀(0)=𝐮₀,𝑝_𝜀(0)=𝑝₀, (2.4) where 𝐮₀ and 𝑝₀ are initial conditions, and 𝜎(𝑑𝕎/𝑑𝑡) is the noise term. Here (1/2)(∇⋅𝐮_𝜀)𝐮_𝜀 is called the stabilization term.

If a terminal time 𝑇 is given and the terminal conditions are specified as 𝐮_𝜀(𝑇)=𝜉 and 𝑝_𝜀(𝑇)=𝜂, one obtains a backward system: 𝑑𝐮_𝜀(𝑡)+𝜈𝐀𝐮_𝜀(𝑡)𝑑𝑡+𝐁(𝐮_𝜀(𝑡))𝑑𝑡+∇𝑝_𝜀(𝑡)𝑑𝑡=𝐟(𝑡)𝑑𝑡+𝐙_𝜀(𝑡)𝑑𝕎(𝑡),𝜀𝑑𝑝_𝜀(𝑡)+∇⋅𝐮_𝜀(𝑡)𝑑𝑡=𝑍_𝜀𝐮(𝑡)𝑑𝑊(𝑡),_𝜀(𝑇)=𝜉,𝑝_𝜀(𝑇)=𝜂 (2.5) for 0≤𝑡≤𝑇, where 𝐀𝐮≜−Δ𝐮 and 𝐁(𝐮,𝐯)≜(𝐮⋅∇)𝐯+(1/2)(∇⋅𝐮)𝐯, with the notation 𝐁(𝐮)≜𝐁(𝐮,𝐮). The processes 𝐙_𝜀 and 𝑍_𝜀 are in spaces 𝐿_ℚ and 𝐿_𝑄, respectively.

Let 𝜏 be a ℱ_𝑡-stopping time when the observations are available. Suppose that the observed velocity and pressure at 𝜏 are 𝐮_𝜀(𝜏)=𝜉∈𝐿²_ℱ𝜏(Ω;𝕃²) and 𝑝(𝜏)=𝜂∈𝐿²_ℱ𝜏(Ω;𝐿²), respectively. Then we introduce the backward stochastic Navier-Stokes equation with artificial compressibility and stabilization in random duration:𝑑𝐮_𝜀(𝑡)+𝜈𝐀𝐮_𝜀(𝑡)𝑑𝑡+𝐁(𝐮_𝜀(𝑡))𝑑𝑡+∇𝑝_𝜀(𝑡)𝑑𝑡=𝐟(𝑡)𝑑𝑡+𝐙_𝜀(𝑡)𝑑𝕎(𝑡),𝜀𝑑𝑝_𝜀(𝑡)+∇⋅𝐮_𝜀(𝑡)𝑑𝑡=𝑍_𝜀𝐮(𝑡)𝑑𝑊(𝑡),_𝜀(𝜏)=𝜉,𝑝_𝜀(𝜏)=𝜂 (2.6) for 0≤𝑡≤𝜏, where the ℱ_𝑡-stopping time 𝜏 is assumed to be bounded by a time 𝑇>0. Note that processes 𝐙_𝜀 and 𝑍_𝜀 measure the randomness that is inherent in the hydrodynamical system. It is this randomness that has possibly led us to the observations at time 𝜏. For instance, in wind tunnel experiments, the form and the magnitude of the randomness has to be ascertained from the velocity observations. This backward system helps us to make an attempt at uncertainty quantification. Here 𝐟 is taken to be deterministic and is always assumed to be in 𝐿²(0,𝑇;ℍ⁻¹).

Definition 2.1. A quaternion of ℱ_𝑡-Adapted processes (𝐮_𝜀,𝐙_𝜀,𝑝_𝜀,𝑍_𝜀) is called a solution of backward Navier-Stokes equation (2.6) if it satisfies the integral form of the system 𝐮_𝜀(𝑡∧𝜏)=𝜉+^𝜏_𝑡∧𝜏{𝜈𝐀𝐮_𝜀(𝑠)+𝐁(𝐮_𝜀(𝑠))+∇𝑝_𝜀(𝑠)−𝐟(𝑠)}𝑑𝑠−^𝜏_𝑡∧𝜏𝐙_𝜀(𝑠)𝑑𝕎(𝑠),𝜀𝑝_𝜀(𝑡∧𝜏)=𝜀𝜂+^𝜏_𝑡∧𝜏∇⋅𝐮_𝜀(𝑠)𝑑𝑠−^𝜏_𝑡∧𝜏𝑍_𝜀(𝑠)𝑑𝑊(𝑠), (2.7) P-a.s., and the following holds: (a)𝐮_𝜀∈𝐿²_ℱ(Ω;𝐿^∞(0,𝜏;𝕃²))∩𝐿²_ℱ(Ω;𝐿²(0,𝜏;ℍ¹₀));(b)𝐙_𝜀∈𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_ℚ));(c)𝑝_𝜀∈𝐿²_ℱ(Ω;𝐿^∞(0,𝜏;𝐿²))∩𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐻¹₀));(d)𝑍_𝜀∈𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_𝑄)).

The following simple results are frequently used and given as lemmas. Readers may refer to Temam [3] for similar proofs.

Lemma 2.2. For any 𝐮,𝐯,𝐰∈ℍ¹₀ and 𝑝∈𝐿², one has (1)⟨𝐀𝐮,𝐰⟩=∑_𝑖,𝑗∫_𝐺𝜕_𝑖𝑢_𝑗𝜕_𝑖𝑤_𝑗𝑑𝑥=⟨𝐀𝐰,𝐮⟩=(𝐮,𝐰)_ℍ¹₀,(2)⟨(𝐮⋅∇)𝐯,𝐰⟩=∑_𝑖,𝑗∫_𝐺𝑢_𝑖(𝜕_𝑖𝑣_𝑗)𝑤_𝑗𝑑𝑥,(3)⟨(𝐮⋅∇)𝐯,𝐰⟩= −⟨(∇⋅𝐮)𝐰,𝐯⟩−⟨(𝐮⋅∇)𝐰,𝐯⟩,(4)⟨𝐁(𝐮,𝐯),𝐰⟩=−⟨𝐁(𝐮,𝐰),𝐯⟩,(5)⟨−∇𝑝,𝐮⟩=−∑_𝑖∫_𝐺𝜕_𝑖𝑝𝑢_𝑖𝑑𝑥=∫_𝐺𝑝𝜕_𝑖𝑢_𝑖𝑑𝑥=⟨𝑝,∇⋅𝐮⟩.

Remark 2.3. Sometimes ⟨𝐁(𝐮,𝐯),𝐰⟩ is denoted by 𝑏̂(𝐮,𝐯,𝐰).

Lemma 2.4. The following results hold for any real-valued smooth functions 𝜙 and 𝜓 with compact support in ℝ²: |𝜙𝜓|²≤𝐶‖𝜙𝜕₁𝜙‖_𝕃1‖𝜓𝜕₂𝜓‖_𝕃1,‖𝜙‖⁴_𝕃4≤𝐶|𝜙|²|∇𝜙|². (2.8)

Proposition 2.5. For any 𝐮 and 𝐯 in ℍ¹₀ and 𝐰∈𝕃⁴, one has |𝑏̂(𝐮,𝐯,𝐰)|≤‖𝐮‖_𝕃⁴‖𝐯‖‖𝐰‖_𝕃⁴+12‖𝐮‖‖𝐯‖_𝕃⁴‖𝐰‖_𝕃⁴. (2.9)

Below is a backward version of the Gronwall inequality used frequently in this paper, and the proof is straightforward.

Lemma 2.6. Suppose that 𝑔(𝑡),𝛼(𝑡),𝛽(𝑡), and 𝛾(𝑡) are integrable functions, and 𝛽(𝑡),𝛾(𝑡) are nonnegative functions. For 0≤𝑡≤𝑇, if 𝑔(𝑡)≤𝛼(𝑡)+𝛽(𝑡)^𝑇_𝑡𝛾(𝜌)𝑔(𝜌)𝑑𝜌, (2.10) then 𝑔(𝑡)≤𝛼(𝑡)+𝛽(𝑡)^𝑇_𝑡𝛼(𝜂)𝛾(𝜂)𝑒^{∫𝜂𝑡𝛽(𝜌)𝛾(𝜌)𝑑𝜌}𝑑𝜂. (2.11) In particular, if 𝛼(𝑡)≡𝛼, 𝛽(𝑡)≡𝛽 and 𝛾(𝑡)≡1, then 𝑔(𝑡)≤𝛼𝑒^{𝛽(𝑇−𝑡)}. (2.12)

3. A Priori Estimates

The purpose of the this paper is to show the existence and uniqueness of the randomly stopped backward stochastic Navier-Stokes equation (2.6). We employ Galerkin's method by defining orthogonal projections 𝑃_𝑁∶𝕃²→𝕃²_𝑁, where 𝕃²_𝑁=span{𝐞₁,𝐞₂,…,𝐞_𝑁}, for all 𝑁∈ℕ. An important result is that the Galerkin-type approximations converge weakly to the solution of the Navier-Stokes equation.

First of all, let us establish some a priori estimates. Let us define the projected operators 𝐀^𝑁≜𝑃_𝑁𝐀 and 𝐁^𝑁≜𝑃_𝑁𝐁. Under projection 𝑃_𝑁, let us construct a finite dimensional system. Let 𝕎^𝑁(𝑡)≜𝑃_𝑁𝕎(𝑡)=^𝑁_𝑖=1√𝑞_𝑖𝑏^𝑖(𝑡)𝐞_𝑖,𝑊^𝑁(𝑡)≜𝑃_𝑁𝑊(𝑡)=^𝑁_𝑖=1𝑞^′_𝑖𝑏^𝑖(𝑡)𝑒_𝑖,𝐟^𝑁(𝑡)≜𝑃_𝑁𝐟(𝑡),𝜉^𝑁≜𝐸(𝑃_𝑁𝜉∣ℱ^𝑁_𝜏),𝜂^𝑁≜𝐸(𝑃_𝑁𝜂∣ℱ^𝑁_𝜏), (3.1) where {ℱ^𝑁_𝑡} is the natural filtration of {𝕎^𝑁(𝑡)} and {𝑊^𝑁(𝑡)}. The projected system with solution (𝐮^𝑁_𝜀,𝐙^𝑁_𝜀,𝑝^𝑁_𝜀,𝑍^𝑁_𝜀) is defined as follows:𝑑𝐮^𝑁_𝜀(𝑡)=−𝜈𝐀^𝑁𝐮^𝑁_𝜀(𝑡)𝑑𝑡−𝐁^𝑁(𝐮^𝑁_𝜀(𝑡))𝑑𝑡−∇𝑝^𝑁_𝜀(𝑡)𝑑𝑡+𝐟^𝑁(𝑡)𝑑𝑡+𝐙^𝑁_𝜀(𝑡)𝑑𝕎^𝑁(𝑡),𝜀𝑑𝑝^𝑁_𝜀(𝑡)+∇⋅𝐮^𝑁_𝜀(𝑡)𝑑𝑡=𝑍^𝑁_𝜀(𝑡)𝑑𝑊^𝑁𝐮(𝑡),_𝑁𝜀(𝜏)=𝜉^𝑁,𝑝^𝑁_𝜀(𝜏)=𝜂^𝑁 (3.2) for 0≤𝑡≤𝜏.

Proposition 3.1. Let 𝜉∈𝐿^∞_ℱ𝜏(Ω;𝕃²), 𝜂∈𝐿^∞_ℱ𝜏(Ω;𝐿²), and 𝐟∈𝐿²(0,𝑇;ℍ⁻¹). Then for any solution of system (3.2), the following is true: (𝐮^𝑁_𝜀,𝐙^𝑁_𝜀)∈{𝐿^∞_ℱ([0,𝜏]×Ω;𝕃²)∩𝐿²_ℱ(Ω;𝐿²(0,𝜏;ℍ¹₀))}×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_ℚ)),(𝑝^𝑁_𝜀,𝑍^𝑁_𝜀)∈{𝐿^∞_ℱ([0,𝜏]×Ω;𝐿²)∩𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐻¹₀))}×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_𝑄)). (3.3)

Proof. Applying the Itô formula to |𝑝^𝑁_𝜀(𝑡)|² to get 𝑑|𝑝^𝑁_𝜀(𝑡)|²2=−𝜀⟨∇⋅𝐮^𝑁_𝜀(𝑡),𝑝^𝑁_𝜀2(𝑡)⟩𝑑𝑡+𝜀⟨𝑍^𝑁_𝜀(𝑡)𝑑𝑊^𝑁(𝑡),𝑝^𝑁_𝜀1(𝑡)⟩+𝜀²tr[𝑍^𝑁_𝜀(𝑡)𝑄(𝑍^𝑁_𝜀(𝑡))^∗=2]𝑑𝑡𝜀⟨∇𝑝^𝑁_𝜀(𝑡),𝐮^𝑁_𝜀2(𝑡)⟩𝑑𝑡+𝜀⟨𝑍^𝑁_𝜀(𝑡)𝑑𝑊^𝑁(𝑡),𝑝^𝑁_𝜀1(𝑡)⟩+𝜀²tr[𝑍^𝑁_𝜀(𝑡)𝑄(𝑍^𝑁_𝜀(𝑡))^∗]𝑑𝑡, (3.4) thus we get 2⟨∇𝑝^𝑁_𝜀(𝑡),𝐮^𝑁_𝜀(𝑡)⟩𝑑𝑡=𝜀𝑑|𝑝^𝑁_𝜀(𝑡)|²−2⟨𝑍^𝑁_𝜀(𝑡)𝑑𝑊^𝑁(𝑡),𝑝^𝑁_𝜀1(𝑡)⟩−𝜀‖𝑍^𝑁_𝜀(𝑡)‖²_𝐿𝑄𝑑𝑡. (3.5) By means of the Itô formula, one has |𝐮^𝑁_𝜀(𝑡∧𝜏)|²=|𝜉^𝑁|²+2^𝜏_𝑡∧𝜏⟨𝜈𝐀^𝑁𝐮^𝑁_𝜀(𝑠)+𝐁^𝑁(𝐮^𝑁_𝜀(𝑠))+∇𝑝^𝑁_𝜀(𝑠)−𝐟^𝑁(𝑠),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠−2^𝜏_𝑡∧𝜏⟨𝐙^𝑁_𝜀(𝑠)𝑑𝕎^𝑁(𝑠),𝐮^𝑁_𝜀(𝑠)⟩−^𝜏_𝑡∧𝜏‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠. (3.6) Clearly, ⟨𝐁^𝑁(𝐮^𝑁_𝜀(𝑠)),𝐮^𝑁_𝜀(𝑠)⟩=0, (3.7) and Lemma 2.2 yields |2⟨𝐟^𝑁(𝑠),𝐮^𝑁_𝜀(𝑠)⟩|≤‖𝐟^𝑁(𝑠)‖²_ℍ⁻¹+‖𝐮^𝑁_𝜀(𝑠)‖²=|𝐟^𝑁(𝑠)|²_ℍ⁻¹+⟨𝐀^𝑁𝐮^𝑁_𝜀(𝑠),𝐮^𝑁_𝜀(𝑠)⟩. (3.8) For 0<𝑟≤𝑡, taking the conditional expectation with respect to ℱ_𝑟∧𝜏, and by (3.5), the above two equation and along with the fact that ‖𝐮_𝜀(𝑠)‖²=⟨𝐀𝐮_𝜀(𝑠),𝐮_𝜀(𝑠)⟩, one gets 𝐸^{ℱ𝑟∧𝜏}|𝐮^𝑁_𝜀(𝑡∧𝜏)|²+𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠+𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏‖𝐮^𝑁_𝜀(𝑠)‖²𝑑𝑠≤𝐸^{ℱ𝑟∧𝜏}|𝜉^𝑁|²+2(𝜈+1)𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏⟨𝐀^𝑁𝐮^𝑁_𝜀(𝑠),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠+𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏‖𝐟^𝑁(𝑠)‖²_ℍ⁻¹𝑑𝑠+𝜀𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏𝑑|𝑝^𝑁_𝜀(𝑠)|²−1𝜀𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏‖𝑍^𝑁_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠, (3.9) P-a.s. Since 𝐀𝐞_𝑗=𝜆_𝑗𝐞_𝑗 and 𝜆_𝑖≤𝜆_𝑗 for 𝑖<𝑗, one gets ⟨𝐀^𝑁𝐮^𝑁_𝜀(𝑠),𝐮^𝑁_𝜀(𝑠)⟩≤𝜆_𝑁|𝐮^𝑁_𝜀(𝑠)|². (3.10) Thus 𝐸^{ℱ𝑟∧𝜏}|𝐮^𝑁_𝜀(𝑡∧𝜏)|²+𝜀𝐸^{ℱ𝑟∧𝜏}|𝑝^𝑁_𝜀(𝑡∧𝜏)|²+𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏‖𝐮^𝑁_𝜀(𝑠)‖²𝑑𝑠+𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ1𝑑𝑠+𝜀𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏‖𝑍^𝑁_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠≤𝐸^{ℱ𝑟∧𝜏}|𝜉^𝑁|²+𝜀𝐸^{ℱ𝑟∧𝜏}|𝜂^𝑁|²+2(𝜈+1)𝜆_𝑁^𝑇_𝑡𝐸^{ℱ𝑟∧𝜏}|𝐮^𝑁_𝜀(𝑠∧𝜏)|²𝑑𝑠+𝐸^{ℱ𝑟∧𝜏𝜏}₀‖𝐟^𝑁(𝑠)‖²_ℍ⁻¹𝑑𝑠, (3.11) P-a.s., and by Lemma 2.6, the backward Gronwall inequality, and letting 𝑟=𝑡, we get |𝐮^𝑁_𝜀(𝑡∧𝜏)|²+𝜀|𝑝^𝑁_𝜀(𝑡∧𝜏)|²+𝐸^{ℱ𝑡∧𝜏𝜏}_𝑡∧𝜏‖𝐮^𝑁_𝜀(𝑠)‖²𝑑𝑠+𝐸^{ℱ𝑡∧𝜏𝜏}_𝑡∧𝜏‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ1𝑑𝑠+𝜀𝐸^{ℱ𝑡∧𝜏𝜏}_𝑡∧𝜏‖𝑍^𝑁_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠≤(𝐸^{ℱ𝑡∧𝜏}|𝜉^𝑁|²+𝜀𝐸^{ℱ𝑡∧𝜏}|𝜂^𝑁|²+𝐸^{ℱ𝑡∧𝜏𝜏}₀‖𝐟^𝑁(𝑠)‖²_ℍ⁻¹𝑑𝑠)𝑒^{2(𝜈+1)𝜆𝑁(𝑇−𝑡)}, (3.12) P-a.s. Because of the integrability of 𝜉, 𝜂, and 𝐟, there exists a constant 𝐾_𝑁, depending on 𝑁 only, s.t. |𝐮^𝑁_𝜀(𝑡)|²+𝜀|𝑝^𝑁_𝜀(𝑡)|²+𝐸^𝜏₀‖𝐮^𝑁_𝜀(𝑠)‖²𝑑𝑠+𝐸^𝜏₀‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠+𝐸^𝜏₀‖𝑍^𝑁_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠≤𝐾_𝑁, (3.13) for all 𝑡∈[0,𝜏], P-a.s.
Similarly, making use of (3.4), it follows that 𝑝^𝑁_𝜀∈𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐻¹₀)).

Proposition 3.2. Let 𝜉∈𝐿^𝑛_ℱ𝜏(Ω;𝕃²), 𝜂∈𝐿^𝑛_ℱ𝜏(Ω;𝐿²), and 𝐟∈𝐿²(0,𝑇;ℍ⁻¹), for all 𝑛∈ℕ and 𝑛≥2. The following is true for any solution of system (3.2): 𝐮^𝑁_𝜀∈𝐿^∞(0,𝜏;𝐿^𝑛_ℱ(Ω;𝕃²))∩𝐿^𝑛_ℱ(Ω;𝐿^𝑛(0,𝜏;ℍ¹₀𝑝)),_𝑁𝜀∈𝐿^∞(0,𝜏;𝐿^𝑛_ℱ(Ω;𝐿²))∩𝐿^𝑛_ℱ(Ω;𝐿^𝑛(0,𝜏;𝐻¹₀)). (3.14)

Proof. Let us prove it by the method of mathematical induction. Similar to Proposition 3.1, it is easy to obtain the result for 𝑛=2. Suppose that it is true for all 𝑚≤𝑛−1. Let us show that the proposition holds for 𝑚=𝑛.
An application of the Itô formula to |𝐮^𝑁_𝜀(𝑡)|^𝑛 yields|𝐮^𝑁_𝜀(𝑡∧𝜏)|^𝑛=|𝜉^𝑁|^𝑛+𝑛^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛−2⟨𝜈𝐀^𝑁𝐮^𝑁_𝜀(𝑠)+𝐁^𝑁(𝐮^𝑁_𝜀(𝑠))+∇𝑝^𝑁_𝜀(𝑠)−𝐟^𝑁(𝑠),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠−𝑛^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛−2⟨𝐙^𝑁_𝜀(𝑠)𝑑𝕎^𝑁(𝑠),𝐮^𝑁_𝜀𝑛(𝑠)⟩−²−𝑛2^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛−2‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠. (3.15) Clearly |∇𝑝^𝑁_𝜀(𝑠)|≤𝐶‖𝑝^𝑁_𝜀(𝑠)‖≤𝐶√𝜅_𝑁|𝑝^𝑁_𝜀(𝑠)|, where 𝜅_𝑁, as stated in Section 2, is the eigenvalue of −Δ for 𝑒_𝑁. Taking the expectation, one obtains 𝐸|𝐮^𝑁_𝜀(𝑡∧𝜏)|^𝑛+𝐸^𝜏_𝑡∧𝜏‖𝐮^𝑁_𝜀(𝑠)‖^𝑛𝑑𝑠≤𝐸|𝜉^𝑁|^𝑛+𝜆_𝑁^𝑛/2𝐸^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛𝑑𝑠+𝑛𝐸^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛−2⟨𝜈𝐀^𝑁𝐮^𝑁_𝜀(𝑠)+∇𝑝^𝑁_𝜀(𝑠)−𝐟^𝑁(𝑠),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠≤𝐸|𝜉^𝑁|^𝑛+(𝜈𝜆_𝑁𝑛+𝜆_𝑁^𝑛/2)𝐸^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛𝑑𝑠+𝑛𝐸^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛−1|∇𝑝^𝑁_𝜀+𝑛(𝑠)|𝑑𝑠2𝐸^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛−2(‖𝐟^𝑁(𝑠)‖²_ℍ⁻¹+𝜆_𝑁|𝐮^𝑁_𝜀(𝑠)|²)𝑑𝑠≤𝐸|𝜉^𝑁|^𝑛+(𝜈𝜆_𝑁𝑛+𝜆_𝑁^𝑛/2+𝑛2𝜆_𝑁)𝐸^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛√𝑑𝑠+𝑛𝐶𝜅_𝑁{𝐸^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛𝑑𝑠}^{(𝑛−1)/𝑛}{𝐸^𝜏_𝑡∧𝜏|𝑝^𝑁_𝜀(𝑠)|^𝑛𝑑𝑠}^1/𝑛+𝑛2^𝑇_𝑡‖𝐟^𝑁(𝑠)‖²_ℍ⁻¹𝐸{1_{[𝑡∧𝜏,𝜏]}|𝐮^𝑁_𝜀(𝑠)|^𝑛−2}𝑑𝑠≤𝐸|𝜉^𝑁|^𝑛+𝑛2{sup_{0≤𝑡≤𝜏}𝐸|𝐮^𝑁_𝜀(𝑡)|^𝑛−2}^𝑇_𝑡‖𝐟^𝑁(𝑠)‖²_ℍ⁻¹𝑑𝑠+(𝜈𝜆_𝑁𝑛+𝜆_𝑁^𝑛/2+𝑛2𝜆_𝑁+(𝑛−1)𝐶√𝜅_𝑁)^𝑇_𝑡𝐸|𝐮^𝑁_𝜀(𝑠∧𝜏)|^𝑛√𝑑𝑠+𝐶𝜅_𝑁^𝑇_𝑡𝐸|𝑝^𝑁_𝜀(𝑠∧𝜏)|^𝑛𝑑𝑠≤𝐾+𝐾(𝑛,𝑁)^𝑇_𝑡𝐸|𝐮^𝑁_𝜀(𝑠∧𝜏)|^𝑛𝑑𝑠+𝐾(𝑛,𝑁)^𝑇_𝑡𝐸|𝑝^𝑁_𝜀(𝑠∧𝜏)|^𝑛𝑑𝑠, (3.16)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10) where 𝐾 is a constant, and 𝐾(𝑛,𝑁) is a constant depending on 𝑛 and 𝑁. Both constants may vary throughout the proof. But we keep the same notations for simplicity. Applying the Itô formula to |𝑝^𝑁_𝜀(𝑡)|^𝑛, one obtains 𝜀𝐸|𝑝^𝑁_𝜀(𝑡∧𝜏)|^𝑛+𝐸^𝜏_𝑡∧𝜏‖𝑝^𝑁_𝜀(𝑠)‖^𝑛𝑑𝑠≤𝜀𝐸|𝜂^𝑁|^𝑛+𝜅_𝑁^𝑛/2𝐸^𝜏_𝑡∧𝜏|𝑝^𝑁_𝜀(𝑠)|^𝑛𝑑𝑠−𝑛𝐸^𝜏_𝑡∧𝜏|𝑝^𝑁_𝜀(𝑠)|^𝑛−2⟨∇𝑝^𝑁_𝜀(𝑠),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠≤𝜀𝐸|𝜂^𝑁|^𝑛+𝜅_𝑁^𝑛/2𝐸^𝜏_𝑡∧𝜏|𝑝^𝑁_𝜀(𝑠)|^𝑛𝑑𝑠+𝑛𝐶√𝜅_𝑁𝐸^𝜏_𝑡∧𝜏|𝑝^𝑁_𝜀(𝑠)|^𝑛−1|𝐮^𝑁_𝜀(𝑠)|𝑑𝑠≤𝜀𝐸|𝜂^𝑁|^𝑛+𝜅_𝑁^𝑛/2𝐸^𝜏_𝑡∧𝜏|𝑝^𝑁_𝜀(𝑠)|^𝑛𝑑𝑠+𝑛𝐶√𝜅_𝑁{𝐸^𝜏_𝑡∧𝜏|𝑝^𝑁_𝜀(𝑠)|^𝑛𝑑𝑠}^{(𝑛−1)/𝑛}×{𝐸^𝜏_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠)|^𝑛𝑑𝑠}^1/𝑛≤𝐾+𝐾(𝑛,𝑁)^𝑇_𝑡𝐸|𝑝^𝑁_𝜀(𝑠∧𝜏)|^𝑛𝑑𝑠+𝐾(𝑛,𝑁)^𝑇_𝑡𝐸|𝐮^𝑁_𝜀(𝑠∧𝜏)|^𝑛𝑑𝑠. (3.17) Adding up (3.16) and (3.17), one gets 𝐸{|𝐮^𝑁_𝜀(𝑡∧𝜏)|^𝑛+𝜀|𝑝^𝑁_𝜀(𝑡∧𝜏)|^𝑛}+𝐸^𝜏_𝑡∧𝜏{‖𝐮^𝑁_𝜀(𝑠)‖^𝑛+‖𝑝^𝑁_𝜀(𝑠)‖^𝑛}𝑑𝑠≤𝐾+𝐾(𝑛,𝑁)^𝑇_𝑡𝐸{|𝐮^𝑁_𝜀(𝑠∧𝜏)|^𝑛+|𝑝^𝑁_𝜀(𝑠∧𝜏)|^𝑛}𝑑𝑠. (3.18) An application of the Gronwall inequality (2.11) yields the result.

4. Existence of Solutions

The following lemma states the monotonicity of drift coefficients. The proof involves Proposition 2.5 and is straightforward.

Lemma 4.1. Assume 𝐮,𝐯∈ℍ¹₀ and 𝐰∈𝕃⁴. The following inequalities are true: (a)|⟨𝐁(𝐮),𝐰⟩|≤2‖𝐮‖^3/2|𝐮|^1/2‖𝐰‖_𝕃⁴,(b)|⟨𝐁(𝐮)−𝐁(𝐯),𝐮−𝐯⟩|≤(𝜈/2)‖𝐮−𝐯‖²+(27/2𝜈³)|𝐮−𝐯|²‖𝐯‖⁴_𝕃4,(c)⟨𝜈𝐀(𝐮−𝐯)+𝐁(𝐮)−𝐁(𝐯),𝐮−𝐯⟩+(27/2𝜈³)‖𝐯‖⁴_𝕃4|𝐮−𝐯|²≥(𝜈/2)‖𝐮−𝐯‖². Furthermore, if 𝐰∈ℍ¹₀, then there exists a constant 𝐶 depending on 𝜈, such that (d)⟨𝜈𝐀(𝐮−𝐯)+𝐁(𝐮)−𝐁(𝐯),𝐮−𝐯⟩+𝐶‖𝐯‖²|𝐮−𝐯|²≥(𝜈/2)‖𝐮−𝐯‖².

Corollary 4.2. For any 𝐮 and 𝐯∈𝕃⁴, let ℎ₁(𝑡)=27𝜈^3𝑡₀‖𝐮(𝑠)‖⁴_𝕃4ℎ𝑑𝑠,₂(𝑡)=27𝜈^3𝑡₀‖𝐯(𝑠)‖⁴_𝕃4𝑑𝑠. (4.1) Then 1⟨𝜈𝐀(𝐮−𝐯)+𝐁(𝐮)−𝐁(𝐯)+2ℎ̇_𝑖(𝑡)(𝐮−𝐯),𝐮−𝐯⟩≥0,𝑖=1,2. (4.2)

The proposition below is used in the proof of the existence, and we provide a brief proof. Readers may refer to [14, 16] for a similar and detailed proof.

Proposition 4.3. Let 𝜉∈𝐿^∞_ℱ𝜏(Ω;𝕃²), 𝜂∈𝐿^∞_ℱ𝜏(Ω;𝐿²), and 𝐟∈𝐿²(0,𝑇;ℍ⁻¹). Then the projected system (3.2) admits a unique adapted solution (𝐮^𝑁_𝜀,𝐙^𝑁_𝜀,𝑝^𝑁_𝜀,𝑍^𝑁_𝜀) in {𝐿^∞_ℱ([0,𝜏]×Ω;𝕃²)∩𝐿²_ℱ(Ω;𝐿²(0,𝜏;ℍ¹₀))}×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_ℚ))×{𝐿^∞_ℱ([0,𝜏]×Ω;𝐿²)∩𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐻¹₀))}×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_𝑄)). (4.3)

Proof. For every 𝑀∈ℕ, let 𝐿_𝑀 be a Lipschitz 𝐶^∞ function which has the following property: 𝐿_𝑀(‖𝐮‖)=1if‖𝐮‖<𝑀,0if‖𝐮‖>𝑀+1,0≤𝐿_𝑀(‖𝐮‖)≤1otherwise. (4.4) Applying the truncation 𝐿_𝑀 to 𝐁, it is easy to show that 𝐿_𝑀𝐁 is Lipschitz and |𝐿_𝑀(‖𝐱‖)𝐁^𝑁(𝐱)−𝐿_𝑀(‖𝐲‖)𝐁^𝑁(𝐲)|≤𝐶_𝑁,𝑀‖𝐱−𝐲‖ (4.5) for any 𝐱,𝐲∈𝕃²_𝑁 and 𝑀∈ℕ. Let us define a truncated projected system: 𝑑𝐮_𝜀^𝑁,𝑀(𝑡)=−𝜈𝐀^𝑁𝐮_𝜀^𝑁,𝑀(𝑡)𝑑𝑡−𝐿_𝑀(‖𝐮_𝜀^𝑁,𝑀(𝑡)‖)𝐁^𝑁(𝐮_𝜀^𝑁,𝑀(𝑡))𝑑𝑡−∇𝑝_𝜀^𝑁,𝑀(𝑡)𝑑𝑡+𝐟^𝑁(𝑡)𝑑𝑡+𝐙_𝜀^𝑁,𝑀(𝑡)𝑑𝕎^𝑁(𝑡),𝜀𝑑𝑝_𝜀^𝑁,𝑀(𝑡)+∇⋅𝐮_𝜀^𝑁,𝑀(𝑡)𝑑𝑡=𝑍_𝜀^𝑁,𝑀(𝑡)𝑑𝑊^𝑁𝐮(𝑡),_{𝜀𝑁,𝑀}(𝜏)=𝜉^𝑁,𝑝_𝜀^𝑁,𝑀(𝜏)=𝜂^𝑁. (4.6) For fixed 𝑝∈𝐿^∞_ℱ([0,𝜏]×Ω;𝐿²)∩𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐻¹₀)), let us map 𝑑𝐮_𝜀^𝑁,𝑀(𝑡)=−𝜈𝐀^𝑁𝐮_𝜀^𝑁,𝑀(𝑡)𝑑𝑡−𝐿_𝑀(‖𝐮_𝜀^𝑁,𝑀(𝑡)‖)𝐁^𝑁(𝐮_𝜀^𝑁,𝑀(𝑡))𝑑𝑡−∇𝑝^𝑁,𝑀(𝑡)𝑑𝑡+𝐟^𝑁(𝑡)𝑑𝑡+𝐙_𝜀^𝑁,𝑀(𝑡)𝑑𝕎^𝑁𝐮(𝑡),_{𝜀𝑁,𝑀}(𝜏)=𝜉^𝑁 (4.7) to ℝ^𝑁, and the image of the system is equivalent to the system. Since the coefficients in the image system are Lipschitz, a well-known result in ℝ^𝑁 (see [14, page 355]) guarantees the existence of a unique adapted solution. Let the solution be (𝐮_𝜀^𝑁,𝑀,𝐙_𝜀^𝑁,𝑀). Then for 𝜀𝑑𝑝_𝜀^𝑁,𝑀(𝑡)+∇⋅𝐮_𝜀^𝑁,𝑀(𝑡)𝑑𝑡=𝑍_𝜀^𝑁,𝑀(𝑡)𝑑𝑊^𝑁𝑝(𝑡)_{𝜀𝑁,𝑀}(𝜏)=𝜂^𝑁, (4.8) there is a unique adapted solution (𝑝_𝜀^𝑁,𝑀,𝑍_𝜀^𝑁,𝑀). Thus we can define an operator Ψ, such that Ψ(𝑝)=𝑝_𝜀^𝑁,𝑀. It can be shown that Ψ is a contraction mapping. Thus the unique adapted solution of (4.6) can be obtained. Let us take the limit of the solution as 𝑀 approaches infinity. It can be shown that the limit is the unique solution of the projected system (3.2).

From now on, let us assume the external body force to be an operator and denote it by 𝐅. We also assume the following coercivity and monotonicity hypotheses in this paper. Such an approach is commonly used in studying the stochastic Euler equations so that a dissipative effect arises. Also they are standard hypotheses in the theory of stochastic PDEs in infinite dimensional spaces (see Chow [15], Kallianpur and Xiong [17], Prévôt and Röckner [18]).

Assumption A. (A.1) 𝐅: ℍ¹₀→ℍ⁻¹ is a continuous operator.
(A.2) There exist positive constants 𝛼 and 𝛽, such that⟨𝜈𝐀𝐮−𝐅(𝐮),𝐮⟩≤𝛼|𝐮|²−𝛽‖𝐮‖²;⟨𝜈𝐀𝐮−𝐅(𝐮),𝐀𝐮⟩≤𝛼‖𝐮‖²−𝛽‖𝐀𝐮‖². (4.9)
(A.3) For any 𝐮 and 𝐯 in ℍ¹₀, a constant 𝜅>𝜈, and a positive constant 𝛼,⟨𝜅𝐀(𝐮−𝐯)−(𝐅(𝐮)−𝐅(𝐯)),𝐮−𝐯⟩≤𝛼|𝐮−𝐯|². (4.10)
(A.4) For any 𝐮∈ℍ¹₀ and some positive constant 𝛼,|⟨𝐅(𝐮),𝐮⟩|≤𝛼‖𝐮‖². (4.11)

Remark 4.4. Assumption (A.2) is usually called the coercivity condition of the dissipative term and the external body force. Assumption (A.3) is the monotonicity condition of dissipative term and the external body force. The first half of the inequality is used in the proof of the uniqueness in Section 5. The second half of the inequality is used in the proof of the existence in Section 4. Assumption (A.4) is the linear growth condition of the external body force.

Under above assumptions, we adjust systems (2.6) and (3.2) to the following two systems:𝑑𝐮_𝜀(𝑡)+𝜈𝐀𝐮_𝜀(𝑡)𝑑𝑡+𝐁(𝐮_𝜀(𝑡))𝑑𝑡+∇𝑝_𝜀(𝑡)𝑑𝑡=𝐅(𝐮_𝜀(𝑡))𝑑𝑡+𝐙_𝜀(𝑡)𝑑𝕎(𝑡),𝜀𝑑𝑝_𝜀(𝑡)+∇⋅𝐮_𝜀(𝑡)𝑑𝑡=𝑍_𝜀𝐮(𝑡)𝑑𝑊(𝑡),_𝜀(𝜏)=𝜉,𝑝_𝜀(𝜏)=𝜂,𝑑𝐮^𝑁_𝜀(𝑡)=−𝜈𝐀^𝑁𝐮^𝑁_𝜀(𝑡)𝑑𝑡−𝐁^𝑁(𝐮^𝑁_𝜀(𝑡))𝑑𝑡−∇𝑝^𝑁_𝜀(𝑡)𝑑𝑡+𝐅^𝑁(𝐮^𝑁_𝜀(𝑡))𝑑𝑡+𝐙^𝑁_𝜀(𝑡)𝑑𝕎^𝑁(𝑡),𝜀𝑑𝑝^𝑁_𝜀(𝑡)+∇⋅𝐮^𝑁_𝜀(𝑡)𝑑𝑡=𝑍^𝑁_𝜀(𝑡)𝑑𝑊^𝑁𝐮(𝑡),_𝑁𝜀(𝜏)=𝜉^𝑁,𝑝^𝑁_𝜀(𝜏)=𝜂^𝑁 (4.12)(4.13) for 0≤𝑡≤𝜏. The existence and uniqueness of an adapted solution of (4.13) can be easily checked in the same fashion as in Proposition 4.3.

Lemma 4.5. Assume 𝐮 and 𝐯∈𝕃⁴. Then the following inequality is true: |⟨𝐁(𝐮)−𝐁(𝐯),𝐮−𝐯⟩|≤(𝜅−𝜈)‖𝐮−𝐯‖²+2716(𝜅−𝜈)³|𝐮−𝐯|²‖𝐯‖⁴_𝕃4. (4.14)

Corollary 4.6. Let 𝐮 and 𝐯∈𝕃⁴. Define 𝑙₁(𝑡)≜^𝑇_𝑡{2𝛼+278(𝜅−𝜈)³‖𝐮(𝑠)‖⁴_𝕃4𝑙}𝑑𝑠,₂(𝑡)≜^𝑇_𝑡{2𝛼+278(𝜅−𝜈)³‖𝐯(𝑠)‖⁴_𝕃4}𝑑𝑠. (4.15) Then 1⟨𝜈𝐀(𝐮−𝐯)+𝐁(𝐮)−𝐁(𝐯)−(𝐅(𝐮)−𝐅(𝐯))+2𝑙_𝑖̇(𝑡)(𝐮−𝐯),𝐮−𝐯⟩≤0,𝑖=1,2. (4.16)

Remark 4.7. To prove Corollary 4.6, the monotonicity assumption (A.3) is used.

Proposition 4.8. (i) Let 𝜉∈𝐿^∞_ℱ𝜏(Ω;𝕃²) and 𝜂∈𝐿^∞_ℱ𝜏(Ω;𝐿²). Then for any solution of system (4.13), the following is true: (𝐮^𝑁_𝜀,𝐙^𝑁_𝜀)∈{𝐿^∞_ℱ([0,𝜏]×Ω;𝕃²)∩𝐿²_ℱ(Ω;𝐿²(0,𝜏;ℍ¹₀))}×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_ℚ)),(𝑝^𝑁_𝜀,𝑍^𝑁_𝜀)∈𝐿^∞_ℱ([0,𝜏]×Ω;𝐿²)×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_𝑄)). (4.17) Moreover, there exists a constant 𝐾, independent of 𝑁, such that sup_{𝑡∈[0,𝜏]}|𝐮^𝑁_𝜀(𝑡)|²+𝐸^𝜏₀‖𝐮^𝑁_𝜀(𝑠)‖²𝑑𝑠+𝜀sup_{𝑡∈[0,𝜏]}|𝑝^𝑁_𝜀(𝑡)|²+𝐸^𝜏₀‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠+𝐸^𝜏₀‖𝑍^𝑁_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠≤𝐾, (4.18) P-a.s.
(ii) Let 𝜉∈𝐿²_ℱ𝜏(Ω;𝕃²) and 𝜂∈𝐿²_ℱ𝜏(Ω;𝐿²). The following is true for any solution of system (4.13):(𝐮^𝑁_𝜀,𝐙^𝑁_𝜀)∈{𝐿^∞(0,𝜏;𝐿²_ℱ(Ω;𝕃²))∩𝐿²_ℱ(Ω;𝐿²(0,𝜏;ℍ¹₀))}×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_ℚ)),(𝑝^𝑁_𝜀,𝑍^𝑁_𝜀)∈𝐿^∞(0,𝜏;𝐿²_ℱ(Ω;𝐿²))×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_𝑄)). (4.19) Moreover, there exists a constant 𝐾, independent of 𝑁, such that sup_{𝑡∈[0,𝜏]}𝐸|𝐮^𝑁_𝜀(𝑡)|²+𝐸^𝜏₀‖𝐮^𝑁_𝜀(𝑠)‖²𝑑𝑠+𝜀sup_{𝑡∈[0,𝜏]}𝐸|𝑝^𝑁_𝜀(𝑡)|²+𝐸^𝜏₀‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠+𝐸^𝜏₀‖𝑍^𝑁_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠≤𝐾. (4.20)

Proof. (i) Similar to the proof of Proposition 3.1, utilizing Assumption (A.2), (3.6) becomes |𝐮^𝑁_𝜀(𝑡∧𝜏)|²=|𝜉^𝑁|²+2^𝜏_𝑡∧𝜏⟨𝜈𝐀^𝑁𝐮^𝑁_𝜀(𝑠)+𝐁^𝑁(𝐮^𝑁_𝜀(𝑠))+∇𝑝^𝑁_𝜀(𝑠)−𝐅^𝑁(𝐮^𝑁_𝜀(𝑠)),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠−2^𝜏_𝑡∧𝜏⟨𝐙^𝑁_𝜀(𝑠)𝑑𝕎^𝑁(𝑠),𝐮^𝑁_𝜀(𝑠)⟩−^𝜏_𝑡∧𝜏‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠≤|𝜉^𝑁|²+2^𝜏_𝑡∧𝜏{𝛼|𝐮^𝑁_𝜀(𝑠)|²−𝛽‖𝐮^𝑁_𝜀(𝑠)‖²+⟨∇𝑝^𝑁_𝜀(𝑠),𝐮^𝑁_𝜀(𝑠)⟩}𝑑𝑠−2^𝜏_𝑡∧𝜏⟨𝐙^𝑁_𝜀(𝑠)𝑑𝕎^𝑁(𝑠),𝐮^𝑁_𝜀(𝑠)⟩−^𝜏_𝑡∧𝜏‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠. (4.21) For 0<𝑟≤𝑡, taking the conditional expectation with respect to ℱ_𝑟∧𝜏, one gets 𝐸^{ℱ𝑟∧𝜏}|𝐮^𝑁_𝜀(𝑡∧𝜏)|²+𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠+2𝛽𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏‖𝐮^𝑁_𝜀(𝑠)‖²𝑑𝑠≤𝐸^{ℱ𝑟∧𝜏}|𝜉^𝑁|²+2𝛼𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏|𝐮^𝑁_𝜀(𝑠∧𝜏)|²𝑑𝑠+𝜀𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏𝑑|𝑝^𝑁_𝜀(𝑠)|²−1𝜀𝐸^{ℱ𝑟∧𝜏𝜏}_𝑡∧𝜏‖𝑍^𝑁_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠, (4.22) P-a.s. By the backward Gronwall inequality, and letting 𝑟=𝑡, we get (4.18).
(ii) The proof is similar to (i).

Proposition 4.9. Suppose that 𝜉∈𝐿^∞_ℱ𝜏(Ω;ℍ¹₀) and 𝜂∈𝐿^∞_ℱ𝜏(Ω;𝐻¹₀). Then for any solution (𝐮_𝜀,𝐙_𝜀,𝑝_𝜀,𝑍_𝜀) of system (4.13), there exists a constant 𝐾₀, such that sup_{𝑡∈[0,𝜏]}‖𝐮_𝜀(𝑡)‖²+sup_{𝑡∈[0,𝜏]}‖𝑝_𝜀‖²≤𝐾₀. (4.23)

Proof. The proof involves an application of the Itô formula to ‖𝐮_𝜀(𝑡)‖², and the second half of the coercivity assumption. We skip the proof since it is similar to Proposition 3.1.

Theorem 4.10. Let 𝜉∈𝐿^∞_ℱ𝜏(Ω;ℍ¹₀) and 𝜂∈𝐿^∞_ℱ𝜏(Ω;𝐻¹₀). For system (4.12), there exists a solution (𝐮_𝜀,𝐙_𝜀,𝑝_𝜀,𝑍_𝜀) in 𝐿^∞_ℱ([0,𝜏]×Ω;ℍ¹₀)×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_ℚ))×𝐿^∞_ℱ([0,𝜏]×Ω;𝐻¹₀)×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_𝑄)). (4.24)

Proof. We have the following steps.Step 1 (The limits). Clearly, by Proposition 4.8, there exist 𝐮_𝜀, 𝑝_𝜀, 𝐙_𝜀, and 𝑍_𝜀, such that 𝐮−→^𝑁𝑘_𝜀^𝑤𝐮_𝜀in𝐿²_ℱ(Ω;𝐿²(0,𝜏;ℍ¹₀𝑝)),_{𝑁𝑘𝜀𝑤}−→𝑝_𝜀in𝐿^∞_ℱ([0,𝜏]×Ω;𝐿²𝐙),_{𝑁𝑘𝜀𝑤}−→𝐙_𝜀in𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_ℚ𝑍)),_{𝑁𝑘𝜀𝑤}−→𝑍_𝜀in𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_𝑄)), (4.25) for a subsequence 𝑁_𝑘. Since 𝐀 is a continuous map from ℍ¹₀ to ℍ⁻¹, ‖𝐀𝐮_𝜀‖_ℍ−1≤𝐶‖𝐮_𝜀‖ (4.26) for all 𝐮_𝜀∈ℍ¹₀ and some constant 𝐶. Thus combined with the assumptions on 𝐅, one knows that 𝜈𝐀^𝑁𝑘𝐮^𝑁𝑘_𝜀−𝐅^𝑁𝑘(𝐮^𝑁𝑘_𝜀)−→^𝑤𝐀in𝐿²_ℱ(Ω;𝐿²(0,𝜏;ℍ⁻¹)) (4.27) for some function 𝐀 and some subsequence 𝑁_𝑘. By Lemma 4.1, ‖𝐁^𝑁(𝐮^𝑁_𝜀(𝑡))‖_ℍ−1=sup_‖𝐰‖=1|⟨𝐁^𝑁(𝐮^𝑁_𝜀(𝑡)),𝐰⟩|≤sup_‖𝐰‖=12‖𝐮^𝑁_𝜀(𝑡)‖^3/2|𝐮^𝑁_𝜀(𝑡)|^1/2‖𝐰‖_𝕃⁴≤𝐾‖𝐮^𝑁_𝜀(𝑡)‖^3/2. (4.28) Thus 𝐁^𝑁𝑘(𝐮^𝑁𝑘_𝜀)−→^𝑤𝐁in𝐿_ℱ^4/3(Ω;𝐿^4/3(0,𝜏;ℍ⁻¹)) (4.29) for some function 𝐁 and some subsequence 𝑁_𝑘. For every 𝑡, we define ℒ_𝑡∶𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_ℚ))→𝐿²_ℱ(Ω;𝐿²(0,𝜏;ℍ⁻¹))𝑀↦^𝜏_𝑡∧𝜏𝑀(𝑠)𝑑𝕎(𝑠). (4.30) It can be shown that ℒ_𝑡 is a bounded linear operator. Hence ^𝜏_𝑡∧𝜏𝐙^𝑁𝑘_𝜀(𝑠)𝑑𝕎^𝑁𝑘(𝑠)−→^𝑤𝜏_𝑡∧𝜏𝐙_𝜀(𝑠)𝑑𝕎(𝑠)in𝐿²_ℱ(Ω;𝐿²(0,𝜏;ℍ⁻¹)). (4.31) Similarly, one can prove that {𝜈𝐀^𝜏_𝑡∧𝜏^𝑁𝑘𝐮^𝑁𝑘_𝜀(𝑠)−𝐅^𝑁𝑘(𝐮^𝑁𝑘_𝜀(𝑠))+𝐁^𝑁𝑘(𝐮^𝑁𝑘_𝜀(𝑠))}𝑑𝑠−→^𝑤𝜏_𝑡∧𝜏{𝐀(𝑠)+𝐁(𝑠)}𝑑𝑠 (4.32) in 𝐿_ℱ^4/3(Ω;𝐿^4/3(0,𝜏;ℍ⁻¹)) and ^𝜏_𝑡∧𝜏𝑍^𝑁𝑘_𝜀(𝑠)𝑑𝑊^𝑁𝑘(𝑠)−→^𝑤𝜏_𝑡∧𝜏𝑍_𝜀(𝑠)𝑑𝑊(𝑠)in𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐻⁻¹)). (4.33) Also ℒ_𝑝∶𝐿^∞_ℱ([0,𝜏]×Ω;𝐿²)→𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐻⁻¹)),𝑝↦^𝜏_𝑡∧𝜏∇𝑝(𝑠)𝑑𝑠 (4.34) is a bounded linear operator. Since 𝑝^𝑁𝑘_𝜀∈𝐿^∞_ℱ([0,𝜏]×Ω;𝐿²), we have ^𝜏_𝑡∧𝜏∇𝑝^𝑁𝑘_𝜀(𝑠)𝑑𝑠−→^𝑤𝜏_𝑡∧𝜏∇𝑝_𝜀(𝑠)𝑑𝑠in𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐻⁻¹)). (4.35) Similarly, ^𝜏_𝑡∧𝜏∇⋅𝐮^𝑁𝑘_𝜀(𝑠)𝑑𝑠−→^𝑤𝜏_𝑡∧𝜏∇⋅𝐮_𝜀(𝑠)𝑑𝑠in𝐿²_ℱ(Ω;𝐿²(0,𝜏;ℍ⁻¹)). (4.36) To sum up, 𝐮_𝜀(𝑡∧𝜏)=𝜉+^𝜏_𝑡∧𝜏{𝐀(𝑠)+𝐁(𝑠)+∇𝑝_𝜀(𝑠)}𝑑𝑠−^𝜏_𝑡∧𝜏𝐙_𝜀(𝑡)𝑑𝕎(𝑠),𝜀𝑝_𝜀(𝑡∧𝜏)=𝜀𝜂+^𝜏_𝑡∧𝜏∇⋅𝐮_𝜀(𝑠)𝑑𝑠−^𝜏_𝑡∧𝜏𝑍_𝜀(𝑠)𝑑𝑊(𝑠) (4.37) hold P-a.s.Step 2 (The Itô formula). For convenience, let us denote 𝑁_𝑘 by 𝑁 again. Let 𝑀≤𝑁 and (ℍ¹₀)_𝑀=𝑃_𝑀(ℍ¹₀). For any 𝐯∈𝐿^∞_ℱ([0,𝜏]×Ω;(ℍ¹₀)_𝑀) and some constant 𝐾, such that ‖𝐯‖≤𝐾 uniformly, define 𝑟(𝑡)=^𝑇_𝑡{2𝛼+278(𝜅−𝜈)³𝐾⁴}𝑑𝑠. (4.38) Applying the Itô formula to 𝑒^{−𝑟(𝑡)}|𝐮^𝑁_𝜀(𝑡)|², we get |𝜉^𝑁|²−𝑒^−𝑟(0)|𝐮^𝑁_𝜀(0)|²=^𝜏₀−𝑟(̇𝑠)𝑒^{−𝑟(𝑠)}|𝐮^𝑁_𝜀(𝑠)|²𝑑𝑠+2^𝜏₀𝑒^{−𝑟(𝑠)}⟨−𝜈𝐀^𝑁𝐮^𝑁_𝜀(𝑠)−𝐁^𝑁(𝐮^𝑁_𝜀(𝑠))−∇𝑝^𝑁_𝜀(𝑠)+𝐅^𝑁(𝐮^𝑁_𝜀(𝑠)),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠+2^𝜏₀𝑒^{−𝑟(𝑠)}⟨𝐙^𝑁_𝜀(𝑠)𝑑𝕎^𝑁(𝑠),𝐮^𝑁_𝜀(𝑠)⟩+^𝜏₀𝑒^{−𝑟(𝑠)}‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠. (4.39) By taking the expectation, we get 𝐸|𝜉^𝑁|²−𝐸𝑒^−𝑟(0)|𝐮^𝑁_𝜀(0)|²+2𝐸^𝜏₀𝑒^{−𝑟(𝑠)}⟨∇𝑝^𝑁_𝜀(𝑠),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠=𝐸^𝜏₀𝑒^{−𝑟(𝑠)}‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠−2𝐸^𝜏₀𝑒^{−𝑟(𝑠)}⟨𝜈𝐀^𝑁𝐮^𝑁_𝜀(𝑠)−𝐅^𝑁(𝐮^𝑁_𝜀(𝑠))+𝐁^𝑁(𝐮^𝑁_𝜀1(𝑠))+2𝑟(̇𝑠)𝐮^𝑁_𝜀(𝑠),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠. (4.40) Clearly, lim_𝑁→∞𝐸|𝜉^𝑁|²=𝐸|𝜉|². By (3.5), it is clear that 2𝐸^𝜏₀𝑒^{−𝑟(𝑠)}⟨∇𝑝^𝑁_𝜀(𝑠),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠=𝜀𝐸|𝜂^𝑁|²−𝜀𝐸𝑒^−𝑟(0)|𝑝^𝑁_𝜀(0)|²−1𝜀𝐸^𝜏₀𝑒^{−𝑟(𝑠)}‖𝑍^𝑁_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠. (4.41) Because of (4.40) and (4.41), one gets the following: lim_𝑁→∞2𝐸^𝜏₀𝑒^{−𝑟(𝑠)}⟨𝜈𝐀^𝑁𝐮^𝑁_𝜀(𝑠)−𝐅^𝑁(𝐮^𝑁_𝜀(𝑠))+𝐁^𝑁(𝐮^𝑁_𝜀1(𝑠))+2𝑟(̇𝑠)𝐮^𝑁_𝜀(𝑠),𝐮^𝑁_𝜀(𝑠)⟩𝑑𝑠=−𝐸|𝜉|²+lim_𝑁→∞𝐸𝑒^−𝑟(0)|𝐮^𝑁_𝜀(0)|²+lim_𝑁→∞𝐸^𝜏₀𝑒^{−𝑟(𝑠)}‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠−𝜀𝐸|𝜂|²+𝜀lim_𝑁→∞𝐸𝑒^−𝑟(0)|𝑝^𝑁_𝜀(0)|²+1𝜀lim_𝑁→∞𝐸^𝜏₀𝑒^{−𝑟(𝑠)}‖𝑍^𝑁_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠≥2𝐸^𝜏₀𝑒^{−𝑟(𝑠)}1⟨𝐀(𝑠)+𝐁(𝑠)+2𝑟(̇𝑠)𝐮_𝜀(𝑠),𝐮_𝜀(𝑠)⟩𝑑𝑠. (4.42) Note that one gets the last inequality by applications of the Itô formula to (4.37), and the fact that lim_𝑁→∞𝐸𝑒^−𝑟(0)|𝐮^𝑁_𝜀(0)|²≥𝐸𝑒^−𝑟(0)|𝐮_𝜀(0)|²,lim_𝑁→∞𝐸𝑒^−𝑟(0)|𝑝^𝑁_𝜀(0)|²≥𝐸𝑒^−𝑟(0)|𝑝_𝜀(0)|²,lim_𝑁→∞𝐸^𝜏₀𝑒^{−𝑟(𝑠)}‖𝐙^𝑁_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠≥𝐸^𝜏₀𝑒^{−𝑟(𝑠)}‖𝐙_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠,lim_𝑁→∞𝐸^𝜏₀𝑒^{−𝑟(𝑠)}‖𝑍^𝑁_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠≥𝐸^𝜏₀𝑒^{−𝑟(𝑠)}‖𝑍_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠. (4.43)Step 3 (Monotonicity). By Corollary 4.6, we get 𝐸^𝜏₀𝑒^{−𝑟(𝑠)}⟨𝜈𝐀𝐮^𝑁_𝜀(𝑠)+𝐁(𝐮^𝑁_𝜀(𝑠))−𝐅(𝐮^𝑁_𝜀1(𝑠))+2𝑟(̇𝑠)𝐮^𝑁_𝜀(𝑠),𝐮^𝑁_𝜀(𝑠)−𝐯(𝑠)⟩𝑑𝑠≤𝐸^𝜏₀𝑒^{−𝑟(𝑠)}1⟨𝜈𝐀𝐯(𝑠)+𝐁(𝐯(𝑠))−𝐅(𝐯(𝑠))+2𝑟(̇𝑠)𝐯(𝑠),𝐮^𝑁_𝜀(𝑠)−𝐯(𝑠)⟩𝑑𝑠. (4.44) Note that 𝐯∈𝐿^∞_ℱ([0,𝜏]×Ω;(ℍ¹₀)_𝑀) where 𝑀≤𝑁. An application of (4.42) yields 𝐸^𝜏₀𝑒^{−𝑟(𝑠)}1⟨𝐀(𝑠)+𝐁(𝑠)+2𝑟(̇𝑠)𝐮_𝜀(𝑠),𝐮_𝜀(𝑠)−𝐯(𝑠)⟩𝑑𝑠≤𝐸^𝜏₀𝑒^{−𝑟(𝑠)}1⟨𝜈𝐀𝐯(𝑠)+𝐁(𝐯(𝑠))−𝐅(𝐯(𝑠))+2𝑟(̇𝑠)𝐯(𝑠),𝐮_𝜀(𝑠)−𝐯(𝑠)⟩𝑑𝑠. (4.45) Since the above inequality is true for all 𝑀∈ℕ and 𝐾>0, it remains true for all 𝐯∈𝐿^∞_ℱ([0,𝜏]×Ω;ℍ¹₀). Thus let 𝐯=𝐮_𝜀+𝜆𝐰 where 𝐰∈𝐿^∞_ℱ([0,𝜏]×Ω;ℍ¹₀) and 𝜆>0, and 𝐸^𝜏₀𝑒^{−𝑟(𝑠)}⟨𝐀(𝑠)+𝐁(𝑠)−𝜈𝐀𝐮_𝜀(𝑠)−𝐁(𝐮_𝜀(𝑠)+𝜆𝐰(𝑠))+𝐅(𝐮_𝜀(𝑠)+𝜆𝐰(𝑠)),𝜆𝐰(𝑠)⟩𝑑𝑠≥𝐸^𝜏₀𝑒^{−𝑟(𝑠)}𝜆⟨𝜆𝜈𝐀𝐰(𝑠)+2𝑟(̇𝑠)𝐰(𝑠),𝜆𝐰(𝑠)⟩𝑑𝑠. (4.46) By the fact that ⟨𝐁(𝐮_𝜀(𝑡)+𝜆𝐰(𝑡)),𝐰(𝑡)⟩=−⟨𝐁(𝐮_𝜀(𝑡)+𝜆𝐰(𝑡),𝐰(𝑡)),𝐮_𝜀(𝑡)+𝜆𝐰(𝑡)⟩=−⟨𝐁(𝐮_𝜀(𝑡)+𝜆𝐰(𝑡),𝐰(𝑡)),𝐮_𝜀(𝑡)⟩=−⟨𝐁(𝐮_𝜀(𝑡),𝐰(𝑡)),𝐮_𝜀(𝑡)⟩−𝜆⟨𝐁(𝐰(𝑡),𝐰(𝑡)),𝐮_𝜀(𝑡)⟩=⟨𝐁(𝐮_𝜀(𝑡)),𝐰(𝑡)⟩+𝜆⟨𝐁(𝐰(𝑡),𝐮_𝜀(𝑡)),𝐰(𝑡)⟩, (4.47) we have 𝐸^𝜏₀𝑒^{−𝑟(𝑠)}⟨𝐀(𝑠)+𝐁(𝑠)−𝜈𝐀𝐮_𝜀(𝑠)−𝐁(𝐮_𝜀(𝑠))+𝐅(𝐮_𝜀(𝑠)+𝜆𝐰(𝑠)),𝐰(𝑠)⟩𝑑𝑠≥𝜆𝐸^𝜏₀𝑒^{−𝑟(𝑠)}⟨𝜈𝐀𝐰(𝑠)+𝐁(𝐰(𝑠),𝐮_𝜀1(𝑠))+2𝑟(̇𝑠)𝐰(𝑠),𝐰(𝑠)⟩𝑑𝑠. (4.48) Letting 𝜆→0, and by the arbitrariness of 𝐰 and the fact that 𝐅 is continuous, we know that 𝐀(𝑠)+𝐁(𝑠)=𝜈𝐀𝐮_𝜀(𝑠)+𝐁(𝐮_𝜀(𝑠))−𝐅(𝐮_𝜀(𝑠))P-a.s., (4.49) and this completes the proof.

5. Uniqueness, Continuity and Convergence of Solutions

5.1. Uniqueness and Continuity

The backward Navier-Stokes equation is well-posed if the regularity of the terminal condition in Proposition 4.9 is imposed. Only the uniqueness and continuity are left to check. Let us first prove the following lemma.

Lemma 5.1. For any 𝐮 and 𝐯 in ℍ¹₀ and 𝐰∈𝕃⁴, one has |⟨𝐁(𝐮)−𝐁(𝐯),𝐰⟩|≤𝐶(‖𝐮‖_𝕃⁴+‖𝐯‖_𝕃⁴)‖𝐮−𝐯‖‖𝐰‖_𝕃⁴+𝐶(‖𝐮‖+‖𝐯‖)‖𝐮−𝐯‖_𝕃⁴‖𝐰‖_𝕃⁴. (5.1)

Proof. By Proposition 2.5, |⟨𝐁(𝐮)−𝐁(𝐯),𝐰⟩|=|−⟨𝐁(𝐮,𝐰),𝐮⟩+⟨𝐁(𝐯,𝐰),𝐯⟩|=|−⟨𝐁(𝐮,𝐰),𝐮−𝐯⟩−⟨𝐁(𝐮,𝐰),𝐯⟩+⟨𝐁(𝐯,𝐰),𝐯⟩|=|−⟨𝐁(𝐮,𝐰),𝐮−𝐯⟩−⟨𝐁(𝐮−𝐯,𝐰),𝐯⟩|=|⟨𝐁(𝐮,𝐮−𝐯),𝐰⟩+⟨𝐁(𝐮−𝐯,𝐯),𝐰⟩|≤‖𝐮‖_𝕃⁴‖𝐮−𝐯‖‖𝐰‖_𝕃⁴+12‖𝐮‖‖𝐮−𝐯‖_𝕃⁴‖𝐰‖_𝕃⁴+‖𝐮−𝐯‖_𝕃⁴‖𝐯‖‖𝐰‖_𝕃⁴+12‖𝐮−𝐯‖‖𝐯‖_𝕃⁴‖𝐰‖_𝕃⁴≤𝐶(‖𝐮‖_𝕃⁴+‖𝐯‖_𝕃⁴)‖𝐮−𝐯‖‖𝐰‖_𝕃⁴+𝐶(‖𝐮‖+‖𝐯‖)‖𝐮−𝐯‖_𝕃⁴‖𝐰‖_𝕃⁴. (5.2)

Theorem 5.2. Let 𝜉∈𝐿^∞_ℱ𝜏(Ω;ℍ¹₀) and 𝜂∈𝐿^∞_ℱ𝜏(Ω;𝐻¹₀). System (4.12) admits a unique adapted solution in 𝐿^∞_ℱ([0,𝜏]×Ω;ℍ¹₀)×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_ℚ))×𝐿^∞_ℱ([0,𝜏]×Ω;𝐻¹₀)×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_𝑄)). (5.3) Also the solution is continuous with respect to the terminal conditions in 𝐿^∞([0,𝜏];𝐿²_ℱ(Ω;𝕃²))×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_ℚ))×𝐿^∞([0,𝜏];𝐿²_ℱ(Ω;𝐿²))×𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿_𝑄)). (5.4)

Proof. The existence of an adapted solution is shown in Theorem 4.10. Suppose that (𝐮_𝜀1,𝐙_𝜀1,𝑝_𝜀1,𝑍_𝜀1) and (𝐮_𝜀2,𝐙_𝜀2,𝑝_𝜀2,𝑍_𝜀2) are solutions of system (4.12) according to terminal conditions (𝜉₁,𝜂₁) and (𝜉₂,𝜂₂), respectively. The regularity of the solutions is guaranteed by Proposition 4.9. Denote 𝐮_𝜀=𝐮_𝜀1−𝐮_𝜀2,𝐙_𝜀=𝐙_𝜀1−𝐙_𝜀2,𝑍_𝜀=𝑍_𝜀1−𝑍_𝜀2,𝑝_𝜀=𝑝_𝜀1−𝑝_𝜀2,𝜉=𝜉₁−𝜉₂,𝜂=𝜂₁−𝜂₂. (5.5) Then one has 𝑑𝐮_𝜀(𝑡)+𝜈𝐀𝐮_𝜀(𝑡)𝑑𝑡+(𝐁(𝐮_𝜀1(𝑡))−𝐁(𝐮_𝜀2(𝑡)))𝑑𝑡+∇𝑝_𝜀(𝑡)𝑑𝑡,=(𝐅(𝐮_𝜀1(𝑡))−𝐅(𝐮_𝜀2(𝑡)))𝑑𝑡+𝐙_𝜀(𝑡)𝑑𝕎(𝑡),𝜀𝑑𝑝_𝜀(𝑡)+∇⋅𝐮_𝜀(𝑡)𝑑𝑡=𝑍_𝜀(𝑡)𝑑𝑊(𝑡),𝐮_𝜀(𝜏)=𝜉,𝑝_𝜀(𝜏)=𝜂. (5.6) Similar to Corollary 4.6, let us define 𝑙(𝑡)=^𝑇_𝑡{2𝛼+278(𝜅−𝜈)³𝐾²₀}𝑑𝑠, (5.7) where 𝐾₀ is the constant in Proposition 4.9. An application of the Itô formula to 𝑒^{−𝑙(𝑡)}|𝐮_𝜀(𝑡)|² and Corollary 4.6 imply 𝑒^{−𝑙(𝑡∧𝜏)}|𝐮_𝜀(𝑡∧𝜏)|²+^𝜏_𝑡∧𝜏𝑒^{−𝑙(𝑠)}‖𝐙_𝜀(𝑠)‖²_𝐿ℚ𝑑𝑠=|𝜉|²+2^𝜏_𝑡∧𝜏𝑒^{−𝑙(𝑠)}⟨𝜈𝐀𝐮_𝜀(𝑠)−(𝐁(𝐮_𝜀1(𝑠))−𝐁(𝐮_𝜀2(𝑠)))−(𝐅(𝐮_𝜀1(𝑠))−𝐅(𝐮_𝜀2+1(𝑠)))2𝑙(̇𝑠)𝐮_𝜀(𝑠),𝐮_𝜀(𝑠)⟩𝑑𝑠+2^𝜏_𝑡∧𝜏𝑒^{−𝑙(𝑠)}⟨∇𝑝_𝜀(𝑠),𝐮_𝜀(𝑠)⟩𝑑𝑠−2^𝜏_𝑡∧𝜏𝑒^{−𝑙(𝑠)}⟨𝐙_𝜀(𝑠)𝑑𝕎(𝑠),𝐮_𝜀(𝑠)⟩≤|𝜉|²+𝜀^𝜏_𝑡∧𝜏𝑒^{−𝑙(𝑠)}𝑑|𝑝_𝜀(𝑠)|²−1𝜀^𝜏_𝑡∧𝜏𝑒^{−𝑙(𝑠)}‖𝑍_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠−2^𝜏_𝑡∧𝜏𝑒^{−𝑙(𝑠)}⟨𝑍_𝜀(𝑡)𝑑𝑊(𝑠),𝑝_𝜀(𝑠)⟩𝑑𝑠−2^𝜏_𝑡∧𝜏𝑒^{−𝑙(𝑠)}⟨𝐙_𝜀(𝑠)𝑑𝕎(𝑠),𝐮_𝜀(𝑠)⟩. (5.8) Taking the expectation, the above inequality becomes 𝐸|𝐮_𝜀(𝑡∧𝜏)|²+𝜀𝐸|𝑝_𝜀(𝑡∧𝜏)|²+𝐸^𝜏_𝑡∧𝜏‖𝐙_𝜀(𝑠)‖²_𝐿ℚ1𝑑𝑠+𝜀𝐸^𝜏_𝑡∧𝜏‖𝑍_𝜀(𝑠)‖²_𝐿𝑄𝑑𝑠≤𝑒^−𝑙(0){𝐸|𝜉|²+𝜀𝐸|𝜂|²}. (5.9) Thus we have proved the uniqueness and continuity of system (4.12).

Remark 5.3. The uniqueness and continuity with weaker terminal conditions, such as when the terminal conditions are uniformly bounded in 𝐿² sense, are still open. The difficulty lies in the nonadaptiveness nature of the backward system. For instance, the function 𝑙₁ defined in Corollary 4.6 is not ℱ_𝑡 adapted. This is why we defined another function 𝑙(𝑡) in the proof of the uniqueness based on the ℍ¹₀-bound of the solution. Fortunately, 𝑙(𝑡) is ℱ_𝑡 adapted and has similar properties as 𝑙₁(𝑡). One can also show the uniqueness and continuity using Lemma 5.1, without introducing the function 𝑙(𝑡).

5.2. The Convergence of the Solution As ɛ Approaches Zero

It is very interesting to study the asymptotic behavior of stochastic Navier-Stokes system with artificial compressibility. We are going to show that as artificial compressibility vanishes, the limit of the solution becomes the solution of the corresponding Navier-Stokes system for a viscous incompressible flow given below:𝑑𝐮(𝑡)=−𝜈𝐀𝐮(𝑡)𝑑𝑡−𝐁(𝐮(𝑡))𝑑𝑡−∇𝑝(𝑡)+𝐅(𝑡)𝑑𝑡+𝐙(𝑡)𝑑𝐖(𝑡),∇⋅𝐮(𝑡)=0,𝐮(𝜏)=𝜉,𝑝(𝜏)=𝜂, (5.10) where 𝐀𝐮≜−(Δ𝐮) and 𝐁(𝐮,𝐯)≜((𝐮⋅∇)𝐯) with the notation 𝐁(𝐮)=𝐁(𝐮,𝐮) (see Temam [3]).

Theorem 5.4. Assume the conditions in Theorem 4.10(ii). Then as 𝜀 approaches 0, the first three elements in the solution of (4.12), (𝐮_𝜀,𝐙_𝜀,𝑝_𝜀), converge to (𝐮,𝐙,𝑝), the solution of (5.10).

Proof. Similar to Step 1 of the proof of Theorem 4.10, we know that there exist 𝐮, 𝑝, 𝐙 and a sequence of positive numbers {𝜀_𝑖}^∞_𝑖=1 such that 𝜀_𝑖→0, 𝐮−→_𝜀𝑖^𝑤𝐮, 𝑝−→_𝜀𝑖^𝑤𝑝 and 𝐙−→_𝜀𝑖^𝑤𝐙 in corresponding spaces.
From (4.18) and (4.20), one knows that along a subsequence,𝐸⟨√𝜀_𝑖𝑑𝑝_𝜀𝑖𝑑𝑡,ℎ⟩→𝐸⟨𝑑𝐷𝑑𝑡,ℎ⟩ (5.11) for some 𝐷∈𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿²)) and for all ℎ∈𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿²)). Thus we get 𝐸⟨𝜀_𝑖𝑑𝑝_𝜀𝑖𝑑𝑡,ℎ⟩→0 (5.12) in the sense of distribution. Since ⟨𝜀𝑑𝑝_𝜀𝑖(𝑡)+∇⋅𝐮_𝜀𝑖(𝑡)𝑑𝑡,ℎ(𝑡)⟩=⟨𝑍_𝜀𝑖(𝑡)𝑑𝑊(𝑡),ℎ(𝑡)⟩, (5.13) we know that 𝐸^𝜏₀⟨∇⋅𝐮(𝑡),ℎ(𝑡)⟩𝑑𝑡=lim_𝑖→∞𝐸^𝜏₀⟨∇⋅𝐮_𝜀𝑖(𝑡),ℎ(𝑡)⟩𝑑𝑡=−lim_𝑖→∞𝐸^𝜏₀⟨𝜀𝑑𝑝_𝜀𝑖(𝑡),ℎ(𝑡)⟩=0 (5.14) for all ℎ∈𝐿²_ℱ(Ω;𝐿²(0,𝜏;𝐿²)). So ∇⋅𝐮= 0 P-a.s. This shows that the limiting system is incompressible.
Similar to Steps 2 and 3 in the proof of Theorem 4.10, we are able to show that (𝐮,𝐙,𝑝) solves (5.10).

Acknowledgments

The author thanks Professor P. Sundar for his helpful discussion and insightful suggestions. The author also thanks the anonymous referee for offering valuable comments and suggestions on the earlier version of the paper.