# Stochastic Navier-Stokes Equations with Artificial Compressibility in Random Durations

Authors: Hong Yin
Journal: International Journal of Stochastic Analysis
 (2010)

---

## 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.


---

## Body

## 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 (1.1)∂u+(u·∇)udt-νΔudt=-∇pdt+f(t)dt+σ(t,u)d𝕎(t),
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 ɛ: (1.2)∂tuɛ-νΔuɛ+(uɛ·∇)uɛ+12(∇·uɛ)uɛ+∇pɛ=f,ɛ∂tpɛ+∇·uɛ=0.
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 boundedL4(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 ℍ01 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 Section2. 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 ɛ approaches zero are also studied.


## 2. Preliminaries

Suppose thatG is a domain bounded in ℝ2 with smooth boundary conditions. Let ɛ be a positive parameter which vanishes to 0. The artificial state equation for a slightly compressible medium is defined as (2.1)ρ=ρ0+ɛp,
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 ɛ: (2.2)∂tuɛ-νΔuɛ+(uɛ·∇)uɛ+12(∇·uɛ)uɛ+∇pɛ=f,ɛ∂tpɛ+∇·uɛ=0,
where uɛ∈𝕃2=𝕃2(G) is the velocity, pɛ∈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, (·,·)ℍ01 the inner product of ℍ01=ℍ01(G), ℍ-1 the dual space of ℍ01, and 〈·,·〉 the duality pairing between ℍ01 and ℍ-1. Let |·| be the norm of 𝕃2 and let ∥·∥ be the norm of ℍ01. Without causing any confusion, we also use the same notations to denote the norms of L2 and H01=H01(G). For any x∈𝕃2 and y∈ℍ01, 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 {ej}j=1∞∈𝕃2∩ℍ01∩𝕃4 be a complete orthonormal system in 𝕃2 such that there exists a nondecreasing sequence of positive numbers {λj}j=1∞, limj→∞λj=∞ and -Δej=λjej for all j. Let ℚek=qkek with ∑k=1∞qk<∞, and {bk(t)} be a sequence of independent standard Brownian motions in ℝ. Then Wiener process 𝕎(t) is taken as 𝕎(t)=∑k=1∞qkbk(t)ek.LetQ be a trace class operator on L2. Similarly, we can define a complete orthonormal system {ej}j=1∞, a nondecreasing sequence of positive numbers {κj}j=1∞ such that -Δej=κjej, and positive numbers qj′ such that Qej=qj′ej and ∑j=1∞qj′<∞. Let W(t)=∑j=1∞qj′bj(t)ej. Then W(t) is an L2-valued Q-Wiener process. From now on, let {ℱt} 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(2.3)〈F,G〉Lℚ=tr(FℚG*)=tr(GℚF*)
for all F and G∈Lℚ, let Lℚ denote the space of linear operators E such that Eℚ1/2 is a Hilbert-Schmidt operator from 𝕃2 to 𝕃2. Similarly, we define LQ for Q, the trace class operator on L2.To be realistic in nature, let us introduce randomness into the system to obtain(2.4)∂uɛ(t)∂t-νΔuɛ(t)+(uɛ(t)·∇)uɛ(t)+12(∇·uɛ(t))uɛ(t)+∇pɛ(t)=f(t)+σ(t)d𝕎(t)dt,ɛ∂pɛ(t)+∇·uɛ(t)dt=0,uɛ(0)=u0,pɛ(0)=p0,
where u0 and p0 are initial conditions, and σ(d𝕎/dt) is the noise term. Here (1/2)(∇·uɛ)uɛ is called the stabilization term.If a terminal timeT is given and the terminal conditions are specified as uɛ(T)=ξ and pɛ(T)=η, one obtains a backward system: (2.5)duɛ(t)+νÂuɛ(t)dt+B̂(uɛ(t))dt+∇pɛ(t)dt=f(t)dt+Zɛ(t)d𝕎(t),ɛdpɛ(t)+∇·uɛ(t)dt=Zɛ(t)dW(t),uɛ(T)=ξ,pɛ(T)=η
for 0≤t≤T, where Âu≜-Δu and B̂(u,v)≜(u·∇)v+(1/2)(∇·u)v, with the notation B̂(u)≜B̂(u,u). The processes Zɛ and Zɛ are in spaces Lℚ and LQ, respectively.Letτ be a ℱt-stopping time when the observations are available. Suppose that the observed velocity and pressure at τ are uɛ(τ)=ξ∈Lℱτ2(Ω;𝕃2) and p(τ)=η∈Lℱτ2(Ω;L2), respectively. Then we introduce the backward stochastic Navier-Stokes equation with artificial compressibility and stabilization in random duration:(2.6)duɛ(t)+νÂuɛ(t)dt+B̂(uɛ(t))dt+∇pɛ(t)dt=f(t)dt+Zɛ(t)d𝕎(t),ɛdpɛ(t)+∇·uɛ(t)dt=Zɛ(t)dW(t),uɛ(τ)=ξ,pɛ(τ)=η
for 0≤t≤τ, where the ℱt-stopping time τ is assumed to be bounded by a time T>0. Note that processes Zɛ and Zɛ 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 ofℱt-Adapted processes (uɛ,Zɛ,pɛ,Zɛ) is called a solution of backward Navier-Stokes equation (2.6) if it satisfies the integral form of the system
(2.7)uɛ(t∧τ)=ξ+∫t∧ττ{νÂuɛ(s)+B̂(uɛ(s))+∇pɛ(s)-f(s)}ds-∫t∧ττZɛ(s)d𝕎(s),ɛpɛ(t∧τ)=ɛη+∫t∧ττ∇·uɛ(s)ds-∫t∧ττZɛ(s)dW(s),
P-a.s., and the following holds:  (a)
uɛ∈Lℱ2(Ω;L∞(0,τ;𝕃2))∩Lℱ2(Ω;L2(0,τ;ℍ01));(b)
Zɛ∈Lℱ2(Ω;L2(0,τ;Lℚ));(c)
pɛ∈Lℱ2(Ω;L∞(0,τ;L2))∩Lℱ2(Ω;L2(0,τ;H01));(d)
Zɛ∈Lℱ2(Ω;L2(0,τ;LQ)).The following simple results are frequently used and given as lemmas. Readers may refer to Temam [3] for similar proofs.Lemma 2.2.
For anyu,v,w∈ℍ01 and p∈L2, one has  (1)
〈Âu,w〉=∑i,j∫G∂iuj∂iwjdx=〈Âw,u〉=(u,w)ℍ01,(2)
〈(u·∇)v,w〉=∑i,j∫Gui(∂ivj)wjdx,(3)
〈(u·∇)v,w〉= -〈(∇·u)w,v〉-〈(u·∇)w,v〉,(4)
〈B̂(u,v),w〉=-〈B̂(u,w),v〉,(5)
〈-∇p,u〉=-∑i∫G∂ipuidx=∫Gp∂iuidx=〈p,∇·u〉.Remark 2.3.
Sometimes〈B̂(u,v),w〉 is denoted by b̂(u,v,w).Lemma 2.4.
The following results hold for any real-valued smooth functionsϕ and ψ with compact support in ℝ2:
(2.8)|ϕψ|2≤C∥ϕ∂1ϕ∥𝕃1∥ψ∂2ψ∥𝕃1,∥ϕ∥𝕃44≤C|ϕ|2|∇ϕ|2.Proposition 2.5.
For anyu and v in ℍ01 and w∈𝕃4, one has
(2.9)|b̂(u,v,w)|≤∥u∥𝕃4∥v∥∥w∥𝕃4+12∥u∥∥v∥𝕃4∥w∥𝕃4.Below is a backward version of the Gronwall inequality used frequently in this paper, and the proof is straightforward.Lemma 2.6.
Suppose thatg(t),α(t),β(t), and γ(t) are integrable functions, and β(t),γ(t) are nonnegative functions. For 0≤t≤T, if
(2.10)g(t)≤α(t)+β(t)∫tTγ(ρ)g(ρ)dρ,
then
(2.11)g(t)≤α(t)+β(t)∫tTα(η)γ(η)e∫tηβ(ρ)γ(ρ)dρdη.
In particular, if α(t)≡α, β(t)≡β and γ(t)≡1, then
(2.12)g(t)≤αeβ(T-t).


## 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 PN:𝕃2→𝕃N2, where 𝕃N2=span{e1,e2,…,eN}, 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 operatorsÂN≜PNÂ and B̂N≜PNB̂. Under projection PN, let us construct a finite dimensional system. Let (3.1)𝕎N(t)≜PN𝕎(t)=∑i=1Nqibi(t)ei,WN(t)≜PNW(t)=∑i=1Nqi′bi(t)ei,fN(t)≜PNf(t),ξN≜E(PNξ∣ℱτN),ηN≜E(PNη∣ℱτN),
where {ℱtN} is the natural filtration of {𝕎N(t)} and {WN(t)}. The projected system with solution (uɛN,ZɛN,pɛN,ZɛN) is defined as follows:(3.2)duɛN(t)=-νÂNuɛN(t)dt-B̂N(uɛN(t))dt-∇pɛN(t)dt+fN(t)dt+ZɛN(t)d𝕎N(t),ɛdpɛN(t)+∇·uɛN(t)dt=ZɛN(t)dWN(t),uɛN(τ)=ξN,pɛN(τ)=ηN
for 0≤t≤τ.Proposition 3.1.
Letξ∈Lℱτ∞(Ω;𝕃2), η∈Lℱτ∞(Ω;L2), and f∈L2(0,T;ℍ-1). Then for any solution of system (3.2), the following is true:
(3.3)(uɛN,ZɛN)∈{Lℱ∞([0,τ]×Ω;𝕃2)∩Lℱ2(Ω;L2(0,τ;ℍ01))}×Lℱ2(Ω;L2(0,τ;Lℚ)),(pɛN,ZɛN)∈{Lℱ∞([0,τ]×Ω;L2)∩Lℱ2(Ω;L2(0,τ;H01))}×Lℱ2(Ω;L2(0,τ;LQ)).Proof.
Applying the Itô formula to|pɛN(t)|2 to get
(3.4)d|pɛN(t)|2=-2ɛ〈∇·uɛN(t),pɛN(t)〉dt+2ɛ〈ZɛN(t)dWN(t),pɛN(t)〉+1ɛ2tr[ZɛN(t)Q(ZɛN(t))*]dt=2ɛ〈∇pɛN(t),uɛN(t)〉dt+2ɛ〈ZɛN(t)dWN(t),pɛN(t)〉+1ɛ2tr[ZɛN(t)Q(ZɛN(t))*]dt,
thus we get
(3.5)2〈∇pɛN(t),uɛN(t)〉dt=ɛd|pɛN(t)|2-2〈ZɛN(t)dWN(t),pɛN(t)〉-1ɛ∥ZɛN(t)∥LQ2dt.
By means of the Itô formula, one has
(3.6)|uɛN(t∧τ)|2=|ξN|2+2∫t∧ττ〈νÂNuɛN(s)+B̂N(uɛN(s))+∇pɛN(s)-fN(s),uɛN(s)〉ds-2∫t∧ττ〈ZɛN(s)d𝕎N(s),uɛN(s)〉-∫t∧ττ∥ZɛN(s)∥Lℚ2ds.
Clearly,
(3.7)〈B̂N(uɛN(s)),uɛN(s)〉=0,
and Lemma 2.2 yields
(3.8)|2〈fN(s),uɛN(s)〉|≤∥fN(s)∥ℍ-12+∥uɛN(s)∥2=|fN(s)|ℍ-12+〈ÂNuɛN(s),uɛN(s)〉.
For 0<r≤t, taking the conditional expectation with respect to ℱr∧τ, and by (3.5), the above two equation and along with the fact that ∥uɛ(s)∥2=〈Âuɛ(s),uɛ(s)〉, one gets
(3.9)Eℱr∧τ|uɛN(t∧τ)|2+Eℱr∧τ∫t∧ττ∥ZɛN(s)∥Lℚ2ds+Eℱr∧τ∫t∧ττ∥uɛN(s)∥2ds≤Eℱr∧τ|ξN|2+2(ν+1)Eℱr∧τ∫t∧ττ〈ÂNuɛN(s),uɛN(s)〉ds+Eℱr∧τ∫t∧ττ∥fN(s)∥ℍ-12ds+ɛEℱr∧τ∫t∧ττd|pɛN(s)|2-1ɛEℱr∧τ∫t∧ττ∥ZɛN(s)∥LQ2ds,
P-a.s. Since Âej=λjej and λi≤λj for i<j, one gets
(3.10)〈ÂNuɛN(s),uɛN(s)〉≤λN|uɛN(s)|2.
Thus
(3.11)Eℱr∧τ|uɛN(t∧τ)|2+ɛEℱr∧τ|pɛN(t∧τ)|2+Eℱr∧τ∫t∧ττ∥uɛN(s)∥2ds+Eℱr∧τ∫t∧ττ∥ZɛN(s)∥Lℚ2ds+1ɛEℱr∧τ∫t∧ττ∥ZɛN(s)∥LQ2ds≤Eℱr∧τ|ξN|2+ɛEℱr∧τ|ηN|2+2(ν+1)λN∫tTEℱr∧τ|uɛN(s∧τ)|2ds+Eℱr∧τ∫0τ∥fN(s)∥ℍ-12ds,
P-a.s., and by Lemma 2.6, the backward Gronwall inequality, and letting r=t, we get
(3.12)|uɛN(t∧τ)|2+ɛ|pɛN(t∧τ)|2+Eℱt∧τ∫t∧ττ∥uɛN(s)∥2ds+Eℱt∧τ∫t∧ττ∥ZɛN(s)∥Lℚ2ds+1ɛEℱt∧τ∫t∧ττ∥ZɛN(s)∥LQ2ds≤(Eℱt∧τ|ξN|2+ɛEℱt∧τ|ηN|2+Eℱt∧τ∫0τ∥fN(s)∥ℍ-12ds)e2(ν+1)λN(T-t),
P-a.s. Because of the integrability of ξ, η, and f, there exists a constant KN, depending on N only, s.t. 							(3.13)|uɛN(t)|2+ɛ|pɛN(t)|2+E∫0τ∥uɛN(s)∥2ds+E∫0τ∥ZɛN(s)∥Lℚ2ds+E∫0τ∥ZɛN(s)∥LQ2ds≤KN,
for all t∈[0,τ], P-a.s.
Similarly, making use of (3.4), it follows that pɛN∈Lℱ2(Ω;L2(0,τ;H01)).Proposition 3.2.
Letξ∈Lℱτn(Ω;𝕃2), η∈Lℱτn(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):
(3.14)uɛN∈L∞(0,τ;Lℱn(Ω;𝕃2))∩Lℱn(Ω;Ln(0,τ;ℍ01)),pɛN∈L∞(0,τ;Lℱn(Ω;L2))∩Lℱn(Ω;Ln(0,τ;H01)).Proof.
Let us prove it by the method of mathematical induction. Similar to Proposition3.1, 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|uɛN(t)|n yields(3.15)|uɛN(t∧τ)|n=|ξN|n+n∫t∧ττ|uɛN(s)|n-2〈νÂNuɛN(s)+B̂N(uɛN(s))+∇pɛN(s)-fN(s),uɛN(s)〉ds-n∫t∧ττ|uɛN(s)|n-2〈ZɛN(s)d𝕎N(s),uɛN(s)〉-n2-n2∫t∧ττ|uɛN(s)|n-2∥ZɛN(s)∥Lℚ2ds.
Clearly |∇pɛN(s)|≤C∥pɛN(s)∥≤CκN|pɛN(s)|, where κN, as stated in Section 2, is the eigenvalue of -Δ for eN. Taking the expectation, one obtains
(3.16)E|uɛN(t∧τ)|n+E∫t∧ττ∥uɛN(s)∥nds≤E|ξN|n+λNn/2E∫t∧ττ|uɛN(s)|nds+nE∫t∧ττ|uɛN(s)|n-2〈νÂNuɛN(s)+∇pɛN(s)-fN(s),uɛN(s)〉ds≤E|ξN|n+(νλNn+λNn/2)E∫t∧ττ|uɛN(s)|nds+nE∫t∧ττ|uɛN(s)|n-1|∇pɛN(s)|ds+n2E∫t∧ττ|uɛN(s)|n-2(∥fN(s)∥ℍ-12+λN|uɛN(s)|2)ds≤E|ξN|n+(νλNn+λNn/2+n2λN)E∫t∧ττ|uɛN(s)|nds+nCκN{E∫t∧ττ|uɛN(s)|nds}(n-1)/n{E∫t∧ττ|pɛN(s)|nds}1/n+n2∫tT∥fN(s)∥ℍ-12E{1[t∧τ,τ]|uɛN(s)|n-2}ds≤E|ξN|n+n2{sup0≤t≤τE|uɛN(t)|n-2}∫tT∥fN(s)∥ℍ-12ds+(νλNn+λNn/2+n2λN+(n-1)CκN)∫tTE|uɛN(s∧τ)|nds+CκN∫tTE|pɛN(s∧τ)|nds≤K+K(n,N)∫tTE|uɛN(s∧τ)|nds+K(n,N)∫tTE|pɛN(s∧τ)|nds,
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 |pɛN(t)|n, one obtains
(3.17)ɛE|pɛN(t∧τ)|n+E∫t∧ττ∥pɛN(s)∥nds≤ɛE|ηN|n+κNn/2E∫t∧ττ|pɛN(s)|nds-nE∫t∧ττ|pɛN(s)|n-2〈∇pɛN(s),uɛN(s)〉ds≤ɛE|ηN|n+κNn/2E∫t∧ττ|pɛN(s)|nds+nCκNE∫t∧ττ|pɛN(s)|n-1|uɛN(s)|ds≤ɛE|ηN|n+κNn/2E∫t∧ττ|pɛN(s)|nds+nCκN{E∫t∧ττ|pɛN(s)|nds}(n-1)/n×{E∫t∧ττ|uɛN(s)|nds}1/n≤K+K(n,N)∫tTE|pɛN(s∧τ)|nds+K(n,N)∫tTE|uɛN(s∧τ)|nds.
Adding up (3.16) and (3.17), one gets
(3.18)E{|uɛN(t∧τ)|n+ɛ|pɛN(t∧τ)|n}+E∫t∧ττ{∥uɛN(s)∥n+∥pɛN(s)∥n}ds≤K+K(n,N)∫tTE{|uɛN(s∧τ)|n+|pɛN(s∧τ)|n}ds.
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 Proposition2.5 and is straightforward.Lemma 4.1.
Assumeu,v∈ℍ01 and w∈𝕃4. The following inequalities are true:  (a)
|〈B̂(u),w〉|≤2∥u∥3/2|u|1/2∥w∥𝕃4,(b)
|〈B̂(u)-B̂(v),u-v〉|≤(ν/2)∥u-v∥2+(27/2ν3)|u-v|2∥v∥𝕃44,(c)
〈νÂ(u-v)+B̂(u)-B̂(v),u-v〉+(27/2ν3)∥v∥𝕃44|u-v|2≥(ν/2)∥u-v∥2.Furthermore, if w∈ℍ01, then there exists a constant C depending on ν, such that  (d)
〈νÂ(u-v)+B̂(u)-B̂(v),u-v〉+C∥v∥2|u-v|2≥(ν/2)∥u-v∥2.Corollary 4.2.
For anyu and v∈𝕃4, let
(4.1)h1(t)=27ν3∫0t∥u(s)∥𝕃44ds,h2(t)=27ν3∫0t∥v(s)∥𝕃44ds.
Then
(4.2)〈νÂ(u-v)+B̂(u)-B̂(v)+12ḣi(t)(u-v),u-v〉≥0,i=1,2.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.
Letξ∈Lℱτ∞(Ω;𝕃2), η∈Lℱτ∞(Ω;L2), and f∈L2(0,T;ℍ-1). Then the projected system (3.2) admits a unique adapted solution (uɛN,ZɛN,pɛN,ZɛN) in
(4.3){Lℱ∞([0,τ]×Ω;𝕃2)∩Lℱ2(Ω;L2(0,τ;ℍ01))}×Lℱ2(Ω;L2(0,τ;Lℚ))×{Lℱ∞([0,τ]×Ω;L2)∩Lℱ2(Ω;L2(0,τ;H01))}×Lℱ2(Ω;L2(0,τ;LQ)).Proof.
For everyM∈ℕ, let LM be a Lipschitz C∞ function which has the following property:
(4.4)LM(∥u∥)={1if∥u∥<M,0if∥u∥>M+1,0≤LM(∥u∥)≤1otherwise.
Applying the truncation LM to B̂, it is easy to show that LMB̂ is Lipschitz and
(4.5)|LM(∥x∥)B̂N(x)-LM(∥y∥)B̂N(y)|≤CN,M∥x-y∥
for any x,y∈𝕃N2 and M∈ℕ. Let us define a truncated projected system:
(4.6)duɛN,M(t)=-νÂNuɛN,M(t)dt-LM(∥uɛN,M(t)∥)B̂N(uɛN,M(t))dt-∇pɛN,M(t)dt+fN(t)dt+ZɛN,M(t)d𝕎N(t),ɛdpɛN,M(t)+∇·uɛN,M(t)dt=ZɛN,M(t)dWN(t),uɛN,M(τ)=ξN,pɛN,M(τ)=ηN.
For fixed p∈Lℱ∞([0,τ]×Ω;L2)∩Lℱ2(Ω;L2(0,τ;H01)), let us map
(4.7)duɛN,M(t)=-νÂNuɛN,M(t)dt-LM(∥uɛN,M(t)∥)B̂N(uɛN,M(t))dt-∇pN,M(t)dt+fN(t)dt+ZɛN,M(t)d𝕎N(t),uɛN,M(τ)=ξN
to ℝN, 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 ℝN (see [14, page 355]) guarantees the existence of a unique adapted solution. Let the solution be (uɛN,M,ZɛN,M). Then for
(4.8)ɛdpɛN,M(t)+∇·uɛN,M(t)dt=ZɛN,M(t)dWN(t)pɛN,M(τ)=ηN,
there is a unique adapted solution (pɛN,M,ZɛN,M). Thus we can define an operator Ψ, such that Ψ(p)=pɛN,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 byF. 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: ℍ01→ℍ-1 is a continuous operator.
(A.2) There exist positive constantsα and β, such that(4.9)〈νÂu-F(u),u〉≤α|u|2-β∥u∥2;〈νÂu-F(u),Âu〉≤α∥u∥2-β∥Âu∥2.
(A.3) For anyu and v in ℍ01, a constant κ>ν, and a positive constant α,(4.10)〈κÂ(u-v)-(F(u)-F(v)),u-v〉≤α|u-v|2.
(A.4) For anyu∈ℍ01 and some positive constant α,(4.11)|〈F(u),u〉|≤α∥u∥2.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 Section5. 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:(4.12)duɛ(t)+νÂuɛ(t)dt+B̂(uɛ(t))dt+∇pɛ(t)dt=F(uɛ(t))dt+Zɛ(t)d𝕎(t),ɛdpɛ(t)+∇·uɛ(t)dt=Zɛ(t)dW(t),uɛ(τ)=ξ,pɛ(τ)=η,(4.13)duɛN(t)=-νÂNuɛN(t)dt-B̂N(uɛN(t))dt-∇pɛN(t)dt+FN(uɛN(t))dt+ZɛN(t)d𝕎N(t),ɛdpɛN(t)+∇·uɛN(t)dt=ZɛN(t)dWN(t),uɛN(τ)=ξN,pɛN(τ)=ηN
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.
Assumeu and v∈𝕃4. Then the following inequality is true:
(4.14)|〈B̂(u)-B̂(v),u-v〉|≤(κ-ν)∥u-v∥2+2716(κ-ν)3|u-v|2∥v∥𝕃44.Corollary 4.6.
Letu and v∈𝕃4. Define
(4.15)l1(t)≜∫tT{2α+278(κ-ν)3∥u(s)∥𝕃44}ds,l2(t)≜∫tT{2α+278(κ-ν)3∥v(s)∥𝕃44}ds.
Then
(4.16)〈νÂ(u-v)+B̂(u)-B̂(v)-(F(u)-F(v))+12l̇i(t)(u-v),u-v〉≤0,i=1,2.Remark 4.7.
To prove Corollary4.6, the monotonicity assumption (A.3) is used.Proposition 4.8.
(i) Letξ∈Lℱτ∞(Ω;𝕃2) and η∈Lℱτ∞(Ω;L2). Then for any solution of system (4.13), the following is true:
(4.17)(uɛN,ZɛN)∈{Lℱ∞([0,τ]×Ω;𝕃2)∩Lℱ2(Ω;L2(0,τ;ℍ01))}×Lℱ2(Ω;L2(0,τ;Lℚ)),(pɛN,ZɛN)∈Lℱ∞([0,τ]×Ω;L2)×Lℱ2(Ω;L2(0,τ;LQ)).
Moreover, there exists a constant K, independent of N, such that
(4.18)supt∈[0,τ]|uɛN(t)|2+E∫0τ∥uɛN(s)∥2ds+ɛsupt∈[0,τ]|pɛN(t)|2+E∫0τ∥ZɛN(s)∥Lℚ2ds+E∫0τ∥ZɛN(s)∥LQ2ds≤K,
P-a.s.
(ii) Letξ∈Lℱτ2(Ω;𝕃2) and η∈Lℱτ2(Ω;L2). The following is true for any solution of system (4.13):(4.19)(uɛN,ZɛN)∈{L∞(0,τ;Lℱ2(Ω;𝕃2))∩Lℱ2(Ω;L2(0,τ;ℍ01))}×Lℱ2(Ω;L2(0,τ;Lℚ)),(pɛN,ZɛN)∈L∞(0,τ;Lℱ2(Ω;L2))×Lℱ2(Ω;L2(0,τ;LQ)).
Moreover, there exists a constant K, independent of N, such that
(4.20)supt∈[0,τ]E|uɛN(t)|2+E∫0τ∥uɛN(s)∥2ds+ɛsupt∈[0,τ]E|pɛN(t)|2+E∫0τ∥ZɛN(s)∥Lℚ2ds+E∫0τ∥ZɛN(s)∥LQ2ds≤K.Proof.
(i) Similar to the proof of Proposition3.1, utilizing Assumption (A.2), (3.6) becomes
(4.21)|uɛN(t∧τ)|2=|ξN|2+2∫t∧ττ〈νÂNuɛN(s)+B̂N(uɛN(s))+∇pɛN(s)-FN(uɛN(s)),uɛN(s)〉ds-2∫t∧ττ〈ZɛN(s)d𝕎N(s),uɛN(s)〉-∫t∧ττ∥ZɛN(s)∥Lℚ2ds≤|ξN|2+2∫t∧ττ{α|uɛN(s)|2-β∥uɛN(s)∥2+〈∇pɛN(s),uɛN(s)〉}ds-2∫t∧ττ〈ZɛN(s)d𝕎N(s),uɛN(s)〉-∫t∧ττ∥ZɛN(s)∥Lℚ2ds.
For 0<r≤t, taking the conditional expectation with respect to ℱr∧τ, one gets
(4.22)Eℱr∧τ|uɛN(t∧τ)|2+Eℱr∧τ∫t∧ττ∥ZɛN(s)∥Lℚ2ds+2βEℱr∧τ∫t∧ττ∥uɛN(s)∥2ds≤Eℱr∧τ|ξN|2+2αEℱr∧τ∫t∧ττ|uɛN(s∧τ)|2ds+ɛEℱr∧τ∫t∧ττd|pɛN(s)|2-1ɛEℱr∧τ∫t∧ττ∥ZɛN(s)∥LQ2ds,
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 thatξ∈Lℱτ∞(Ω;ℍ01) and η∈Lℱτ∞(Ω;H01). Then for any solution (uɛ,Zɛ,pɛ,Zɛ) of system (4.13), there exists a constant K0, such that
(4.23)supt∈[0,τ]∥uɛ(t)∥2+supt∈[0,τ]∥pɛ∥2≤K0.Proof.
The proof involves an application of the Itô formula to∥uɛ(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.
Letξ∈Lℱτ∞(Ω;ℍ01) and η∈Lℱτ∞(Ω;H01). For system (4.12), there exists a solution (uɛ,Zɛ,pɛ,Zɛ) in
(4.24)Lℱ∞([0,τ]×Ω;ℍ01)×Lℱ2(Ω;L2(0,τ;Lℚ))×Lℱ∞([0,τ]×Ω;H01)×Lℱ2(Ω;L2(0,τ;LQ)).Proof.
We have the following steps.Step 1 (The limits).
Clearly, by Proposition4.8, there exist uɛ, pɛ, Zɛ, and Zɛ, such that
(4.25)uɛNk→wuɛinLℱ2(Ω;L2(0,τ;ℍ01)),pɛNk→wpɛinLℱ∞([0,τ]×Ω;L2),ZɛNk→wZɛinLℱ2(Ω;L2(0,τ;Lℚ)),ZɛNk→wZɛinLℱ2(Ω;L2(0,τ;LQ)),
for a subsequence Nk. Since Â is a continuous map from ℍ01 to ℍ-1,
(4.26)∥Âuɛ∥ℍ-1≤C∥uɛ∥
for all uɛ∈ℍ01 and some constant C. Thus combined with the assumptions on F, one knows that
(4.27)νÂNkuɛNk-FNk(uɛNk)→wAinLℱ2(Ω;L2(0,τ;ℍ-1))
for some function A and some subsequence Nk. By Lemma 4.1,
(4.28)∥B̂N(uɛN(t))∥ℍ-1=sup∥w∥=1|〈B̂N(uɛN(t)),w〉|≤sup∥w∥=12∥uɛN(t)∥3/2|uɛN(t)|1/2∥w∥𝕃4≤K∥uɛN(t)∥3/2.
Thus
(4.29)B̂Nk(uɛNk)→wBinLℱ4/3(Ω;L4/3(0,τ;ℍ-1))
for some function B and some subsequence Nk. For every t, we define
(4.30)ℒt:Lℱ2(Ω;L2(0,τ;Lℚ))→Lℱ2(Ω;L2(0,τ;ℍ-1))M↦∫t∧ττM(s)d𝕎(s).
It can be shown that ℒt is a bounded linear operator. Hence
(4.31)∫t∧ττZɛNk(s)d𝕎Nk(s)→w∫t∧ττZɛ(s)d𝕎(s)inLℱ2(Ω;L2(0,τ;ℍ-1)).
Similarly, one can prove that
(4.32)∫t∧ττ{νÂNkuɛNk(s)-FNk(uɛNk(s))+B̂Nk(uɛNk(s))}ds→w∫t∧ττ{A(s)+B(s)}ds
in Lℱ4/3(Ω;L4/3(0,τ;ℍ-1)) and
(4.33)∫t∧ττZɛNk(s)dWNk(s)→w∫t∧ττZɛ(s)dW(s)inLℱ2(Ω;L2(0,τ;H-1)).
Also
(4.34)ℒp:Lℱ∞([0,τ]×Ω;L2)→Lℱ2(Ω;L2(0,τ;H-1)),p↦∫t∧ττ∇p(s)ds
is a bounded linear operator. Since pɛNk∈Lℱ∞([0,τ]×Ω;L2), we have
(4.35)∫t∧ττ∇pɛNk(s)ds→w∫t∧ττ∇pɛ(s)dsinLℱ2(Ω;L2(0,τ;H-1)).
Similarly,
(4.36)∫t∧ττ∇·uɛNk(s)ds→w∫t∧ττ∇·uɛ(s)dsinLℱ2(Ω;L2(0,τ;ℍ-1)).
To sum up,
(4.37)uɛ(t∧τ)=ξ+∫t∧ττ{A(s)+B(s)+∇pɛ(s)}ds-∫t∧ττZɛ(t)d𝕎(s),ɛpɛ(t∧τ)=ɛη+∫t∧ττ∇·uɛ(s)ds-∫t∧ττZɛ(s)dW(s)
hold P-a.s.Step 2 (The Itô formula).
For convenience, let us denoteNk by N again. Let M≤N and (ℍ01)M=PM(ℍ01). For any v∈Lℱ∞([0,τ]×Ω;(ℍ01)M) and some constant K, such that ∥v∥≤K uniformly, define
(4.38)r(t)=∫tT{2α+278(κ-ν)3K4}ds.
Applying the Itô formula to e-r(t)|uɛN(t)|2, we get
(4.39)|ξN|2-e-r(0)|uɛN(0)|2=∫0τ-ṙ(s)e-r(s)|uɛN(s)|2ds+2∫0τe-r(s)〈-νÂNuɛN(s)-B̂N(uɛN(s))-∇pɛN(s)+FN(uɛN(s)),uɛN(s)〉ds+2∫0τe-r(s)〈ZɛN(s)d𝕎N(s),uɛN(s)〉+∫0τe-r(s)∥ZɛN(s)∥Lℚ2ds.
By taking the expectation, we get
(4.40)E|ξN|2-Ee-r(0)|uɛN(0)|2+2E∫0τe-r(s)〈∇pɛN(s),uɛN(s)〉ds=E∫0τe-r(s)∥ZɛN(s)∥Lℚ2ds-2E∫0τe-r(s)〈νÂNuɛN(s)-FN(uɛN(s))+B̂N(uɛN(s))+12ṙ(s)uɛN(s),uɛN(s)〉ds.
Clearly, limN→∞E|ξN|2=E|ξ|2. By (3.5), it is clear that
(4.41)2E∫0τe-r(s)〈∇pɛN(s),uɛN(s)〉ds=ɛE|ηN|2-ɛEe-r(0)|pɛN(0)|2-1ɛE∫0τe-r(s)∥ZɛN(s)∥LQ2ds.
Because of (4.40) and (4.41), one gets the following:
(4.42)lim̲N→∞2E∫0τe-r(s)〈νÂNuɛN(s)-FN(uɛN(s))+B̂N(uɛN(s))+12ṙ(s)uɛN(s),uɛN(s)〉ds=-E|ξ|2+lim̲N→∞Ee-r(0)|uɛN(0)|2+lim̲N→∞E∫0τe-r(s)∥ZɛN(s)∥Lℚ2ds-ɛE|η|2+ɛlim̲N→∞Ee-r(0)|pɛN(0)|2+1ɛlim̲N→∞E∫0τe-r(s)∥ZɛN(s)∥LQ2ds≥2E∫0τe-r(s)〈A(s)+B(s)+12ṙ(s)uɛ(s),uɛ(s)〉ds.
Note that one gets the last inequality by applications of the Itô formula to (4.37), and the fact that
(4.43)lim̲N→∞Ee-r(0)|uɛN(0)|2≥Ee-r(0)|uɛ(0)|2,lim̲N→∞Ee-r(0)|pɛN(0)|2≥Ee-r(0)|pɛ(0)|2,lim̲N→∞E∫0τe-r(s)∥ZɛN(s)∥Lℚ2ds≥E∫0τe-r(s)∥Zɛ(s)∥Lℚ2ds,lim̲N→∞E∫0τe-r(s)∥ZɛN(s)∥LQ2ds≥E∫0τe-r(s)∥Zɛ(s)∥LQ2ds.Step 3 (Monotonicity).
By Corollary4.6, we get
(4.44)E∫0τe-r(s)〈νÂuɛN(s)+B̂(uɛN(s))-F(uɛN(s))+12ṙ(s)uɛN(s),uɛN(s)-v(s)〉ds≤E∫0τe-r(s)〈νÂv(s)+B̂(v(s))-F(v(s))+12ṙ(s)v(s),uɛN(s)-v(s)〉ds.
Note that v∈Lℱ∞([0,τ]×Ω;(ℍ01)M) where M≤N. An application of (4.42) yields
(4.45)E∫0τe-r(s)〈A(s)+B(s)+12ṙ(s)uɛ(s),uɛ(s)-v(s)〉ds≤E∫0τe-r(s)〈νÂv(s)+B̂(v(s))-F(v(s))+12ṙ(s)v(s),uɛ(s)-v(s)〉ds.
Since the above inequality is true for all M∈ℕ and K>0, it remains true for all v∈Lℱ∞([0,τ]×Ω;ℍ01). Thus let v=uɛ+λw where w∈Lℱ∞([0,τ]×Ω;ℍ01) and λ>0, and
(4.46)E∫0τe-r(s)〈A(s)+B(s)-νÂuɛ(s)-B̂(uɛ(s)+λw(s))+F(uɛ(s)+λw(s)),λw(s)〉ds≥E∫0τe-r(s)〈λνÂw(s)+λ2ṙ(s)w(s),λw(s)〉ds.
By the fact that
(4.47)〈B̂(uɛ(t)+λw(t)),w(t)〉=-〈B̂(uɛ(t)+λw(t),w(t)),uɛ(t)+λw(t)〉=-〈B̂(uɛ(t)+λw(t),w(t)),uɛ(t)〉=-〈B̂(uɛ(t),w(t)),uɛ(t)〉-λ〈B̂(w(t),w(t)),uɛ(t)〉=〈B̂(uɛ(t)),w(t)〉+λ〈B̂(w(t),uɛ(t)),w(t)〉,
we have
(4.48)E∫0τe-r(s)〈A(s)+B(s)-νÂuɛ(s)-B̂(uɛ(s))+F(uɛ(s)+λw(s)),w(s)〉ds≥λE∫0τe-r(s)〈νÂw(s)+B̂(w(s),uɛ(s))+12ṙ(s)w(s),w(s)〉ds.
Letting λ→0, and by the arbitrariness of w and the fact that F is continuous, we know that
(4.49)A(s)+B(s)=νÂuɛ(s)+B̂(uɛ(s))-F(uɛ(s))P-a.s.,
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 Proposition4.9 is imposed. Only the uniqueness and continuity are left to check. Let us first prove the following lemma.Lemma 5.1.
For anyu and v in ℍ01 and w∈𝕃4, one has
(5.1)|〈B̂(u)-B̂(v),w〉|≤C(∥u∥𝕃4+∥v∥𝕃4)∥u-v∥∥w∥𝕃4+C(∥u∥+∥v∥)∥u-v∥𝕃4∥w∥𝕃4.Proof.
By Proposition2.5,
(5.2)|〈B̂(u)-B̂(v),w〉|=|-〈B̂(u,w),u〉+〈B̂(v,w),v〉|=|-〈B̂(u,w),u-v〉-〈B̂(u,w),v〉+〈B̂(v,w),v〉|=|-〈B̂(u,w),u-v〉-〈B̂(u-v,w),v〉|=|〈B̂(u,u-v),w〉+〈B̂(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.Theorem 5.2.
Letξ∈Lℱτ∞(Ω;ℍ01) and η∈Lℱτ∞(Ω;H01). System (4.12) admits a unique adapted solution in
(5.3)Lℱ∞([0,τ]×Ω;ℍ01)×Lℱ2(Ω;L2(0,τ;Lℚ))×Lℱ∞([0,τ]×Ω;H01)×Lℱ2(Ω;L2(0,τ;LQ)).
Also the solution is continuous with respect to the terminal conditions in
(5.4)L∞([0,τ];Lℱ2(Ω;𝕃2))×Lℱ2(Ω;L2(0,τ;Lℚ))×L∞([0,τ];Lℱ2(Ω;L2))×Lℱ2(Ω;L2(0,τ;LQ)).Proof.
The existence of an adapted solution is shown in Theorem4.10. Suppose that (uɛ1,Zɛ1,pɛ1,Zɛ1) and (uɛ2,Zɛ2,pɛ2,Zɛ2) 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
(5.5)u¯ɛ=uɛ1-uɛ2,Z¯ɛ=Zɛ1-Zɛ2,Z¯ɛ=Zɛ1-Zɛ2,p¯ɛ=pɛ1-pɛ2,ξ¯=ξ1-ξ2,η¯=η1-η2.
Then one has
(5.6)du¯ɛ(t)+νÂu¯ɛ(t)dt+(B̂(uɛ1(t))-B̂(uɛ2(t)))dt+∇p¯ɛ(t)dt,=(F(uɛ1(t))-F(uɛ2(t)))dt+Z¯ɛ(t)d𝕎(t),ɛdp¯ɛ(t)+∇·u¯ɛ(t)dt=Z¯ɛ(t)dW(t),u¯ɛ(τ)=ξ¯,p¯ɛ(τ)=η¯.
Similar to Corollary 4.6, let us define
(5.7)l(t)=∫tT{2α+278(κ-ν)3K02}ds,
where K0 is the constant in Proposition 4.9. An application of the Itô formula to e-l(t)|u¯ɛ(t)|2 and Corollary 4.6 imply
(5.8)e-l(t∧τ)|u¯ɛ(t∧τ)|2+∫t∧ττe-l(s)∥Z¯ɛ(s)∥Lℚ2ds=|ξ¯|2+2∫t∧ττe-l(s)〈12l̇(s)u¯ɛνÂu¯ɛ(s)-(B̂(uɛ1(s))-B̂(uɛ2(s)))-(F(uɛ1(s))-F(uɛ2(s)))+12l̇(s)u¯ɛ(s),u¯ɛ(s)〉ds+2∫t∧ττe-l(s)〈∇p¯ɛ(s),u¯ɛ(s)〉ds-2∫t∧ττe-l(s)〈Z¯ɛ(s)d𝕎(s),u¯ɛ(s)〉≤|ξ¯|2+ɛ∫t∧ττe-l(s)d|p¯ɛ(s)|2-1ɛ∫t∧ττe-l(s)∥Z¯ɛ(s)∥LQ2ds-2∫t∧ττe-l(s)〈Z¯ɛ(t)dW(s),p¯ɛ(s)〉ds-2∫t∧ττe-l(s)〈Z¯ɛ(s)d𝕎(s),u¯ɛ(s)〉.
Taking the expectation, the above inequality becomes
(5.9)E|u¯ɛ(t∧τ)|2+ɛE|p¯ɛ(t∧τ)|2+E∫t∧ττ∥Z¯ɛ(s)∥Lℚ2ds+1ɛE∫t∧ττ∥Z¯ɛ(s)∥LQ2ds≤e-l(0){E|ξ¯|2+ɛE|η|2}.
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 inL2 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 ℱt adapted. This is why we defined another function l(t) in the proof of the uniqueness based on the ℍ01-bound of the solution. Fortunately, l(t) is ℱt 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 Asɛ 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:(5.10)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,
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 Theorem4.10(ii). Then as ɛ approaches 0, the first three elements in the solution of (4.12), (uɛ,Zɛ,pɛ), converge to (u,Z,p), the solution of (5.10).Proof.
Similar to Step1 of the proof of Theorem 4.10, we know that there exist u, p, Z and a sequence of positive numbers {ɛi}i=1∞ such that ɛi→0, uɛi→wu, pɛi→wp and Zɛi→wZ in corresponding spaces.
From (4.18) and (4.20), one knows that along a subsequence,(5.11)E〈ɛidpɛidt,h〉→E〈dDdt,h〉
for some D∈Lℱ2(Ω;L2(0,τ;L2)) and for all h∈Lℱ2(Ω;L2(0,τ;L2)). Thus we get
(5.12)E〈ɛidpɛidt,h〉→0
in the sense of distribution. Since
(5.13)〈ɛdpɛi(t)+∇·uɛi(t)dt,h(t)〉=〈Zɛi(t)dW(t),h(t)〉,
we know that
(5.14)E∫0τ〈∇·u(t),h(t)〉dt=limi→∞E∫0τ〈∇·uɛi(t),h(t)〉dt=-limi→∞E∫0τ〈ɛdpɛi(t),h(t)〉=0
for all h∈Lℱ2(Ω;L2(0,τ;L2)). So ∇·u= 0 P-a.s. This shows that the limiting system is incompressible.
Similar to Steps2 and 3 in the proof of Theorem 4.10, we are able to show that (u,Z,p) solves (5.10).


## 5.1. Uniqueness and Continuity

The backward Navier-Stokes equation is well-posed if the regularity of the terminal condition in Proposition4.9 is imposed. Only the uniqueness and continuity are left to check. Let us first prove the following lemma.Lemma 5.1.
For anyu and v in ℍ01 and w∈𝕃4, one has
(5.1)|〈B̂(u)-B̂(v),w〉|≤C(∥u∥𝕃4+∥v∥𝕃4)∥u-v∥∥w∥𝕃4+C(∥u∥+∥v∥)∥u-v∥𝕃4∥w∥𝕃4.Proof.
By Proposition2.5,
(5.2)|〈B̂(u)-B̂(v),w〉|=|-〈B̂(u,w),u〉+〈B̂(v,w),v〉|=|-〈B̂(u,w),u-v〉-〈B̂(u,w),v〉+〈B̂(v,w),v〉|=|-〈B̂(u,w),u-v〉-〈B̂(u-v,w),v〉|=|〈B̂(u,u-v),w〉+〈B̂(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.Theorem 5.2.
Letξ∈Lℱτ∞(Ω;ℍ01) and η∈Lℱτ∞(Ω;H01). System (4.12) admits a unique adapted solution in
(5.3)Lℱ∞([0,τ]×Ω;ℍ01)×Lℱ2(Ω;L2(0,τ;Lℚ))×Lℱ∞([0,τ]×Ω;H01)×Lℱ2(Ω;L2(0,τ;LQ)).
Also the solution is continuous with respect to the terminal conditions in
(5.4)L∞([0,τ];Lℱ2(Ω;𝕃2))×Lℱ2(Ω;L2(0,τ;Lℚ))×L∞([0,τ];Lℱ2(Ω;L2))×Lℱ2(Ω;L2(0,τ;LQ)).Proof.
The existence of an adapted solution is shown in Theorem4.10. Suppose that (uɛ1,Zɛ1,pɛ1,Zɛ1) and (uɛ2,Zɛ2,pɛ2,Zɛ2) 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
(5.5)u¯ɛ=uɛ1-uɛ2,Z¯ɛ=Zɛ1-Zɛ2,Z¯ɛ=Zɛ1-Zɛ2,p¯ɛ=pɛ1-pɛ2,ξ¯=ξ1-ξ2,η¯=η1-η2.
Then one has
(5.6)du¯ɛ(t)+νÂu¯ɛ(t)dt+(B̂(uɛ1(t))-B̂(uɛ2(t)))dt+∇p¯ɛ(t)dt,=(F(uɛ1(t))-F(uɛ2(t)))dt+Z¯ɛ(t)d𝕎(t),ɛdp¯ɛ(t)+∇·u¯ɛ(t)dt=Z¯ɛ(t)dW(t),u¯ɛ(τ)=ξ¯,p¯ɛ(τ)=η¯.
Similar to Corollary 4.6, let us define
(5.7)l(t)=∫tT{2α+278(κ-ν)3K02}ds,
where K0 is the constant in Proposition 4.9. An application of the Itô formula to e-l(t)|u¯ɛ(t)|2 and Corollary 4.6 imply
(5.8)e-l(t∧τ)|u¯ɛ(t∧τ)|2+∫t∧ττe-l(s)∥Z¯ɛ(s)∥Lℚ2ds=|ξ¯|2+2∫t∧ττe-l(s)〈12l̇(s)u¯ɛνÂu¯ɛ(s)-(B̂(uɛ1(s))-B̂(uɛ2(s)))-(F(uɛ1(s))-F(uɛ2(s)))+12l̇(s)u¯ɛ(s),u¯ɛ(s)〉ds+2∫t∧ττe-l(s)〈∇p¯ɛ(s),u¯ɛ(s)〉ds-2∫t∧ττe-l(s)〈Z¯ɛ(s)d𝕎(s),u¯ɛ(s)〉≤|ξ¯|2+ɛ∫t∧ττe-l(s)d|p¯ɛ(s)|2-1ɛ∫t∧ττe-l(s)∥Z¯ɛ(s)∥LQ2ds-2∫t∧ττe-l(s)〈Z¯ɛ(t)dW(s),p¯ɛ(s)〉ds-2∫t∧ττe-l(s)〈Z¯ɛ(s)d𝕎(s),u¯ɛ(s)〉.
Taking the expectation, the above inequality becomes
(5.9)E|u¯ɛ(t∧τ)|2+ɛE|p¯ɛ(t∧τ)|2+E∫t∧ττ∥Z¯ɛ(s)∥Lℚ2ds+1ɛE∫t∧ττ∥Z¯ɛ(s)∥LQ2ds≤e-l(0){E|ξ¯|2+ɛE|η|2}.
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 inL2 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 ℱt adapted. This is why we defined another function l(t) in the proof of the uniqueness based on the ℍ01-bound of the solution. Fortunately, l(t) is ℱt 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 Asɛ 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:(5.10)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,
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 Theorem4.10(ii). Then as ɛ approaches 0, the first three elements in the solution of (4.12), (uɛ,Zɛ,pɛ), converge to (u,Z,p), the solution of (5.10).Proof.
Similar to Step1 of the proof of Theorem 4.10, we know that there exist u, p, Z and a sequence of positive numbers {ɛi}i=1∞ such that ɛi→0, uɛi→wu, pɛi→wp and Zɛi→wZ in corresponding spaces.
From (4.18) and (4.20), one knows that along a subsequence,(5.11)E〈ɛidpɛidt,h〉→E〈dDdt,h〉
for some D∈Lℱ2(Ω;L2(0,τ;L2)) and for all h∈Lℱ2(Ω;L2(0,τ;L2)). Thus we get
(5.12)E〈ɛidpɛidt,h〉→0
in the sense of distribution. Since
(5.13)〈ɛdpɛi(t)+∇·uɛi(t)dt,h(t)〉=〈Zɛi(t)dW(t),h(t)〉,
we know that
(5.14)E∫0τ〈∇·u(t),h(t)〉dt=limi→∞E∫0τ〈∇·uɛi(t),h(t)〉dt=-limi→∞E∫0τ〈ɛdpɛi(t),h(t)〉=0
for all h∈Lℱ2(Ω;L2(0,τ;L2)). So ∇·u= 0 P-a.s. This shows that the limiting system is incompressible.
Similar to Steps2 and 3 in the proof of Theorem 4.10, we are able to show that (u,Z,p) solves (5.10).



---

*Source: 730492-2010-06-30.xml*