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

(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 ɛ:

(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 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 ɛ approaches zero are also studied.

2. Preliminaries

Suppose that G 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)

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)

where u_ɛ ∈ 𝕃² = 𝕃²(G) is the velocity, p_ɛ ∈ 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, the inner product of , ℍ⁻¹ the dual space of , and 〈·, ·〉 the duality pairing between and ℍ⁻¹. Let |·| be the norm of 𝕃² and let ∥·∥ be the norm of . Without causing any confusion, we also use the same notations to denote the norms of L² and . For any x∈𝕃² and y∈, 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 be a complete orthonormal system in 𝕃² such that there exists a nondecreasing sequence of positive numbers , lim _j→∞λ_j = ∞ and −Δe_j = λ_je_j for all j. Let ℚe_k = q_ke_k with , and {b^k(t)} be a sequence of independent standard Brownian motions in ℝ. Then Wiener process 𝕎(t) is taken as 𝕎(t)=.

Let Q be a trace class operator on L². Similarly, we can define a complete orthonormal system , a nondecreasing sequence of positive numbers such that −Δe_j = κ_je_j, and positive numbers such that and . Let W(t)=. 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

(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

(2.4)

where u₀ and p₀ are initial conditions, and σ(d𝕎/dt) is the noise term. Here (1/2)(∇·u_ɛ)u_ɛ is called the stabilization term.

If a terminal time T is given and the terminal conditions are specified as u_ɛ(T) = ξ and p_ɛ(T) = η, one obtains a backward system:

(2.5)

for 0 ≤ t ≤ T, where

and

, with the notation

. The processes Z_ɛ and Z_ɛ 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

and p(τ) = η∈

, respectively. Then we introduce the backward stochastic Navier-Stokes equation with artificial compressibility and stabilization in random duration:

(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_ɛ and Z_ɛ 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_ɛ, Z_ɛ, p_ɛ, Z_ɛ) is called a solution of backward Navier-Stokes equation (2.6) if it satisfies the integral form of the system

(2.7)

P-a.s., and the following holds:

(a)
u_ɛ∈∩;
(b)
Z_ɛ∈;
(c)
p_ɛ∈∩;
(d)
Z_ɛ∈.

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 and p ∈ L², one has

1 = ∑_i,j∫_G∂_iu_j∂_iw_jdx ==,
2 〈(u · ∇)v, w〉 = ∑_i,j∫_Gu_i(∂_iv_j)w_jdx,
3 〈(u · ∇)v, w〉 = −〈(∇·u)w, v〉 − 〈(u · ∇)w, v〉,
4 =,
5 〈−∇p, u〉 = − ∑_i∫_G∂_ip u_idx=∫_Gp∂_iu_idx = 〈p, ∇·u〉.

Remark 2.3. Sometimes is denoted by .

Lemma 2.4. The following results hold for any real-valued smooth functions ϕ and ψ with compact support in ℝ²:

(2.8)

Proposition 2.5. For any u and v in and w ∈ 𝕃⁴, one has

(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

(2.10)

then

(2.11)

In particular, if α(t) ≡ α, β(t) ≡ β and γ(t) ≡ 1, then

(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 , where 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

and

. Under projection P_N, let us construct a finite dimensional system. Let

(3.1)

where

is the natural filtration of {𝕎^N(t)} and {W^N(t)}. The projected system with solution

is defined as follows:

(3.2)

for 0 ≤ t ≤ τ.

Proposition 3.1. Let , , and f ∈ L²(0, T; ℍ⁻¹). Then for any solution of system (3.2), the following is true:

(3.3)

Proof. Applying the Itô formula to to get

(3.4)

thus we get

(3.5)

By means of the Itô formula, one has

(3.6)

Clearly,

(3.7)

and Lemma 2.2 yields

(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

, one gets

(3.9)

P-a.s. Since

and λ_i ≤ λ_j for i < j, one gets

(3.10)

Thus

(3.11)

P-a.s., and by Lemma 2.6, the backward Gronwall inequality, and letting r = t, we get

(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.

(3.13)

for all t ∈ [0, τ], P-a.s.

Similarly, making use of (3.4), it follows that ∈.

Proposition 3.2. Let , , 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)

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 yields

(3.15)

Clearly

, where κ_N, as stated in Section 2, is the eigenvalue of −Δ for e_N. Taking the expectation, one obtains

(3.16)

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

, one obtains

(3.17)

Adding up (3.16) and (3.17), one gets

(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 and w ∈ 𝕃⁴. The following inequalities are true:

(a)
,
(b)
+,
(c)
+.

Furthermore, if

, then there exists a constant C depending on ν, such that

(d)
+ C∥v∥² | u − v|² ≥ (ν/2)∥u−v∥².

Corollary 4.2. For any u and v ∈ 𝕃⁴, let

(4.1)

Then

(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 , , and f ∈ L²(0, T; ℍ⁻¹). Then the projected system (3.2) admits a unique adapted solution in

(4.3)

Proof. For every M ∈ ℕ, let L_M be a Lipschitz C^∞ function which has the following property:

(4.4)

Applying the truncation L_M to

, it is easy to show that

is Lipschitz and

(4.5)

for any

and M ∈ ℕ. Let us define a truncated projected system:

(4.6)

For fixed

, let us map

(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

. Then for

(4.8)

there is a unique adapted solution

. Thus we can define an operator Ψ, such that

. 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: →ℍ⁻¹ is a continuous operator.

(A.2) There exist positive constants α and β, such that

(4.9)

(A.3) For any u and v in , a constant κ > ν, and a positive constant α,

(4.10)

(A.4) For any u∈ and some positive constant α,

(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:

(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:

(4.14)

Corollary 4.6. Let u and v ∈ 𝕃⁴. Define

(4.15)

Then

(4.16)

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

Proposition 4.8. (i) Let and . Then for any solution of system (4.13), the following is true:

(4.17)

Moreover, there exists a constant K, independent of N, such that

(4.18)

P-a.s.

(ii) Let and . The following is true for any solution of system (4.13):

(4.19)

Moreover, there exists a constant K, independent of N, such that

(4.20)

Proof. (i) Similar to the proof of Proposition 3.1, utilizing Assumption (A.2), (3.6) becomes

(4.21)

For 0 < r ≤ t, taking the conditional expectation with respect to ℱ_r∧τ, one gets

(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 and . Then for any solution (u_ɛ, Z_ɛ, p_ɛ, Z_ɛ) of system (4.13), there exists a constant K₀, such that

(4.23)

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

Theorem 4.10. Let and . For system (4.12), there exists a solution (u_ɛ, Z_ɛ, p_ɛ, Z_ɛ) in

(4.24)

Proof. We have the following steps.

Step 1 (The limits). Clearly, by Proposition 4.8, there exist u_ɛ, p_ɛ, Z_ɛ, and Z_ɛ, such that

(4.25)

for a subsequence N_k. Since

is a continuous map from

to ℍ⁻¹,

(4.26)

for all

and some constant C. Thus combined with the assumptions on F, one knows that

(4.27)

for some function A and some subsequence N_k. By Lemma 4.1,

(4.28)

Thus

(4.29)

for some function B and some subsequence N_k. For every t, we define

(4.30)

It can be shown that ℒ_t is a bounded linear operator. Hence

(4.31)

Similarly, one can prove that

(4.32)

and

(4.33)

Also

(4.34)

is a bounded linear operator. Since

∈

, we have

(4.35)

Similarly,

(4.36)

To sum up,

(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 . For any v∈ and some constant K, such that ∥v∥ ≤ K uniformly, define

(4.38)

Applying the Itô formula to

, we get

(4.39)

By taking the expectation, we get

(4.40)

Clearly, lim _N→∞E | ξ^N|² = E | ξ|². By (3.5), it is clear that

(4.41)

Because of (4.40) and (4.41), one gets the following:

(4.42)

Note that one gets the last inequality by applications of the Itô formula to (4.37), and the fact that

(4.43)

Step 3 (Monotonicity). By Corollary 4.6, we get

(4.44)

Note that v∈

where M ≤ N. An application of (4.42) yields

(4.45)

Since the above inequality is true for all M ∈ ℕ and K > 0, it remains true for all v∈

. Thus let v = u_ɛ + λw where

and λ > 0, and

(4.46)

By the fact that

(4.47)

we have

(4.48)

Letting λ → 0, and by the arbitrariness of w and the fact that F is continuous, we know that

(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 and w ∈ 𝕃⁴, one has

(5.1)

Proof. By Proposition 2.5,

(5.2)

Theorem 5.2. Let and . System (4.12) admits a unique adapted solution in

(5.3)

Also the solution is continuous with respect to the terminal conditions in

(5.4)

Proof. The existence of an adapted solution is shown in Theorem 4.10. Suppose that (u_ɛ1, Z_ɛ1, p_ɛ1, Z_ɛ1) and (u_ɛ2, Z_ɛ2, p_ɛ2, Z_ɛ2) 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)

Then one has

(5.6)

Similar to Corollary 4.6, let us define

(5.7)

where K₀ is the constant in Proposition 4.9. An application of the Itô formula to

and Corollary 4.6 imply

(5.8)

Taking the expectation, the above inequality becomes

(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 -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:

(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 ɛ approaches 0, the first three elements in the solution of (4.12), (u_ɛ, Z_ɛ, p_ɛ), 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 such that ɛ_i → 0, , and in corresponding spaces.

From (4.18) and (4.20), one knows that along a subsequence,

(5.11)

for some D∈

and for all h∈

. Thus we get

(5.12)

in the sense of distribution. Since

(5.13)

we know that

(5.14)

for all h∈

. 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.

References

1 Ladyzhenskaya O. A., The Mathematical Theory of Viscous Incompressible Flow, 1969, Gordon and Breach Science Publishers, New York, NY, USA, MR0254401.
Google Scholar
2 Constantin P. and Foias C., Navier-Stokes Equations, 1988, The University of Chicago Press, Chicago, Ill, USA, MR972259.
10.7208/chicago/9780226764320.001.0001
Google Scholar
3 Temam R., Navier-Stokes Equations, 1979, North-Holland, New York, NY, USA, MR603444.
Web of Science® Google Scholar
4 Flandoli F. and Gatarek D., Martingale and stationary solutions for stochastic Navier-Stokes equations, Probability Theory and Related Fields. (1995) 102, no. 3, 367–391, MR1339739, https://doi.org/10.1007/BF01192467, ZBL0831.60072.
10.1007/BF01192467
Web of Science® Google Scholar
5 Mikulevicius R. and RozovskyB. L.II, Global L2-solutions of stochastic Navier-Stokes equations, Annals of Probability. (2005) 33, no. 1, 137–176, MR2118862, https://doi.org/10.1214/009117904000000630.
10.1214/009117904000000630
Web of Science® Google Scholar
6 Menaldi J.-L. and Sritharan S. S., Stochastic 2-D Navier-Stokes equation, Applied Mathematics and Optimization. (2002) 46, no. 1, 31–53, https://doi.org/10.1007/s00245-002-0734-6, MR1922695, ZBL1016.35072.
10.1007/s00245-002-0734-6
Web of Science® Google Scholar
7 Lions J. L., Sentinels and stealthy perturbations. Semicomplete set of sentinels, Mathematical and Numerical Aspects of Wave Propagation Phenomena, 1991, SIAM, Philadelphia, Pa, USA, 239–251, MR1105998.
Google Scholar
8 Lions J.-L., Distributed systems with incomplete data and problems of environment: some remarks, Mathematics, Climate and Environment, 1993, 27, Masson, Paris, France, 58–101, RMA Research Notes in Applied Mathematics, MR1263043, ZBL0829.92019.
Google Scholar
9 Bismut J.-M., Conjugate convex functions in optimal stochastic control, Journal of Mathematical Analysis and Applications. (1973) 44, no. 2, 384–404, MR0329726, https://doi.org/10.1016/0022-247X(73)90066-8, ZBL0276.93060.
10.1016/0022-247X(73)90066-8
Web of Science® Google Scholar
10 Pardoux E. and Peng S. G., Adapted solution of a backward stochastic differential equation, Systems and Control Letters. (1990) 14, no. 1, 55–61, MR1037747, https://doi.org/10.1016/0167-6911(90)90082-6, ZBL0692.93064.
10.1016/0167-6911(90)90082-6
Web of Science® Google Scholar
11 Ma J. and Yong J., On linear, degenerate backward stochastic partial differential equations, Probability Theory and Related Fields. (1999) 113, no. 2, 135–170, MR1676768, https://doi.org/10.1007/s004400050205, ZBL0922.60053.
10.1007/s004400050205
Web of Science® Google Scholar
12 Hu Y., Ma J., and Yong J., On semi-linear degenerate backward stochastic partial differential equations, Probability Theory and Related Fields. (2002) 123, no. 3, 381–411, 2-s2.0-0036022589, https://doi.org/10.1007/s004400100193.
10.1007/s004400100193
Web of Science® Google Scholar
13 Ma J. and Yong J., Adapted solution of a degenerate backward spde, with applications, Stochastic Processes and Their Applications. (1997) 70, no. 1, 59–84, MR1472959, https://doi.org/10.1016/S0304-4149(97)00057-4, ZBL0911.60048.
10.1016/S0304-4149(97)00057-4
Web of Science® Google Scholar
14 Yong J. and Zhou X. Y., Stochastic Controls, 1999, Springer, New York, NY, USA.
10.1007/978-1-4612-1466-3
Google Scholar
15 Chow P. L., Stochastic Partial Differential Equations, 2007, Taylor & Francis, Boca Raton, Fla, USA.
10.1201/9781420010305
Google Scholar
16 Sundar P. and Yin H., Existence and uniqueness of solutions to the backward 2D stochastic Navier-Stokes equations, Stochastic Processes and Their Applications. (2009) 119, no. 4, 1216–1234, MR2508571, https://doi.org/10.1016/j.spa.2008.06.007, ZBL1161.60322.
10.1016/j.spa.2008.06.007
Web of Science® Google Scholar
17 Kallianpur G. and Xiong J., Stochastic Differential Equations in Infinite Dimensional Spaces, 1995, 26, Institute of Mathematical Statistics, IMS Lecture Notes-Monograph Series, MR1918539, https://doi.org/10.1007/s004400100193, ZBL1011.60046.
10.1214/lnms/1215451864
Google Scholar
18 Prévôt C. and Röckner M., A Concise Course on Stochastic Partial Differential Equations, 2007, Springer, Berlin, Germany.
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Stochastic Navier-Stokes Equations with Artificial Compressibility in Random Durations

Abstract

1. Introduction

2. Preliminaries

3. A Priori Estimates

4. Existence of Solutions

5. Uniqueness, Continuity and Convergence of Solutions

5.1. Uniqueness and Continuity

5.2. The Convergence of the Solution As ɛ Approaches Zero

Acknowledgments

References

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Stochastic Navier-Stokes Equations with Artificial Compressibility in Random Durations

Abstract

1. Introduction

2. Preliminaries

3. A Priori Estimates

4. Existence of Solutions

5. Uniqueness, Continuity and Convergence of Solutions

5.1. Uniqueness and Continuity

5.2. The Convergence of the Solution As ɛ Approaches Zero

Acknowledgments

References

Citing Literature

References

Related

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