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International Journal of Stochastic Analysis
Volume 2010 (2010), Article ID 730492, 24 pages
Stochastic Navier-Stokes Equations with Artificial Compressibility in Random Durations
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA
Received 1 December 2009; Accepted 11 May 2010
Academic Editor: Jiongmin M. Yong
Copyright © 2010 Hong Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 , Constantin and Foias , and Temam , 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 , Mikulevicius and Rozovsky , Menaldi and Sritharan , 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 , the two-dimensional stochastic Navier-Stokes equation in a bounded domain with no-slip condition reads where is the constant viscosity, is the velocity, is the pressure, is the external body force and 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  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 : Backward stochastic Navier-Stokes equations (BSNSEs for short) arise as an inverse problem wherein the velocity profile at a time 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 , and the systematic study of general backward stochastic differential equations (BSDEs for short) were put forward first by Pardoux and Peng , Ma, Protter, Yong, Zhou, and several other authors in a finite-dimensional setting. Ma and Yong  have studied linear degenerate backward stochastic differential equations motivated by stochastic control theory. Later, Hu et al.  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 . A nice introduction to backward stochastic differential equations is presented in the book by Yong and Zhou , 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 balls in , which was first observed by Menaldi and Sritharan . 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 sense is given in Section 5. The continuity of solutions and the convergence as approaches zero are also studied.
Suppose that is a domain bounded in with smooth boundary conditions. Let be a positive parameter which vanishes to 0. The artificial state equation for a slightly compressible medium is defined as where is the density, is the pressure, and 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 : where is the velocity, is the pressure, is the external body force, and is the kinematic viscosity. Readers may refer to Temam  for details.
Denote by the inner product of , the inner product of , the dual space of , and the duality pairing between and . Let be the norm of and let be the norm of . Without causing any confusion, we also use the same notations to denote the norms of and . For any and , there exists , such that = . Then the mapping is linear, injective, compact and continuous. A similar result holds for and .
Suppose that is a complete probability space. Let be an -valued -Wiener process, where is a trace class operator on . Let be a complete orthonormal system in such that there exists a nondecreasing sequence of positive numbers , and for all . Let with , and be a sequence of independent standard Brownian motions in . Then Wiener process is taken as =.
Let be a trace class operator on . Similarly, we can define a complete orthonormal system , a nondecreasing sequence of positive numbers such that , and positive numbers such that and . Let =. Then is an -valued -Wiener process. From now on, let be the natural filtration of and , augmented by all the -null sets of . A complete definition of Hilbert space-valued Wiener processes can be found in .
With inner product for all and , let denote the space of linear operators such that is a Hilbert-Schmidt operator from to . Similarly, we define for , the trace class operator on .
To be realistic in nature, let us introduce randomness into the system to obtain where and are initial conditions, and is the noise term. Here is called the stabilization term.
If a terminal time is given and the terminal conditions are specified as and , one obtains a backward system: for , where and , with the notation . The processes and are in spaces and , respectively.
Let be a -stopping time when the observations are available. Suppose that the observed velocity and pressure at are and , respectively. Then we introduce the backward stochastic Navier-Stokes equation with artificial compressibility and stabilization in random duration: for , where the -stopping time is assumed to be bounded by a time . Note that processes and measure the randomness that is inherent in the hydrodynamical system. It is this randomness that has possibly led us to the observations at time . 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 is taken to be deterministic and is always assumed to be in .
Definition 2.1. A quaternion of -Adapted processes is called a solution of backward Navier-Stokes equation (2.6) if it satisfies the integral form of the system P-a.s., and the following holds: (a);(b);(c);(d).
The following simple results are frequently used and given as lemmas. Readers may refer to Temam  for similar proofs.
Lemma 2.2. For any and , one has ()===,()=,()= ,()=,()===.
Remark 2.3. Sometimes is denoted by .
Lemma 2.4. The following results hold for any real-valued smooth functions and with compact support in :
Proposition 2.5. For any and in and , one has
Below is a backward version of the Gronwall inequality used frequently in this paper, and the proof is straightforward.
Lemma 2.6. Suppose that , and are integrable functions, and , are nonnegative functions. For , if then In particular, if , and , then
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, for all . 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 , let us construct a finite dimensional system. Let where is the natural filtration of and . The projected system with solution is defined as follows: for .
Proposition 3.1. Let , , and . Then for any solution of system (3.2), the following is true:
Proof. Applying the Itô formula to to get
thus we get
By means of the Itô formula, one has
and Lemma 2.2 yields
For , taking the conditional expectation with respect to , and by (3.5), the above two equation and along with the fact that , one gets
P-a.s. Since and for , one gets
P-a.s., and by Lemma 2.6, the backward Gronwall inequality, and letting , we get
P-a.s. Because of the integrability of , , and , there exists a constant , depending on only, s.t.
for all , P-a.s.
Similarly, making use of (3.4), it follows that .
Proposition 3.2. Let , , and , for all and . The following is true for any solution of system (3.2):
Proof. Let us prove it by the method of mathematical induction. Similar to Proposition 3.1, it is easy to obtain the result for . Suppose that it is true for all . Let us show that the proposition holds for .
An application of the Itô formula to yields Clearly , where , as stated in Section 2, is the eigenvalue of for . Taking the expectation, one obtains where is a constant, and is a constant depending on and . Both constants may vary throughout the proof. But we keep the same notations for simplicity. Applying the Itô formula to , one obtains Adding up (3.16) and (3.17), one gets 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 . The following inequalities are true: (a),(b)+,(c)+. Furthermore, if , then there exists a constant depending on , such that (d)+.
Corollary 4.2. For any and , let Then
Proposition 4.3. Let , , and . Then the projected system (3.2) admits a unique adapted solution in
Proof. For every , let be a Lipschitz function which has the following property: Applying the truncation to , it is easy to show that is Lipschitz and for any and . Let us define a truncated projected system: For fixed , let us map to , 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 (see [14, page 355]) guarantees the existence of a unique adapted solution. Let the solution be . Then for there is a unique adapted solution . Thus we can define an operator , such that . It can be shown that is a contraction mapping. Thus the unique adapted solution of (4.6) can be obtained. Let us take the limit of the solution as 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 . 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 , Kallianpur and Xiong , Prévôt and Röckner ).
Assumption A. (A.1) : is a continuous operator.
(A.2) There exist positive constants and , such that
(A.3) For any and in , a constant , and a positive constant ,
(A.4) For any and some positive constant ,
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: for . 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 and . Then the following inequality is true:
Corollary 4.6. Let and . Define Then
Remark 4.7. To prove Corollary 4.6, the monotonicity assumption (A.3) is used.
(i) Let and . Then for any solution of system (4.13), the following is true:
Moreover, there exists a constant , independent of , such that
(ii) Let and . The following is true for any solution of system (4.13): Moreover, there exists a constant , independent of , such that
Proof. (i) Similar to the proof of Proposition 3.1, utilizing Assumption (A.2), (3.6) becomes
For , taking the conditional expectation with respect to , one gets
P-a.s. By the backward Gronwall inequality, and letting , we get (4.18).
(ii) The proof is similar to (i).
Proposition 4.9. Suppose that and . Then for any solution of system (4.13), there exists a constant , such that
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 in
Proof. We have the following steps.Step 1 (The limits). Clearly, by Proposition 4.8, there exist , , , and , such that for a subsequence . Since is a continuous map from to , for all and some constant . Thus combined with the assumptions on , one knows that for some function and some subsequence . By Lemma 4.1, Thus for some function and some subsequence . For every , we define It can be shown that is a bounded linear operator. Hence Similarly, one can prove that in and Also is a bounded linear operator. Since , we have Similarly, To sum up, hold P-a.s.Step 2 (The Itô formula). For convenience, let us denote by again. Let and . For any and some constant , such that uniformly, define Applying the Itô formula to , we get By taking the expectation, we get Clearly, . By (3.5), it is clear that Because of (4.40) and (4.41), one gets the following: Note that one gets the last inequality by applications of the Itô formula to (4.37), and the fact that Step 3 (Monotonicity). By Corollary 4.6, we get Note that where