Hindawi Publishing Corporation
International Journal of Stochastic Analysis
Volume 2010, Article ID 730492, 24 pages
doi:10.1155/2010/730492
Research Article
Stochastic Navier-Stokes Equations with
Artificial Compressibility in Random Durations
Hong Yin
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA
Correspondence should be addressed to Hong Yin, hongyin@usc.edu
Received 1 December 2009; Accepted 11 May 2010
Academic Editor: Jiongmin M. Yong
Copyright q 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 MintyBrowder 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
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domain G ⊂ R2 with no-slip condition reads
∂u u · ∇udt − νΔudt −∇pdt ftdt σt, udWt,
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:
1
∂t uε − νΔuε uε · ∇uε ∇ · uε uε ∇pε f,
2
ε∂t pε ∇ · uε 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 BlackScholes 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 H 1
sense is given in Section 5. The continuity of solutions and the convergence as ε approaches
zero are also studied.
International Journal of Stochastic Analysis
3
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:
1
∂t uε − νΔuε uε · ∇uε ∇ · uε uε ∇pε f,
2
2.2
ε∂t pε ∇ · uε 0,
where uε ∈ L2 L2 G is the velocity, pε ∈ 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 H01 H01 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 Wt be an L2 -valued Q1
2
4
Wiener process, where Q is a trace class operator on L2 . Let {ej }∞
j1 ∈ L ∩ H0 ∩ L be
2
a complete orthonormal system in L such that there exists a nondecreasing sequence of
positive numbers {λj }∞
j1 , limj → ∞ λj ∞ and −Δej λj ej for all j. Let Qek qk ek with
∞
k
independent standard Brownian motions in R.
k1 qk < ∞, and {b t} be a sequence of √ k
Then Wiener process Wt is taken as Wt ∞
k1 qk b tek .
Let Q be a trace class operator on L2 . Similarly, we can define a complete orthonormal
∞
system {ej }∞
j1 , a nondecreasing sequence of positive numbers {κj }j1 such that −Δej κj ej ,
∞ j
and positive numbers qj such that Qej qj ej and ∞
qj b tej .
j1 qj < ∞. Let Wt j1
Then Wt is an L2 -valued Q-Wiener process. From now on, let {Ft } be the natural filtration
of {Wt} and {Wt}, 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, GLQ trFQG∗ trGQF∗ 2.3
for all F and G ∈ LQ , let LQ denote the space of linear operators E such that EQ1/2 is a HilbertSchmidt operator from L2 to L2 . Similarly, we define LQ for Q, the trace class operator on
L2 .
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International Journal of Stochastic Analysis
To be realistic in nature, let us introduce randomness into the system to obtain
dWt
∂uε t
1
− νΔuε t uε t · ∇uε t ∇ · uε tuε t ∇pε t ft σt
,
∂t
2
dt
2.4
ε∂pε t ∇ · uε tdt 0,
uε 0 u0 ,
pε 0 p0 ,
where u0 and p0 are initial conditions, and σdW/dt is the noise term. Here 1/2∇ · uε uε
is called the stabilization term.
If a terminal time T is given and the terminal conditions are specified as uε T ξ and
pε T η, one obtains a backward system:
ε tdt Bu
ε tdt ∇pε tdt ftdt Zε tdWt,
duε t νAu
2.5
εdpε t ∇ · uε tdt Zε tdWt,
uε T ξ,
pε T η
−Δu and Bu,
for 0 ≤ t ≤ T , where Au
v u · ∇v 1/2∇ · uv, with the notation
Bu Bu, u. The processes Zε and Zε are in spaces LQ and LQ , respectively.
Let τ be a Ft -stopping time when the observations are available. Suppose that
the observed velocity and pressure at τ are uε τ ξ ∈ L2Fτ Ω; L2 and pτ η ∈
L2Fτ Ω; L2 , respectively. Then we introduce the backward stochastic Navier-Stokes equation
with artificial compressibility and stabilization in random duration:
ε tdt Bu
ε tdt ∇pε tdt ftdt Zε tdWt,
duε t νAu
2.6
εdpε t ∇ · uε tdt Zε tdWt,
uε τ ξ,
pε 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 Zε and Zε 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 Ft -Adapted processes uε , Zε , pε , Zε is called a solution of
backward Navier-Stokes equation 2.6 if it satisfies the integral form of the system
uε t ∧ τ ξ τ
τ ε s Bu
ε s ∇pε s − fs ds −
νAu
Zε sdWs,
t∧τ
εpε t ∧ τ εη τ
t∧τ
∇ · uε sds −
τ
t∧τ
t∧τ
Zε sdWs,
2.7
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P-a.s., and the following holds:
a uε ∈ L2F Ω; L∞ 0, τ; L2 ∩ L2F Ω; L2 0, τ; H10 ;
b Zε ∈ L2F Ω; L2 0, τ; LQ ;
c pε ∈ L2F Ω; L∞ 0, τ; L2 ∩ L2F Ω; L2 0, τ; H01 ;
d Zε ∈ L2F 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 any u, v, w ∈ H10 and p ∈ L2 , one has
w = i,j ∂i uj ∂i wj dx = Aw,
u = u, wH1 ,
1 Au,
G
0
2 u · ∇v, w = i,j G ui ∂i vj wj dx,
3 u · ∇v, w = −∇ · uw, v − u · ∇w, v,
4 Bu,
v, w = −Bu,
w, v,
5 −∇p, u = − i G ∂i p ui dx= G p∂i ui dx = p, ∇ · u.
v, w.
Remark 2.3. Sometimes Bu,
v, w is denoted by bu,
Lemma 2.4. The following results hold for any real-valued smooth functions φ and ψ with compact
support in R2 :
2
φψ ≤ C
φ∂1 φ
1 ψ∂2 ψ 1 ,
L
L
2 2
4
φ
4 ≤ Cφ ∇φ .
L
2.8
Proposition 2.5. For any u and v in H10 and w ∈ L4 , one has
1
bu, v, w ≤ uL4 vwL4 uvL4 wL4 .
2
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 gt, αt, βt, and γt are integrable functions, and βt, γt are
nonnegative functions. For 0 ≤ t ≤ T , if
T
γ ρ g ρ dρ,
2.10
η
α η γ η e t βργρdρ dη.
2.11
gt ≤ αt βt
t
then
gt ≤ αt βt
T
t
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International Journal of Stochastic Analysis
In particular, if αt ≡ α, βt ≡ β and γt ≡ 1, then
gt ≤ αeβT −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 PN : L2 → L2N , where L2N 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 operators
and B
N PN B.
Under projection PN , let us construct a finite dimensional system.
N PN A
A
Let
WN t PN Wt N
i1
fN t PN ft,
qi bi tei ,
W N t PN Wt ξN E PN ξ | FτN ,
N qi bi tei ,
i1
3.1
ηN E PN η | FτN ,
where {FtN } is the natural filtration of {WN t} and {W N t}. The projected system with
N
N
N
solution uN
ε , Zε , pε , Zε is defined as follows:
N N
N
N
N
dt − ∇pεN tdt fN tdt ZN
duN
u
u
−ν
A
−
B
t
tdt
t
ε
ε
ε
ε tdW t,
N
N
εdpεN t ∇ · uN
ε tdt Zε tdW t,
N
uN
ε τ ξ ,
3.2
pεN τ ηN
for 0 ≤ t ≤ τ.
Proposition 3.1. Let ξ ∈ L∞
Ω; L2 , η ∈ L∞
Ω; L2 , and f ∈ L2 0, T ; H−1 . Then for any solution
Fτ
Fτ
of system 3.2, the following is true:
N
2
∈ L∞
∩ L2F Ω; L2 0, τ; H10
× L2F Ω; L2 0, τ; LQ ,
uN
ε , Zε
F 0, τ × Ω; L
2
pεN , ZεN ∈ L∞
∩ L2F Ω; L2 0, τ; H01
× L2F Ω; L2 0, τ; LQ .
F 0, τ × Ω; L
3.3
Proof. Applying the Itô formula to |pεN t|2 to get
2
1 ∗ 2 N
2
N
Zε tdW N t, pεN t 2 tr ZεN tQ ZεN t dt
dpεN t − ∇ · uN
ε t, pε t dt ε
ε
ε
∗ 2
2
1 ∇pεN t, uN
ZεN tdW N t, pεN t 2 tr ZεN tQ ZεN t dt,
ε t dt ε
ε
ε
3.4
International Journal of Stochastic Analysis
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thus we get
1
N 2
N 2
N
N
N
2 ∇pεN t, uN
ε t dt εdpε t − 2 Zε tdW t, pε t − Zε t
dt.
LQ
ε
3.5
By means of the Itô formula, one has
τ 2 2
N
N
N
N
N uN
N uN
νA
uε t ∧ τ ξN 2
ε s B
ε s ∇pε s − f s, uε s ds
−2
τ
t∧τ
t∧τ
N
N
ZN
ε sdW s, uε s
3.6
τ N 2
−
Zε s
ds.
LQ
t∧τ
Clearly,
N
N uN
B
ε s , uε s 0,
3.7
and Lemma 2.2 yields
N 2
N 2 N 2
N
N N
2 fN s, uN
ε s ≤ f s
−1 uε s
f s −1 A uε s, uε s.
H
H
3.8
For 0 < r ≤ t, taking the conditional expectation with respect to Fr∧τ , and by 3.5, the above
ε s, uε s, one gets
two equation and along with the fact that uε s2 Au
EFr∧τ uN
ε t
τ τ 2
N 2
N 2
Fr∧τ
Fr∧τ
∧ τ E
Zε s
ds E
uε s
ds
t∧τ
2
≤ EFr∧τ ξN 2ν 1EFr∧τ
εEFr∧τ
LQ
τ t∧τ
τ
t∧τ
τ N 2
N N
N
Fr∧τ
A uε s, uε s ds E
f s
−1 ds
τ 2 1
N 2
dpεN s − EFr∧τ
Zε s
ds,
LQ
ε
t∧τ
t∧τ
t∧τ
H
3.9
j λj ej and λi ≤ λj for i < j, one gets
P-a.s. Since Ae
N 2
N
N uN
A
ε s, uε s ≤ λN uε s .
3.10
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International Journal of Stochastic Analysis
Thus
EFr∧τ uN
ε t
τ 2
2
N 2
Fr∧τ N
Fr∧τ
∧ τ εE pε t ∧ τ E
uε s
ds
t∧τ
τ 1 Fr∧τ τ N 2
N 2
Fr∧τ
E
Zε s
ds E
Zε s
ds
LQ
LQ
ε
t∧τ
t∧τ
T
2
2
2
EFr∧τ uN
≤ EFr∧τ ξN εEFr∧τ ηN 2ν 1λN
ε s ∧ τ ds
EFr∧τ
3.11
t
τ N 2
f s
−1 ds,
H
0
P-a.s., and by Lemma 2.6, the backward Gronwall inequality, and letting r t, we get
τ 2
2
N 2
N
N
Ft∧τ
uε s
ds
uε t ∧ τ εpε t ∧ τ E
t∧τ
τ 1 Ft∧τ τ N 2
N 2
Ft∧τ
E
Zε s
ds E
Zε s
ds
LQ
LQ
ε
t∧τ
t∧τ
τ 2
2
N 2
≤ EFt∧τ ξN εEFt∧τ ηN EFt∧τ
f s
−1 ds e2ν1λN T −t ,
3.12
H
0
P-a.s. Because of the integrability of ξ, η, and f, there exists a constant KN , depending on N
only, s.t.
τ τ τ 2
2
N 2
N 2
N 2
s
ds
E
s
ds
E
Z
Zε s
ds ≤ KN ,
uε t εpεN t E uN
ε
ε
0
LQ
0
LQ
0
3.13
for all t ∈ 0, τ, P-a.s.
Similarly, making use of 3.4, it follows that pεN ∈ L2F Ω; L2 0, τ; H01 .
Proposition 3.2. Let ξ ∈ LnFτ Ω; L2 , η ∈ LnFτ 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:
∞
0, τ; LnF Ω; L2 ∩ LnF Ω; Ln 0, τ; H10 ,
uN
ε ∈L
pεN
∈L
∞
0, τ; LnF
Ω; L
2
∩
LnF
Ω; L
n
0, τ; H01
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.
International Journal of Stochastic Analysis
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n
An application of the Itô formula to |uN
ε t| yields
n
N
uε t ∧ τ
τ n
N n−2 N N
N
N
N
N
N uN
νA uε s B
ξ n
uε s
ε s ∇pε s − f s, uε s ds
t∧τ
τ n2 − n τ N n−2 N
N n−2 N 2
N
N
−n
Zε sdW s, uε s −
uε s
uε s Zε s
ds.
LQ
2
t∧τ
t∧τ
3.15
√
Clearly |∇pεN s| ≤ CpεN s ≤ C κN |pεN s|, where κN , as stated in Section 2, is the
eigenvalue of −Δ for eN . Taking the expectation, one obtains
τ τ n
n
N n
N n
N
n/2
EuN
E
s
ds
≤
E
λ
E
∧
τ
t
u
uε s ds
ξ
ε
ε
N
t∧τ
t∧τ
τ N n−2 N N
ds
nE
νA uε s ∇pεN s − fN s, uN
s
uε s
ε
t∧τ
τ n τ N
N n
N n−1 N n/2
≤ Eξ νλN n λN E
uε s ds nE
uε s ∇pε sds
t∧τ
t∧τ
τ 2 n
N n−2 N 2
E
s
ds
uε s
f s
−1 λN uN
ε
H
2
t∧τ
τ n n N n
≤ EξN νλN n λn/2
E
λ
s
ds
u
N
ε
N
2
t∧τ
n−1/n τ 1/n
τ √
N n
N n
nC κN E
E
uε s ds
pε s ds
t∧τ
t∧τ
n−2 n T
N 2
ds
s
f s
−1 E 1t∧τ,τ uN
ε
H
2 t
T
n n
N 2
N n−2
N
sup Euε t
≤ Eξ f s
−1 ds
H
2 0≤t≤τ
t
n
√ T N
n
νλN n λn/2
λ
−
1C
κ
Euε s ∧ τ ds
n
N
N
N
2
t
√
C κN
T n
EpεN s ∧ τ ds
t
≤ K Kn, N
T T n
n
EuN
ds
Kn,
N
EpεN s ∧ τ ds,
∧
τ
ε s
t
t
3.16
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International Journal of Stochastic Analysis
where K is a constant, and Kn, 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 |pεN t|n , one obtains
τ n
N n
εEpεN t ∧ τ E
pε s
ds
t∧τ
τ τ n
N n
N n−2
∇pεN s, uN
E
ds
−
nE
≤ εEηN κn/2
s
p
pε s
ε
ε s ds
N
t∧τ
t∧τ
τ τ n
√
N
N n
N n−1 N n/2
≤ εEη κN E
pε s ds nC κN E
pε s uε sds
t∧τ
t∧τ
τ n−1/n
τ n
√
N n
N n
≤ εEηN κn/2
E
ds
nC
κ
ds
E
s
s
p
p
N
ε
ε
N
t∧τ
3.17
t∧τ
τ 1/n
N n
× E
uε s ds
t∧τ
T T n
n
N
≤ K Kn, N
Epε s ∧ τ ds Kn, N
EuN
ε s ∧ τ ds.
t
t
Adding up 3.16 and 3.17, one gets
τ n
n N n N n
N
E
ε
E uN
∧
τ
∧
τ
t
t
uε s
pε s
ds
p
ε
ε
t∧τ
T n n N
≤ K Kn, N
E uN
∧
τ
∧
τ
s
s
ds.
p
ε
ε
3.18
t
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 |Bu,
w| ≤ 2u3/2 |u|1/2 wL4 ,
b |Bu
− Bv,
u − v| ≤ ν/2u − v2 + 27/2ν3 |u − v|2 v4L4 ,
− v Bu
c νAu
− Bv,
u − v + 27/2ν3 v4L4 |u − v|2 ≥ ν/2u − v2 .
Furthermore, if w ∈ H10 , then there exists a constant C depending on ν, such that
− v Bu
d νAu
− Bv,
u − v + Cv2 |u − v|2 ≥ ν/2u − v2 .
International Journal of Stochastic Analysis
11
Corollary 4.2. For any u and v ∈ L4 , let
27
h1 t 3
ν
27
h2 t 3
ν
t
0
t
0
us4L4 ds,
4.1
vs4L4 ds.
Then
1
− v Bu
νAu
− Bv
ḣi tu − v, u − v ≥ 0,
2
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.
Ω; L2 , η ∈ L∞
Ω; L2 , and f ∈ L2 0, T ; H−1 . Then the projected
Proposition 4.3. Let ξ ∈ L∞
Fτ
Fτ
N
N
N
system 3.2 admits a unique adapted solution uN
ε , Zε , pε , Zε in
2
2
2
1
2
2
∩
L
Ω;
L
0,
τ;
H
×
L
Ω;
L
L∞
τ;
L
τ
×
Ω;
L
0,
0,
Q
0
F
F
F
2
× L∞
∩ L2F Ω; L2 0, τ; H01
× L2F Ω; L2 0, τ; LQ .
F 0, τ × Ω; L
4.3
Proof. For every M ∈ N, let LM be a Lipschitz C∞ function which has the following property:
⎧
⎪
1
if u < M,
⎪
⎪
⎨
LM u 0
if u > M 1,
⎪
⎪
⎪
⎩
0 ≤ LM u ≤ 1 otherwise.
4.4
it is easy to show that LM B
is Lipschitz and
Applying the truncation LM to B,
N x − LM yB
N y ≤ CN,M x − y
LM xB
4.5
for any x, y ∈ L2N and M ∈ N. Let us define a truncated projected system:
N N,M
N uN,M
duN,M
uε t dt − ∇pεN,M tdt
t −νA
tdt − LM uN,M
t
B
ε
ε
ε
fN tdt ZN,M
tdWN t,
ε
4.6
εdpεN,M t ∇ · uN,M
tdt ZεN,M tdW N t,
ε
uN,M
τ ξN ,
ε
pεN,M τ ηN .
12
International Journal of Stochastic Analysis
0, τ × Ω; L2 ∩ L2F Ω; L2 0, τ; H01 , let us map
For fixed p ∈ L∞
F
N N,M
N uN,M
duN,M
uε t dt − ∇pN,M tdt
t −νA
tdt − LM uN,M
t
B
ε
ε
ε
fN tdt ZN,M
tdWN t,
ε
4.7
uN,M
τ ξN
ε
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 uN,M
, ZN,M
. Then for
ε
ε
εdpεN,M t ∇ · uN,M
tdt ZεN,M tdW N t
ε
pεN,M τ ηN ,
4.8
there is a unique adapted solution pεN,M , ZεN,M . Thus we can define an operator Ψ, such
that Ψp pεN,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
− Fu, u ≤ α|u|2 − βu2 ;
νAu
2
− Fu, Au
νAu
≤ αu2 − β
Au
.
4.9
A.3 For any u and v in H10 , a constant κ > ν, and a positive constant α,
− v − Fu − Fv, u − v ≤ α|u − v|2 .
κAu
4.10
A.4 For any u ∈ H10 and some positive constant α,
|Fu, u| ≤ αu2 .
4.11
International Journal of Stochastic Analysis
13
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:
ε tdt Bu
ε tdt ∇pε tdt Fuε tdt Zε tdWt,
duε t νAu
4.12
εdpε t ∇ · uε tdt Zε tdWt,
N N
N
duN
ε t −ν A uε tdt − B
pε τ η,
uε τ ξ,
N
N
N
N
uN
uN
ε t dt − ∇pε tdt F
ε t dt Zε tdW t,
4.13
N
N
εdpεN t ∇ · uN
ε tdt Zε tdW t,
N
uN
ε τ ξ ,
pεN 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. Assume u and v ∈ L4 . Then the following inequality is true:
u − v ≤ κ − νu − v2 Bu − Bv,
27
16κ − ν3
|u − v|2 v4L4 .
4.14
Corollary 4.6. Let u and v ∈ L4 . Define
T
2α l1 t t
T
l2 t 2α t
27
8κ − ν
us4L4
3
27
8κ − ν
ds,
4.15
vs4L4
3
ds.
Then
1
− v Bu
νAu
− Bv
− Fu − Fv l̇i tu − v, u − v ≤ 0,
2
i 1, 2.
Remark 4.7. To prove Corollary 4.6, the monotonicity assumption A.3 is used.
4.16
14
International Journal of Stochastic Analysis
Ω; L2 and η ∈ L∞
Ω; L2 . Then for any solution of system 4.13,
Proposition 4.8. i Let ξ ∈ L∞
Fτ
Fτ
the following is true:
N
2
uN
∈ L∞
∩ L2F Ω; L2 0, τ; H10
× L2F Ω; L2 0, τ; LQ ,
ε , Zε
F 0, τ × Ω; L
2
2
2
pεN , ZεN ∈ L∞
.
×
L
Ω;
L
0,
τ;
L
τ
×
Ω;
L
0,
Q
F
F
4.17
Moreover, there exists a constant K, independent of N, such that
τ 2
N 2
N 2
sup uN
E
s
ds
ε
sup
t
t
u
p
ε
ε
ε
0
t∈0,τ
t∈0,τ
τ τ 2
N 2
E ZN
s
ds
E
s
Z
ds ≤ K,
ε
ε
LQ
0
4.18
LQ
0
P-a.s.
4.13:
ii Let ξ ∈ L2Fτ Ω; L2 and η ∈ L2Fτ Ω; L2 . The following is true for any solution of system
N
∈ L∞ 0, τ; L2F Ω; L2 ∩ L2F Ω; L2 0, τ; H10
× L2F Ω; L2 0, τ; LQ ,
uN
ε , Zε
pεN , ZεN ∈ L∞ 0, τ; L2F Ω; L2 × L2F Ω; L2 0, τ; LQ .
4.19
Moreover, there exists a constant K, independent of N, such that
τ 2
N 2
N 2
sup EuN
E
s
ds
ε
sup
E
t
t
u
p
ε
ε
ε
0
t∈0,τ
t∈0,τ
τ τ 2
N 2
E Zε s
ds E ZεN s
ds ≤ K.
LQ
0
4.20
LQ
0
Proof. i Similar to the proof of Proposition 3.1, utilizing Assumption A.2, 3.6 becomes
2
N
uε t ∧ τ
τ 2
N
N
N
N
N
N
N uN
∇p
,
u
ds
νA
u
u
ξN 2
B
−
F
s
s
s
s
s
ε
ε
ε
ε
ε
−2
τ t∧τ
t∧τ
τ N 2
N
N
−
ZN
s
u
sdW
s,
s
Z
ds
ε
ε
ε
t∧τ
LQ
τ 2
2
N 2
N
N
ds
≤ ξN 2
αuN
−
β
∇p
u
s
s
s,
s
u
ε
ε
ε
ε
−2
τ t∧τ
t∧τ
N
N
ZN
ε sdW s, uε s
τ N 2
−
Zε s
ds.
t∧τ
LQ
4.21
International Journal of Stochastic Analysis
15
For 0 < r ≤ t, taking the conditional expectation with respect to Fr∧τ , one gets
EFr∧τ uN
ε t
τ τ 2
N 2
N 2
Fr∧τ
Fr∧τ
∧ τ E
Zε s
ds 2βE
uε s
ds
LQ
t∧τ
2
≤ EFr∧τ ξN 2αEFr∧τ
t∧τ
τ τ 2
2
N
Fr∧τ
dpεN s
uε s ∧ τ ds εE
t∧τ
1
− EFr∧τ
ε
4.22
t∧τ
τ N 2
Zε s
ds,
LQ
t∧τ
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∞
Ω; H10 and η ∈ L∞
Ω; H01 . Then for any solution
Fτ
Fτ
uε , Zε , pε , Zε of system 4.13, there exists a constant K0 , such that
2
sup uε t2 sup pε ≤ K0 .
t∈0,τ
t∈0,τ
4.23
Proof. The proof involves an application of the Itô formula to uε t2 , 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∞
Ω; H10 and η ∈ L∞
Ω; H01 . For system 4.12, there exists a solution
Fτ
Fτ
uε , Zε , pε , Zε in
1
2
2
∞
1
2
2
.
L∞
F 0, τ × Ω; H0 × LF Ω; L 0, τ; LQ × LF 0, τ × Ω; H0 × LF Ω; L 0, τ; LQ
4.24
Proof. We have the following steps.
Step 1 The limits. Clearly, by Proposition 4.8, there exist uε , pε , Zε , and Zε , such that
w
k
uN
ε −→ uε
w
pεNk −→ pε
in L2F Ω; L2 0, τ; H10 ,
2
in L∞
F 0, τ × Ω; L ,
w
in L2F Ω; L2 0, τ; LQ ,
w
in L2F Ω; L2 0, τ; LQ ,
k
ZN
ε −→ Zε
ZεNk −→ Zε
4.25
is a continuous map from H1 to H−1 ,
for a subsequence Nk . Since A
0
Auε H−1
≤ Cuε 4.26
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International Journal of Stochastic Analysis
for all uε ∈ H10 and some constant C. Thus combined with the assumptions on F, one knows
that
w
Nk
k
k
Nk uN
uN
−→ A in L2F Ω; L2 0, τ; H−1
νA
ε −F
ε
4.27
for some function A and some subsequence Nk . By Lemma 4.1,
N N
B uε t
H−1
N N
uε t , w sup B
w1
3/2 N 1/2
≤ sup 2
uN
uε t wL4
ε t
w1
4.28
3/2
≤ K uN
.
ε t
Thus
w
k
Nk uN
B
−→ B
ε
4/3
−1
Ω;
L
0,
τ;
H
in L4/3
F
4.29
for some function B and some subsequence Nk . For every t, we define
Lt : L2F Ω; L2 0, τ; LQ −→ L2F Ω; L2 0, τ; H−1
M −→
τ
4.30
MsdWs.
t∧τ
It can be shown that Lt is a bounded linear operator. Hence
τ
w
Nk
k
ZN
ε sdW s −→
t∧τ
τ
t∧τ
Zε sdWs
in L2F Ω; L2 0, τ; H−1 .
4.31
Similarly, one can prove that
τ τ
w
Nk
Nk
Nk Nk
Nk
Nk
νA uε s − F
uε s B
uε s ds −→
{As Bs}ds
t∧τ
4.32
t∧τ
Ω; L4/3 0, τ; H−1 and
in L4/3
F
τ
t∧τ
w
ZεNk sdW Nk s −→
τ
t∧τ
Zε sdWs in L2F Ω; L2 0, τ; H −1 .
4.33
International Journal of Stochastic Analysis
17
Also
2
−→ L2F Ω; L2 0, τ; H −1 ,
Lp : L ∞
F 0, τ × Ω; L
p −→
τ
4.34
∇psds
t∧τ
is a bounded linear operator. Since pεNk ∈ L∞
0, τ × Ω; L2 , we have
F
τ
w
∇pεNk sds −→
τ
t∧τ
in L2F Ω; L2 0, τ; H −1 .
∇pε sds
t∧τ
4.35
Similarly,
τ
w
k
∇ · uN
ε sds −→
τ
t∧τ
in L2F Ω; L2 0, τ; H−1 .
∇ · uε sds
t∧τ
4.36
To sum up,
uε t ∧ τ ξ τ
"
!
As Bs ∇pε s ds −
t∧τ
εpε t ∧ τ εη τ
∇ · uε sds −
t∧τ
τ
Zε tdWs,
t∧τ
τ
4.37
Zε sdWs
t∧τ
hold P-a.s.
Step 2 The Itô formula. For convenience, let us denote Nk by N again. Let M ≤ N and
0, τ × Ω; H10 M and some constant K, such that v ≤ K
H10 M PM H10 . For any v ∈ L∞
F
uniformly, define
T
2α rt t
27
8κ − ν3
K
4
4.38
ds.
2
Applying the Itô formula to e−rt |uN
ε t| , we get
2 τ
2
2
N
−r0 N
−ṙse−rs uN
ξ − e
uε 0 ε s ds
0
2
τ
0
N
N uN
N uN
e−rs −νA
ε s − B
ε s − ∇pε s
F
N
2
τ
e
0
−rs
uN
ε s
, uN
ε s
4.39
ds
N
N
ZN
ε sdW s, uε s
τ
0
2
e−rs ZN
ε s
ds.
LQ
18
International Journal of Stochastic Analysis
By taking the expectation, we get
τ
2
2
N
−r0 N
Eξ − Ee
uε 0 2E e−rs ∇pεN s, uN
ε s ds
E
0
τ
2
e−rs ZN
ε s
ds
LQ
0
− 2E
τ
0
1
N
N
N
N
N
N
N uN
B
u
u
ṙsu
e−rs νA
−
F
u
ds.
s
s
s
s,
s
ε
ε
ε
ε
ε
2
4.40
Clearly, limN → ∞ E|ξN |2 E|ξ|2 . By 3.5, it is clear that
τ
2E
e
−rs
0
∇pεN s, uN
ε s
2
2 1 τ
2
N
−r0 N
ds εEη − εEe
pε 0 − E e−rs ZεN s
ds.
LQ
ε
0
4.41
Because of 4.40 and 4.41, one gets the following:
τ
1
N
N
N
N uN
N uN
uN
e−rs νA
ε s − F
ε s B
ε s ṙsuε s, uε s ds
2
0
N →∞
τ
2
2
e−rs ZN
−E|ξ|2 lim Ee−r0 uN
ε 0 lim E
ε s
ds
lim 2E
N →∞
N →∞
0
LQ
τ
2 1
2
2
− εEη ε lim Ee−r0 pεN 0 lim E e−rs ZεN s
ds
LQ
εN →∞ 0
N →∞
τ
1
≥ 2E e−rs As Bs ṙsuε s, uε s ds.
2
0
4.42
Note that one gets the last inequality by applications of the Itô formula to 4.37, and the fact
that
2
−r0
lim Ee−r0 uN
|uε 0|2 ,
ε 0 ≥ Ee
N →∞
2
2
lim Ee−r0 pεN 0 ≥ Ee−r0 pε 0 ,
N →∞
τ
lim E
N →∞
2
e−rs ZN
ε s
ds
LQ
0
τ
lim E
N →∞
0
≥E
4.43
τ
e
0
−rs
Zε s2LQ ds,
τ
2
e−rs ZεN s
ds ≥ E e−rs Zε s2LQ ds.
LQ
0
International Journal of Stochastic Analysis
19
Step 3 Monotonicity. By Corollary 4.6, we get
τ
E
0
1
N
N
N
N
N
−
F
u
ṙsu
e−rs νAu
B
u
u
−
vs
ds
s
s
s
s,
s
ε
ε
ε
ε
ε
2
τ
1
−rs
N
νAvs Bvs − Fvs ṙsvs, uε s − vs ds.
≤E e
2
0
4.44
0, τ × Ω; H10 M where M ≤ N. An application of 4.42 yields
Note that v ∈ L∞
F
τ
E
0
1
e−rs As Bs ṙsuε s, uε s − vsds
2
τ
1
Bvs
− Fvs ṙsvs, uε s − vsds.
≤ E e−rs νAvs
2
0
4.45
Since the above inequality is true for all M ∈ N and K > 0, it remains true for all
0, τ × Ω; H10 . Thus let v uε λw where w ∈ L∞
0, τ × Ω; H10 and λ > 0, and
v ∈ L∞
F
F
τ
E
ε s − Bu
ε s λws Fuε s λws, λwsds
e−rs As Bs − νAu
0
≥E
τ
e
−rs
0
λ
λνAws
ṙsws, λwsds.
2
4.46
By the fact that
ε t λwt, wt
Bu
ε t λwt, wt, uε t λwt
−Bu
ε t λwt, wt, uε t
−Bu
ε t, wt, uε t − λ Bwt,
−Bu
wt, uε t
4.47
ε t, wt λ Bwt,
Bu
uε t, wt ,
we have
τ
E
ε s − Bu
ε s Fuε s λws, ws ds
e−rs As Bs − νAu
0
≥ λE
τ
0
e−rs
1
νAws Bws, uε s ṙsws, ws ds.
2
4.48
20
International Journal of Stochastic Analysis
Letting λ → 0, and by the arbitrariness of w and the fact that F is continuous, we know that
ε s Bu
ε s − Fuε s P-a.s.,
As Bs νAu
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
w ≤ CuL4 vL4 u − vwL4 Cu vu − vL4 wL4 .
Bu − Bv,
5.1
Proof. By Proposition 2.5,
w Bu − Bv,
w, v − Bu,
w, u Bv,
w, v − Bu,
w, u − v − Bu,
w, v Bv,
− v, w, v − Bu,
w, u − v − Bu
− v, v, w u − v, w Bu
Bu,
5.2
1
≤ uL4 u − vwL4 uu − vL4 wL4 u − vL4 vwL4
2
1
u − vvL4 wL4
2
≤ CuL4 vL4 u − vwL4 Cu vu − vL4 wL4 .
Theorem 5.2. Let ξ ∈ L∞
Ω; H10 and η ∈ L∞
Ω; H01 . System 4.12 admits a unique adapted
Fτ
Fτ
solution in
1
2
2
∞
1
2
2
.
L∞
F 0, τ×Ω; H0 ×LF Ω; L 0, τ; LQ ×LF 0, τ×Ω; H0 ×LF Ω; L 0, τ; LQ
5.3
International Journal of Stochastic Analysis
21
Also the solution is continuous with respect to the terminal conditions in
L∞ 0, τ; L2F Ω; L2 × L2F Ω; L2 0, τ; LQ × L∞ 0, τ; L2F Ω; L2 × L2F Ω; L2 0, τ; LQ .
5.4
Proof. The existence of an adapted solution is shown in Theorem 4.10. Suppose that
uε1 , Zε1 , pε1 , Zε1 and uε2 , Zε2 , pε2 , Zε2 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
uε uε1 − uε2 ,
Zε Zε1 − Zε2 ,
pε pε1 − pε2 ,
ξ ξ1 − ξ2 ,
Z ε Zε1 − Zε2 ,
η η1 − η2 .
5.5
Then one has
ε tdt Bu
ε1 t − Bu
ε2 t dt ∇p tdt,
duε t νAu
ε
Fuε1 t − Fuε2 tdt Zε tdWt,
εdpε t ∇ · uε tdt Zε tdWt,
uε τ ξ,
5.6
pε τ η.
Similar to Corollary 4.6, let us define
T
lt 2α t
27
8κ − ν
K2
3 0
ds,
5.7
where K0 is the constant in Proposition 4.9. An application of the Itô formula to e−lt |uε t|2
and Corollary 4.6 imply
e
−lt∧τ
2
|uε t ∧ τ| τ
t∧τ
2
e−ls Zε s
ds
LQ
τ
2
ε s − Bu
ε1 s − Bu
ε2 s − Fuε1 s − Fuε2 s
e−ls νAu
ξ 2
t∧τ
2
τ
t∧τ
1
l̇suε s, uε s ds
2
τ
$
#
e−ls ∇pε s, uε s ds − 2
e−ls Zε sdWs, uε s
t∧τ
τ
2
2
2 1 τ −ls ≤ ξ ε
e−ls dpε s −
e
Z ε s
ds
LQ
ε
t∧τ
t∧τ
τ
τ
e−ls Z ε tdWs, pε sds − 2
e−ls Zε sdWs, uε s.
−2
t∧τ
t∧τ
5.8
22
International Journal of Stochastic Analysis
Taking the expectation, the above inequality becomes
2
E|uε t ∧ τ|2 εEpε t ∧ τ E
≤e
−l0
2
2
Eξ εEη .
τ τ 2
2
1
Zε s
ds E
Z ε s
ds
LQ
LQ
ε
t∧τ
t∧τ
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 Ft adapted. This is why we defined another function lt in the proof of
the uniqueness based on the H10 -bound of the solution. Fortunately, lt 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 lt.
5.2. The Convergence of the Solution As ε 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:
dut −νAutdt − Butdt − ∇pt Ftdt ZtdWt,
∇ · ut 0,
uτ ξ,
pτ η,
5.10
where Au −Δu and Bu, v u · ∇v with the notation Bu Bu, 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, uε , Zε , pε , 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
w
w
w
sequence of positive numbers {εi }∞
→ u, pεi −→ p and Zεi −→ Z in
i1 such that εi → 0, uεi −
corresponding spaces.
From 4.18 and 4.20, one knows that along a subsequence,
√ dpεi
dD
εi
, h −→ E
,h
E
dt
dt
5.11
for some D ∈ L2F Ω; L2 0, τ; L2 and for all h ∈ L2F Ω; L2 0, τ; L2 . Thus we get
Eεi
dpεi
, h −→ 0
dt
5.12
International Journal of Stochastic Analysis
23
in the sense of distribution. Since
#
$
εdpεi t ∇ · uεi tdt, ht Zεi tdWt, ht,
5.13
we know that
τ
E
∇ · ut, htdt lim E
0
i→∞
τ
0
∇ · uεi t, htdt − lim E
i→∞
τ
$
#
εdpεi t, ht 0
5.14
0
for all h ∈ L2F 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.
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