Existence of a martingale weak solution to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with a stochastic perturbation

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by  Zhong Tan

EXISTENCE OF A MARTINGALE WEAK SOLUTION TO THE
EQUATIONS OF NON-STATIONARY MOTION OF NON-NEWTONIAN
FLUIDS WITH A STOCHASTIC PERTURBATION
7
1
ZHONGTAN,HUAQIAOWANG,ANDYUCONGWANG
0
2
Abstract. In this paper, we consider the stochastic incompressible non-Newtonian
n
fluidsdrivenbyacylindricalWienerprocessW withshearratedependentonviscosity
a
inabounded LipschitzdomainD∈Rn duringthe timeinterval (0,T). Forq> 2n+2
J n+2
in the growth conditions (1.2), we prove the existence of a martingale weak solution
5
with ∇·u = 0 by using a pressure decomposition which is adapted to the stochastic
setting,thestochasticcompactness methodandtheL∞-truncation.
]
P
A
.
h
t 1. Introduction
a
m Let D ∈ Rn (n ≥ 2) be a bounded Lipschitz domain. For the time interval (0,T), we set
[ Q:=(0,T)×D. In this paper, we consider the following equations:

1 du+∇·(u⊗u−S+pI)dt=fdt+Φ(u)dW,
v
∇·u=0,
2  (1.1)
4 u|∂D =0,
2
u| =u ,
t=0 0
1
0 whereS ={Sij}isthedeviatoricstresstensor,pthepressure,uthevelocity,f theexternal
.
1 forceandW acylindricalWienerprocesswithvaluesinaHilbertspace. Φsatisfythelinear
0 growth assumption (see Sect.2 for details).
7 ThestressS maydependonboth(x,t)andthe“rateofstraintensor”D={D },which
ij
1 is defined by D =D (u):= 1(∂ ui+∂ uj), i,j =1,··· ,n. We refer to [2], [4] and [27]
: ij ij 2 xj xi
v about the continuum mechanical background. As far as we know, the fluids with shear
i
X dependent viscosity are often used in engineering practice. So it’s meaningful to study
this kind of fluid. In this paper, S is assumed to be a function of the shear rate and the
r
a constitutive relations reads as
S =ν(D )D,
II
where D = 1D : D is the second invariant of D. Here are some examples of precise
II 2
constructions of S: for q ∈(1,+∞), constant ν ,
0
S =ν0(DII)q−22D,
S =ν0(1+DII)q−22D.
For detials, see [2, 6, 42]. If q ∈ (1,2), we say the non-Newtonian fluids is pseudoplastic
or shear thinning (for example, ketchup); if q = 2, it’s Newtonian fluids; if q ∈ (2,+∞),
we saythe non-Newtonianfluids is dilatantorshear thickening(for example, batter). The
Date:January6,2017.
2010 Mathematics Subject Classification. 35R60,35D30, 60H15,35K55,76D03.
Key words and phrases. L∞-truncation, martingale weak solution, Non-Newtonian fluids, pressure
decompsition.
1
2 ZHONGTAN,HUAQIAOWANG,ANDYUCONGWANG
following two constitutive laws which are also of interest in engineering practice are given
by
S =ν0(DII)q−22D+ν D,
∞
S =ν0(1+DII)q−22D+ν D,
∞
where ν and ν are positive constants and q ∈ [1,+∞). An extensive list for specific
0
∞
q-values for different fluids can be found in [6].
Forq ∈[1,+∞),weassumethedeviatoricstresstensorS satisfythefollowingconditions
inthis paper: S :Q×Mn →Mn is aCarath´eodoryfunction. ∀ξ ∈Mn (vectorspace
sym sym sym
of all symmetric n×n matrices ξ = {ξ }. We equip Mn with scalar product ξ : η and
ij sym
1
norm kξk:=(ξ :ξ)2.), for almost all (x,t)∈Q,
|S(x,t,ξ)|≤C0kξkq−1+η1, (1.2)
where C >0, η ≥0, η ∈Lq′(Q), 1/q+1/q =1; ∀ξ ∈Mn , for almost all (x,t)∈Q,
0 1 1 ′ sym
S(x,t,ξ):ξ ≥C kξkq−η , (1.3)
0 2
where C >0, η ≥0, η ∈L1(Q); ∀ξ,η ∈Mn (ξ 6=η), for almost all (x,t)∈Q,
0 2 2 sym
(S(x,t,ξ)−S(x,t,η):(ξ−η)>0. (1.4)
The flow of a homogenous incompressible fluid without stochastic part is described by
the following equations:
∂ u+∇·(u⊗u−S+pI)=−∇·f,
t
∇·u=0,
 (1.5)
u|∂D =0,
u| =u .
t=0 0
Inthe latesixties,LionsandLadyshenskayain[28,29,30,31]startedthe mathematical
discussion of power-lawmodel. In [28], Ladyzhenskayaachievedthe existence and unique-
ness of weak solutions and in [31] Lions achieved these results for q ≥ 3n+2. They showed
n+2
theexistenceofaweaksolutioninthespaceLq(0,T;W1,q (D))∩L (0,T;L2(D)). Inthis
0,div ∞
particular case that u⊗u : D(u) ∈ L1(Q) follows from parabolic interpolation, the proof
of existence is based on monotone operators and compactness arguments. In [33], Ma´lek,
Neˇcas and Ruˇziˇcka proved the existence for q ∈[2,11) under the assumption that D ∈R3
5
is a bounded domain with C3-boundary and that S(D) has the form S(D)=∂DΦ(DII). In
[43], Wolf improved this result to the case p> 2n+2 by using L -truncation.
n+2 ∞
In the fluid motion, apart from the force f, there might be further quantities with a
influence on the motion. This influence usually is small and can be shown by adding a
stochastic part to the equation. The stochastic part to the equation can be understood
as a turbulence. This type of equation is often used in fluid mechanics since they model
the phenomenon of perturbation. So it’s very interesting to study the stochastic fluids.
In SPDES, we consider two concepts: strong (pathwise) solutions and weak (martingale)
solutions. Strong solutions means that the underlying probability space and the Wiener
process are given in advance. While martingale solutions means that the combination
of these stochastic elements and the fluid variables is the solution of the problem and
the original equations are satisfied in the sense of distributions. Clearly, the existence of
strong solutions implies the existence of martingale solutions. There are many research
results on the stochastic Newton flow dating back to the 1970’s with the initial work
of Bensoussan and Temam [5]. For example, the existence of strong solutions and mar-
tingale solutions to the stochastic incompressible Navier-Stokes equations is established
by Da Prato-Zabczyk [15], Breckner [8], Menaldi-Sritharan [34], Glatt-Holtz-Ziane [22],
EXISTENCE OF MARTINGALE SOLUTIONS TO THE STOCHASTIC NON-NEWTONIAN FLUIDS 3
Taniguchi[41],Cap´ınski-Peszat[13,14],Kim[26],Cap´ınski-Gatarek[12],Flandoli-Gatarek
[20], Mikulevicius-Rozovskii [35, 36], Brze´zniak-Motyl [11] and the references therein; for
the stochastic incompressible MHD equations, the existence of solutions is considered in
[40] and the references therein. For the stochastic incompressible non Newtonian flow,
there areonly afew results. Recently, Breit[17]provedthe existence ofa martingaleweak
solutionof the stochastic Navier-Stokesequations ofthe model S(D(u))=(1+Du)p 2Du.
−
Inthispaper,wewillprovetheexistenceofmartingalesolutionsofthestochasticequations
(1.1) with S =ν(D )D, which is the general form of S(D(u))=(1+Du)p 2Du.
II −
Comparingwiththeworkin[43],wefacetheessentialchallengeofestablishingsufficient
compactness in order to be able to pass to the limit in the class of solutions. In general
it is not possible to get any compactness in ω as no topological structure on the sample
space Ω. That is, even if a space X is compactly embedded in another space Y, it is not
usually the case that L2(Ω,X) is compactly embedded in L2(Ω,Y). As such, Aubin-Lions
Lemma orArzela`-AscoliTheorem,whichclassicallymakepossible the passageto the limit
in the nonlinear terms, cannot be directly applied in the stochastic setting. To overcome
this difficulty, it is classical to rather concentrate on compactness of the set of laws of
the approximations (the Prokhorov Theorem, which is used to obtain compactness in the
collection of probability measures associated to the approximate solutions) and apply the
Skorokhod embedding Theorem, which provides almost sure convergences of a sequence
of random variables that have the same laws as the original ones, but relative to a new
underlying stochastic basis. However, the Skorokhod embedding Theorem is restricted to
metric spaces but the structure of the stochastic non Newtonian equations naturally leads
to weakly converging sequences. For this, we apply the Jakubowski-Skorokhod Theorem
which is valid on a large class of topological spaces (including separable Banach spaces
with weak topology). Compared with the work in [17], the biggest difference is that we
use the cut-off function to prove the approximated equations for S = ν(D )D, which is
II
the general form of S in [17] also hold on the new probability space, rather than using a
general and elementary method that was recently introduced in [38].
The rest of the paper is organized as follows. In Sect 2, we formulate some stochastic
background and give our main Theorem. In Sect 3, we reconstructed the pressure which
disappears in the weak formulation. In Sect 4, we use Galerkin method adding a large
power of u to study auxiliary problem. In Sect 5, we prove the main theorem.
2. Hypotheses and Main Theorem
Let (Ω,F,F ,P) be a stochastic basis, where F is a nondecreasing family of sub-σ-
t t
fields of F, i.e., F ⊂ F for 0 ≤ s ≤ t ≤ T. Assume that filtration {F ,0 ≤ t ≤ T} is
s t t
right-continuous and F contains all the P-negligible events in F.
0
TheprocessW isacylindricalWiener process,i.e., W(t)= β (t)e ,with(β )
k 1 k k k k 1
being mutually independent real-valued standard Wiener proces≥ses relative to F an≥d
t
P
{e } a complete orthonormal system in a separable Hilbert space U. Since W don’t
k k 1
actua≥lly converge on U, we define U ⊃ U by U = {v = α e ; α2/k2 < ∞}.
0 0 k 1 k k k 1 k
The norm of U is given by kvk2 = α2/k2,v = ≥α e . Then≥the embedding
0 U0 k 1 k Pk 1 k k P
U ֒→U is Hilbert-Schmidt and the trajec≥tories of W are P≥-a.s. continuous with values in
0
P P
U . Note that
0
t
ψ(r)dW(r)
Z0
where ψ ∈ L2(Ω;L (U,L2(D))) is progressively measurable, defines a P-almost surely
2
continuous L2(Ω) valued F -martingale. Furthermore, we can multiply the Itoˆ’s integral
t
4 ZHONGTAN,HUAQIAOWANG,ANDYUCONGWANG
with a test-function since
t t
∞
ψ(r)·ϕdxdW(r) = ψ(r)e ·ϕdxdβ (r), ϕ∈L2(D)
k k
Z0 ZD k=1Z0 ZD
X
is well-defined.
In this paper, the mapping Φ(z) : U → L2(D) is defined by Φ(z)e = g (z(·)), ∀z ∈
k k
L2(D). We assume that g ∈C(R×D) and satisfy the following condition:
k
|g (ξ)|≤c(1+|ξ|), ∀ξ ∈Rn,
k (2.1)
k 1
X≥
|∇g (ξ)|2 ≤c, ∀ξ ∈Rn,
k (2.2)
k 1
X≥
and additionally implies
supk2|g (ξ)|2 ≤c(1+|ξ|2), ∀ξ ∈Rn.
k (2.3)
k 1
≥
Now, we are ready to give a precise definition of the martingale weak solutions.
Definition 2.1. Let µ , µ be Borel probability measures on L2 (D) and L2(Q) respec-
0 f div
tively. A system
((Ω,F,F ,P),u,u ,f,W)
t 0
is called a martingale weak solution to (1.1), and S satisfy (1.2), (1.3), and (1.4) with the
initial datum µ and µ if the following conditions are satisfy:
0 f
(1) (Ω,F,F ,P) is a stochastic basis with a complete right-continuous filtration,
t
(2) W is an F -cylindrical Wiener process,
t
(3)u∈L2(Ω;L (0,T;L2(D)))∩Lq(Ω;Lq(0,T;W1,q (D)))isprogressivelymeasurable,
∞ 0,div
(4) u0 ∈L2(Ω;L2(D)) with µ0 =P◦u−01,
(5) f ∈L2(Ω;L2(Q)) is adapted to F and µ =P◦f 1,
t f −
(6) ∀ϕ∈C (D) and ∀t∈[0,T], it holds that P-a.s.
0∞,div
t
(u(t)−u )·ϕdx= u⊗u:D(ϕ)−S(x,r,D(u)):D(ϕ)dxdr
0
ZD Z0ZD
t t
+ f ·ϕdxdr+ Φ(u)·ϕdxdW(r).
Z0ZD Z0ZD
Next, we state our main result.
Theorem 2.1. Assume that q > 2n+2, S satisfies (1.2), (1.3) and (1.4). Φ satisfies (2.1)
n+2
and (2.2). And further suppose that
kvkβL2(D)dµ0(v)<∞, kgkβL2(Q)dµf(g)<∞ (2.4)
ZL2div(D) ZL2(Q)
with β :=max{2n+2,qn+2q}. Then there exists a martingale weak solution to (1.1) in the
n n
sense of Definition 2.1.
3. Pressure Decomposition
In the present section we are going to introduce a pressure method generalizes [43] to
the stochastic case. Here the pressure p will be decomposed into four part p , p , p and
1 2 h
p . We show a-priori estimates for the components p , p , p and p .
Φ 1 2 h Φ
EXISTENCE OF MARTINGALE SOLUTIONS TO THE STOCHASTIC NON-NEWTONIAN FLUIDS 5
Theorem3.1. Let(Ω,F,F ,P)beastochasticbasis,v ∈L2(Ω;L (0,T;L2(D)))adapted
t ∞
to F . Assume H +H ∈ Lα(Ω;Lα(Q)) adapted to F for some α > 1, H ∈ Lα1(Ω×
t 1 2 t 1
Q,P⊗Ln+1) and H ,∇H ∈ Lα2(Ω×Q,P⊗Ln+1). Moreover, let v ∈ L2(Ω;L2 (D)),
2 2 0 div
and Φ∈L2(Ω;L (0,T;L (U,L2(D)))) progressively measurable such that
∞ 2
t t
(v(t)−v )·ϕdx+ (H +H ):∇ϕdxdr = Φ·ϕdxdW(r) (3.1)
0 1 2
ZD Z0ZD Z0ZD
holds for all ϕ∈C (D). Then there are functions p , p , p and p adapt to F such
0∞,div 1 2 h Φ t
that
(1) ∆p =0 and the following estimates are satisfied for θ :=min{2,α}:
h
β β
E |p |α1dxdt ≤cE |H |α1dxdt ,
1 1
(cid:18)ZQ (cid:19) (cid:18)ZQ (cid:19)
β β
E |p |α2dxdt ≤cE |H |α2dxdt ,
2 2
(cid:18)ZQ (cid:19) (cid:18)ZQ (cid:19)
β
T β
E |∇p |α2dxdt ≤cE |H |α2 +|∇H |α2dxdt ,
2 2 2
Z0 ZD′ ! (cid:18)ZQ (cid:19)
E |p +p |αdxdt ≤cE |H +H |αdxdt ,
1 2 1 2
(cid:18)ZQ (cid:19) (cid:18)ZQ (cid:19)
E sup |p |2dx ≤cE sup kΦk2 ,
Φ L2(U,L2(D))
t∈(0,T)ZD ! t∈(0,T) !
E sup |p |θdx ≤cE 1+ sup |v|2dx+ sup kΦk2
h L2(U,L2(D))
t∈(0,T)ZD ! (cid:18) t∈(0,T)ZD t∈(0,T)
+ |v |2dx+ |H +H |αdxdt ,
0 1 2
ZD ZQ (cid:19)
β β
E sup |p |θdx ≤cE sup |v|2dx+ sup kΦk2
h L2(U,L2(D))
t∈(0,T)ZD ! t∈(0,T)ZD t∈(0,T) !
β
+cE 1+ |v |2dx+ |H +H |αdxdt ,
0 1 2
(cid:18) ZD ZQ (cid:19)
for all 1≤β <∞ and D ⊂⊂D.
′
(2) for all ϕ∈C (D), it holds that
0∞
t
(v(t)−v −∇p (t))·ϕdx+ (H +H ):∇ϕdxdr
0 h 1 2
ZD Z0ZD
t t
= (p +p )divϕdxdr+ p (t)divϕdx+ Φ·ϕdxdW(r).
1 2 Φ
Z0ZD ZD Z0ZD
Moreover, we have p(t) = p (t)+p (t)+ t(p +p )dr ∈ Lθ(Ω;L (0,T;Lθ(D))) and
h Φ 0 1 2 ∞
p (0)=p (0)=p (0)=p (0)=0 P-a.s..
1 2 h Φ
R
Proof. Let v be a weak solution to (3.1) for all ϕ ∈ W1,θ′ (D),1/θ+1/θ = 1. Then by
0,div ′
De Rahm’s theorem (see [21]), there exists a unique function p(t)∈Lθ(D) with p(0)=0,
0
such that
t t
(v(t)−v )·ϕdx− p(t)divϕdx+ (H +H ):∇ϕdxdr = Φ·ϕdxdW(r),
0 1 2
ZD ZD Z0ZD Z0ZD
6 ZHONGTAN,HUAQIAOWANG,ANDYUCONGWANG
for all ϕ∈W1,θ′(D).
0
By using the Bogovski˘i-operator Bog (see [7]) and let B = Bog (ϕ−(ϕ) ) where
D D D
(ϕ) = 1 ϕdx, then we can get
D D D
| |
R t t
p(t)ϕdx= (v(t)−v )·B(ϕ)dx+ (H +H ):∇B(ϕ)dxdr− Φ·B(ϕ)dxdW(r).
0 1 2
ZD ZD Z0ZD Z0ZD
Hence, we have
t t
p(t)=B (v(t)−v )+ (∇B) (H +H )dr− B ΦdW(r),
∗ 0 ∗ 1 2 ∗
Z0 Z0
where B denotes the adjoint of B with respect to the L2(D) inner product.
∗
Since θ := min{2,α}, using the continuity of B on L2(D), (∇B) on Lα(D) and the
∗ ∗
Burkholder-Davis-Gundy inequality, one has
t
E sup |p|θ .E sup |B (v(t)−v )|2+ |(∇B) (H +H )|α+ (B Φ)2dW(r)
∗ 0 ∗ 1 2 ∗
(0,T)ZD ! (0,T)ZD ZQ Z0 !
T
.E 1+sup |v|2+|v |2+ |H +H |α+ kΦk2 dt .
0 1 2 L2(U,L2(D))
(0,T)ZD ZQ Z0 !
(3.2)
Then p∈Lθ(Ω;L (0,T;Lθ(D))).
∞
Let∆−D2 bethesolutionoperatortothebi-Laplaceequationwithrespecttozerobound-
ary values for function and gradient. Let p0 = ∆∆−D2∆p and ph = p−p0. Using the
continuity of the operator ∆∆−D2∆ from Lθ(D) to Lθ(D) (see [37]), we have
E sup |p |θdx
0
t∈(0,T)ZD !
T
.E 1+ sup |v|2dx+ |v |2dx+ |H +H |αdxdt+ kΦk2 dt .
0 1 2 L2(U,L2(D))
t∈(0,T)ZD ZD ZQ Z0 !
(3.3)
E sup |p |θdx
h
t∈(0,T)ZD !
T
.E 1+ sup |v|2dx+ |v |2dx+ |H +H |αdxdt+ kΦk2 dt .
0 1 2 L2(U,L2(D))
t∈(0,T)ZD ZD ZQ Z0 !
(3.4)
Note that p (t) ∈ ∆W2,θ(D) is uniquely determined as the solution to the following
0 0
equation:
t t
p (t)∆ϕdx= (H +H ):∇2ϕdxdr− Φ·∇ϕdxdW(r), (3.5)
0 1 2
ZD Z0ZD Z0ZD
for all ϕ∈C (D).
0∞
From[37],weknowthatp ∈∆W2,α1(D)andp ∈∆W2,α2(D)aretheuniquesolutions
1 0 2 0
(defined P⊗L1-a.e.) such that
p (t)∆ϕdx= H :∇2ϕdx, (3.6)
1 1
ZD ZD
p (t)∆ϕdx= H :∇2ϕdx, (3.7)
2 2
ZD ZD
EXISTENCE OF MARTINGALE SOLUTIONS TO THE STOCHASTIC NON-NEWTONIAN FLUIDS 7
for all ϕ∈C (D). Then we have
0∞
(p (t)+p (t))∆ϕdx= (H +H ):∇2ϕdx,
1 2 1 2
ZD ZD
for all ϕ∈C (D) and p +p ∈∆W2,q˜(D). From Lemma 2.3 in [43], it follows that
0∞ 1 2 0
|p |α1dx≤c |H |α1dx,
1 1
ZD ZD
|p |α2dx≤c |H |α2dx,
2 2
ZD ZD
|p +p |αdx≤c |H +H |αdx P⊗L1−a.e..
1 2 1 2
ZD ZD
These imply
β β
E |p |α1dxdt ≤cE |H |α1dxdt ,
1 1
(cid:18)ZQ (cid:19) (cid:18)ZQ (cid:19)
β β
E |p |α2dxdt ≤cE |H |α2dxdt ,
2 2
(cid:18)ZQ (cid:19) (cid:18)ZQ (cid:19)
β
T β
E |∇p |α2dxdt ≤cE |H |α2 +|∇H |α2dxdt ,
2 2 2
Z0 ZD′ ! (cid:18)ZQ (cid:19)
E |p +p |αdxdt ≤cE |H +H |αdxdt .
1 2 1 2
(cid:18)ZQ (cid:19) (cid:18)ZQ (cid:19)
Letp :=p (t)− t(p +p )dr ∈∆W2,θ(D). From(3.5),(3.6)and(3.7),itfollowsthat
Φ 0 0 1 2 0
p is the unique solution to
Φ
R
t
p (t)∆ϕdx= Φ·∇ϕdxdW(r),
Φ
ZD Z0ZD
for all ϕ ∈ C (D). Since p (t) ∈ ∆W2,θ(D), by Weyl’s Lemma, for all ϕ ∈ C (D), we
0∞ Φ 0 0∞
have
t
p (t)ϕdx= Φ·∇(∆ 2∆ϕ)dxdW(r).
Φ −
ZD Z0ZD
Then pΦ = 0tD∗ΦdW(r), P⊗Ln+1-a.e., where D = ∇∆−D2∆ : L2(D) → W01,2(D), D∗ :
L2(D)→L2(D). Using the Burkholder-Davis-Gundy inequality, we obtain
R
E sup |p |2dx ≤cE sup kD Φk2 dt
Φ ∗ L2(U,L2(D))
t∈(0,T)ZD ! t∈(0,T) ! (3.8)
≤cE sup kΦk2 dt .
L2(U,L2(D))
t (0,T) !
∈
t
Finally, we can infer that p˜ (t) := p (t)+ (p +p )dr solves (3.5) and there holds
0 Φ 0 1 2
p˜ (t)∈∆W2,θ(D)whichimpliesp (t):=p (t)+ t(p +p )dr. Then,wegettheequation
Φ 0 0 Φ R 0 1 2
claimed in (2) of Theorem 3.1.
R
(cid:3)
Corollary 3.1. Let the assumptions of Theorem 3.1 be satisfied. There exists Φ ∈
p
L2(Ω;L (0,T;L (U,L2 (D)))) progressivelymeasurable such that
∞ 2 loc
t
p (t)divϕdx= Φ ·ϕdxdW(r), ∀ϕ∈C (D).
Φ p 0∞
ZD Z0ZD
8 ZHONGTAN,HUAQIAOWANG,ANDYUCONGWANG
Let D′ ⊂⊂D, then Φp satisfies kΦpekkL2(D′) ≤c(D′)kΦekkL2(D), ∀k,that is,it holds that
P⊗L1-a.e.
kΦpkL2(U,L2(D′)) ≤c(D′)kΦkL2(U,L2(D)).
If we assume that Φ satisfies (2.1) and (2.2), then there holds
kΦp(v1)−Φp(v2)kL2(U,L2(D′)) ≤c(D′)kv1−v2kL2(D), ∀v1,v2 ∈L2(D).
Proof. From the proof of Theorem 3.1, it follows that
t
p (t)divϕdx= Φ·∇(∆ 2∆divϕ)dxdW(r)
Φ −
ZD Z0ZD
t
= Φe ·∇(∆ 2∆divϕ)dxdβ
k − k
k Z0ZD
X (3.9)
t
= ∇∆∆−2divΦek·ϕdxdβk
k Z0ZD
X
t
= ∇∆∆ 2divΦ·ϕdxdW(r),
−
Z0ZD
for all ϕ∈C (D). Let Φ =∇∆∆ 2divΦ. Then we can get the first claim. By using the
0∞ p −
localregularitytheoryforthebi-Laplaceequationin[37],wecanprovetherestresults. (cid:3)
4. The approximated System
Let us consider the following approximate system:
du+∇·(u⊗u−S+pI)dt+ε|u|q˜ 2udt=fdt+Φ(u)dW,
−
(4.1)
(u|t=0 =u0,
for ε>0, depending on the law µ on L2 (D) and µ on L2(Q).
0 div f
Assume that f is adapted to F (otherwise enlarge it) and f ∈ L2(Ω;L2 (Q)) with
t div
µf = P◦f−1 and u0 ∈ L2(Ω;L2div(D)) with µ0 = P◦u−01. For the purpose of control the
nonlineartermu⊗u:∇u,weaddthetermε|u|q˜ 2uandchooseq˜≥max{2q,3}suchthat
− ′
the solution is an admissible test function. Notice that 2 + 1 ≤1 and 1 + 1 ≤1. Let
q˜ p q˜ 1 2
−
V =L2(Ω;L (0,T;L2(D)))∩Lq˜(Ω×Q;P⊗Ln+1)∩Lq(Ω;Lq(0,T;W1,q (D))).
q,q˜ ∞ 0,div
Fromtheappendixof[32],weknowthatthereexistasequence{λ }⊂Randasequence
k
of functions {w }⊂Wℓ,2 (D), ℓ∈N such that
k 0,div
(a) w is an eigenvector to the eigenvalue λ of the Stokes-operator in the sense that
k k
hwk,ϕiW0ℓ,2 =λk wk·ϕdx, ∀ϕ∈W0ℓ,,d2iv(D),
ZD
(b) w w dx=δ , ∀k,m∈N,
D k m km
(c) 1≤λ ≤λ ≤··· and λ →∞,
1 2 k
(d) Rh√wλkk,√wλmmiW0ℓ,2 =δkm, ∀k,m∈N,
(e) {w } is a basis of Wℓ,2 (D).
k 0,div
Now, we use Galerkin approximation to separate space and time. Then approximate
equations(4.1)becomesanordinarystochasticdifferentialequation. Byusingtheclassical
existence theorems for SDEs from [3], [18] and [19], we canprovethe existence of approxi-
mated solution. To this end, choosing ℓ>1+ n, suchthat Wℓ,2(D)֒→W1, (D). We are
2 0 ∞
finding an approximated solution:
N
uN = cNw =CN ·wN,
k k
k=1
X
EXISTENCE OF MARTINGALE SOLUTIONS TO THE STOCHASTIC NON-NEWTONIAN FLUIDS 9
where CN =(cN):Ω×(0,T)→RN and wN =(w ,w ,··· ,w ).
i 1 2 N
Let PN :L2 (D)→X :=span{w ,w ,··· ,w } be the orthogonal projection, i.e.,
div 1 2 N
N
PNv = hv,wkiL2 ·wk.
k=1
X
Therefore, we would like to solve the system
duN ·wkdx+ S(x,t,D(uN)):D(wk)dxdt+ε |uN|q˜−2uN ·wkdxdt
ZD ZD ZD
= uN ⊗uN :∇w dxdt+ f ·w dxdt+ Φ(uN)·w dxdW(t), (4.2)
k k k
ZD ZD ZD
uN(0)=Pu ,
0
P-a.s. for k =1,2,··· ,N and for a.e. t.
Assume that WN(s) = N β e (s) = βN(s)·eN. Then it turned out to solving the
k=1 k k
following ordinary stochastic differential equation:
P
dCN =A(t,CN)dt+B(CN)dβN,
t (4.3)
(CN(0)=C0,
where
N
A(t,CN)= − S(x,t,CN ·D(wN)):D(w )dx+ (CN ·wN)⊗(CN ·wN):∇w dx
k k
(cid:18) ZD ZD (cid:19)k=1
N N
− ε |CN ·wN|q˜ 2(cN ·wN)·w dxdt + f ·w dx ,
− k k
(cid:18) ZD (cid:19)k=1 (cid:18)ZD (cid:19)k=1
N
B(CN)= Φ(CN ·wN)e ·w dx ,
l k
(cid:18)ZD (cid:19)k,l=1
N
C0 = hv0,wkiL2(D) k=1.
In (cid:0)order to make(cid:1)use of the classical existence theorems for SDEs, we need to prove
that A and B satisfy globally Lipschitz continuous condition and growth condition in the
following. Note that
(A(t,CN)−A(t,CˆN))·(CN −CˆN)
=− (S(x,t,D(uN))−S(x,t,D(uˆN))):(D(uN)−D(uˆN))dx
ZD
+ (uN ⊗uN −uˆN ⊗uˆN):(D(uN)−D(uˆN))dx
ZD
−ε (|uN|q˜ 2uN −|uˆN|q˜ 2uˆN)(uN −uˆN)dx
− −
ZD
≤ (uN ⊗uN −uˆN ⊗uˆN):(D(uN)−D(uˆN))dx.
ZD
Here we have used the monotonicity assumption (1.4). If |CN|≤R and |CˆN|≤R, then
(A(t,CN)−A(t,CˆN))·(CN −CˆN)≤c(R,N)|CN −CˆN|2.
This implies weakmonotonicityinthe senseof(3.1.3)in [39]byusing Lipschitz continuity
B for CN, cf (2.1) and (2.2). By virtue of uN ⊗uN :D(uN)dx=0, (1.3) and Ho¨lder’s
D
inequality, we have
R
A(t,CN)·CN =− S(x,t,D(uN)):D(uN)dx−ε |uN|q˜dx+ f(t)·uNdx
ZD ZD ZD
10 ZHONGTAN,HUAQIAOWANG,ANDYUCONGWANG
≤ η dx+ f(t)·uNdx
2
ZD ZD
≤c(1+kf(t)k kvNk )
2 2
≤c(1+kf(t)k )(1+kCNk ).
2 2
By using Ho¨lder’s inequality and (2.1), one has
A(t,CN)·CN +|B(CN)|2 ≤c(1+kf(t)k )(1+|CN|2).
2
Since t(1+kf(t)k )dt<∞P-a.s.,thisyieldsweakgrowthconditioninthesenseof(3.1.4)
0 2
in [39]. Then we obtaina unique strongsolutionCN ∈L2(Ω;C0([0,T]))to the SDE (4.3).
R
Next, we will get a priori estimate.
Lemma 4.1. Under the assumption of (1.2), (1.3) and (1.4) with q ∈(1,∞), (2.1), (2.2),
q˜≥{2q,3} and
′
kvk2 dµ (v)<∞, kgk2 dµ (g)<∞, (4.4)
L2(D) 0 L2(Q) f
ZL2div(D) ZL2(Q)
then there holds uniformly in N:
E sup |uN(t)|2dx+ |∇uN|qdxdt+ε |uN|q˜dxdt
t∈(0,T)ZD ZQ ZQ !
≤c 1+ kvk2 dµ (v)+ kgk2 dµ (g) ,
L2(D) 0 L2(Q) f
ZL2div(D) ZL2(Q) !
where c is independent of ε.
Proof. Since duN = N dcN ·w , uN⊗uN :DuNdx=0, w w dx=δ ,∀k,m∈
k=1 k k D D k m km
N, and
P R R
dcNk =− S(x,t,D(uN)):D(wk)dxdt−ε |uN|q˜−2uN ·wkdxdt
ZD ZD
+ uN ⊗uN :∇w dxdt+ f ·w dxdt+ Φ(uN)·w dxdWN(t),
k k k
ZD ZD ZD
Itoˆ’s formula f(X)= 1|X|2 yields
2
1 1 N t 1 N t
kuN(t)k2 = kCN(0)k2 + CNd(CN)(r)dxdr+ dhCNi(r)dxdr
2 L2(D) 2 L2(D) k k 2 k
k=1Z0ZD k=1Z0ZD
X X
1 t t
= kPNu k2 − S(x,r,D(uN)):D(uN)dxdr−ε |uN|q˜dxdr
2 0 L2(D)
Z0ZD Z0ZD
t t
+ f ·uNdxdr+ uN ·Φ(uN)dxdWN(r) (4.5)
Z0ZD Z0ZD
1 N t
+ |Φ(uN)e |2dxdr.
i
2
i=1Z0ZD
X
From (1.3), (4.5) and Korn’s inequality, it follows that
t t
E |uN(t)|2dx+ |∇uN|qdxdr+ε |uN|q˜dxdr
(cid:18)ZD Z0ZD Z0ZD (cid:19)
≤c 1+I +I +I +E kv k2 ,
1 2 3 0 L2(D)
h (cid:16) (cid:17)i