Hindawi Publishing Corporation
Boundary Value Problems
Volume 2010, Article ID 636140, 47 pages
doi:10.1155/2010/636140
Research Article
Weak Solutions of a Stochastic Model for
Two-Dimensional Second Grade Fluids
P. A. Razafimandimby1 and M. Sango1, 2
1
2
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA
Correspondence should be addressed to P. A. Razafimandimby, paulrazafi@gmail.com
Received 3 July 2009; Accepted 28 February 2010
Academic Editor: Robert Finn
Copyright q 2010 P. A. Razafimandimby and M. Sango. 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.
We initiate the investigation of a stochastic system of evolution partial differential equations
modelling the turbulent flows of a second grade fluid filling a bounded domain of R2 . We establish
the global existence of a probabilistic weak solution.
1. Introduction
The study of turbulence either in Newtonian flows or Non-Newtonian flows is one of the
greatest unsolved and still not well-understood problem in contemporary applied sciences.
For indepth coverage of the deep and fascinating investigations undertaken in this field,
the abundant wealth of results obtained, and remarkable advances achieved we refer to
the monographs in 1–4 and references therein. The hypothesis relating the turbulence
to the “randomness of the background field” is one of the motivations of the study of
stochastic version of equations governing the motion of fluids flows. The introduction of
random external forces of noise type reflects small irregularities that give birth to a new
random phenomenon, making the problem more realistic. Such approach in the mathematical
investigation for the understanding of the turbulence phenomenonwas pioneered by
Bensoussan and Temam in 5 where they studied the Stochastic Navier-Stokes Equation
SNSE excited by random forces. Since then, stochastic partial differential equations and
stochastic models of fluid dynamics have been the object of intense investigations which have
generated several important results. We refer, for instance, to 6–22. Similar investigations for
Non-Newtonian fluids have almost not been undertaken except in very few work; we refer,
for instance, to 23–25 for some computational studies of stochastic models of polymeric
2
Boundary Value Problems
fluids. It is worth to note that in the Non-Newtonian case the study of stochastic models is
relevant not only for the analytical approach to turbulent flows but also for practical needs
related to the physics of the corresponding fluids 2.
In the present work, we initiate the mathematical analysis for the stochastic model of
incompressible second grade fluids. An incompressible fluid of second grade with a velocity
field u is a special example of a differential Rivlin-Ericksen fluid. It was shown in 26 that its
stress tensor T is given by
1.1
T −
p1 νA1 α1 A2 α2 A21 ,
where p is the scalar pressure field, ν is the kinematic viscosity, and A1 and A2 are the first
two Rivlin-Ericksen tensors defined by
A1 ∂ui
∂xj
DA1
A1
A2 Dt
i,j
∂ui
∂xj
∂uj
∂xi
i,j
,
i,j
∂uj
∂xi
1.2
A1 ,
i,j
where D/Dt denotes the material derivative. The constants α1 and α2 represent the normal
stress moduli. The incompressibility requires that
div u 0.
1.3
Taking into account some thermodynamical aspects, Dunn and Fosdick proved in 27 that
the kinematic viscosity ν must be nonnegative. In addition, they found that the free energy
must be a quadratic function of A1 . This implies in particular that the Clausius-Duhem
inequality is satisfied and the Helmholtz free energy is minimum at equilibrium if and only
if
α1 α2 0,
α1 ≥ 0.
1.4
In what follows we assume that α1 α > 0 and ν > 0. We also refer to 28, 29 for more recent
works concerning those conditions.
Those thermodynamical conditions imply that the stress tensor T can be written in the
following form:
1
∂
1 T
T
A1 A1 L − L −
L − L A1 ,
T −
p1 νA1 α
∂t
2
2
1.5
where
L
∂ui
∂xj
.
i,j
1.6
Boundary Value Problems
3
We can check that
div T −∇
p νΔu α
1
∂Δu
α curlΔu × u ∇ u · Δu |A1 |2
.
∂t
4
1.7
For a given external force f the dynamical equation for a second grade fluid is
∂u
1
curlu × u ∇ |u|2 div T f.
∂t
2
1.8
Making use of the latter equation and 1.7, we obtain the system of partial differential
equations
∂
u − αΔu − νΔu curlu − αΔu × u ∇P f,
∂t
1.9
div u 0,
where
1
1
P p − α u · Δu |A1 | |u|2
4
2
1.10
is the modified pressure. For a given connected and bounded domain D of R2 and finite time
horizon 0, T we complete the above system with the initial value
u0 u0
in D,
1.11
and the Dirichlet boundary value condition
u 0 on ∂D × 0, T .
1.12
The interest in the investigation of problem 1.9 arises from the fact that it is an admissible
model of slow flow fluids. Furthermore, once the above thermodynamical compatibility
conditions are satisfied “the second grade fluid has general and pleasant properties such
as boundedness, stability, and exponential decay” see again 27. It also can be taken as a
generalization of the Navier-Stokes Equation NSE. Indeed they reduce to NSE when α 0;
moreover recent work 30 shows that it is a good approximation of the NSE. See also 31–36
for interesting discussions to their relationship with other fluid models.
Due to the above nice properties, the mathematical analysis of the second grade
fluid has attracted many prominent researchers in the deterministic case. The first relevant
analysis was done by Ouazar in his 1981 thesis; together with Cioranescu, they published
the related result in 37, 38. Their method was based on the Galerkin approximation
scheme involving a priori estimates for the approximating solutions using a special basis
consisting of eigenfunctions corresponding to the scalar product associated with the operator
curlu − αΔu. They proved global existence and uniqueness without restriction on the initial
4
Boundary Value Problems
data for the two-dimensional case. Cioranescu and Girault 39, as well as Bernard 40
extended this method to the three dimensional case; global existence was obtained with some
reasonable restrictions on the initial data. For another approach to global existence using
Schauder’s fixed point technics, we refer to 41 and some relevant references therein.
As already mentioned, in this work we propose a stochastic version of the problem
1.9, 1.11-1.12. More precisely, we assume that a connected and bounded open set D of
R2 with boundary ∂D of class C3 , a finite time horizon 0, T , and a nonrandom initial value
u0 are given. We consider the problem
du − αΔu −νΔu curlu − αΔu × u ∇Pdt Fu, tdt Gu, tdW
in D × 0, T ,
div u 0 in D × 0, T ,
1.13
u 0 in ∂D × 0, T ,
u0 u0
in D,
where u u1 , u2 and P represent the random velocity and pressure, respectively. The
system is to be understood in the Ito sense. It is the equation of motion of an incompressible
second grade fluid driven by random external forces Fu, t and Gu, tdW, where W is a
Rm -valued standard Wiener process.
As far as we know, this paper is the first dealing with the stochastic version of the
equation governing the motion of a second grade fluid filling a connected and bounded
domain D of R2 . Consequently, we could by no means exhaust the mathematical analysis
of the problem; many questions are still open but we hope that this pioneering work will
find its applications elsewhere. We limited ourselves to the discussion of a global existence
result of a probabilistic weak solution in the two-dimensional case. In forthcoming papers we
will address other questions such as the existence probabilistic strong solutions under more
stringent conditions, the uniqueness of those solutions, and their behaviour when α → 0. It
should be noted that solving this problem is not easy even in the deterministic case, the nature
of the nonlinearities being one of the main difficulties in addition to the complex structure of
the equations. Besides the obstacles encountered in the deterministic case, the introduction of
the noise term Gu, tdW in the stochastic version induces the appearance of expressions that
are very hard to control when proving some crucial estimates. Overcoming these problems
will require a-tour-de force in the work.
The rest of the paper is organized as follows. In Section 2, we give some notations,
necessary background of probabilistic or analytical nature. Section 3 is devoted to the
formulation of the hypotheses and the main result. We introduce a Galerkin approximation of
the problem and derive crucial a priori estimates for its solution in Section 4; a compactness
result is also derived. We prove the main result in Section 5.
2. Notations and Preliminaries
Let us start with some informationsabout some functional spaces needed in this work. Let D
be an open subset of R2 , let 1 ≤ p ≤ ∞, and let k be a nonnegative integer. We consider the
well-known Lebesgue and Sobolev spaces Lp D and W k,p D, respectively. When p 2, we
k,p
write W k,2 D H k D. We denote by W0 D the closure in W k,p D of Cc∞ D the space
Boundary Value Problems
5
k,p
of infinitely differentiable function with compact support in D. If p 2, we denote W0 D
by H0k D. We assume that the Hilbert space H01 D is endowed with the scalar product
∇u · ∇v dx u, v D
2 ∂u ∂v
dx,
∂x
i ∂xi
i1 D
2.1
where ∇ is the gradient operator. The norm · generated by this scalar product is equivalent
to the usual norm of W 1,2 D in H01 D. If the domain D is smooth enough and bounded,
then for any m and p such that mp > 2 the embedding
2.2
W jm,p ⊂ W j,q
is compact for any 1 ≤ q ≤ ∞. More Sobolev embedding theorems can be found in 42 and
references therein.
Next we define some probabilistic evolution spaces necessary throughout the paper.
Let Ω, F, Ft 0≤t≤T , P be a given stochastic basis; that is, Ω, F, P is complete probability
space and Ft 0≤t≤T is an increasing sub-σ-algebras of F such that F0 contains every P -null
subset of Ω. For any reflexive separable real Banach space X endowed with the norm · X ,
for any p ≥ 1, Lp 0, T ; X is the space of X-valued measurable functions u defined on 0, T such that
uLp 0,T ;X T
0
1/p
p
uX dt
2.3
< ∞.
For any r, p ≥ 1 we denote by Lp Ω, P; Lr 0, T ; X the space of processes u uω, t with
values in X defined on Ω × 0, T such that
1 u is measurable with respect to ω, t and, for each t, u is Ft measurable,
2 ut, ω ∈ X for almost all ω, t and
uLp Ω,P;Lr 0,T ;X
⎛ p/r ⎞1/p
T
⎠ < ∞,
⎝E
urX dt
2.4
0
where E denotes the mathematical expectation with respect to the probability
measure P.
When r ∞, we write
uLp Ω,P;L∞ 0,T ;X 1/p
p
E ess sup uX dt
0≤t≤T
< ∞.
2.5
Next we give some compactness results of probabilistic nature due to Prokhorov and
Skorokhod. Let us consider Ω as a separable and complete metric space and F its Borel σfield. A family Pk of probability measures on Ω, F is relatively compact if every sequence of
6
Boundary Value Problems
elements of Pk contains a subsequence Pkj which converges weakly to a probability measure
P; that is, for any φ bounded and continuous function on Ω,
φωdPkj dω −→
φωdPdω.
2.6
The family Pk is said to be tight if, for any ε > 0, there exists a compact set Kε ⊂ Ω such that
PKε ≥ 1 − ε, for every P ∈ Pk .
We frequently use the following two theorems. We refer to 43 for their proofs.
Theorem 2.1 see Prokhorov. The family Pk is relatively compact if and only if it is tight.
Theorem 2.2 see Skorokhod. For any sequence of probability measures Pk on Ω which converges
to a probability measure P, there exist a probability space Ω
, F
, P
and random variables Xk , X with
values in Ω such that the probability law of Xk (resp., X) is Pk (resp., P) and limk → ∞ Xk XP
-a.s.
We proceed now with the definitions of additional spaces frequently used in this work.
In what follows we denote by X the space of R2 -valued functions such that each component
belongs to X. A simply-connected bounded domain D with boundary of class C3 is given. We
introduce the spaces
2
V u ∈ C∞
c such that div u 0 ,
V closure of V
in H1 D,
H closure of V
in L2 D.
2.7
We denote by ·, · and | · | the inner product and the norm induced by the inner product and
the norm in L2 D on H, respectively. The inner product and the norm induced by that of
H10 D on V are denoted respectively by ·, · and · . In the space V, the latter norm is
equivalent to the norm generated by the following scalar product see, e.g., 37 u, vV u, v αu, v,
for any u, v ∈ V.
2.8
We also introduce the following space:
W u ∈ V such that curlu − αΔu ∈ L2 D .
2.9
The following lemma tells us that the norm generated by the scalar product
u, vW u, vV curlu − αΔu, curlv − αΔv,
2.10
is equivalent to the usual H3 D-norm on W. Its proof can be found, for example, in 37, 39.
Boundary Value Problems
7
Lemma 2.3. The following (algebraic and topological) identity holds:
W W,
2.11
v ∈ H3 D such that div v 0 and v|∂D 0 .
W
2.12
where
Moreover, there is a positive constant C such that
|v|2H3 D ≤ C |v|2V |curlv − αΔv|2 ,
2.13
for any v ∈ W.
By this lemma we can endow the space W with norm | · |W which is generated by the
scalar product 2.10.
From now on, we identify the space V with its dual space V via the Riesz
representation, and we have the Gelfand chain
W ⊂ V ⊂ W ,
2.14
where each space is dense in the next one and the inclusions are continuous.
The following inequalities will be used frequently.
Lemma 2.4. For any u ∈ W, v ∈ W, and w ∈ W one has
|curlu − αΔu × v, w| ≤ C|u|H3 |v|V |w|W .
2.15
|curlu − αΔu × u, w| ≤ C|u|2V |w|W ,
2.16
One also has
for any u ∈ W and w ∈ W.
Proof. We introduce the well-known trilinear form b used in the study of the Navier-Stokes
equation by setting
bu, v, w 2 i,j1 D
ui
∂vj
wj dx.
∂xi
2.17
We give the following identity whose proof can be found in 37, 40. This equation is valid
for any smooth solenoidal functions Φ, v, and w as
curl Φ × v, w bv, Φ, w − bw, Φ, v.
2.18
8
Boundary Value Problems
We derive from this that for any u ∈ W, v ∈ W, and w ∈ W
|curlu − αΔu × v, w| ≤ C|v|L2 D |∇u − αΔu|L2 D |w|L∞ D ,
2.19
where Hölder’s inequality was used. The Sobolev embedding 2.2 and the equivalence of
the norms | · |W and | · |3H D imply 2.15.
By 2.18 we deduce
curlu − αΔu × u, w bu, u, w − αbu, Δu, w αbw, Δu, u.
2.20
With the help of integration by parts and using the fact that u and w are elements of W we
derive that
2
b
bu, Δu, w j1
∂u
∂u
, w,
∂xj
∂xj
2
b
bw, Δu, u j1
2
∂w ∂u
b u,
,
,
∂xj ∂xj
i1
∂u
∂w
, u,
∂xj
∂xj
2.21
2.22
.
We use these results to derive the following estimate. For any elements u ∈ V and w ∈ L4 D,
we obtain by Hölder’s inequality
|bu, u, w| ≤ C|u|L4 D u|w|L4 D .
2.23
And since the space V and W are, respectively, continuously embedded in L4 D and V, then
2.24
|bu, u, w| ≤ C|u|2V |w|W .
We also have
2
2 ∂u |bu, Δu, w| ≤ |∇w|L∞ D ∂xj 2
j1
L D
⎛ 2
2 ∂w
|u|L4 D ⎝ ∂x
j 4
j1
L D
⎞1/2 ⎛ 2
2 ∂u
⎠ ⎝ ∂x
j 2
j1
2.25
⎞1/2
⎠
.
L D
We derive from this and the Sobolev embedding 2.2 that
|bu, Δu, w| ≤ C|u|2V |w|W .
2.26
By an analogous argument we have
|bw, Δu, u| ≤ C|w|W |u|2V .
2.27
Boundary Value Problems
9
The estimates 2.20, 2.24–2.27 yield
|curlu − αΔu × u, w| ≤ C|u|2V |w|W ,
2.28
for any u ∈ W and w ∈ W. This completes the proof of the lemma.
Next we give some results on which most of the proofs in forthcoming sections rely.
We start by stating a theorem on solvability of the “generalized Stokes equations”
v − αΔv ∇q f
in D,
div v 0 in D,
2.29
v 0 on ∂D.
By a solution of this system we mean a function v ∈ V which satisfies
v, h αv, h f, h ,
2.30
for any h ∈ V.
The proof of the following result can be derived from an adaptation of the results
obtained by Solonnikov in 44, 45.
Theorem 2.5. Let D be a connected, bounded open set of Rn n ≥ 2 with boundary ∂D of class Cl
and let f be a function in Hl , l ≥ 1. Then 2.29 has a unique solution v. Moreover if f is an element
of V, v ∈ Hl2 ∩ V, and the following hold:
v, hV v, h,
|v|W ≤ Cf V ,
2.31
for any h ∈ V.
Next we formulate Aubin-Lions’s compactness theorem; its proof can be found in 46.
Theorem 2.6. Let X, B, Y be three Banach spaces such that the following embedding is continuous:
X ⊂ B ⊂ Y.
2.32
Moreover, assume that the embedding X ⊂ B is compact, then the set F consisting of functions v ∈
Lq 0, T ; B, 1 ≤ q ≤ ∞, such that
t2
sup
0≤h≤1 t1
p
|vt h − vt|Y dt −→ 0,
for any 0 < t1 < t2 < T , is compact in Lp 0, T ; B for any p.
as h −→ 0,
2.33
10
Boundary Value Problems
Last but not least we present the famous Kolmogorov-Čentsov continuity criterion for
stochastic processes. We refer to 47, 48 for its proof and some of its extension.
Theorem 2.7 see Kolmogorv-Čentsov. Suppose that a real-valued process X {Xt , 0 ≤ t ≤ T }
on a probability space Ω, P satisfies the condition
E|Xth − Xt |γ ≤ Ch1β ,
0 ≤ t, h ≤ T,
2.34
{X
t, 0 ≤ t ≤
for some positive constants γ, β, and C. Then there exists a continuous modification X
T } of X, which is locally Hölder continuous with exponent κ ∈ 0, β/γ.
3. Hypotheses and the Main Result
We state on our problem the following.
3.1. Hypotheses
1 We assume that
F : V × 0, T −→ V
3.1
is continuous in both variables. We also assume that, for any t ∈ 0, T and any v ∈ V
|Fv, t|V ≤ C1 |v|V .
3.2
2 We also define a nonlinear operator G as follows:
G : V × 0, T −→ V⊗
m
3.3
is continuous in both variables. We require that, for any t ∈ 0, T , Gv, t satisfy
|Gv, t|V⊗m ≤ C1 |v|V .
3.4
3.2. Statement of the Main Theorem
We introduce the concept of solution of the problem 1.13 that is of interest to us.
Definition 3.1. By a solution of the problem 1.13, we mean a system
Ω, F, P, Ft , W, u ,
3.5
Boundary Value Problems
11
where
1 Ω, F, P is a complete probability space; Ft is a filtration on Ω, F, P,
2 Wt is an m-dimensional Ft standard Wiener process,
3 for a.e. t, ut ∈ Lp Ω, P; L∞ 0, T ; W ∩ Lp Ω, P; L∞ 0, T ; V, 2 ≤ p < ∞,
4 for almost all t, ut is Ft measurable,
5 P-a.s the following integral equation of Itô type holds:
ut − u0, vV t
t
νu, v curlus − αΔus × u, vds
0
Fus, s, vds 0
3.6
t
Gus, s, vdWs
0
for any t ∈ 0, T and v ∈ W.
Remark 3.2. In the above definition the quantity
as:
t
Gus, s, vdWs 0
t
0
Gus, s, vdWs should be understood
m t
k1
Gk us, s, vdWk s,
0
3.7
where Gk and Wk denote the kth component of G and W, respectively.
Now we state our main result.
Theorem 3.3. Assume that u0 ∈ W; assume also that all the assumptions, namely, 3.2 and 3.4,
on the operators F, G are satisfied; then the problem 1.13 has a solution in the sense of the above
definition. Moreover, almost surely the paths of the process u are W-valued weakly continuous.
4. Auxiliary Results
In this section we derive crucial a priori estimates from the Galerkin approximation. They
will serve as a toolkit for the proof of Theorem 3.3.
4.1. The Approximate Solution
The following statement is a consequence of the spectral theorem for self-adjoint compact
operator stated in 49. The injection of W into V is compact. Let I be the isomorphism of W onto
W, then the restriction of I to V is a continuous compact operator into itself. Thus, there exists a
sequence ei of elements of W which forms an orthonormal basis in W, and an orthogonal basis in V.
This sequence verifies:
for any v ∈ W v, ei W λi v, ei V ,
where λi1 > λi > 0, i 1, 2, . . . .
4.1
12
Boundary Value Problems
We have the following important result due to 39 about the regularity of the ei -s.
Lemma 4.1. Let D be a bounded, simply-connected open set of R2 with a boundary of class C3 , then
the eigenfunctions of 4.1 belong to H4 D.
We consider the subset WN Spane1 , . . . , eN ⊂ W and we look for a finitedimensional approximation of a solution of our problem as a vector uN ∈ WN that can be
written as the Fourier series:
uN t N
4.2
ciN tei x.
i1
t
Let us consider a complete probabilistic system Ω, F, P, F , W such that the filtration {Ft }
satisfies the usual condition and W is an m-dimensional standard Wiener process taking
values in Rm . We require uN to satisfy the following system:
d uN , ei ν uN , ei dt b uN , uN , ei dt − αb uN , ΔuN , ei dt αb ei , ΔuN , uN dt
V
F t, uN , ei dt G t, uN , ei dW,
4.3
where uN
0 as the orthogonal projection of u0 in the space WN is given as
N
uN
0 or u 0 −→ u0
strongly in V
4.4
as N → ∞. The Fourier coefficients ciN in 4.2 are solutions of a system of stochastic
ordinary differential equations which satisfy the conditions of the existence theorem of
Skorokhod 50 see also 47. Therefore the sequence of functions uN exists at least on a
short interval 0, TN . Global existence will follow from a priori estimates for uN .
4.2. A Priori Estimates
From now on C is a constant depending only on the data, and may change from one line to
the next one. We start by proving the following lemma.
Lemma 4.2. For any N ≥ 1 one has
2
E sup uN t < ∞.
0≤t≤T
V
4.5
One also has
2
E sup uN t < ∞.
0≤t≤T
W
4.6
Boundary Value Problems
13
Proof. From now on we denote by |v|∗ the quantity | curlv − αΔv| for any v ∈ W. For any
integer M ≥ 1 we introduce the stopping time
τM
⎧ ⎨inf 0 ≤ t; uN t uN t ≥ M
V
∗
⎩∞ if 0 ≤ t; uN t uN t ≥ M ∅.
V
∗
4.7
We will use a modification of the argument used in 7.
For any 0 ≤ s ≤ t ∧ τM , t ∈ 0, TN , we may apply Itô’s formula see, e.g., 7, 12 for
φuN s, ei V uN s, ei 2V to 4.3 and obtain
uN s, ei
2
V
s
uN r, ei
ν uN r, ei b uN r, uN r − αΔuN r, ei dr
2
V
0
s
2
uN r, ei
−αb ei , ΔuN r, uN r F r, uN , ei dr
V
0
4.8
s s
2
N
G r, u , ei dW uN r, ei
G r, uN , ei dr.
0
V
0
N
2
N
We note that |uN |2V i1 λi u , ei V . Then, multiplying by λi the above equation and
summing over i from 1 to N give us
s
s s N
2
2
N 2
N 2
N
N
N
F r, u , u dr λi
Gr, uN , ei dr
u s 2ν u dr u0 2
V
V
0
0
i1
s G r, uN , uN dW,
2
0
4.9
0
where we have used the fact that buN , uN , uN 0.
We obtain from 4.9 that
s s N
2 2
N
N 2
N
N
dr
≤
u
G uN r, r , ei dr
s
2ν
λ
u
r, r
u0 u
i
V
0
V
i1
0
s 2 F uN r, r , uN r dr
4.10
0
s 2
G uN r, r , uN r dW ,
0
for any 0 ≤ s ≤ t ∧ τM , t ∈ 0, TN . For any u ∈ V we have
|u| ≤ Pu,
4.11
14
Boundary Value Problems
where P is the so called Poincaré’s constant. The last inequality implies that
F uN s, s , uN ≤ P2 uN F uN s, s .
4.12
−1
P2 α |v|2V ≤ v2 ≤ α−1 |v|2V ,
4.13
We also mention that
for any v ∈ V.
From the former equation and this one we find
P2
N 2
N
N
1 u s .
F u s, s , u s ≤ 2C
V
α
4.14
2
N
To find a uniform estimate for the corrector term N
i1 λi Gu s, ei is not straightforward;
this is the difficulty already mentioned in the introduction. Since the corrector term is
explicitly written as function depending on the scalar product in L2 D ·, · and the ei -s
form an orthonormal basis resp., orthogonal basis of W resp, V, then the usual Bessel’s
inequality see, e.g., 6 does not apply anymore. To circumvent this difficulty we consider
the following generalized Stokes equation:
− αΔG
∇q G uN s, s
G
in D,
4.15
0 in D,
div G
0 on ∂D,
G
in W⊗m when ∂D is of class C3 and
for any s ∈ 0, T . By Theorem 2.5, 4.15 has a solution G
N
⊗m
Gu s, s ∈ V . Moreover, there exists a positive constant C0 such that
G
H3 D⊗m
≤ C0 G uN s, s V⊗m
,
4.16
ei V GuN s, s, ei for any i ≥ 1.
and G,
Since the norms |·|H3 D and |·|W are equivalent on W, then there exists another positive
constant C∗ such that
G
W⊗m
≤ C∗ C0 G uN s, s V⊗m
.
4.17
Boundary Value Problems
15
depends continuously on the data GuN s, s. Therefore, we
Equation 4.17 implies that G
N
as Gu
s, s. We find from 4.17 that
note the above G
N
N
2 2
uN s, s , ei
λi G uN s, s , ei λi G
i1
V
i1
N
2
1 N
G u s, s , ei .
W
λ
i1 i
4.18
We deduce from this that
N
2
2
1 N
1 N
≤ Gu
s, s ⊗m .
G u s, s , ei
W
W
λ
λ
1
i1 i
4.19
By 4.17 and the assumption on G, we have
N
2 2
λi G uN s, s , ei ≤ C 1 uN s .
V
i1
4.20
Collecting this information, we obtain from 4.10 that
s
2
N 2
u s 2ν uN r dr
V
0
s
s 2
G uN r, r , uN r dW .
≤ C C uN r dr 2
V
0
4.21
0
Taking the sup over s ≤ t ∧ τM in both sides of this inequality and passing to the
mathematical expectation in the resulting relation and finally applying Burkhölder-DavisGundy’s inequality see, e.g., 48 to the stochastic term, we get
t∧τM 2
N 2
E sup uN s 2νE
u s ds
s≤t∧τM
V
0
t∧τ
1/2
t∧τM 2
M
N 2
N
N
Gu s, s, u s ds
.
≤ C CE
u s ds 2C1 E
V
0
4.22
0
Now, we estimate
t∧τ
M
γ E
0
1/2
2
N
N
G u s, s , u s ds
.
4.23
16
Boundary Value Problems
With the same argument that we have used for the term |FuN s, s, uN s|, we have
⎡
γ ≤ CE⎣ sup uN s
t∧τ
M
V
s≤t∧τM
0
1/2 ⎤
2
⎦.
G uN s, s ×m ds
4.24
V
By ε-Young’s inequality
t∧τM 2
2
γ ≤ CεE sup uN s Cε E
GuN s, s ×m ds
V
s≤t∧τM
4.25
V
0
Using the assumption on G one has
t∧τM 2 2
1 uN s .
γ ≤ CεE sup uN s Cε E
V
s≤t∧τM
4.26
V
0
With convenient choice of ε 1 − 2C1 Cε 1/2, the estimates 4.22 and 4.26 allow us to
write
t∧τM t∧τM 2
N 2
N 2
E sup uN s 4νE
u s ds ≤ C CE
u s ds.
s≤t∧τM
V
0
V
0
4.27
We derive from this and Gronwall’s inequality that
2
E sup uN s ≤ C.
0≤s≤t∧τM
4.28
V
We recall the following relationship which is very important in the sequel:
uN s, s , ei ,
λi G uN s, s , ei G
W
i ≥ 1,
4.29
N s, s is the solution in W of GS.
where Gu
To alleviate notation, we only write uN when we mean uN ·. Let us set
φ uN −νΔuN curl uN − αΔuN × uN − F uN , t .
4.30
By Lemma 4.1, φuN ∈ H1 D. We have
d uN , ei φ uN , ei dt G uN , t , ei dW.
V
4.31
Boundary Value Problems
17
By Theorem 2.5 a solution vN ∈ W of the following system exists:
vN − αΔvN ∇q φ uN
in D,
4.32
div vN 0 in D,
vN 0 on ∂D.
Moreover,
v N , ei
V
φ uN , ei ,
4.33
for any i. Thus,
d uN , ei φ uN , ei dt d uN , ei vN , ei dt
V
V
G uN , t , ei dW.
V
4.34
The following follows by multiplying the latter equation by λi and using the relationship
4.1:
d uN , ei
vN , ei dt λi G uN , t , ei dW.
W
4.35
W
Recalling 4.29, we obtain
uN , t , ei dW.
vN , ei dt G
d uN , ei
W
W
4.36
W
Now applying the Itô’s formula see, e.g., 7 to ϕuN , ei W uN , ei 2W , we have
2
2
uN , t , ei dW.
uN , t , ei dt 2 uN , ei
d uN , ei
vN , ei dt G
2 uN , ei
G
W
W
W
W
W
W
4.37
Summing both sides of the last equation from 1 to N yields
N 2
2
uN , t , ei dt 2 G
uN , t , uN dW.
duN 2 uN , vN dt G
W
W
i1
W
W
4.38
18
Boundary Value Problems
Using the definition of | · |W and the scalar product ·, ·W , we can rewrite the above equation
in the form
# 2 2 $
dt
d uN uN 2 vN , uN curl vN − αΔvN curl uN − αΔuN
V
∗
V
uN , t , curl uN − αΔuN dW
uN , t − αΔG
2 curl G
4.39
N
2
uN , t , ei dt 2 G
uN , t , uN dW.
λ2i G
V
V
i1
In view of Remark 3.2, we have to make the convention that in the sequel
dW
curl G uN , t , curl uN − αΔuN
dt
m N
curl Gk u , t
, curl u − αΔu
N
N
dW
k1
dt
4.40
k
.
we obtain
Using the definition of vN and G,
# $
2 2
dt
d uN uN 2 φ uN , uN curl φ uN , curl uN − αΔuN
V
∗
N
2
λ2i GuN , t, ei dt 2 G uN , t , uN dW
4.41
i1
2 curl G uN , t , curl uN − αΔuN dW.
With the help of 4.9, 4.41 can be rewritten in the following way:
2
duN 2 curl φ uN , curl uN − αΔuN dt
∗
N 2
2 curl G uN , t , curl uN − αΔuN dW λi λ2i G uN , t , ei dt.
4.42
i1
We infer from the definition of φuN that
curl φ uN −ν curl ΔuN F uN , t curl curl uN − αΔuN × uN .
4.43
In taking advantage of the dimension we get
curl curl uN − αΔuN × uN uN · ∇ curl uN − αΔuN .
4.44
Boundary Value Problems
19
This yields
uN · ∇ curl uN − αΔuN − ν curl ΔuN F uN , t , curl uN − αΔuN
curl φ uN , curl uN − αΔuN .
4.45
Owing to Lemma 4.1 we readily check that
uN · ∇ β, β 0,
4.46
where β curluN − αΔuN . Consequently,
ν curl ΔuN , curl uN − αΔuN curl F uN , t , curl uN − αΔuN
− curl φ uN , curl uN − αΔuN .
4.47
Or equivalently
ν N ν
α
N
N
N
N
curl u curl F u , t , curl u − αΔu
u −
∗
α
α
ν
curl φ uN , curl uN − αΔuN .
4.48
We derive from 4.42 and the last equation that
d N 2 2ν N 2 2ν
α
curl uN curl F uN , t , curl uN − αΔuN
u u −
∗
∗
dt
α
α
ν
N λi i1
λ2i
dW
2
G uN , t , ei 2 curl G uN , t , curl uN − αΔuN
.
dt
4.49
We argue as before in considering the stopping time τM . We derive from 4.49 that
s
N 2
u s ∗
0
s
N 2
2ν N 2 2
N
dr
λi λi G u r, r , ei
u r −
∗
α
i1
#
α
$
2ν
dr
curl uN r − curl F uN r, r , curl uN r − αΔuN r
ν
0 α
s
2
curl G uN r, r , curl uN r − αΔuN r W.
0
4.50
20
Boundary Value Problems
Hence,
N s 2ν s 2
N 2
N 2
λi λ2i
G uN r, r , ei dr
u s u r dr −
∗
∗
0 α
0
i1
s
2
s 2ν N N
≤ uN
u
r
r
u
curl
2curl F uN r, r uN r dr
0 ∗
∗
∗
α
0
0
s curl G uN r, r , curl uN r − αΔuN r dW .
2
4.51
0
Taking the supremum over s ≤ t ∧ τM in the last estimate, and taking the mathematical
expectation in the resulting relation yields
t∧τM N 2
t∧τM 2
2ν N 2
λi λ2i E
G uN s, s , ei ds
E sup uN s E
u s ds −
∗
∗
α
s≤t∧τM
0
0
i1
t∧τM
t∧τM 2
2ν N N N
N
≤ uN
u
E
s
E
2
F
u
s
s
s
s,
u
u
curl
curl
ds
0 ∗
∗
∗
α
0
0
s∧τM curl G uN s, s , curl uN s − αΔuN s dW .
2E sup s≤t∧τM 0
4.52
For any ε1 ≥ 0 and ε2 ≥ 0, we have
t∧τM N t∧τM 2
2ν N 2
N 2
2
λi λi E
G uN s, s , ei ds
E sup u s E
u s ds −
∗
∗
α
s≤t∧τM
0
0
i1
t∧τM
2
2 2 2 2ν N
N
N
≤ u0 E
curl u s curl F u s, s ds
∗
αε
ε
1
2
0
s∧τM N
N
N
curl G u s, s , curl u s − αΔu s dW 2E sup s≤t∧τM
0
t∧τM 2νε1
N 2
2ε2
u s ds.
∗
α
0
4.53
We choose ε1 1/4 and ε2 ν/4α and we deduce from the last inequality the following
estimate,
t∧τM N 2
t∧τM 2
ν N 2
λi λ2i E
G uN s, s , ei ds
E sup uN s E
u s ds −
∗
∗
α
s≤t∧τM
0
0
i1
t∧τM
2
2 2α 2 8ν N
N
N
≤ u0 E
curl u s curl F u s, s ds
∗
αε
ν
1
0
s∧τM N
N
N
curl G u s, s , curl u s − αΔu s dW .
2E sup s≤t∧τM 0
4.54
Boundary Value Problems
21
Thanks to 4.20, 4.29 and 4.17 we see that
N λi s∧τM t∧τM 2
N 2
N
Gu s, s, ei ds ≤ C CE
E
u s ds.
λ2i
0
i1
V
0
4.55
Let us estimate
t∧τM N
N
N
curl G u s, s , curl u s − αΔu s dW .
γ 2E sup s≤t∧τM
0
4.56
By Fubini’s theorem and the Burkhölder-Davis-Gundy’s inequality we obtain
γ ≤ 6E
t∧τ
M
2
curl G uN s, s , curl uN s − αΔuN s
dW
1/2
0
⎛
≤ 6E⎝ sup uN s
∗
s≤t∧τM
t∧τ
M
2
curl G uN s, s 1/2 ⎞
⎠.
4.57
0
Making use of an ε-Young’s inequality, the following holds:
t∧τM 2
N 2 6
γ ≤ 6εE sup u s E
curl G uN s, s ds.
∗
ε
s≤t∧τM
0
4.58
Choosing ε 1/12, we write
t∧τM 2
1
N 2
γ ≤ E sup u s 72E
curl G uN s, s ds.
∗
2 s≤t∧τM
0
4.59
Combining 4.54, 4.55, and 4.59, we obtain
t∧τM ν N 2
N 2
E sup u s E
u s ds
∗
∗
α
s≤t∧τM
0
t∧τM t∧τM 2
2
N
≤ CE
curl F u s, s CE
curl G uN s, s ds
0
4.60
0
t∧τM t∧τM 2
2
N 2
N
u
CE
s
ds
CE
s
CuN
u
curl
ds.
0
∗
0
V
0
By a straightforward calculation we have
curlφ2 ≤ 2 φ2
V
α
for any φ ∈ V.
4.61
22
Boundary Value Problems
Owing to 4.61 and the assumptions on F and G, we derive from 4.60 that
t∧τM t∧τM 2
ν N 2
N 2
N 2
N
E sup u s E
u s ds ≤ C u0 CE
u s ds.
∗
∗
∗
V
α
s≤t∧τM
0
0
4.62
This and the estimate 4.28 imply
t∧τM 2
ν N 2
E sup uN s E
u s ds ≤ C.
∗
∗
α
s≤t∧τM
0
4.63
It is easy to check that, as M → ∞, t ∧ τM → t almost surely for any t ∈ 0, TN . Since the
constant C is independent of N, the estimates 4.28, 4.63 and the Dominated Lebesgue’s
Convergence Theorem complete the proof of the lemma.
Lemma 4.3. For any 4 ≤ p < ∞ one has
p
E supuN s < ∞,
s≤T
V
s≤T
W
4.64
p
E supuN s < ∞.
Proof. We recall that
2
2
duN t 2νuN t dt − 2 Fun t, t, uN t dt
V
4.65
N
2
λi G uN t, t , ei dt 2 G uN t, t , uN t dW.
i1
2p/4
For a fixed p ≥ 4 the application of Itô’s formula to the function φ|uN t|2V |uN t|V
yields
%
2 N p/2 p N p/2−2
− νuN t Fun t, t, uN t
du t u t
V
V
2
&
N 2 p − 4 GuN t, t, uN t2
1
N
dt
G u t, t , ei 2 i1
4
uN t2
p/2−2 p uN t
G uN t, t , uN t dW.
V
2
V
4.66
Boundary Value Problems
23
Hence
t N p/2 N p/2 p N p/2−2
u s
u t u0 V
V
V
2 0
%
2 × − νuN s Fun s, s, uN s
&
N 2 p − 4 GuN s, s, uN s2
1
N
ds
G u s, s , ei 2 i1
4
uN s2
4.67
V
p t N p/2−2 N
G u s, s , uN s dW,
u s
V
2 0
for any t ∈ 0, T . In squaring the last equation and in making use of some elementary
inequalities we obtain
t
p
N p/2−2
N
N p
u s
u t ≤ Cu0 C
V
V
V
0
#
2 × − νuN s Fun s, s, uN s
& 2
N
p − 4 GuN s, s, uN s2
1
2
N
ds
G u s, s , ei 2 i1
4
|uN s|2V
C
t
N p/2−2
G uN s, s , uN s dW
u s
V
0
2
.
4.68
We deduce from 4.14, 4.20, and 4.68 that
t
p
2
N p
N 2
N p/2−2
N
1 u s
u t ≤ Cu0 C
u s
V
V
C
t
V
0
V
N p/2−2
G uN s, s , uN s dW
u s
V
0
4.69
2
.
We find from this that
t p
p
N 4
N p−4
1
E supuN s ≤ CuN
CE
s
s
ds
u
u
0
s≤t
V
V
V
0
V
2
s N p/2−2
G uN r, r , uN r dW .
CE sup
u r
s≤t
0
V
4.70
24
Boundary Value Problems
It is clear that
p−4
N p−4
.
u s ≤ 1 uN s
4.71
V
V
Hence,
t p
p
p
1 uN s
E supuN s ≤ CuN
CE
ds
0
s≤t
V
V
V
0
2
s N p/2−2
N
N
dW
G
u
CE sup
.
r
,
u
r
r,
r
u
s≤t
4.72
V
0
Now let us denote by γ1 the stochastic term in 4.70. As before, we use the Burkhölder-DavisGundy’s inequality and get
t p−4 2
G uN s, s , uN s ds
γ1 ≤ CE uN s
0
V
0
V
t p−4 2
2 ≤ CE uN s G uN s, s uN s .
V
4.73
V
The following follows from the same arguments as used before and by the assumption on G:
t p
1 uN s
ds.
γ1 ≤ CE
V
0
4.74
This, the estimate 4.70, and Gronwall’s inequality imply
p
E supuN s < ∞,
s≤t
V
4.75
which completes the proof of the first estimate of the lemma.
Let us now proceed to the proof of the second estimate of Lemma 4.3. We rewrite 4.49
in the form
N 2 2 2ν curl uN s, curl uN s − αΔuN s ds
λi λ2i G uN s, s , ei duN s ∗
α
i1
2ν N 2
N
N
N
− u s 2 curl F u s, s , curl u s − αΔu s
∗
α
2 curl G uN s, s , curl uN s − αΔuN s dW.
4.76
Boundary Value Problems
25
2p/4
Applying Itô formula to the function ϕ|uN s|2∗ |uN s|∗
we have
p/2−2
p/2 p duN s − uN s
∗
∗
2
×
2 curl F uN s, s , curl uN s − αΔuN s
N 2 2ν 1
λi λ2i G uN s, s , ei curl uN s, curl uN s − αΔuN s
2 i1
α
2 2ν N 2 p − 4 curlG uN s, s , curl uN s − αΔuN s
ds
−
u s ∗
α
4
uN s2
∗
p N p/2−2 curl G uN s, s , curl uN s − αΔuN s dW.
u s
∗
2
4.77
Hence,
t N p/2 N p/2 p N p/2−2
u t u0 u s
∗
∗
∗
2 0
2 curl F uN s, s , curl uN s − αΔuN s
×
N 2
1
λi λ2i G uN s, s , ei
2 i1
2ν curl uN s, curl uN s − αΔuN s
α
2 2ν N 2 p − 4 curl G uN s, s , curl uN s − αΔuN s
ds
−
u s ∗
α
4
uN s2
∗
p
2
t N p/2−2
curl G uN s, s , curl uN s − αΔuN s dW,
u s
0
∗
4.78
26
Boundary Value Problems
for any t ∈ 0, T . The following follows in squaring both sides of the last inequality:
% t
p
N p
N p/2−2
N
u t ≤ Cu0 C
u s
∗
∗
∗
0
×
2 curl F uN s, s , curl uN s − αΔuN s
N 2
1
λi λ2i G uN s, s , ei
2 i1
2ν 2ν N 2
curl uN s, curl uN s − αΔuN s −
u s
∗
α
α
&2
N
N
2
p − 4 curl G u s, s , curl u s − αΔuN s
ds
4
uN s2
C
t
∗
N p/2−2
curl G uN s, s , curl uN s − αΔuN s dW
u s
2
∗
0
.
4.79
For almost all s ∈ 0, T , we note that
1 uN s .
curl uN s, curl uN s − αΔuN s ≤ C 1 uN s
V
4.80
W
We also check readily that
1 uN s ,
curl F uN s, s , curl uN s − αΔuN s ≤ C 1 uN s
V
W
curl GuN s, s, curluN s − αΔuN s2 2
≤ C 1 uN sV .
2
uN s
4.81
4.82
∗
Thanks to the continuous injection of W into V, all the above estimates still hold with |uN ·|V
replaced by |uN ·|W . It follows from this argument and 4.79 that
t
2
N p N p
N p/2−2
N 2
1 u s
ds
u t ≤ u0 C
u s
∗
∗
C
∗
0
t
0
W
N p/2−2
curl G uN s, s , curl uN s − αΔuN s dW
u s
∗
2
.
4.83
Boundary Value Problems
27
Taking the supremum over s ≤ t followed by the mathematical expectation yields
p
E supuN s
s≤t
∗
t
2
p
N 2
N p/2−2
N
1 u s
≤ u0 CE
ds
u s
∗
∗
0
CE sup
s≤t
W
2
s N p/2−2
curl G uN r, r , curl uN r−αΔuN r dW .
u r
∗
0
4.84
Applying the Martingale inequality and Hölder’s inequality in the last estimate we obtain
t p p
4
N p−4
1 uN s
E supuN s ≤ uN
CE
ds
u s
0
s≤t
∗
∗
∗
0
W
t p−4 2
curl G uN s, s , curl uN s − αΔuN s
ds.
CE uN s
∗
0
4.85
We can use the same idea we have used to find 4.81 to get an upper bound of the form
C1|uN s|W 4 for |curlGuN s, s, curluN s−αΔuN s2 |· Then, we derive from 4.85
that
t p
p
N p
N
E supu s ≤ Cu0 C
1 uN s
ds.
s≤t
We obviously have
∗
N u s
∗
W
W
0
p p ≤ C uN s uN s .
V
∗
4.86
4.87
p
Finally, using a previous result concerning E sups≤t |uN s|V , 4.86 and Gronwall’s inequality
we obtain
p
E supuN s < ∞.
4.88
s≤t
∗
This completes the proof of the lemma.
Remark 4.4. Lemmas 4.2 and 4.3 imply in particular that
p
E supuN t < ∞,
t≤T
V
t≤T
W
p
E supuN t < ∞,
for any 1 ≤ p < ∞.
4.89
28
Boundary Value Problems
The following result is central in the proof of the forthcoming crucial estimate of the
finite difference of our approximating solution.
Lemma 4.5. Let t, s ∈ 0, T such that s ≤ t. For a fixed t ∈ 0, T , let
vN t N
λi vN t, ei ei
4.90
V
i1
be an element of WN which satisfies Lemmas 4.2 and 4.3. The following holds:
t #
2 2
$
N
N
N 2
N
N
N
N
dr
u t − v s − u s − v s 2 ν u r − u r, v r
V
V
s
t t N
2
2
G uN r, r , uN r − vN s dW λi
G uN r, r , ei dr
s
i1
s
t t N
N
N
F uN r, r , uN r − vN s dr
2 b v s, Δu r, u r dr 2
s
4.91
s
t t − 2 b uN r, uN r, vN s dr − 2 b uN r, ΔuN r, vN s dr.
s
s
Proof. For vN , for any s, t such that 0 ≤ s ≤ t ≤ T , we have
d N
u t − vN s, ei ν uN t, ei b uN t, uN t, ei
V
dt
− αb uN t, ΔuN t, ei αb ei , ΔuN t, uN t
d
F uN t, t , ei G uN t, t , ei
W,
dt
4.92
1 ≤ i ≤ N.
This relation can be rewritten as the following Itô equation:
d uN t − vN s, ei ν uN t, ei dt b uN t, uN t, ei dt
V
− αb uN t, ΔuN t, ei dt αb ei , ΔuN t, uN t dt
F uN t, t , ei dt G uN t, t , ei dW.
4.93
Boundary Value Problems
29
Applying Itô’s formula to the function uN t, vN s, ei 2V , multiplying the result by λi , and
then summing over i from 1 to N yield
d|w|2V 2ν
uN t, w
dt 2b uN t, uN t, w dt − 2αb uN t, ΔuN t, w dt
2αb w, ΔuN t, uN t dt − 2 F uN t, t , w dt
4.94
N
2
λi G uN t, t , ei 2 G uN t, t , w dW,
i1
where w uN t−vN s. Using the trilinearity of b and the well-known identity bu, u, u 0,
u ∈ V, we find that
b uN t, uN t, w − αb uN t, ΔuN t, w αb w, ΔuN t, uN t
b uN t, uN t, vN s − αb uN t, ΔuN t, vN s αb vN s, ΔuN t, uN t .
4.95
The lemma follows in combining this relation with 4.94, and integrating the resulting
equation between s and t.
The following result can be proved by a similar argument used in 15, but we prefer
to give our own proof which is interesting in itself.
Lemma 4.6. There exists a positive constant C > 0 such that for all 0 ≤ δ < 1 and N ∈ N, the
following inequality holds:
T −δ 2
N
E sup
u s θ − uN s ∗ ≤ Cδ1/2 .
|θ|≤δ
W
0
4.96
Proof. Since uN s ∈ WN , s ∈ 0, T and it satisfies Lemmas 4.2 and 4.3 then we can take
vN s uN s and t s θ, 0 ≤ θ ≤ δ ≤ 1 and apply Lemma 4.5. We obtain
sθ #
2
$
N
N 2
N
N
N
dr
s
θ
−
u
s
2
ν
r
−
u
u
r, r
u
u
V
s
sθ sθ N
2
2
G uN r, r , uN r − uN s dW λi
G uN r, r , ei dr
s
i1
s
sθ sθ N
N
N
F uN r, r , uN r − uN s dr
b u s, Δu r, u r dr 2
2
s
s
sθ sθ −2
b uN r, uN r, uN s dr − 2
b uN r, ΔuN r, uN s dr.
s
s
4.97
30
Boundary Value Problems
We derive that
T −δ
2
E
sup uN s θ − uN s ≤ I1 I2 I3 I4 I5 I6 I7 ,
0
4.98
V
0≤θ≤δ
where
I1 2E
T −δ
0
I2
I3
I4
I5
I6
I7
sθ N
N
N
G u r, r , u r − u s dW ,
sup s
0≤θ≤δ
sθ T −δ
N
2 G uN r, r , ei dr ,
E
sup λi
0 0≤θ≤δ i1
s
T −δ
sθ N N
N
dr ,
E
sup 2
ν u r u r, u r
0 0≤θ≤δ
s
T −δ
sθ 2E
sup b uN s, ΔuN r, uN r dr ,
0 0≤θ≤δ
s
T −δ
sθ N
N
N
2E
sup b u r, Δu r, u s dr ,
0 0≤θ≤δ
s
T −δ
sθ N
N
N
2E
sup b u r, u r, u s dr ,
0 0≤θ≤δ s
T −δ
sθ N
N
N
F u r, r , u r − u s dr .
2E
sup 0 0≤θ≤δ
s
4.99
The proof of the lemma will consist of the following five steps.
Step 1 estimate of I3 . Owing to the equivalence of the two norms · and | · |V we see that
I3 is dominated by
sδ
T −δ
N
N N
sup u s θ − u s E
2CE
u r dr ds.
0
V
0≤θ≤δ
V
s
4.100
We find from this and by a successive application of Cauchy-Schwarz’s inequality that
T −δ
1/2 T −δ
1/2
2
N
N 2
N
I3 ≤ Cδ E
E
sup u s θ − u s ds
sup u r dr
.
0
0≤θ≤δ
V
0
0≤r≤T
V
4.101
Since s θ ∈ 0, T for any s ∈ 0, T − δ and any 0 ≤ θ ≤ δ, we get using the triangle inequality
and Lemma 4.2 that
I3 ≤ Cδ.
4.102
Boundary Value Problems
31
Step 2 estimates for I4 , I5 and I6 . The estimate 2.16 and a successive application of CauchySchwarz’s inequality yield
I4 ≤ Cδ
1/2
T −δ
1/2 T −δ sδ
1/2
N 2
N 4
E
E
.
u s ds
u r drds
W
0
0
4.103
V
s
By Cauchy’s inequality, we have
I4 ≤ Cδ
1/2
T −δ
T −δ sδ N 2
N 4
E
u s ds E
u r drds .
0
W
0
4.104
V
s
Thanks to Lemmas 4.2 and 4.3, we derive from the latter estimate that
4.105
I4 ≤ Cδ1/2 .
Similar estimates hold for I5 and I6 .
Step 3 estimate for I7 . Thanks to the idea used in the proof of the estimate 4.102, we have
that the quantity
Cδ
1/2
T −δ
1/2 T −δ sδ
1/2
2
2
N
N
N
E
sup u s θ − u s ds
E
,
F u r, r drds
0
V
0≤θ≤δ
0
V
s
4.106
dominates I7 . By the assumption on F and the argument used in deriving 4.102 we have
I7 ≤ Cδ.
4.107
Step 4 estimate for I2 . We use the same argument as used in the proof of Lemma 4.2 to get
an estimate of the form
N
N 2
N
λi G u r, r , ei ≤ C 1 u r .
V
i1
4.108
We derive from the definition of I2 , the latter estimate and Lemma 4.2 that
I2 ≤ Cδ.
4.109
Step 5 estimate for I1 . Thanks to Fubini’s Theorem and the Burkhölder-Davis-Gundy
inequality we have
1/2
T −δ sδ 2
N
N
N
G u r, r , u r − u s dr
E
ds.
I1 ≤ 6
0
s
4.110
32
Boundary Value Problems
By a sequence of Cauchy-Schwarz’s inequality we find that I1 is bounded from above by
T −δ
1/2 T −δ sδ
1/2
2
2
N
N
N
C E
sup u s θ − u s ds
E
.
G u r, r drds
0
V
0≤θ≤δ
0
V
s
4.111
By the assumption on G and by Lemma 4.2 we get from the latter equation that
4.112
I1 ≤ Cδ1/2 .
Combining all the estimates in Steps 1–5 we get,
T −δ
2
E
sup uN s θ − uN s ds ≤ Cδ1/2 .
0
4.113
V
0≤θ≤δ
Since V is continuously embedded in W∗ ,
T −δ
T −δ
2
2
E
sup uN s θ − uN s ∗ ds ≤ CE
sup uN s θ − uN s ds
0
W
0≤θ≤δ
0
V
0≤θ≤δ
4.114
≤ Cδ1/2 .
The lemma follows readily from this last inequality and noting that a similar argument can
be carried out to find a similar estimate for negative values of θ.
4.3. Tightness Property and Application of
Prokhorov’s and Skorokhod’s Theorems
We denote by Z the following subset of L2 0, T ; V:
'
Z
∞
∞
z ∈ L 0, T ; W ∩ L 0, T ; V; sup
|θ|≤μM
T −μM
(
|zt θ −
0
zt|2W∗
≤ CνM ,
4.115
for any sequences νM , μM such that νM , μM → 0 as M → ∞. The following result is a version
of Theorem 2.6 due to Bensoussan 51.
Lemma 4.7. The set Z is compact in L2 0, T ; V.
Next we consider the space S C0, T ; Rm × L2 0, T ; V endowed with its Borel σalgebra BS and the family of probability measures PN on S, which is the probability
measure induced by the following mapping:
φ : w −→ Ww, ·, uN w, · ,
that is, for any A ∈ BS, PN A Pφ−1 A.
4.116
Boundary Value Problems
33
We have the following lemma.
Lemma 4.8. The family PN N≥1 is tight.
Proof. For any ε > 0 and M ≥ 1, we claim that there exists a compact subset Kε of S such that
PN Kε ≥ 1 − ε. To prove our claim we define the sets
⎧
⎪
⎪
⎨
⎫
⎪
⎪
⎬
M/8
Wε W : sup 2
|Wt − Ws| ≤ Jε , ∀M ,
⎪
⎪
⎪
⎪
t,s∈0,T ⎩
⎭
M
|t−s|<T/2
'
z; sup|zt|2V
t≤T
Zε ≤ Kε ,
sup|zt|2W
t≤T
T −μM
≤ Lε , sup
|θ|≤μM
(
|zt θ −
0
zt|2W∗
≤ Rε νM ,
4.117
where the sequences νM and μM are chosen so that they are independent of ε, νM , μM → 0 as
√
M → ∞ and M μM /νM < ∞. It is clear by Ascoli-Arzela’s Theorem that Wε is a compact
subset of C0, T ; Rm , and by Lemma 4.7, Zε is a compact subset of L2 0, T ; V. We have to
∈ Wε × Zε < ε. Indeed, we have
show that Aε PN W, uN /
⎡
2M
∞ -
Aε ≤ P⎣
1
sup Wt − Ws ≥ Jε M/8
2
t,s∈Ij
M1 j1
P
'
M
⎤
2
N
⎦ P supu t ≥ Kε
t≤T
V
(
T −μM 2
N
N 2
N
sup
P supu t ≥ Lε ,
u t θ − u t ∗ ≥ Rε νM
|θ|≤μM
W
0
t≤T
W
4.118
where {Ij : 1 ≤ j ≤ 2M } is a family of intervals of length T/2M which forms a partition of the
interval 0, T . It is well-known that for any Wiener process B
E|Bt − Bs|2m Cm |t − s|m
for any m ≥ 1,
4.119
where Cm is a constant depending only on m. From this and Markov’s Inequality
Pω : ζw ≥ α ≤
1 k
E
,
|ζω|
αk
4.120
where ζω is a random variable on Ω, F, P and positive numbers k and α, we obtain
Aε ≤
2m 1 T m
1
1
N 2
N 2
Cm 2M/8
E
sup
t
E
sup
t
u
u
V
W
Kε t≤T
Lε t≤T
Jε2m 2M
M1 j1
2M
∞ M
1
E sup
Rε νM |θ|≤μM
T −μM 2
N
u t θ − uN t ∗ .
0
W
4.121
34
Boundary Value Problems
Owing to Lemmas 4.2, 4.6 and by choosing m 2, we have
√
∞
1
C2 T 2 1
1 μM
−1/2M
Aε ≤
2
C
Kε Lε Rε M νM
L4ε M1
√
√ C2 T 2 1
1
1 μM
.
2 2 C
≤
Kε Lε Rε M νM
Jε4
4.122
A convenient choice of Jε , Kε , Lε , Rε completes the proof of the claim, and hence the proof of
the lemma.
It follows by Prokhorov’s Theorem Theorem 2.1 that the family PN N≥1 is relatively
compact in the set of probability measures equipped with the weak convergence topology
on S. Then, we can extract a subsequence PNμ that weakly converges to a probability measure
P. By Skorokhod’s Theorem Theorem 2.2, there exists a probability space Ω, F, P and
random variables W Nμ , uNμ and W, u on Ω, F, P with values in S such that
W Nμ −→ W
uNμ −→ u
in C0, T ; Rm P-a.s,
in L2 0, T ; VP-a.s.
4.123
4.124
Moreover,
the probability law of W Nμ , uNμ is PNμ and that of W, u is P.
4.125
For the filtration Ft , it is enough to choose Ft σWs, us : 0 ≤ s ≤ t, t ∈ 0, T .
It remains to prove that the limit process W is a Wiener process. To fix this, it is
sufficient to show that for any 0 < t1 < t2 < · · · < tm T , the increments process
Wtj − Wtj−1 are independent with respect to Ftj−1 , distributed normally with mean 0
and variance tj − tj−1 . That is, to show that for any λj ∈ Rm and i2 −1
⎛
⎞
m
m
.
1 exp − λ2j tj − tj−1 .
E exp⎝i λj W tj − W tj−1 ⎠ 2
j1
j1
4.126
Equation 4.126 will follow if we have
$
#
λθ2
iλWt θ − Wt
.
exp −
E exp
2
Ft
4.127
We rely on the fact that for any random variables X and Y on any probability space Ω, F, P
such that X is F-measurable and E|Y | < ∞, E|XY | < ∞, we have
Y
XY
XE
,
E
F
F
Y
EE
EY ,
F
4.128
Boundary Value Problems
35
that is,
Y
EXY E XE
.
F
4.129
Now, let us consider an arbitrary bounded continuous functional ϑt W, v on S depending
only on the values of W and v on 0, T . Owing to the independence of Wt to ϑt W, v and
the fact that W is a Wiener process, we have
E exp iλ Wt θ − Wt ϑt W, v
E ϑt W, v
E exp iλ Wt θ − Wt
exp −
λθ2
2
4.130
E ϑt W, v .
In view of 4.125, this implies that
E exp iλ W Nμ t θ − W Nμ t ϑt W Nμ , v
E ϑt W Nμ , v
E exp iλ W Nμ t θ − W Nμ t
exp −
λθ2
2
4.131
E ϑt W Nμ , v .
Now, the convergences 4.123 and 4.124 and the continuity of ϑ allow us to pass to the
limit in this latter equation and obtain
/
0
λθ2
Eϑt W, v,
E expiλWt θ − Wtϑt W, v exp −
2
4.132
which, in view of 4.129, implies 4.127. The choice of the above filtration implies then that
W is a Ft -standard m-dimensional Wiener process.
Theorem 4.9. The pair uNμ , W Nμ satisfies the equation
uNμ s, ei
V
ν
t uNμ s, ei
0
ds t curl uNμ s − αΔuNμ s × uNμ s, ei ds
0
t t N
μ
Nμ
F u s, s , ei ds G uNμ s, s , ei dW Nμ ,
u0 , ei V
for any i ≥ 1.
0
0
4.133
36
Boundary Value Problems
Proof. Let i ≥ 1 be an arbitrary fixed integer. We set
X
N
t T t N
N
N
u s, ei ds −
F uN s, s , ei ds
u s, ei − u0 , ei ν
V
V
0
0
0
2
t curl uN s − αΔuN s × uN s, ei ds −
G uN s, s , ei dW dt.
0
t 0
4.134
Obviously
XN 0 P-a.s,
4.135
XN
0.
1 XN
4.136
which implies in particular that
E
Now we let
Y
Nμ
t T t uNμ s, ei ds −
F uNμ s, s , ei ds
uNμ s, ei ν
V
0
0
0
t N
μ
− u0 , ei curl uNμ s − αΔuNμ s × uN s, ei ds
V
4.137
0
2
t Nμ
Nμ G u s, s , ei dW dt.
0
We will prove that
E
YNμ
0.
1 YNμ
4.138
The difficulty we encounter is that XN is not a deterministic functional of uN and W because
of the stochastic term. To overcome this obstacle we introduce
Gε ut, t 1
ε
T t−s
φ −
Gus, sds,
ε
0
4.139
where φ is a mollifier. It is clear that
T
T
2
ε
E |G ut, t| dt ≤ E |Gut, t|2 dt.
0
0
4.140
Boundary Value Problems
37
Moreover,
in L2 Ω, P; L2 0, T ; V ,
Gε u·, · −→ Gu·, ·
4.141
which implies in particular that
Gε u·, ·, ei −→ Gu·, ·, ei in L2 Ω, P; L2 0, T for any i ≥ 1.
4.142
Let us denote by XN,ε and YNμ ,ε the analog of XN and YNμ with G replaced by Gε .
Introduce the mapping
ϕN,ε : C0, T ; Rm × L2 0, T ; V −→ Ω, F, P ,
ϕN,ε W, v
N
XN,ε
.
1 XN,ε
4.143
Now, it is seen that ϕN,ε is a bounded continuous functional on S. Next, let us define
ϕNμ ,ε W, uNμ XNμ ,ε
.
1 XNμ ,ε
4.144
We have
E
YNμ ,ε
Nμ
Nμ
W
.
Eϕ
,
u
N
,ε
μ
1 YNμ ,ε
4.145
Since ϕNμ ,ε W Nμ , uNμ is a bounded functional on S and since the law of W Nμ , uNμ is PNμ
see 4.125, then
E
YNμ ,ε
1 YNμ ,ε
S
ϕw, vdPNμ .
4.146
We note that law W, uNμ PNμ , so
S
ϕw, vdPNμ Eϕ W, uNμ
XNμ ,ε
.
E
1 XNμ ,ε
4.147
That is,
E
YNμ ,ε
XNμ ,ε
E
.
1 YNμ ,ε
1 XNμ ,ε
4.148
38
Boundary Value Problems
Note that
YNμ ,ε
XNμ
YNμ
YNμ ,ε
XNμ ,ε
YNμ
E
−
E
E
−
−
E
E
1 YNμ
1 XNμ
1 YNμ 1 YNμ ,ε
1 YNμ ,ε
1 XNμ ,ε
XNμ ,ε
XNμ
.
E
−
1 XNμ ,ε 1 XNμ
4.149
We can check that
YNμ
YNμ − YNμ ,ε
YNμ ,ε E
−
E ,
N
N
,ε
N
N
,ε
1Y μ 1Y μ 1 Y μ 1 Y μ 4.150
and it implies that
T
1/2
YNμ
2
YNμ ,ε ε Nμ
Nμ
E
−
.
≤ C E G u t, t − G u t, t , ei dt
1 YNμ 1 YNμ ,ε 0
4.151
We also have
T
1/2
XNμ
2
XNμ ,ε ε Nμ
Nμ
−
.
E
≤ C E G u t, t − G u t, t , ei dt
1 XNμ 1 XNμ ,ε 0
4.152
The above estimates and 4.145 yield
T
1/2
2
YNμ XNμ ε Nμ
Nμ
.
E
− E
≤ C E G u t, t − G u t, t , ei dt
1 YNμ 1 XNμ 0
4.153
Passing to limit to the above relation implies 4.138 and, hence, 4.133.
5. Proof of the Main Result
5.1. Passage to the Limits
From the tightness property we have
uNμ −→ u
in L2 0, T ; V P-a.s.
5.1
Since uNμ satisfies the two equivalent equations 4.133, then it verifies the same estimates as
uN .
Boundary Value Problems
39
Let us consider the positive nondecreasing function ϕx xp , p ≥ 4 defined on R .
We have
lim
x→∞
φx
∞.
x
5.2
p
Thanks to the estimate E supt∈0,T |uNμ |V ≤ C, we have
P φ uNμ L2 0,T ;V
< ∞.
5.3
Thanks to the uniform integrability criteria in 52 we see that |uNμ |L2 0,T ;V is uniform
integrable with respect to the probability measure.
We can deduce from Vitali’s Theorem that
in L2 Ω, P, L2 0, T ; V .
5.4
in L2 Ω, P, L2 0, T ; L2 D ,
5.5
uNμ −→ u
This implies in particular that
uNμ −→ u
∂uNμ
∂u
−→
∂xi
∂xi
in L2 Ω, P, L2 0, T ; L2 D ,
i 1, 2.
5.6
Thanks to 5.4, we can still extract a new subsequence from uNμ denoted again by uNμ so
that
uNμ −→ u
in V dt × dP-almost everywhere.
5.7
It is readily seen that
uNμ , ei
−→ u, ei strongly in L2 Ω, P; L2 0, T .
5.8
Let χ be an element of L∞ Ω × 0, T , dP ⊗ dt.
Since ei ∈ H3 D ⊂ L∞ D, then χei ∈ L∞ Ω × 0, T × D, dP ⊗ dt ⊗ dx. Thanks to 5.5,
5.6 we have that
Nμ
∂uk ∂uk χei k −→ uj
χei k
∂xj
∂xj
in L1 Ω × 0, T × D,
5.9
T
T Nμ
Nμ
E b u , u , χei dt −→ E b u, u, χei dt for any i.
5.10
Nμ
uj
which implies that
0
0
40
Boundary Value Problems
Since in view of Lemma 4.1 ei ∈ H4 D, then
∂
∂xk
∂ei
χ
∂xj
∈ L∞ Ω, P, L∞ 0, T ; H2 D .
5.11
∈ L∞ Ω, P, L∞ 0, T ; L∞ D.
5.12
l
Since H2 D ⊂ L∞ D, then
∂
∂xk
∂ei
χ
∂xj
l
With the help of 5.5 we obtain
Nμ
u
∂
k ∂xk
∂ei
χ
∂xj
l
∂ei
χ
∂xj
∂
−→ uk
∂xk
in L2 Ω, P; L2 0, T ; L2 D .
5.13
l
We derive from this and 5.6 that
T E
uNμ
0
D
∂
k ∂xk
∂ei
χ
∂xj
l
∂uNμ
∂xj
l
T
∂ei
∂u
∂
χ
dx dt −→ E
uk
dx dt,
∂xk
∂xj
∂xj
0 D
l
l
5.14
for any i, j, k, l. In the above equations fk denotes the kth component of the vector function
f.
We can use the same argument to show that
T T Nμ ∂ei l ∂uNμ
∂ei l ∂u
∂u
∂u
χ
dx dt −→ E
χ
dx dt,
E
∂xj
∂xj
∂xj
∂xj
∂xj
∂xj
0 D
0 D
k
l
k
l
5.15
for any i, j, k, l.
With the help of 2.21, and 5.14, 5.15, we have
T T
Nμ
Nμ
E b u , Δu , ei χ dt −→ E bu, Δ, ei χ dt,
0
∀i.
5.16
0
Thanks to density of L∞ Ω × 0, T , dP ⊗ dt in L2 Ω × 0, T , dP ⊗ dt and by taking χ ∈
L∞ Ω × 0, T , dP ⊗ dt as a test function, we deduce from 5.16 that
b uNμ , ΔuNμ , ei bu, Δu, ei weakly in L2 Ω, P; L2 0, T ,
for any i.
5.17
Boundary Value Problems
41
Using 2.22, we can imitate the argument used above to show that
b ei , ΔuNμ , uNμ bei , Δu, u
weakly in L2 Ω, P; L2 0, T ,
5.18
for any i.
We conclude with 2.18, 5.10, 5.17 and 5.18 that
curl uNμ − αΔuNμ × uNμ , ei curlu − αΔu × u, ei ,
5.19
weakly in L2 Ω, P; L2 0, T for any i.
It follows from 5.7, the Lemma 4.3, the assumption on F, and Vitali’s theorem that
strongly in L2 Ω, P; L2 0, T ; V .
F uNμ ·, · −→ Fu·, ·
5.20
This implies in particular that
F uNμ ·, · , ei −→ Fu·, ·, ei strongly in L2 Ω, P; L2 0, T ,
5.21
for any i.
It remains to prove that
t t
G uNμ , s , ei dW Nμ Gu, s, ei dW
0
weakly- in L2 Ω, P; L∞ 0, T ,
5.22
0
for any t ∈ 0, T and i as μ → ∞. Using a similar argument as in 53, we will just show that
T T
Nμ
Nμ
G u , s , ei dW Gu, s, ei dW
0
weakly in L2 Ω, P,
5.23
0
from 5.22 follows. From now on we fix i ≥ 1. First, Lemma 4.3, the convergence 5.7, the
assumption on G, and Vitali’s theorem imply that
G uNμ , · , ei −→ Gu, ·, ei in L2 Ω, P; L2 0, T 5.24
as μ → ∞. We consider the already introduced regularized function Gε u·, · in 4.139. We
readily check that
Gε u·, ·, ei −→ Gu·, ·, ei in L2 Ω, P; L2 0, T ,
5.25
as ε → 0. We also have
T T 2
2
ε Nμ
ε
E G u , t − G u, t, ei dt ≤ E G uNμ , t − Gu, t, ei dt.
0
0
5.26
42
Boundary Value Problems
The crucial point is to show that
T
T
Gε uNμ , t , ei dW Nμ Gε u, t, ei dW
0
weakly in L2 Ω, P.
5.27
0
Since
2
T T 2
ε
Nμ
Nμ G u , t , ei dW E
Gε uNμ , t , ei dt < ∞,
E
0
0
5.28
T
then 0 Gε uNμ , t, ei dW Nμ weakly converges to a certain β in L2 Ω, P. An integration-byparts yields
T
ε
G u
Nμ
, t , ei dW
Nμ
G u
ε
Nμ
T , T , ei −
0
T
W Nμ t
0
d ε Nμ
G u t, t , ei dt,
dt
5.29
where
1
d ε Nμ
G u t, t , ei dt
ε
T
t − s Nμ
d
φ −
G v s, s , ei ds.
ε
0 dt
5.30
By virtue of the convergence
uNμ , W Nμ −→ u, W in C0, T ; Rm × L2 0, T ; V
5.31
P-almost surely, we have
T
0
T
d
Gε uNμ , t , ei dW Nμ −→ Gε uT , T , ei − Wt Gε ut, t, ei dt
dt
0
5.32
for almost all ω ∈ Ω. The term in the left-hand side of 5.32 is equal to
T
Gε ut, t, ei dW.
5.33
0
Now let us pick an element ζ ∈ L∞ Ω, P. We have
T T
ε
Nμ
Nμ
E
G u t, t , ζei dW −→ E Gε ut, t, ζei dW,
0
0
5.34
Boundary Value Problems
43
that is
β
T
Gε ut, t, ei dW.
5.35
0
Indeed, thanks to the estimate 4.140, Lemma 4.2 the sequence of random variables
T ε N
G u μ t, t, ζei dW Nμ is uniformly integrable. Owing to the convergence 5.32 and the
0
applicability of the Vitali’s Theorem, we get 5.34. We also have 5.27 since L∞ Ω, F, P is
dense in L2 Ω, P.
Let ζ ∈ L∞ Ω, P; we write
T
T G uNμ t, t , ζei dW Nμ − E Gut, t, ζei dW ≤ J1 J2 J3 ,
E
0
0
where
T T ε
Nμ
Nμ
Nμ
Nμ G u t, t , ζei dW − E
G u t, t , ζei dW ,
J1 E
0
0
T
T J2 E
Gε uNμ t, t , ζei dW Nμ − E Gε ut, t, ζei dW ,
0
0
T
T
ε
J3 E G ut, t, ζei dW − E Gut, t, ζei dW .
0
0
5.36
5.37
By Cauchy-Schwarz’s inequality and owing to 5.25, the term J3 of the RHS of 5.36
converges to zero as ε → 0.
By 5.34, the term J2 in the RHS of 5.36 converges to zero as μ → ∞.
By Cauchy-Schwarz’s inequality again, some simple calculations, and making use of
the estimate 5.26 and the convergence 5.24 and 5.25 we see that J1 converges to zero as
ε → 0 and μ → ∞.
In view of these convergences passing to the limit as ε → 0 and μ → ∞ in 5.36 we
get 5.22.
Combining all those results and passing to the limit in 4.133, we see that u satisfies
3.6. This proves the first part of Theorem 3.3. The next subsection addresses the continuity
in time of our solution.
5.2. Proof of Continuity of the Paths of u
We have already shown that for any i ≥ 1 the equation
t
ut, ei V u0 , ei V Fus, s − curlu − αΔu × u, ei − νus, ei ds
t
0
Gus, s, ei dW
0
holds almost surely for any t ∈ 0, T .
5.38
44
Boundary Value Problems
For any i ≥ 1 let ϕ be the mapping
0, T −→ R,
5.39
t −→ ϕt ut, ei V .
Let θ > 0. We have
tθ
ϕt − ϕt θ ≤ Fus, s − curlu − αΔu × u, ei − νus, ei ds
t
tθ
Gus, s, ei dW .
t
5.40
Let p > 4, we obtain by raising both sides of the last inequality to the power p/2
p/2
tθ
p/2
tθ
ϕt − ϕt θp/2 ≤ C
Gus, s, ei dW C
|Fus, s, ei |ds
t
t
C
tθ
p/2
|curlu − αΔu × u, ei |ds
tθ
p/2
C ν
.
|us, ei |ds
t
t
5.41
We infer from this that
tδ
p/2
p/2
Eϕt − ϕt θ ≤ E
|curlu − αΔu × u, ei |ds
t
p/2
tδ
CE sup
Gus, s, ei dW t
0≤δθ
tδ
CE
p/2
|Fus, s, ei |ds
t
tδ
CE
t
p/2
ν|us, ei |ds
,
5.42
Boundary Value Problems
45
which implies by the help of martingale inequality that
p/2
Eϕt − ϕt θ ≤ Cθp−2/2
tδ
|curlu − αΔu × u, ei |p/2 ds
t
Cθ
p−2/2
tδ
|Fus, s, ei |p/2 ds
t
Cθp−2/2 ν
tδ
5.43
|us, ei |p/2 ds
t
tθ
CE
p/2
|Gus, s, ei |
2
.
t
Using previous estimates and some elementary inequalities, the following holds:
p/2
Eϕt − ϕt θ ≤ C θ1p−2/2 θ1p−4/4 ,
5.44
for any θ > 0. We conclude from Kolmogorov-C̆entsov Theorem that the stochastic process
ϕ· u·, ei V has almost surely a continuous modification with respect to the time variable
t. Identifying u with this modification, we see that u has almost surely continuous paths
taking values in V-weak. Since u is also in the class Lp/2 Ω, P; L∞ 0, T ; W, then u· also
has almost surely continuous paths with respect to t taking values in W-weak see 54 for
justification. It follows that the initial condition ux, 0 u0 ∈ W in 1.13 makes sense.
Acknowledgments
The research of the authors is supported by the University of Pretoria and the National
Research Foundation South Africa. The second author acknowledges the support of the
National Science Foundation under the agreement no. DMS-0635607 during his stay at
the Institute for Advanced Study at Princeton. The authors also would like to thank the
anonymous referee for his/her insightful comments.
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