Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise

0.45 MB·

by  Zdzislaw Brzezniak

QUASIPOTENTIAL AND EXIT TIME FOR 2D STOCHASTIC
NAVIER-STOKES EQUATIONS DRIVEN BY SPACE TIME WHITE
NOISE
4
1
0
2 Z. BRZEŹNIAK AND S. CERRAI AND M. FREIDLIN

y
a Abstract. We aredealingwiththeNavier-Stokesequationinaboundedregulardomain
M ofR2, perturbedby anadditive Gaussiannoise ∂wQδ/∂t,whichis white in time andcoloreOd
in space. We assume that the correlation radius of the noise gets smaller and smaller as
0
δ 0, so that the noise converges to the white noise in space and time. For every δ > 0
3
ց
we introduce the large deviation action functional Sδ and the correspondingquasi-potential
T
] U and, by using arguments from relaxation and Γ-convergence we show that U converges
R δ δ
to U =U0, in spite of the fact that the Navier-Stokes equation has no meaning in the space
P
of square integrable functions, when perturbed by space-time white noise. Moreover, in the
.
h case of periodic boundary conditions the limiting functional U is explicitly computed.
t Finally, we apply these results to estimate of the asymptotics of the expected exit time
a
m of the solution of the stochastic Navier-Stokes equation from a basin of attraction of an
asymptotically stable point for the unperturbed system.
[

2
v
9
9
2 Contents
6
. 1. Introduction 1
1
0 Acknowledgments 5
4 2. Notation and preliminaries 5
1 3. The skeleton equation 10
:
v 4. Some basic facts on relaxation and Γ-convergence 18
i 5. The large deviation action functional 19
X
6. The quasi-potential 24
r
a 7. Stochastic Navier Stokes equations with periodic boundary conditions 29
8. Convergence of U to U 30
δ
9. Application to the exit problem 33
Appendix A. Proofs of some auxiliary results 36
Appendix B. Proofs of Lemmas in Section 9 42
References 46
1. Introduction
Date: June 2, 2014.
1
2 Z. BRZEŹNIAKANDS. CERRAIAND M. FREIDLIN
Let be a regular bounded open domain of R2. We consider here the 2-dimensional Navier-
O
Stokes equation in , perturbed by a small Gaussian noise
O
∂u(t,x)
= ∆u(t,x) (u(t,x) u(t,x))u(t,x) p(t,x)+√εη(t,x),
∂t − ·∇ −∇
with the incompressibility condition
divu(t,x) = 0
and initial and boundary conditions
u(t,x) = 0, x ∂ , u(0,x) = u (x).
0
∈ O
Here 0 < ε << 1 and η(t,x) is a Gaussian random field, white in time and colored in space.
In what follows, for any α R we shall denote by V the closure in the space [Hα( )]2
α
∈ O
of the set of infinitely differentiable 2-dimensional vector fields, having zero divergence and
compact support on , and we shall set H = V and V = V . We will also set
0 1
O
D(A) = [H2( )]2 V, Ax = ∆x, x D(A).
O ∩ − ∈
The operator A is positive and self-adjoint, with compact resolvent. We will denote with
0 < λ1 λ2 and ek k∈N the eigenvalues and the eigenfunctions of A, respectively.
≤ ≤ ··· { }
Moreover, we will define the bilinear operator B : V V V by setting
−1
× →
B(u,v),z = z(x) [(u(x) )v(x)] dx.
h i · ·∇
ZO
With these notations, if we apply to each term of the Navier-Stokes equation above the
projection operator into the space of divergence free fields, we formally arrive to the abstract
equation
du(t)+Au(t)+B(u(t),u(t)) = √εdwQ(t), u(0) = u , (1.1)
0
where the noise wQ(t) is assumed to be of the following form
∞
wQ(t) = Qe β (t), t 0, (1.2)
k k
≥
k=1
X
for some sequence of independent standard Brownian motions βk(t) k∈N and a linear oper-
{ }
ator Q defined on H (for all details see Section 2).
As well known, white noise in space and time (that is Q = I) cannot be taken into consider-
ation in order to study equation (1.1) in the space H. But if we assume that Q is a compact
operator satisfying suitable conditions, as for example Q A−α, for some α > 0, we have
∼
that for any u H and T > 0 equation (1.1) is well defined in C([0,T];H) and the validity
0
∈
of a large deviation principle and the problem of the exit of the solution of equation (1.1)
from a domain can be studied.
As in the previous work [11], where a class of reaction-diffusion equations in any space di-
mension perturbed by multiplicative noise has been considered, in the present paper we want
to see how we can describe the small noise asymptotics of equation (1.1), as if the noisy
perturbation were given by a white noise in space and time. This means that, in spite of
the fact that equation (1.1) is not meaningful in H when the noise is white in space, the
relevant quantities for the large deviations and the exit problems associated with it can be
approximated by the analogous quantities that one would get in the case of white noise in
QP 2D SNSES ST WN 3
space. In particular, when periodic boundary conditions are imposed, such quantities can be
explicitly computed and such approximation becomes particularly useful.
In what follows we shall consider a family of positive linear operators Q defined on
δ δ∈(0,1]
{ }
H, such that for any fixed δ (0,1] equation (1.1), with noise
∈
∞
wQδ(t) = Q e β (t), t 0,
δ k k
≥
k=1
X
is well defined in C([0,T];H), and Q is strongly convergent to the identity operator in H, for
δ
δ 0. For each fixed δ (0,1], the family (ux ) satisfies a large deviation principle
ց ∈ {L ε,δ }ε∈(0,1]
in C([0,T];H) with action functional
1 T
Sδ(u) = Q−1(u′(t)+Au(t)+B(u(t),u(t))) 2 dt,
T 2 δ H
Z0
and the corresponding quasi-pote(cid:12)ntial is defined by (cid:12)
(cid:12) (cid:12)
U (φ) = inf Sδ(u) : u C([0,T];H), u(0) = 0, u(T) = φ, T > 0 .
δ T ∈
Our purpose here is to(cid:8)show that, despite we cannot prove any limit for the (cid:9)solution uδ of
equation (1.1), nevertheless, for all φ H such that U (φ) < ,
δ
∈ ∞
limU (φ) = U(φ), (1.3)
δ
δ→0
where U(φ) is defined as U (φ), with the action functional Sδ replaced by
δ T
1 T
S (u) = u′(t)+Au(t)+B(u(t),u(t)) 2 dt.
T 2 | |H
Z0
To this purpose, the key idea consists in characterizing the quasi-potentials U and U as
δ
U (φ) = min Sδ (u) : u and u(0) = φ (1.4)
δ −∞ ∈ X
and
(cid:8) (cid:9)
U(φ) = min S (u) : u and u(0) = φ , (1.5)
−∞
{ ∈ X }
where
= u C(( ,0];H) : lim u(t) = 0
H
X ∈ −∞ t→−∞| |
and the functionals Sδ and(cid:8)S are defined on in a natural wa(cid:9)y, see formulae (5.6) and
−∞ −∞ X
(5.5) later on.
In this way, in the definition of U and U, the infimum with respect to time T > 0 has
δ
disappeared and we have only to take the infimum of suitable functionals in the space :=
φ
X
u : u(0) = φ . In particular, the convergence of U (φ) to U(φ) becomes the convergence
δ
{ ∈ X }
of the infima of Sδ in to the infimum of S in , so that (1.3) follows once we prove
−∞ Xφ −∞ Xφ
that Sδ is Gamma-convergent to S in , as δ 0. Moreover, as a consequence of (1.5),
−∞ −∞ Xφ ց
in the case of the stochastic Navier-Stokes equations with periodic boundary conditions we
can prove, see section 7, that
U(φ) = φ 2. (1.6)
| |V
This means that U(φ) canbeexplicitly computed andthe use of (1.3)in applicationsbecomes
particularly relevant. Let us point out that a similar explicit formula for the quasipotential
4 Z. BRZEŹNIAKANDS. CERRAIAND M. FREIDLIN
has been derived for linear SPDEs by Da Prato, Pritchard and Zabczyk in [17] and in the
recent work by the second andthird authors for stochastic reaction diffusion equations in [11].
A finite dimensional counterpart of our formula (1.6) was first derived in Theorem IV.3.1 in
the monograph [20].
The proofsof characterizations (1.4) and (1.5) and of the Gamma-convergence of Sδ to S
−∞ −∞
are based on a thorough analysis of the Navier-Stokes equation with an external deterministic
force in the domain of suitable fractional powers of the operator A.
One of the main motivation for proving (1.3) comes from the study of the expected exit time
τε,δ of the solution uε,δ from a domain D in L2( ), which is attracted to the zero function.
φ φ O
Actually, in the second part of the paper we prove that, under suitable regularity properties
of D, for any fixed δ > 0
limε log E(τε,δ) = inf U (y). (1.7)
φ δ
ε→0 y∈∂D
This means that, as in finite dimension, the expectation of τε,δ can be described in terms of
φ
the quantity U (φ). Moreover, once we have (1.3), by a general argument introduced in [11]
δ
and based again on Gamma-convergence, we can prove that if D is a domain in H such that
any point φ V ∂D can be approximated in V by a sequence φn n∈N D(A12+α) ∂D
∈ ∩ { } ⊂ ∩
(think for example of D as a ball in H), then
lim inf U (φ) = inf U(φ).
δ
δ→0 φ∈∂D φ∈∂D
According to (1.7), this implies that for 0 < ε << δ << 1
1
Eτε,δ exp inf U(φ) .
φ ∼ ε φ∈∂D
(cid:18) (cid:19)
In particular, if D is the ball of H of radius c and the boundary conditions are periodic, in
view of (1.6) for any φ D we get,
∈
e−c2ελ21 Eτε,δ 1, 0 < ε << δ << 1.
φ ∼
At the end of this long introduction, we would like to point out that although 2-D stochastic
Navier-StokesequationswithperiodicboundaryconditionshavebeeninvestigatedbyFlandoli
and Gozzi in [23] and Da Prato and Debussche in [16] from the point of view of Kolmogorov
equations and the existence of a Markov process, we do not know whether our results (even
in the periodic case) could be derived from these papers. One should bear in mind that
the solution from [16] exists for almost every initial data u from a certain Besov space of
0
negative orderwithrespect toaspecific Gaussianmeasure whileweconstruct aquasipotential
for every u from the space H whose measure is equal to 0. Of course our results are also
0
valid for 2-D stochastic Navier-Stokes equations with Dirichlet boundary conditions.
We have been recently become awareof a work by F.Bouchet et all[3] where somehow related
issues are considered from a physical point of view. We hope to be able to understand the
relationship between our work and this work in a future publication.
QP 2D SNSES ST WN 5
Acknowledgments. Research of the first named author was also supported by the EPSRC
grant EP/E01822X/1, research of the second named author was also supported by NSF grant
DMS0907295 and research of the third named author was also supported by the NSF grants
DMS 0803287 and 0854982.
The first named author would like to thank the University of Maryland for hospitality during
his visit in Summer 2008. The second named author would like to thank Michael Salins for
some useful discussions about Lemmas 9.8 and 9.9.
Finally, the three authors would like to thank the two anonymous referees for reading care-
fully the original manuscript and giving very good suggestions that helped us to improve
considerably the final version of the paper.
2. Notation and preliminaries
Let R2 be an open and bounded set. We denote by Γ = ∂ the boundary of . We
O ⊂ O O
will always assume that the closure of the set is a manifold with boundary of C∞ class,
O O
whose boundary ∂ is denoted by Γ, is a 1-dimensional infinitely differentiable manifold
O
being locally on one side of , see condition (7.10) from [30, chapter I]. Let us also denote
O
by ν the unit outer normal vector field to Γ.
It is known that is a Poincaré domain, i.e. there exists a constant λ > 0 such that the
1
O
following Poincaré inequality is satisfied
λ ϕ2(x)dx ϕ(x) 2dx, ϕ H1( ). (2.1)
1 ≤ |∇ | ∈ 0 O
ZO ZO
In order to formulate our problem in an abstract framework, let us recall the definition of
the following functional spaces. First of all, let ( ) (resp. ( )) be the set of all C∞ class
D O D O
vector fields u : R2 R2 with compact support contained in the set (resp. ). Then, let
→ O O
us define
E( ) = u L2( ,R2) : divu L2( ) ,
O { ∈ O ∈ O }
= u ( ) : divu = 0 ,
V ∈ D O
H = the closure of in L2( ),
(cid:8) (cid:9)
V O
H1( ,R2) = the closure of ( ,R2) in H1( ,R2),
0 O D O O
V = the closure of in H1( ,R2).
V 0 O
The inner products in all the L2 spaces will be denoted by ( , ). The space E( ) is a Hilbert
· · O
space with a scalar product
u,v := (u,v) +(divu,divv) . (2.2)
E(O) L2(O,R2) L2(O,R2)
h i
We endow the set H with the inner product ( , ) and the norm induced by L2( ,R2).
· · H |·|H O
Thus, we have
2
(u,v) = u (x)v (x)dx,
H j j
j=1 ZO
X
The space H can also be characterised in the following way. Let H−1(Γ) be the dual space
2
of H1/2(Γ), the image in L2(Γ) of the trace operator γ : H1( ) L2(Γ) and let γ be the
0 ν
O →
6 Z. BRZEŹNIAKANDS. CERRAIAND M. FREIDLIN
bounded linear map from E( ) to H−1(Γ) such that, see [42, Theorem I.1.2],
2
O
γ (u) = the restriction of u ν to Γ, if u ( ). (2.3)
ν
· ∈ D O
Then, see [42, Theorem I.1.4],
H = u E( ) : divu = 0 and γ (u) = 0 ,
ν
{ ∈ O }
H⊥ = u E( ) : u = p, p H1( ) .
{ ∈ O ∇ ∈ O }
Let us denote by P : L2( ,R2) H the orthogonal projection called usually the Leray-
O →
Helmholtz projection. It is known, see for instance [42, Remark I.1.6] that
Pu = u (p+q), u L2( ,R2), (2.4)
−∇ ∈ O
where, foru L2( ), pistheuniquesolutionofthefollowinghomogenousboundaryDirichlet
∈ O
problem for the Laplace equation
∆p = divu H−1( ), p = 0. (2.5)
Γ
∈ O |
and q H1( ) is the unique solution of the following in-homogenous Neumann boundary
∈ O
problem for the Laplace equation
∂q
∆q = 0, = γ (u p). (2.6)
ν
∂ν Γ −∇
Note that the function p above satisfies p (cid:12)(cid:12) L2( ,R2) and div(u p) = 0. In particular,
∇ ∈(cid:12) O −∇
u p E( ) so that q is well defined.
It−is∇prov∈ed inO[42, Remark I.1.6] that P maps continuously the Sobolev space H1( ,R2) into
O
itself. Below, we will discuss continuity of P with respect to other topologies.
Since the set is a Poincaré domain, the norms on the space V induced by norms from the
Sobolev spaceOs H1( ,R2) and H1( ,R2) are equivalent. The latter norm and the associated
O 0 O
inner product will be denoted by and , , respectively. They satisfy the following
|·|V · · V
equality
(cid:0) (cid:1)
2
∂u ∂v
u,v = j j dx, u,v H1( ,R2).
V ∂x ∂x ∈ 0 O
i,j=1ZO i i
(cid:0) (cid:1) X
Since the space V is densely and continuously embedded into H, by identifying H with its
dual H′, we have the following embeddings
V H = H′ V′. (2.7)
∼
⊂ ⊂
Let us observe here that, in particular, the spaces V, H and V′ form a Gelfand triple.
We will denote by V′ and , the norm in V′ and the duality pairing between V and V′,
|·| h· ·i
respectively.
The presentation of the Stokes operator is standard and we follow here the one given in [9].
We first define the bilinear form a : V V R by setting
× →
a(u,v) := ( u, v) , u,v V. (2.8)
H
∇ ∇ ∈
As obviously the bilinear form a coincides with the scalar product in V, it is V-continuous,
i.e. there exists some C > 0 such that
a(u,u) C u 2, u V
| | ≤ | |V ∈
QP 2D SNSES ST WN 7
Hence, by the Riesz Lemma, there exists a unique linear operator : V V′, such that
A →
a(u,v) = u,v , for u,v V. Moreover, since is a Poincaré domain, the form a is
hA i ∈ O
V-coercive, i.e. it satisfies a(u,u) α u 2 for some α > 0 and all u V. Therefore, in
≥ | |V ∈
view of the Lax-Milgram theorem, see for instance Temam [42, Theorem II.2.1], the operator
: V V′ is an isomorphism.
A →
Next we define an unbounded linear operator A in H as follows
D(A) = u V : u H
{ ∈ A ∈ }
(2.9)

Au = u, u D(A).

A ∈
1
It is now well established that under suitable assumptions related to the regularity of the

domain , the space D(A) can be characterized in terms of the Sobolev spaces. For example,
O
(see [27], where only the 2-dimensional case is studied but the result is also valid in the
3-dimensional case), if R2 is a uniform C2-class Poincaré domain, then we have
O ⊂
D(A) = V H2( ,R2) = H H1( ,R2) H2( ,R2),
∩ O ∩ 0 O ∩ O
(2.10)

Au = P∆u, u D(A).

− ∈
It is also a classical result, see e.g. Cattabriga [13] or Temam [41, p. 56], that A is a positive

self adjoint operator in H and
(Au,u) λ u 2, u D(A). (2.11)
≥ 1| |H ∈
where the constant λ > 0 is from the Poincaré inequality (2.1). Moreover, it is well known,
1
see for instance [41, p. 57] that V = D(A1/2). Moreover, from [44, Theorem 1.15.3, p. 103]
it follows that
D(Aα/2) = [H,D(A)] ,
α
2
where [ , ] is the complex interpolation functor of order α, see e.g. [30], [44] and [38,
α
· · 2 2
Theorem 4.2]. Furthermore, as shown in [44, Section 4.4.3], for α (0, 1)
∈ 2
D(Aα/2) = H Hα( ,R2). (2.12)
∩ O
The above equality leads to the following result.
Proposition 2.1. Assume that α (0, 1). Then the Leray-Helmholtz projection P is a well
∈ 2
defined and continuous map from Hα( ,R2) into D(Aα/2).
O
Proof. Let us fix α (0, 1). Since, by its definition, the range of P is contained in H, it is
∈ 2
sufficient to prove that for every u Hα( ,R2), Pu Hα( ,R2). For this aim, let us fix
u Hα( ,R2). Then divu Hα−1(∈). ThOerefore, by t∈he elliOptic regularity we infer that the
∈ O ∈ O
solution p of the problem (2.5) belongs to the Sobolev space Hα+1( ) H1( ) and therefore
p Hα( ,R2). O ∩ 0 O
∇ ∈ O
Since by [42, Theorem I.1.2], the linear map γ is bounded from E( ) to H−1(Γ) and from
ν 2
O
E( ) H1( ,R2) to H1(Γ), by a standard interpolation argument we infer that γ is a
2 ν
O ∩ O
bounded linear map from E( ) Hα( ,R2) to H−1+α(Γ). Thus we infer that γ (u p)
2 ν
O ∩ O −∇ ∈
1
These assumptions are satisfied in our case
8 Z. BRZEŹNIAKANDS. CERRAIAND M. FREIDLIN
H−1+α(Γ) and by the Stokes formula (I.1.19) from [42], γ (u p),1 = 0. Therefore, again
2 ν
h −∇ i
by the elliptic regularity, see for instance [30], the solution q of the problem (2.6) belongs to
Hα+1( ) and therefore q Hα( ,R2). This proves that Pu Hα( ,R2) as required.
O ∇ ∈ O ∈ O
The proof is complete.
Remark 2.2. We only claim that the above result is true for α < 1. In particular, we do not
2
claim that is P is a bounded linear map from H1( ,R2) to V and we are not aware of such a
O
result. However, if this is true, Proposition 2.1 will hold for any α (0,1) with a simple proof
∈
by complex interpolation. However, it seems to us that the result for α > 1 is not true, since
2
we cannot see how one could prove that Pu = 0. The reason why Proposition 2.1 holds
∂O
|
for any α (0, 1) is that according to identity (2.12) the only boundary conditions satisfied
∈ 2
by functions belonging to D(Aα/2) are those satisfied by functions belonging to the space H.
One can compare with the paper [39] by Temam (or chapter 6 of his book [40]).
Let us finally recall that by a result of Fujiwara–Morimoto [24] the projection P extends to
a bounded linear projection in the space Lq( ,R2), for any q (1, ).
O ∈ ∞
Now, consider the trilinear form b on V V V given by
× ×
2
∂v
j
b(u,v,w) = u w dx, u,v,w V.
i j
∂x ∈
i,j=1ZO i
X
Indeed, b is a continuous trilinear form such that
b(u,v,w) = b(u,w,v), u V,v,w H1( ,R2), (2.13)
− ∈ ∈ 0 O
and
1/2 1/2 1/2 1/2
u u v Av w u V,v D(A),w H
| |H |∇ |H |∇ |H | |H | |H ∈ ∈ ∈
1/2 1/2
u Au v w u D(A),v V,w H
b(u,v,w) C | |H | |H |∇ |H| |H ∈ ∈ ∈ (2.14)
| | ≤  |u|H|∇v|H|w|H1/2|Aw|1H/2 u ∈ H,v ∈ V,w ∈ D(A)
1/2 1/2 1/2 1/2
u u v w w u,v,w V,
| |H |∇ |H |∇ |H| |H |∇ |H ∈

for some constant C> 0 (for a proof see for instance [42, Lemma 1.3, p.163] and [41]).

Define next the bilinear map B : V V V′ by setting
× →
B(u,v),w = b(u,v,w), u,v,w V,
h i ∈
and the homogenous polynomial of second degree B : V V′ by
→
B(u) = B(u,u), u V.
∈
Let us observe that if v D(A), then B(u,v) H and the following inequality follows
∈ ∈
directly from the first inequality in (2.14)
B(u,v) 2 C u u v Av , u V, v D(A). (2.15)
| |H ≤ | |H|∇ |H|∇ |H| |H ∈ ∈
Moreover, the following identity is a direct consequence of (2.13).
B(u,v),v = 0, u,v V. (2.16)
h i ∈
Let us also recall the following fact (see [9, Lemma 4.2]).
QP 2D SNSES ST WN 9
Lemma 2.3. The trilinear map b : V V V R has a unique extension to a bounded
trilinear map from L4( ,R2) (L4( ,R×2) H×) V→and from L4( ,R2) V L4( ,R2) into
R. Moreover, B mapsOL4( ,R×2) HO(and∩so V×) into V′ and O × × O
O ∩
|B(u)|V′ ≤ C1|u|2L4(O,R2) ≤ 21/2C1|u|H|∇u|H ≤ C2|u|2V, u ∈ V. (2.17)
Proof. It it enoughto observe that due to the Hölder inequality, the following inequality holds
b(u,v,w) C u v w , u,v,w H1( ,R2). (2.18)
| | ≤ | |L4(O,R2)|∇ |L2(O)| |L4(O,R2) ∈ 0 O
Thus, our result follows from (2.13).
Let us also recall the following well known result, see [42] for a proof.
Lemma 2.4. For any T (0, ] and for any u L2(0,T;D(A)) with u′ L2(0,T;H), we
∈ ∞ ∈ ∈
have
T
B(u(t),u(t)) 2 dt < .
| |H ∞
Z0
2
Proof. Our assumption implies that u C([0,T];V) (for a proof see for instance [46, Propo-
∈
sition I.3.1]). Then, we can conclude thanks to (2.15)
The restriction of the map B to the space D(A) D(A) has also the following representation
×
B(u,v) = P(u v), u,v D(A), (2.19)
∇ ∈
where P is the Leray-Helmholtz projection operator and u v = 2 ujD v L2( ,R2).
∇ j=1 j ∈ O
This representation together with Proposition 2.1 allows us to prove the following property
of the map B. P
Proposition 2.5. Assume that α (0, 1). Then for any s (1,2] there exists a constant
∈ 2 ∈
c > 0 such that
|B(u,v)|D(Aα/2) ≤ c|u|D(A2s)|v|D(A1+2α), u,v ∈ D(A). (2.20)
Proof. In view of equality (2.19), since the Leray-Helmholtz projection P is a well defined
and continuous map from Hα( ,R2) into D(Aα/2) and since the norms in the spaces D(As)
2
are equivalent to norms in Hs(O,R2), it is enough to show that
O
u v C u v , u,v H2( ,R2).
Hα Hs H1+α
| ∇ | ≤ | | | | ∈ O
Thelast inequality isaconsequence oftheMarcinkiewicz InterpolationTheorem, thecomplex
interpolation and the following two inequalities for scalar functions which can be proved by
using Gagliado-Nirenberg inequalities
uv C u v , u Hs,v L2,
L2 Hs L2
| | ≤ | | | | ∈ ∈
uv C u v , u Hs,v H1.
H1 Hs H1
| | ≤ | | | | ∈ ∈
2
Note that in the case T = one also has limt→∞u(t)=0 in V.
∞
10 Z. BRZEŹNIAKANDS. CERRAIAND M. FREIDLIN
3. The skeleton equation
We are here dealing with the following functional version of the Navier-Stokes equation
u′(t)+νAu(t)+B(u(t),u(t)) = f(t), t (0,T)
∈
(3.1)

u(0) = u ,
 0
where T (0, ] and ν > 0. Let us recall the following definition (see [42, Problem 2, section

∈ ∞
III.3])
Definition 3.1. Given T > 0, f L2(0,T;V′) and u H, a solution to problem (3.1) is a
0
function u L2(0,T;V) such that∈u′ L2(0,T;V′), u(0∈) = u 3 and (3.1) is fulfilled.
0
∈ ∈
It is known (see e.g. [42, Theorems III.3.1/2]) that if T (0, ], then for every f
∈ ∞ ∈
L2(0,T;V′) and u H there exists exactly one solution u to problem (3.1), which satis-
0
∈
fies
u 2 + u 2 u 2 + f 2 . (3.2)
| |C([0,T],H) | |L2(0,T,V) ≤ | 0|H | |L2(0,T,V′)
Moreover, see [42, Theorem III.3.10], if T < and f L2(0,T;H) then
∞ ∈
sup √tu(t) 2 + T √tAu(t) 2 dt
t∈(0,T]| |V 0 | |H
R (3.3)
≤ |u0|2H +|f|2L2(0,T;V′) +T|f|2L2(0,T;H) ec |u0|4H+|f|4L2(0,T;V′) ,
(cid:0) (cid:1)
(cid:16) (cid:17)
for a constant c independent of T, f and u .
0
Finally, if T (0, ], f L2(0,T;H) and u V, then the unique solution u satisfies
0
∈ ∞ ∈ ∈
u L2(0,T;D(A)) C([0,T];V), u′ L2(0,T;H),
∈ ∩ ∈
4
and, for t [0,T] [0, ),
∈ ∩ ∞
d d
u(t) 2 +λ u(t) 2 u(t) 2 + Au(t) 2 (3.4)
dt| |V 1| |V ≤ dt| |V | |H
2 f(t) 2 +108 u(t) 2 u(t) 2 u(t) 2.
≤ | |H | |V| |H| |V
Hence, by the Gronwall Lemma and inequality (3.2), for any t [0,T] [0, )
∈ ∩ ∞
t 2
|u(t)|2V ≤ |u0|2V +2 |f(s)|2Hds e−λ1t+54 |u0|2H+R0t|f(s)|2V′ds . (3.5)
(cid:16) Z0 (cid:17) (cid:0) (cid:1)
so that, in particular,
2
eλ1t u 2 u 2 +2 f 2 e54 |u0|2H+|f|2L2(0,T;V′) . (3.6)
| |C([0,T],V) ≤ | 0|V | |L2(0,T;H)
(cid:0) (cid:1)
(cid:16) (cid:17)
3
It is known, see for instance [42, Lemma III.1.2] that these two properties of u imply that there exists a
unique u¯ C([0,T],H). When we write u(0) later we mean u¯(0).
4 ∈
Please note that [0,T] [0, ) is equal to [0,T] if T < and to [0, ) if T = .
∩ ∞ ∞ ∞ ∞