Nonlinear Analysis 64 (2006) 1018 – 1024
www.elsevier.com/locate/na
Optimal stopping-time problem for stochastic
Navier–Stokes equations and infinite-dimensional
variational inequalities
V. Barbua , S.S. Sritharanb,∗
a Department of Mathematics, University of Iasi, Iasi 6600, Romania
b Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA
Received 29 April 2005; accepted 12 May 2005
Abstract
We prove the existence of a solution for the obstacle problem associated with the Kolmogorov
operator corresponding to the stopping-time problem for stochastic Navier–Stokes equations in 2-D.
䉷 2005 Elsevier Ltd. All rights reserved.
Keywords: Optimal stopping; Stochastic Navier–Stokes equations; Variational inequality
1. Introduction
Optimal stopping problems have well-known applications in stochastic analysis, control
theory and finance. This subject has stimulated lot of interest in analysis because of its
connection to free-boundary problems (obstacle problems) for partial differential equations.
In infinite dimensional case very limited literature is available (see for example [13,7,15]).
In [12] a semigroup formulation of the optimal stopping and impulse control problems was
studied for the stochastic Navier–Stokes equation. Motivation for impulse and stopping
time problems for stochastic fluid dynamics come from control of turbulence intermittency
and sequential initialization methods in geophysical weather prediction. In this paper we
will study the optimal stopping problem for 2-D stochastic fluid dynamics using methods
∗ Corresponding author. Tel.: +1 307 766 4221.
E-mail address: [email protected] (S.S. Sritharan).
0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2005.05.054
V. Barbu, S.S. Sritharan / Nonlinear Analysis 64 (2006) 1018 – 1024
1019
from infinite dimensional analysis and give a sharp existence result for the corresponding
infinite dimensional variational inequality.
Consider the stopping-time problem for the stochastic Navier–Stokes equations
F (s, X(s)) ds + EG(X()) ,
(1.1)
(t, x) := inf E
t
dX(r) + (AX(r) + BX(r)) dr =
Q dW (r),
r t,
X(t) = x,
(1.2)
where F : (0, ∞) × H → R, G : H → R are given functions and
D(A) = V ∩ (H 2 (D))2 ,
A = −P ,
V = (H01 (D))2 ∩ H ,
H = {x ∈ (L (D)) ; ∇ · x = 0, x · n = 0 on jD}.
2
2
Here, P is the Leray projection on H and B is the nonlinear inertia term on V defined by
xi Di xj yj d ∀x, y ∈ V ,
(Bx, y) =
where (·, ·) is the scalar product on H and the pairing between V and V and · is the norm
of V. By | · | we shall denote the norm of H. We note that x2 = |A1/2 x|2 . In stochastic
equation (1.2), Q ∈ L(H ; H ) is a nonnegative symmetric operator with Tr Q < ∞ and W
is a cylindrical Wiener process on H associated with a stochastic basis {, F, P , {Ft }}
with values in H. Finally, D is an open, bounded and smooth domain of R 2 .
It turns out that the value function defined by (1.1) (after a suitable change of time
variable) is formally the solution to the variational inequality (for finite dimensional case
see [5,6,12])
j
(t, x) − 21 Tr[QD 2x (t, x)] + (Ax + Bx, Dx (t, x))
jt
F (t, x) ∀t 0, x ∈ D(A), (t, x) G(x) ∀t 0, x ∈ H ,
(1.3)
j
(t, x) − 21 Tr[QD 2x (t, x)] + (Ax + Bx, Dx (t, x)) = F (t, x)
jt
in {x; (t, x) < G(x)}, (0, x) = 0 (x), x ∈ H .
Here we shall prove an existence and uniqueness result for problem (1.3) viewed as a
nonlinear equation of the form
d(t)
− N(t) + NK (t) F (t),
dt
t ∈ [0, T ), (0) = 0 ,
(1.4)
where N is the infinitesimal generator (the Kolmogorov operator) of the transition semigroup
P (t) associated with the stochastic differential equation (1.2) and NK is the normal cone
to convex, closed subset K ⊂ L2 (H ; ),
K = { ∈ L2 (H ; ); G on H },
(1.5)
where is an invariant measure for P (t). The main result (Theorem 1 below) relies on some
recent results in [4] (see also [3]) on characterization of operator N.
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V. Barbu, S.S. Sritharan / Nonlinear Analysis 64 (2006) 1018 – 1024
The following notations will also be used. Cb (H ) is the space of all uniformly continuous
and bounded functions : H → R and Cbk (H ) of C k functions whose differentials up to
order k are also uniformly bounded and continuous. H01 (D) and H 2 (D) are usual Sobolov
spaces on D ⊂ R 2 and EA (H ) is the linear span of all functions (·) = ei(h,·) , h ∈ D(A).
By D k we shall denote the diferential of order k.
2. The main result
We note first that for each x ∈ H the stochastic differential (1.2) has a unique solution
X ∈ L2 (; C([0, T ]; H ) ∩ L2 (0, T ; V )) [11,8]. If we denote by P (t) : Cb (H ) → Cb (H )
the transition semigroup
(P (t))(x) = E(X(t, x)),
x ∈ H, t 0, ∈ Cb (H ),
where X=X(t, x) is the solution to (1.2), then it follows by the Krylov–Bogoliubov theorem
(see e.g. [4,10]) that there is an invariant measure for P (t), i.e.,
(P (t))(x)(dx) =
(x)(dx) ∀ ∈ Cb (H ).
H
H
Moreover, is unique if is sufficiently large, and support ⊂ V . More precisely, one has
(see [4])
2
x2 e|x| (dx)C for 0 < < 0 .
H
Then P (t) has an extension to a C0 -contraction semigroup on L2 (H, ).
Denote by N : D(N) ⊂ L2 (H ; ) → L2 (H ; ) the infinitesimal generator of P (t) and
let N0 ⊂ N be defined by
(N0 )(x) =
1
2
Tr[QD 2x (x)] − (Ax + Bx, Dx (x))
∀ ∈ EA (H ).
We recall the following result established in [4] (see also [3] for the periodic case).
If 0 (QL(H,H ) + Tr Q + Tr[A
Q]) is sufficiently large and if
Tr[A
Q] < ∞
for > 23 ,
(2.1)
then N0 is dissipative in L2 (H ; ) and its closure N̄0 in L2 (H ; ) coincides with N. Moreover, one has the “Carre du Champ’s” identity
1
[N](x)(x)(dx) = −
| Q Dx (x)|2 (dx) ∀ ∈ D(N ),
(2.2)
2 H
H
The operator N arising in the variational inequality (1.4) is just the operator N defined
2
above. The normal cone NK : L2 (H, ) → 2L (H,) is defined by
( − )(dx) 0, ∀ ∈ K , ∈ K.
(2.3)
NK () = ∈ L2 (H, );
H
Now we are ready to formulate the main existence result for Eq. (1.4)
V. Barbu, S.S. Sritharan / Nonlinear Analysis 64 (2006) 1018 – 1024
1021
Theorem 1. Assume that is sufficiently large and that condition (2.1) holds. Suppose
further that G ∈ L2 (H ; ) and
(P (t)G)(x) G(x) ∀t 0, x ∈ H .
(2.4)
Then for each 0 ∈ D(N) ∩ K and F ∈ W 1,1 ([0, T ]; L2 (H ; )) there is a unique function
∈ W 1,∞ ([0, T ]; L2 (H ; )) such that N ∈ L∞ (0, T ; L2 (H ; )) and
d
(t) − N (t) + (t) = F (t) a.e. t ∈ (0, T ),
dt
(t) ∈ NK ((t)) a.e. t ∈ (0, T ),
(2.5)
(2.6)
(0) = 0 .
(2.7)
Moreover, : [0, T ] → L2 (H ; ) is differentiable from the right and
d+
(t) = F (t) + N (t) − PNK ((t)) (F (t) + N (t))
dt
where PNK () is the projection on the cone NK ().
∀t ∈ [0, T ),
(2.8)
Here W 1,p ([0, T ]; L2 (H ; )), 1 p ∞ is the space of all absolutely continuous functions
d
: [0, T ] → L2 (H ; ) such that
∈ Lp (0, T ; L2 (H ; )).
dt
It must be emphasized that a solution to (2.5) and (2.6) can be indeed viewed as
a solution to variational inequality (1.3). Indeed, let us assume that G, ∈ Cb (H ); by
definition of NK () we have (here we simply write instead of (t, ·))
(2.9)
(x)(x)(dx)0 ∀ ∈ L2 (H ; ), 0 ∀ ∈ NK ()
H
and
H
(x)(x)(dx) = 0,
(2.10)
for all ∈ Cb (H ), support ⊂ {x ∈ H ; (x) < G(x)} = H0 .
By (2.9) it follows that
|− (x)|2 (dx)0,
(2.11)
while (2.10) yields
|(x)|2 (dx) = 0.
(2.12)
H
H0
Taking into account that the transition semigroup P (t) is irreducible (see [8,9]) it follows
that the measure is full, i.e., (Br ) > 0 for any ball Br = {x ∈ H ; |x − x0 | < r}, and so
(2.11) and (2.12) imply that
(x)0
∀x ∈ H,
(x) = 0
∀x ∈ H0
(2.13)
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V. Barbu, S.S. Sritharan / Nonlinear Analysis 64 (2006) 1018 – 1024
as desired. However, if (t, ·), G are in L2 (H ; ) (integrable only) then the second equation
in (1.3) holds in some generalized sense.
By Theorem 1 and formula (2.2) we infer also that:
Corollary 1. Under assumptions of Theorem 1, the solution to (2.5)–(2.7) satisfies also
Q Dx ∈ L∞ (0, T ; L2 (H ; )).
(2.14)
√
√
∞
Here Q Dx (x) = k=1 Dk (x) k ek , a.e. in H , where Qek = k ek .
We note that if G ∈ Cb2 (H ) then by the Ito’s formula for stochastic Navier–Stokes
equation [14],
dG(X(t, x)) = (Dx G(X(t, x)), dX(t, x)) + 21 Tr[QD 2x G(X(t, x))] dt
= − (AX(t, x) + BX(t, x), Dx G(X(t, x))) dt
+ 21 Tr[QD 2x G(X(t, x))] dt +
Q dW, Dx G(X(t, x)) .
and so
t 1
Tr[QD 2x G(X(s, x))] − (AX(s, x)
2
0
+ BX(s, x), Dx G(X(s, x))) ds .
(P (t)G)(x) − G(x) = E
Hence condition (2.4) is implied in this case by the following one:
1
2
Tr[QD 2x G(x)] − (Ax + Bx, Dx G(x))0
∀x ∈ D(A).
(2.15)
More generally, condition (2.4) holds if G ∈ D(N) and
NG 0
on H ,
which means that G is super-harmonic with respect to N.
3. Proof of Theorem 1
We shall prove that under assumptions of Theorem 1 the multi-valued operator
A = −N + NK () ∀ ∈ D(A),
D(A) = D(N) ∩ K
(3.1)
L2 (H ; );
is m-accretive in
i.e.,
¯
¯
(A − A)(
− )(dx)0
H
¯ ∈ D(A)
∀, (3.2)
and
R(I + A) = L2 (H ; )
∀ > 0,
where R stands for the range of the operator and I is the identity operator.
(3.3)
V. Barbu, S.S. Sritharan / Nonlinear Analysis 64 (2006) 1018 – 1024
1023
Since (3.2) is obvious by (2.2) and the definition of NK we shall confine to prove (3.3).
To this end it suffices to show that
(I − N)−1 K ⊂ K
∀ > 0
(3.4)
(see e.g. [1,2,6], in particular, [1, Chapter IV, Theorem 1.8]). Let g ∈ K be arbitrary but
fixed. We note that by resolvent representation formula,
1 ∞ −t/
e
P (t)g(x) dt
(I − N)−1 g(x) =
0
∞
1
e−t/ g(X(t, x)) dt ∀x ∈ H ,
= E
0
where X = X(t, x) is the solution to the stochastic (1.2). Hence,
∞
1
(I − N)−1 g(x) E
e−t/ G(X(t, x)) dt ∀ > 0.
0
(3.5)
Then by assumption (2.4) we obtain that
EG(X(t, x))G(x)
∀t ∈ [0, T ], x ∈ H ,
and so (3.5) yields
(I − N)−1 g(x)G(x) ∀x ∈ H, > 0
and (3.4) follows. Eq. (3.4) also implies that (see [1,2,6])
NL2 (H ;) A0 L2 (H ;)
∀ ∈ D(A),
where A0 is the minimal section of the operator A, i.e.,
A0 L2 (H ;) = inf{L2 (H ;) ;
∈ A}.
Then to obtain the conclusions of Theorem 1 it suffices to apply the standard existence
theorem for nonlinear Cauchy problems in Hilbert spaces of accretive type (see e.g. [1,2,6],
in particular [1, Theorem 1.6, Chapter III] would be adequate).
Remark 1. The specific examples of F and G are of the form
F (t, x) ≡ x2 ,
G(x) = k(|x|2 )|x|2 ,
where k ∈ C 2 (R + ) is such that
0 k(r)r,
k 0, k > 0, k(0) = 0.
Then a simple calculation shows that condition (2.15) holds if is large enough and is
sufficiently small.
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V. Barbu, S.S. Sritharan / Nonlinear Analysis 64 (2006) 1018 – 1024
We also note that (for a simply connected domain D ⊂ R 2 ) F is then the enstrophy (total
vorticity) in the flow field since x2 = |curl(x)|2 , ∀x ∈ V . Moreover, x ∈ L2 (H ; ).
Indeed as noted earlier, from the definition of invariant measure,
(N)(x)(dx) = 0 where (x) = |x|2 .
H
This gives,
2
x2 (dx) = Tr(Q),
H
which implies that the enstropy is integrable with respect to the invariant measure.
Acknowledgements
The work of S.S. Sritharan is supported by the Army Research Office, Probability and
Statistics Program.
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