A stability estimate for robin boundary coefficients in stokes fluid flows
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Vietnam Journal
of Agricultural
Sciences
ISSN 2588-1299
VJAS 2018; 1(4): 289-304
/>
A Stability Estimate for Robin Boundary
Coefficients in Stokes Fluid Flows
Phan Quang Sang and Nguyen Thuy Dung
Faculty of Information Technology, Vietnam National University of Agriculture, Hanoi
131000, Vietnam
Abstract
In this report, we examine the unsteady Stokes equations with nonhomogeneous boundary conditions. As an application of a Carleman
estimate, we first establish log type stabilities for the solution of the
equations from either an interior measurement of the velocity, or a
boundary observation depending on the trace of the velocity and of
the Cauchy stress tensor measurements on a part of the boundary.
We then consider the inverse problem of determining the timeindependent Robin coefficient from a measurement of the solution
and of Cauchy data on a sub-boundary.
Keywords
Inverse problems, Carleman inequality, Stokes equation, Stability
estimate
AMS Classification: 35R30, 76D07
Introduction
Received: May 28, 2018
Accepted: September 19, 2018
Correspondence to
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/>
The Stokes equations are famous equations that describe
incompressible fluid flows where the advective inertial forces are
small compared with the viscous forces (also called creeping flow).
Such a flow is characterized by the property by which the fluid
velocities are very slow, while the viscosities are very large, or the
length-scales of the flow are very small.
The Stokes equations can be applied to many situations
occurring in nature, in technology, and in the modeling of biological
problems, for examples, the swimming flow of microorganisms, the
flow of lava, the motion of paint, or the flow viscous polymers
generally (Dusenbery, 2011), blood flow in the cardiovascular
system (Vignon-Clementel et al., 2006), and airflow in the lungs
(Baffico et al., 2010).
In this paper we consider the unsteady Stokes equations which
can be modeled as following. Let be a bounded open nonempty
subset of N ( N 2 or N 3) .
289
A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows
Q (0,T) x
T 0 , we denote
For some
and
consider
a
velocity-pressure pair
(v, p) L (0,T;H ()) H (0,T; L ()) L (0,T;H ()) solution to the following unsteady Stokes
equations:
2
2
1
2
2
1
vt v p f
div(v) d
v(0, x) v0 ( x),
in
Q,
in
for
Q,
x ,
(1)
where f L2 (Q) is an applied body force and d L2 (0,T;H1 ()) .
We notice that the existence of the solution of the Stokes equation (1) is not guaranteed in
general. However, it is guaranteed in certain Sobolev spaces under specific conditions; see for
example an inf-sup condition (Bramble, 2003; Necas, 2012). In this paper, we will not go into this
issue but will focus on the stability of the solution and on an inverse problem of determining a
friction boundary coefficient.
Moreover, we need an additional observation to ensure the uniqueness of the solution. There are
two main ways of giving such an observation: it is given either by the value of the velocity v in an
(arbitrary small) open nonempty subset , or by the Cauchy data (v, (v, p)n) on a part of the
boundary. That is, either
v(t , x) vobs (t , x) in Q : (0,T) ,
(2)
v gD
or
(v, p)n g N
on
(0,T) obs
on
(0,T) obs
(3)
Here, n is the outward unit normal to which is assumed to be of class C 2 , and the stress
def
tensor is defined by (v, p) D(v) pI , where is a constant which represents the kinematic
def
1
viscosity of the fluid we consider, D(v) (v t v) is the symmetrized gradient, and I is the
2
identity matrix. The uniqueness of the corresponding pair (v, p) is guaranteed by a unique
continuation result for the Stokes equations proven in Fabre and Lebeau (1996).
We show in this work the main following results. The first result is an estimate of the solution
with respect to the initial data. Then, the second result is the global stability of the solution when we
locally change the initial local data. The last result is the stability of a boundary coefficient, called
the Robin coefficient, on the unobservable part of the initial data when we change the local data.
The method that we use in this work is based on the construction of an appropriate Carleman
estimate for the unsteady Stokes Eq. (1). This method is widely used in many works, including
Boulakia et al. (2013) and Badra et al. (2016). However, these works were for steady Stokes
equations, or for two dimensions N 2. The results of this paper are presented for Stokes equations
with time, and in three dimensions N 3 .
In the following and throughout this work, C 0 denotes a generic constant which, unless
otherwise stated, only depends on the geometry of and may change from line to line.
Theorem 1.1. Assume that
(v, p) L2 (0,T;H2 ()) H 1 (0,T; L2 ()) L2 (0,T;H1 ()) is
solution of the Stokes Eq .(1) such that v
L2 (0,T;H2 ( ))
p
L2 (0,T;H1 ( ))
the
M for some M 0 . Then there
exists a constant C 0 such that we have the following estimates:
290
Vietnam Journal of Agricultural Sciences
Phan Quang Sang and Nguyen Thuy Dung (2018)
v
L2 ( Q )
and
C
v
M
ln 1
f
L2 ( Q )
C
M
d
L2 Q
L2 0,T ;H1
v L (Q )
2
,
(4)
M
ln 1
f
M
L2 Q
d
v
L2 Q
L2 (0,T ; H
3
2 (
obs ))
(v , p ) n
L2 (0,T ; H
1
2 (
obs ))
.
(5)
Moreover,
curl (v)
(L2 ( Q )) 2 N 3
p div (v )
L2 ( Q )
M
C
ln 1
f
M
L2 Q
d
L2 Q
v
L2 (0,T ; H
3
2 (
obs ))
(v , p ) n
L2 (0,T ; H
1
2 (
obs ))
1
2
.
(6)
We notice that the (4), (5), and (6) estimates will be further proven by Theorem 3.1.2 and
Theorem 3.2.3.
As an application of the above theorem, we can obtain the stability estimate for the Stokes
equations (1).
Assume that (vi , pi ) L2 (0,T;H2 ()) H 1 (0,T; L2 ()) L2 (0,T;H1 ()), i 1, 2, resulting in two
solutions for Eq. (1) associated to one of two types of observations:
either
(7)
vi (t , x) vi obs (t , x), i 1, 2,
in (0,T) ,
or
vi g i D , i 1,2, on
i
(vi , pi ) n g N , i 1,2, on
(0,T) obs
(0,T) obs
.
(8)
Then we have the following result (proven with equations 43, 44, and 45):
Theorem 1.2. Assume that (vi , pi ) L2 (0,T;H2 ()) H 1 (0,T; L2 ()) L2 (0,T;H1 ()), i 1, 2,
There are two solutions for Eq. (1) associated with one of two additional observations given by
(7) or (8). Moreover, we suppose that v1 v2 L2 (0,T;H2 ()) p1 p2 L2 (0,T;H1 ( )) M for some M 0 .
Then there exists a constant C 0 such that we have the following estimates:
v1 v2
C
L2 ( Q )
M
M
ln 1
v1 v2 L2 (Q )
,
(9)
and
v1 v2
L2 ( Q )
C
/>
M
ln 1
v1 v2
M
L2 (0,T ; H
3
2 (
obs ))
(v1 , p1 ) n (v2 , p2 ) n
L2 (0,T ; H
1
2 (
obs ))
.
(10)
291
A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows
Moreover, we have
curl (v1 v2 )
(L2 ( Q )) 2 N 3
p1 p2
L2 ( Q )
M
C
ln 1
v1 v2
1
M
L2 (0,T ; H
3
2 (
obs ))
(v1 , p1 ) n (v2 , p2 ) n
L2 (0,T ; H
1
2 (
obs ))
2
(11)
We notice that the results of this theorem lead to the uniqueness of the solution (v, p) of Eq. (1): if
v1 v2 in Q then v1 v2 in Q , or if the Cauchy data (v1 , (v1 , p1 ) n) (v2 , (v2 , p2 ) n) on (0,T) obs ,
then v1 v2 in Q. This matches the unique continuation result given in Fabre and Lebeau (1996).
Similar stability estimates were given for the Navier-Stokes equations, as in the paper by Badra
et al. (2016).
An important purpose of this article is to prove the stability in the determination of the Robin
boundary coefficient from the value of velocity v and the Cauchy data (v, (v, p)n) on a part of the
boundary. This kind of inverse problem is very significant in general in corrosion detection: the
determination of the Robin coefficient on the inaccessible portion of the boundary thanks to
electrostatic measurements performed on the accessible boundary part.
We assume that 0 is another open nonempty subset of boundary such that 0 obs .
We suppose that on 0 , corresponding to the previous pairs (v1 , p1 ), (v2 , p2 ) , the fluid has two
friction boundary coefficients given by the conditions
(vi , pi ) n i vi 0, i 1, 2.
(12)
The coefficients i in (12) are called the Robin coefficients. We have the following stability
estimate for the Robin coefficients (proven with equations 46, 47, 48, 49, and 50):
Theorem 1.3. Assume that (vi , pi ) L2 (0,T;H2 ()) H 1 (0,T; L2 ()) L2 (0,T;H1 ()), i 1, 2, are
two solutions of Eq. (1) associated with the additional observation given by (8). Let i , i 1, 2 be the
two Robin coefficients given by (12). Let be the set x 0 , v1 ( x) v2 ( x) 0 and we assume
that is a compact of 0 \ with a nonempty interior, and then let m 0 be a constant such that
max( v1 , v2 ) m on .
Moreover, we suppose that v1 v2
L2 (0,T;H2 ( ))
p1 p2
M for some M 0.
L2 (0,T;H1 ( ))
Then there exists a constant C 0 such that we have the following estimates
1 2
L2 (0,T )
C
m
M
ln 1
v1 v2
1
M
L2 (0,T ; H
3
2 (
obs ))
(v1 , p1 ) n (v2 , p2 ) n
L2 (0,T ; H
1
2 (
obs ))
4
(13)
There is a wide collection of mathematical works dealing with inverse boundary coefficient
problems. Most of them prove a logarithmic stability estimate for boundary coefficients in stationary
Stokes equations (Chaabane et al., 2004; Sincich, 2007; Bellassoued et al., 2008; Cheng et al., 2008)
292
Vietnam Journal of Agricultural Sciences
Phan Quang Sang and Nguyen Thuy Dung (2018)
or in two dimensions (Boulakia et al., 2013). The paper by Badra et al. (2016) presented the inverse
problem of the Robin coefficient for stationary Navier-Stokes equations. The paper by Boulakia et al.
(2013) gave stability estimates for the Robin coefficient but in the two dimensional Stokes equations.
Otherwise, the present inverse problem is for unsteady Stokes equations in two or three
dimensions. It improves upon several of the previously cited works and so it is new.
Notations Through this paper, is a nonempty bounded open subset of
N
for N 2 or N 3,
with a boundary of class C and is a nonempty open subset of .
For some T 0, we denote Q (0,T) x and Q (0,T) .
2
Let v be a vector field, v v1 , v2 ,..., vN , then we define:
the gradient of v is v x vi
the Laplacian of v is v vi i 1, N
j
1i , j N
,
N
j 1
2
v
x 2j i
,
i 1, N
N
the divergence of v is div v xi vi , and
i 1
the curl of v is the vector function is:
x v3 x v2
3
2
curl (v) x1 v2 x2 v1 if N 2 , or curl (v) x3 v1 x1 v3 if N 3.
x1 v2 x2 v1
Carleman Estimate for Unstaedy Stokes Equations
The main aim of this section is to prove a Carleman inequality for the non-homogeneous Stokes
equations. For that, we first prove a Carleman inequality for a velocity-pressure pair in
L2 0, T ;H 02 H10 (0,T;L2 ()) L2 0, T ;H10
and then we use a domain extension argument to
recover the non-homogeneous case.
For T 0 , we recall that Q (0, T ) x and Q (0, T ) x for an open nonempty subset .
Let : be a function satisfying
C 2 (; ), c0 and 0 in \ ,
(14)
c0 on
for some positive constant c0 0 . For the existence of such a function, see Tucsnak and Weiss
(2009), for instance.
Then we introduce the weight functions:
2
e ( x )
1
e ( x ) e
t, x
, ˆ t
, t, x
t T t
t T t
t T t
0 0 t ˆ t ec0 , 0 0 t ˆ t ec0 e
/>
2
C
C
,
(15)
.
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A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows
Carleman estimate in the case of homogeneous boundary data
Due to a result from Badra et al. (2016), we can easily get the following result.
Theorem 2.1.1. Let k 0,1 , F L2 (Q) , and G L2 (Q) , then there exists C 0, 1 , and s 1
such that for all and s s , the solution v L2 (Q) of
v F divG
v0
in
on
Q
,
Q
satisfies the following inequality: k 1 v s 2 k 1 v
2
e
2
2 s
d xdt
Q
C s 1/2 k/2 1 Fe s
2
2
s1/2 k/2 Ge s
L2 Q
L2 Q
s 2 k 1v 2 e2 s dxdt .
Q
We recall here a Carleman estimate for homogeneous Stokes equations cited from Imanuvilov
and Yamamoto (2003).
Theorem 2.1.2. Let F L2 (Q) and G L2 (Q) , then there exists C 0, 1 and s 1 such that
for all and s s , the solution v L2 (Q) of
vt v F divG in Q
v0
s
1
on Q
1 v s v
2
2
e
2 s
Q
, satisfies the following inequality:
d x d t C Fe s
2
L2 Q
Ge s
2
L2 Q
s v 2e 2 s dxdt ,
Q
(16)
where the constant C 0 is dependent continuously on , k and is independent of s .
Using the two previous theorems, we can get a Carleman estimate for the unsteady Stokes
equations with homogenous boundary data.
Theorem 2.1.3. There exists C 0, 1 and s 1 such that for all and s s , and for all
(v, p ) L2 0, T ;H 02 H10 0, T ;L2 L2 0, T ;H10 , the following inequalities hold:
v
2
s curl v s 2 2 v
2
2
e
2 s
d xdt
Q
2
C s 3 3 v e 2 s d x d t vt v p e s
Q
and
s div v p
2
Q
2
L2 (Q)
div v s
1/2
1/2 s 2
e
,
L2 (Q)
2
2
e 2 s dxdt C p v e 2 s dxdt s div v p e 2 s dxdt .
Q
Q
(17)
(18)
def
Proof: We set f vt v p . Easy calculations yield:
curl v t (curl v) curl f
in
Q,
(19)
v curl(curl v) (div v)
in
Q,
(20)
(div v p) div f
294
(21)
Vietnam Journal of Agricultural Sciences
Phan Quang Sang and Nguyen Thuy Dung (2018)
To get (18), we just apply Theorem 2.1.1 for k 0 to Eq. (21).
Now we introduce a relatively compact open subset 0 of and apply Theorem 2.1.2 (the
inequality (16)) for k 0 to Eq. (19) to obtain:
s 1 1 curl v e2 s d x d t s curl v e2 s d x d t
2
Q
2
Q
2
C s curl v e2 s d x d t (vt v p)e s
Q0
,
L2 (Q)
2
(22)
2
2
C s 1 2 e divv 2 p v e 2 se dxdt s 3 2 e3 dxdt .
Q
Q
(23)
In the last inequality, let us estimate the local term in curlv by a local term in v . For that, we
introduce the function C0 ( ) such that 0 1 and 1 in 0 . Using an integration by parts
in Q , we get
s curl v
2
e 2 s d x d t s
Q0
curl v
2
e 2 s d x d t s curl( e 2 s curlv)d x d t
Q
Q
C s 2 2 e2 s v curl v d x d t s e 2 s v curl v d x d t ,
Q
Q
and then with the Cauchy- Schwarz inequality:
s curl v
2
e 2 se d x d t
Q0
s
1 1 2 s
e
curl v s e2 s curl v
2
2
d x dt
Q
C
s 3 3e2 s v d x d t.
2
Q
By combining (22) with the above inequality for small enough values of 0 , we obtain
s
1 curl v e2 s d x d t s curl v e 2 s d x d t
2
1
2
Q
Q
2
C s 3 3 v e 2 s d x d t vt v p e s
Q
.
L2 (Q)
(24)
2
Finally, (17) is obtained by first applying (16) (for k 1 ) to Eq. (20) and then using the estimate
of curlv given by (24).
Carleman estimate in the case of non-homogeneous boundary data
In this section, we prove a Carleman inequality for the Stokes equations with non-homogenous
boundary data. We consider the equation:
vt v p f
div v d
in
Q,
in
Q.
(25)
We recall that C 0 denotes a generic constant depending only on the geometry of the boundary
and is independent of s.
/>
295
A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows
Theory 2.2.1. There exists C 0, 1 and sˆ 1 such that for all and s sˆ , every solution
(v, p) L2 0, T ;H 2 H1 0, T ;L2 L2 0, T ;H1 of (25) satisfies:
v
2
s curl v s 2 2 v
2
2
e
2 s
d xdt
Q
2
C s 3 3 v e 2 s d x d t e s f
Q
Ce2 s0 v
2
L2 (0,T;H 2 ( )
p
d s 1/2 1/2e s
2
L2 (Q)
2
L2 (0,T;H1 ( )
,
2
L2 (Q)
(26)
and
Q
2
s d p e 2 s dxdt C f
Q
Ce 2 s 0
v
2
p
L2 (0,T;H 2 ( )
2
e 2 s dxdt
2
L2 (0,T;H1 ( )
Proof: Let be a bounded domain of
2
s d p e 2 s dxdt
Q
.
(27)
( N 2 or N 3 ) of class C 2 such that is relatively
compact in . We denote Q (0, T ) . We extend to (while keeping the same name) in such a
way that:
C 2 (; ), 0 and
N
0 in \ ,
(28)
c0 on , 0 c0 in \ , c0 in ,
and we denote 0 0 t ˆ t ec0 e
Let
2
C
.
be a linear continuous map from L2 0, T ;H 2 H1 0, T ;L2 L2 0, T ;H1 into
E
such that
L2 0, T ;H02 H10 0, T ;L2 L2 0, T ;H10
E (v, p) (v, p) in Q (given by Stein’s
def
Theorem, see Adams (2003)), and we define v , p E v, p . Then the pair v, p E v, p is the
solution to the system:
vt v p f
div v d
v0
v
0
n
p0
in
Q,
in
Q,
in
Q,
in
Q,
in
Q ,
where f L2 Q and d L2 0, T ;H1 () are given by
f f
and d d in Q , and by
f vt v p and d div v in Q \ Q . From the continuity of the extension operator E , we have:
f
2
2
L Q \Q
d
2
L2 0,T ;H1 \
C v
2
L2 (0,T;H2 ( )
p
2
L2 (0,T;H1 ( )
.
(29)
Next, by applying estimate (17) of Theorem 2.1.3, we have:
296
Vietnam Journal of Agricultural Sciences
Phan Quang Sang and Nguyen Thuy Dung (2018)
v
2
s curl v s 2 2 v
2
2
e
2 s
d xdt
Q
2
C s 3 3 v e 2 s d x d t e s f
Q
2
L2 Q
d s
1/2
1/2 s
2
e
(30)
.
L2 ( Q )
Moreover, for s sˆ 1, applying the estimates (28) and (29), we have:
2
e s f
L2 Q \ Q
e 2 s f
2
d s 1/2 1/2 e s
2
L2 Q \ Q
d
L2 ( Q \ Q )
2
L2 0,T ;H1 \
Ce 2 s0 v
2
L2 (0,T;H 2 ( )
p
2
L2 (0,T;H1 ( )
.
(31)
Using (30) and (31), we have the proof for (26).
To prove (27), we apply (18) of Theorem 2.1.3 to v, p to get:
Q
2
2
2
s div v p e 2 s dxdt C f e 2 s dxdt s div v p e 2 s dxdt
Q
Q
2
2
C f e 2 s dxdt s div v p e 2 s dxdt Ce2 s0 v
Q
Q
2
L2 (0,T;H 2 ( )
p
2
L2 (0,T;H1 ( )
.
Stability Estimates for Unsteady Stokes Equations and The Inverse Problem of the
Robin Coefficient
In this section, we show stability estimates for unsteady Stokes equations corresponding to a
distributed observation or a boundary observation, which allow proving the main results announced
in Theorem 1.1 and Theorem 1.2. Then, we can apply them to the inverse problem of determining the
Robin boundary coefficient presented in Theorem 1.3.
Estimates for the solutions with a distributed observation
In this subsection, we use the Carleman inequalities given in Theorem 2.4 to obtain several
stability estimates with a distributed observation.
Theorem 3.1.1. There exists ˆ 1 and sˆ 1 such that all ˆ and all s sˆ , with large enough
c , result in every solution (v, p) L2 0, T ;H 2 H1 0, T ;L2 L2 0, T ;H1 of Eq. (25)
satisfying:
v
2
L (Q)
e se
c
f
L2 Q
d
v L (Q ) s v L (0,T;H
1
L2 0,T ;H1
2
2
2
( )
p
L2 (0,T;H1 ( )
.
(32)
Proof: Let be the function defined by (14). We define the following:
min min (t, x) c0 , and max max (t, x) ,
(t, x )Q
(t, x )Q
2
e max
e max e
1 1 t
,1 1 t
t T t
t T t
C
0 0 t ˆ t e c0 , 0 0 t ˆ t e c0 e
2
/>
,
(33)
C
.
297
A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows
We apply (26) to v, p to get
s curl v
s 2 2 v
2
2
e
2 s
d xdt
Q
2
C s 313 v e 2 s1 d x d t e s1 f
Q
Ce2 s 0
and then
s
curl v s 20 2 v
2
0
2
e
v
2 s 0
2
L2 (0,T;H 2 ( )
p
2
L2 Q
2
L2 (0,T;H1 ( )
d s 1/ 211/ 2 e s1
2
,
L2 Q
d xdt
Q
2
C s 313 v e 2 s1 d x d t e s1 f
Q
Ce2 s0 v
2
L2 (0,T;H 2 ( )
p
2
L2 Q
2
L2 (0,T;H1 ( )
d s 1/211/2 e s1
.
2
L2 Q
(34)
Then, by dividing inequality (34) by e 2s 0 and using (33), we obtain
s
curl v s 20 2 v
2
0
2
d x dt
Q
2 2 s
2 s
C s 313 v e 1 0 d x d t e 1 0 f
Q
2 s
e 1 0 d s 1/ 211/ 2
L Q
C v
2
L2 (0,T;H 2 ( )
p
2
L2 (0,T;H1 ( )
2
2
.
2
L2 Q
Thus, we have
Q
2 s
3 2 2 s
e 1 0
2
v d x d t C s 12 v e 1 0 d x d t
f
Q 0
s 202
C
2
2
2 2 v L2 (0,T;H 2 ( ) p L2 (0,T;H1 ( )
s 0
e 2 se
c
2
L Q
2
2 s
e 1 0 11
s02
d
f
2
L2 Q
d
2
L2 0,T ;H1
v
2
L2 Q
1
s2
v
2
L2 (0,T;H 2 ( )
p
2
L Q
2
2
L2 (0,T;H1 ( )
,
with large enough c (independent of ).
From the above theorem, we can show a logarithmic estimate for the solutions of the Stokes
equations that prove the inequality (4) of Theorem 1.1.
Theorem 3.1.2. There exists c 0 such that all ˆ 1 results in every solution
(v, p) L2 0, T ;H 2 H1 0, T ;L2 L2 0, T ;H1
of Eq. (25) such that v
L2 (0,T;H2 ( ))
p
L2 (0,T;H1 ( ))
M for some M 0 satisfying:
c
v
298
L2 (Q)
ee M
ln 1
f
M
d L2 0,T ;H1 v L2 Q
L2 Q
.
(35)
Vietnam Journal of Agricultural Sciences
Phan Quang Sang and Nguyen Thuy Dung (2018)
Proof: We introduce A f
L2 Q
then we can write (32) in the form: v
d
L2 0,T ;H1
2
L (Q )
e sC A
v
L2 Q
and apply Theorem 3.1 for s sˆ , and
def
c
M , where C e c .
s
First, if the case A 0 , since the previous inequality is true for all s sˆ , we obtain v
2
L2 (Q )
0 and
then (35) holds.
In the following, we assume A 0 .
1
1
M
M
We suppose that
ln(1 ) sˆ and choose s ln(1 ) . This yields
A
A
2C
2C
v
L2 ( Q )
1/2
M A
2C c
M 1
M
A M
ln 1
A
.
Next, using the fact that
1
1
x ln(1 x)
for 0 x 1, i.e M A, and
1
1
for x 1, i.e M A,
1/2
ln(1 x)
x
we obtain (35) (by choosing large enough values of c 0 ).
c
1
M
ln(1 ) sˆ , we have M ee A for some constant c 0 , and (32) with s sˆ
In the case
A
2C
gives v
c
L2 (Q)
ee A, for some constant c 0 . Then the conclusion follows from
AM
1
1
A
1
(since
, x 0 ).
M
M
M
x
ln(1
x
)
ln(1 )
A
Estimates for the solutions with a boundary observation
Theorem 3.2.1. There exists c 0 such that all ˆ 1 results in every solution
(v, p) L2 0, T ;H 2 H1 0, T ;L2 L2 0, T ;H1 of the Stokes equations (1) associated with
an additional observation given by (3) such that
v
L2 (0,T;H 2 ( )
p
L2 (0,T;H1 ( )
M
for some M 0
satisfies
sec
e
f L2 Q d L2 0,T ;H1 v L2 0,T;H3/2 obs v, p n L2 0,T;H1/2 obs
L (Q)
1
v L2 (0,T;H2 ( ) p L2 (0,T;H1 ( ) ,
s
and
v
2
curl (v)
(L2 ( Q ))2 N 3
p div (v )
L2 ( Q )
c
e se f L2 Q d L2 0,T ;H1 v
1
1/2 v L2 (0,T;H 2 ( ) p L2 (0,T;H1 ( ) .
s
/>
(36)
L2 0,T;H 3/2 obs
v, p n L 0,T;H
2
1/2
obs
(37)
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A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows
Let us begin by proving the following lemma, which is a construction of an extension of the domain
Q and of the solution (v, p ) of Problem (25). It is deduced from a result of Badra et al. (2016).
Lemma 3.2.2. Let be an extension of
obs . We also denote Q 0, T .
There
exists
an
of class C 2 through obs , namely is class C 2 ,
v, p L2 0, T ;H2 H1 0, T ;L2 L2 0, T ;H1 of
extension
v, p L2 0, T ;H 2 H1 0, T ;L2 L2 0, T ;H1 such that
v
v ,
obs
obs
v
v
,p
p ,
obs
n obs n obs obs
with the following estimate
v
2
2
H 1 0,T;H 2 \
p L2 0,T;H1 \
C v
In particular,
v
2
H1 (0,T;H 2 ( )
p
p L2 0,T;H1/2 .
obs
L2 0,T;H1/2 obs
2
L2 (0,T;H1 ( )
C v
(38)
2
v
L2 0,T;H 3/2 obs
n
2
2
2
L2 (0,T;H 2 ( )
p
2
L2 (0,T;H1 ( )
.
(39)
Proof of Theorem 3.2.1. Let v, p be an extension of v, p as in Lemma 3.2.2.
Let us consider \ a non-empty bounded open subset, and denote Q 0, T .
Applying (26) and (27) to v, p we get
s curl v
2
s 2 2 v s p div v
2
2
e
2 s
d x dt
Q
2
2
2
C
s 3 3 v s p div v e 2 s d x d t e s vt v p 2 div v s 1/2 1/2 e s
L Q
Q
Ce 2 s 0
v
2
L2 (0,T;H 2 ( )
p
2
L2 (0,T;H1 ( )
2
L2 Q
,
and then from the estimates (38) and (39) of Lemma 3.2.2 and \ we deduce
s curl v
2
s 2 2 v s p div v
2
2
e
2 s
d xdt
Q
2
2
2
C s 3 3 v s p div v e 2 s1 d x d t e s vt v p 2 div v s 1/2 1/2e s1
L Q
Q
Ce2 s0 v
2
L2 (0,T;H 2 ( )
p
C e s1 f
C13e 2 s1
2
L2 (0,T;H1 ( )
2
L Q
2
s
div v e s1
3
2
L2 Q
v vt v p div v s p div v
2
2
2
2
2
2
2
L2 Q
dxdt
Q \Q
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Vietnam Journal of Agricultural Sciences
Phan Quang Sang and Nguyen Thuy Dung (2018)
Ce 2 s0
v
2
L2 (0,T;H 2 ( )
C e s1 f
2
L Q
2
3 2 s1
1 e
Cs
3
Ce 2 s0 v
p
2
L2 (0,T;H1 ( )
div(v) e s1
2
L2 Q
2
v
v 2
L 0,T;H3/2 obs
n
2
L2 (0,T;H 2 ( )
p
2
L2 0,T;H1/2 obs
2
L2 (0,T;H1 ( )
,
p
2
2
1/2
L 0,T;H
obs
(40)
with 1 being defined by (33).
Thus, by dividing the inequality (40) by e 2s 0 and using that e2 s0 2 v e 2 s d x d t v d x d t ,
2
2
Q
Q
we have
2
v
L2 (Q)
2
2 s
2
e 1 0 s13 2
v
2
2
f
div
v
v
p
L2 Q
2
L2 0,T;H3/2 obs
L2 0,T;H1/2 obs
L Q
n L2 0,T;H1/2 obs
2
C
2
2
2 2 v L2 (0,T;H 2 ( ) p L2 (0,T;H1 ( ) .
s
We note that from the definition of v, p n , it is possible to replace the term
C
v
n
2
2
1/2
L 0,T;H
v
obs
2
2
L (Q)
1
2 v
s
p
2
L2 0,T;H1/2 obs
c
e2 se f
2
L2 (0,T;H 2 ( )
2
in the last inequality with v, p n
L2 0,T;H1/2 obs
, so we get
2
2
v
v
,
p
n
2
3/2
L 0,T ;H
L 0,T;H obs
L2 0,T;H1/2 obs
2
d
L Q
2
p
2
2
2
L2 (0,T;H1 ( )
1
.
(41)
With a similar argument as above, we can prove that
curl (v)
2
(L2 ( Q ))2 N 3
p div(v)
2
L2 ( Q )
c
2
2
2
2
e 2 se f L2 Q d L2 0,T ;H1 v L2 0,T;H3/2 v, p n 2
L 0,T;H1/2 obs
obs
1
2
2
v L2 (0,T;H 2 ( ) p L2 (0,T;H1 ( ) .
s
(42)
The estimates (41) and (42) directly lead to (36) and (37).
With the help of Theorem 3.2.1 and using similar arguments of the proof of Theorem 3.1.2, we
have the following results, which prove the inequalities (5) and (6) of Theorem 1.1.
Theo rem 3.2.3. Assume that (v, p) L2 (0,T;H2 ()) H 1 (0,T; L2 ()) L2 (0,T;H1 ()) is the
solution of the Stokes equation (1) associated with an additional observation given by (3) such that
v L2 (0,T;H2 ( ) p L2 (0,T;H1 ( ) M for some M 0 . Then there exists a constant C 0 such that we
have the following estimates:
/>
301
A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows
v
L2 ( Q )
C
M
ln 1
f
M
L2 Q
d
L2 Q
v
L2 (0,T ; H
3
2 (
obs ))
( v, p ) n
L2 (0,T ; H
1
2 (
obs ))
.
Moreover, we have
curl (v)
(L2 ( Q ))2 N 3
p div(v)
L2 ( Q )
M
C
ln 1
f
M
L2 Q
d
L2 Q
v
L2 (0,T ; H
3
2 (
obs ))
( v, p ) n
L2 (0,T ; H
1
2 (
obs ))
1
2
.
Stability of solutions: the proof of Theorem 1.2
This section focuses on the proof of Theorem 1.2, which shows the stability of the solutions of
the Stokes equations.
Let (v, p) (v1 v2 , p1 p2 ) , then it is the solution to
vt v p f
div(v) 0
v(0, x) v0 ( x)
in
in
for
Q
(43)
Q .
x
This equation is a particular form of Eq. (1). The additional observation for the pair (v, p) is:
v v1 v2 in
(0,T) , (44) or
v v1 v2
on (0,T) obs
.
(v, p) n (v1 , p1 ) n (v2 , p2 ) n on (0,T) obs
(45)
By applying Theorem 1.1 for the pair (v, p) , we directly get the estimates (9), (10), and (11).
Stability of the Robin coefficients: the proof of Theorem 1.3
In this section, we show the stability of the Robin coefficients in the Stokes equations with the
help of the stability of the solutions. We focus on the proof of Theorem 1.3.
Let (vi , pi ), i 1, 2, be two solutions of Eq. (1) associated with the additional observation given
by (8) and let i , i 1, 2 be the two Robin coefficients given by (12).
As in the proof of Theorem 1.2 shown above, let (v, p) (v1 v2 , p1 p2 ) , which is the solution to
Eq. (43) with the additional observation given by (45).
Without a loss of generality, we can assume that v1 m . The Robin coefficients given by (8)
then satisfy the relationship:
1 2 v1 2v (v, p) n 2v D(v) pI n on 0 .
Thus, we get the estimate
C
1 2 L2 (0,T )
v L2 (0,T ) v L2 (0,T ) p L2 (0,T ) .
0
0
0
m
(46)
Using an interpolation inequality borrowed from Badra et al. (2016) L2 () C L2 () H 1 () , we
can get
302
Vietnam Journal of Agricultural Sciences
Phan Quang Sang and Nguyen Thuy Dung (2018)
p
v
C p
L2 (0,T )0
L2 (0,T )0
C v
and then v
1
2
L (Q )
p
2
1
2
L2 ( Q )
L2 (0,T )0
v
1
2
L2 (0,T;H1 ( ))
1
2
L2 (0,T;H1 ( ))
C v
(47)
,
(48)
,
1
2
L2 (0,T;H1 ( ))
v
1
2
.
L2 (0,T;H2 ( ))
(49)
Combining (47), (48), and (49) with the interpolation inequality
H 1 ()
1
2
L2 ( )
1
2
,
the
H 2 ()
inequality (46) then leads to
1 2
L2 (0,T )
C 34
v
v 2
m L (Q )
1
C
v L24(Q ) v
m
C
v
m
1
4
L2 ( Q )
M
3
1
4
L2 (0,T;H 2 ( ))
v
1
4
L2 ( Q )
3
4
L2 (0,T;H 2 ( ))
p
1
2
L2 ( Q )
4
p
1
2
L2 ( Q )
M
1
2
v
p
3
4
L2 (0,T;H 2 ( ))
1
2
L2 (0,T;H1 ( ))
.
p
1
2
L2 ( Q )
p
1
2
L2 (0,T;H1 ( ))
(50)
Hence, the result (13) of Theorem 1.3 is proven by applying Theorem 1.2 (the inequalities (10) and
(11)) to the estimate v L2 (Q ) and p L2 (Q ) in the above inequality (50).
We notice from the result of Theorem 1.3 that if the boundary observations of v1 and v2 on the
observable boundary part are equal, then the corresponding Robin coefficients 1 and 2 are also
equal.
Discussion
The estimates (9), (10), and (11) (stated in
Theorem 1.2) show the stability of the velocity
and of the pair of solutions (v, p) with respect to
a distributed observation or a boundary
observation. These results extend the results of
Boulakia et al. (2013) and Badra et al. (2016)
by building an appropriate Carleman estimate
for the unsteady Stokes Eq. (1).
On the other hand, the uniqueness of the
solution (v, p) of Eq. (1) is guaranteed thanks to
the above results: if v1 v2 in Q then v1 v2 in
Q , or if the Cauchy data (v1 , (v1 , p1 ) n)
(v2 , (v2 , p2 ) n) on (0,T) obs , then v1 v2 in
Q. This matches the unique continuation result
given in Fabre and Lebeau (1996).
The estimate (13) shows the stability in the
determination of the Robin boundary coefficient
from the value of the velocity v and the Cauchy
data (v, (v, p)n) on a part of the boundary.
This means that the Robin coefficient can be
/>
determined from an unobservable boundary part
of the velocity and of the Cauchy data.
Many mathematical works dealing with
boundary coefficient problems have previously
been published (Chaabane et al., 2004; Sincich,
2007; Bellassoued et al., 2008; Cheng et al.,
2008; Boulakia et al., 2013). However, the
present work is for unsteady Stokes equations in
two or three dimensions.
Conclusions
The present work treats, for the first time,
the inverse problem of determining the timeindependent Robin coefficient in unsteady
Stokes equations (in 2 or 3 dimensions) with
non-homogeneous boundary conditions.
The stability estimates of a log type are
established for the solution of the equations
from either an interior measurement of the
velocity, or a boundary observation depending
on the trace of the velocity and of the Cauchy
stress tensor measurements on a part of the
boundary.
303
A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows
A stability estimate for the Robin
coefficient is then established from a
measurement of the solution and of the Cauchy
data on a sub-boundary. It is very significant in
the determination of the Robin coefficient on
the inaccessible portion of the boundary thanks
to electrostatic measurements performed on the
accessible boundary part.
Acknowledgements
We would like to thank the Vietnam
National University of Agriculture and the
Faculty of Information Technology for their
support, which helped us to complete this work.
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Vietnam Journal of Agricultural Sciences
