Technische Universit¨at Berlin
Institut f¨ur Mathematik
Regularity of the adjoint state for the
instationary Navier-Stokes equations
Arnd R¨osch, Daniel Wachsmuth
Preprint 10-2004
Preprint-Reihe des Instituts f¨ur Mathematik
Technische Universit¨at Berlin
Report 10-2004 April 2004
Adjoint state regularity 1
Regularity of the adjoint state for the instationary
Navier-Stokes equations 1
Arnd R¨
osch2,Daniel Wachsmuth3
Abstract. In this article, we are considering imbeddings of abstract functions in spaces of func-
tions being continuous in time. A family of functions depending on certain parameters is discussed in
detail. In particular, this example shows that such functions do not belong to the space C([0, T ], H).
In the second part, we investigate an optimal control problem for the instationary Navier-Stokes
equation. We will answer the question, in which sense the initial value problem for the adjoint
equation can be solved.
Key words. Vector-valued functions, imbeddings, optimal control, Navier-Stokes equations,
regularity of adjoints
AMS subject classifications. Primary 46E40, Secondary 49N60
1. Introduction. In this paper, we will study the regularity of abstract func-
tions. The discussed properties are heavily connected to the optimal control of insta-
tionary Navier-Stokes equations. Here, the gradient of a given objective functional is
evaluated by means of an adjoint state. The adjoint state is itself the solution of an
evolution equation. The discussion of abstract functions in the first part of the paper
will reflect important properties of the adjoint state.
The aim of the present article is two-folded. At first, we want to shed light on
imbeddings of abstract functions in spaces of continuous functions. We have to refer
to the mostly classical results due to Lions, [4]. Given a Gelfand triple V ,→H ,→V0,
the space
W(0, T) = y∈L2(0, T;V) : d
dty∈L2(0, T;V0)
is continuously imbedded in C([0, T ], H). In a recent research paper, the question
of compact imbeddings is considered, [2]. However, to the knowledge of the authors
there are no further results in the literature generalizing the result of Lions substan-
tially except the following one in the book of Dautray and Lions [4, XVIII.3.5, p.
521]. They wrote, that it suffices to require d
dt y∈L1(0, T;V0) to get the continuity
y∈C([0, T], H).
Since the adjoint state of the Navier-Stokes equation does not belong to W(0, T ) in
general, this result in [4] is used in different papers concerning the optimal control of
the instationary Navier-Stokes equation, see [8, 10, 11, 14].
In this paper, we will show that this more general imbedding result cannot be true.
To this aim we discuss a family of functions depending on certain parameters in detail.
Nevertheless, the authors want to point out that in our opinion the incorrectness of
the imbedding result in [4] does not influence the main results in the mentioned papers
[8, 10, 11, 14].
1This work was partially supported by DFG SFB 557 ”Control of complex turbulent shear flows”
at TU Berlin.
2Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian
Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria.
3Institut f¨ur Mathematik, Technische Universit¨at Berlin, Str. des 17. Juni 136, D-10623 Berlin,
Germany.
2Arnd R¨
osch, Daniel Wachsmuth
These examples shows also that the result of Amann [2] is really sharp.
The second part of the article deals with the adjoint state connected with an optimal
control problem for the instationary Navier-Stokes equations. For certain regularity
of the data it belongs to the space
W(2,4/3; V, V 0) := y∈L2(0, T;V) : d
dty∈L4/3(0, T;V0).
As already mentioned, one cannot expect that this space is imbedded in C([0, T ], H).
Naturally, there arises the question: is there an imbedding of W(2,4/3; V, V 0) in
C([0, T], X), where Xis a space of weaker topology than H?
The article is organized as follows: In Section 2, we construct families of functions and
study regularity properties. In the second part, Section 3, we give a brief overview
of the theory of optimal control for instationary Navier-Stokes equations. Finally,
we present a regularity result of the adjoint state. A last example shows that this
regularity cannot be improved by imbedding arguments.
2. Counterexamples. Here, we will deal with imbeddings of abstract functions
in spaces of continuous functions. At first, we state the most classical result in this
field. Let V ,→H ,→V0be a Gelfand triple.
Theorem 2.1. The space
W(p, q;V, V 0) := y∈Lp(0, T;V) : d
dty∈Lq(0, T;V0)
is continuously imbedded in C([0, T], H)if 1/p + 1/q ≤1.
For the proof in the case p=q= 2 we refer to [4]. It can be easily adapted to the
case 1/p + 1/q = 1, cf. [5].
In the sequel, we will construct several functions which are in W(p, q;V, V 0), where
p, q do not meet the assumptions of the previous theorem. We prove that in the case
1/p+1/q > 1 there is no imbedding W(p, q;V, V 0),→C([0, T ], H). We are also looking
for a positive result of the kind: for given p, q the space W(p, q;V, V 0) is continuously
imbedded in C([0, T ], X), where Xis a space with weaker topology than H.
Consider the following example: Let Ω = [0,1] and T > 0. Set V:= H1
0(Ω), H=
L2(Ω), and V0induced by the H-scalar product such that V ,→H ,→V0forms a
Gelfand-triple. Define a function fα,k over Ω ×[0, T] by
fα,k(x, t) =
∞
X
n=1
n−1/2e(−nαt)sin nkπx, (2.1)
where kis a natural number.
Lemma 2.2. The function fα,k given by (2.1) has the following properties:
(i) fα,k 6∈ C([0, T]; H)for α > 0,
(ii) fα,k ∈Lp(0, T;V)for p < α
k+1/2,
(iii) d
dt fα,k ∈Lq(0, T;V0)for q < α
α+1/2−k.
Proof. Set vn(x) := sin nkπx. At first, observe that the functions vnare orthogonal
with respect to the Has well as to the V-scalar product. It holds
|vn|H=1
√2and |vn|V=1
√2nkπ.
Adjoint state regularity 3
Now, we want to derive the V0-norm of vn. Let φ∈Vbe a test function. After partial
integration, we find using the Cauchy-Schwarz inequality
hvn, φiV0,V =Z1
0
sin(nkπx)φ(x)dx =1
nkπZ1
0
cos(nkπx)φ0(x)dx ≤1
√2nkπ|φ|V.
This allows us to conclude
|vn|V0≤1
√2nkπ.
Setting φ(x) := vn(x), we obtain
|vn|V0=1
√2πn−k.
(i) Let fN
α,K be the function defined by the finite series
fN
α,k(x, t) =
N
X
n=1
n−1/2e(−nαt)sin nkπx.
For t= 0, we obtain
fN
α,k(x, 0) =
N
X
n=1
n−1/2sin nkπx.
Consequently, we find for the L2-norm
kfN
α,k(·,0)k2
H=
N
X
n=1
1
nZT
0
sin2nkπx dx =
N
X
n=1
1
2n.
This series grows unboundedly for N→ ∞. Therefore, fα,k(0) cannot belong to H,
which implies that fα,k is not in C([0, T]; H).
(ii) Again, we consider the finite series fN
α,k. We want to estimate the Lp(0, T;V)-
norm of fN
α,k. We derive first using H¨olders inequality
kfN
α,kkLp(V)= ZT
0 N
X
n=1
n−1/2e(−nαt)|vn|V!p
dt!1/p
≤
N
X
n=1 ZT
0n−1/2e(−nαt)|vn|Vpdt!1/p
.
The integral on the right-hand side can be computed by
ZT
0n−1/2e(−nαt)|vn|Vpdt =ZT
0π
√2p
np(k−1/2)e(−pnαt)dt
≤1
pπ
√2p
np(k−1/2)−α.
4Arnd R¨
osch, Daniel Wachsmuth
Hence, we arrive at the estimate
kfN
α,kkLp(V)≤π
√2p
√p
N
X
n=1
nk−1/2−α/p.
This series will be finite for N→ ∞, if
k−1/2−α/p < −1
or equivalently
p < α
k+ 1/2
holds.
(iii) Similarly, the Lq(0, T ;V0)-estimate can be proven. We shall begin with
d
dtfN
α,k
Lq(V0)
= ZT
0 N
X
n=1
nα−1/2e(−nαt)|vn|V0!q
dt!1/q
≤
N
X
n=1 ZT
0nα−1/2e(−nαt)|vn|V0qdt!1/q
.
We find for the time integral
ZT
0nα−1/2e(−nαt)|vn|V0qdt =1
√2πq
nq(α−k−1/2) ZT
0
e(−qnαt)dt
≤1
q1
√2πq
nq(α−k−1/2)−α.
This implies
d
dtfN
α,k
Lq(V0)≤1
√2q
√q π
N
X
n=1
nα−k−1/2−α/q.
The series on the right hand side is uniformly bounded for
α−1/2−k−α/q < −1,
which is equivalent to
q < α
α+ 1/2−k,
and completes the proof.
Remark 2.3. For α=p(k+ 1/2) + εwith some fixed ε > 0, we find that (i) and (ii)
are automatically fulfilled. Moreover, we obtain from (iii)
q < pk +p/2 + ε
(p−1)k+p/2 + ε+ 1/2.
Adjoint state regularity 5
If kis sufficiently large, the value of qis arbitrary close to
p0=p
p−1=1
1−1
p
.
This shows, that the proposition of Theorem 2.1 is sharp.
Further, we can conclude that there is no imbedding of W(2,1; V, V 0)in C([0, T], H)
as stated in [4, p. 521].
In the following, we will denote by Hs(Ω) for −1≤s≤0 the Sobolev-Slobodeckij
spaces of fractional order. We have H−1=V0and H0=Hwith the notation already
introduced.
Corollary 2.4. Let us consider the function
fα,k,l(x, t) =
∞
X
n=1
n−le(−nαt)sin nkπx.
This function satisfies
(i) fα,k,l 6∈ C([0, T ]; H)for α > 0and l≤1
2,
(ii) fα,k,l ∈Lp(0, T ;V)for p < α
k−l+1 ,
(iii) d
dt fα,k,l ∈Lq(0, T;V0)for q < α
α−k−l+1 ,
(iv) fα,k,l ∈C([0, T]; H−s)for s > 1−l
kand l < 1
2,
(v) fα,k,l 6∈ C([0, T]; H−s)for s < 1/2−l
kand l < 1
2.
Proof. The points (i)–(iii) can be shown similarly to Lemma 2.2. To prove (iv) and
(v) we use interpolation theory. Given v∈H, we have
|v|H−s≤c1|v|θ1
V0|v|1−θ1
H
with θ1=s. For a function v∈V, we obtain
|v|H≤c2|v|θ2
H−s|v|1−θ2
V(2.2)
with θ2=1
1+s. Hence, for vn(x) = sin nkπx we find
|vn|H−s≤c1|vn|s
V0|vn|1−s
H=c1
√2π−sn−sk =: c3n−sk.(2.3)
Let us denote by fN
α,k,l the finite series
fN
α,k,l(x, t) =
N
X
n=1
n−le(−nαt)sin nkπx.
Now, we are ready to prove (iv). We want to show that fα,k,l is in C([0, T], H−s). To
this aim, let t∈[0, T] be given. Let the pre-requisite s > 1−l
kbe fulfilled. We derive
using (2.3)
kfα,k,l(t)kH−s=k
∞
X
n=1
n−le(−nαt)vnkH−s≤
∞
X
n=1 kn−lvnkH−s≤c3
∞
X
n=1
n−l−sk.
6Arnd R¨
osch, Daniel Wachsmuth
By assumption, we have −l−sk < −1 for the exponent. Therefore, we obtain uni-
form convergence of fα,k,l(t). This uniform convergence and the fact n−le(−nαt)vn∈
C([0, T]; H−s) for each nand fixed sensures the continuity of the abstract function.
(v) Starting from (2.2), we get
kfN
α,k,lkH−s≥c−s−1
2kfN
α,k,lks+1
HkfN
α,k,lk−s
V,
since obviously fN
α,k,l 6= 0 holds. For sufficiently large Nand l < 1/2, we estimate the
H-Norm of fN
α,k,l by
kfN
α,k,lk2
H=1
2
N
X
n=1
n−2l≥1
2ZN
1
x−2ldx =1
2
1
1−2l(N1−2l−1) ≥1
4
1
1−2lN1−2l.
Note, that the estimate is also correct for negative values of l.
Similarly, we have to derive a bound of the V-norm. For sufficiently large N, we
obtain
kfN
α,k,lk2
V=π2
2
N
X
n=1
n−2ln2k≤π2
2 1 + ZN
0
(x+ 1)2k−2ldx!
=π2
21 + 1
1+2k−2l(N+ 1)1+2k−2l−1
≤π2
1+2k−2lN1+2k−2l.
Since by assumption l < 1/2, it holds 1 + 2k−2l > 0 for k > 0. Altogether, we found
kfN
α,k,lkH−s≥cN s+1
2(1−2l)−s
2(1+2k−2l)=cN1/2−sk−l,
which tends to infinity if s < 1/2−l
k. Hence, fα,k,l cannot be a function continuous in
time with values in such H−sspaces.
This examples shows that under certain conditions it may happens that abstract
functions are continuous with value in some space H−sbut discontinuous with values
in spaces of integrable functions.
3. Application to an optimal control problem. In this section, we will con-
sider optimal control of the instationary Navier-Stokes equations. As model problem
serves the minimization of the quadratic objective functional
min J(y, u) = αT
2ZΩ|y(x, T)−yT(x)|2dx +αQ
2ZQ|y(x, t)−yQ(x, t)|2dxdt
+αR
2ZQ|curl y(x, t)|2dxdt +γ
2ZQ|u(x, t)|2dxdt (3.1)
subject to the instationary Navier-Stokes equations
yt−ν∆y+ (y·∇)y+∇p=u+fin Q,
div y= 0 in Q,
y= 0 on Γ,
y(0) = y0in Ω.
(3.2)
Adjoint state regularity 7
The control has to be in a set of admissible controls, u∈Uad, given by
Uad ={u∈L2(Q)2:ua,i(x, t)≤ui(x, t)≤ub,i(x, t) a.e. on Q, i = 1,2}.
Here, Ω is an open bounded subset of R2with C2-boundary Γ, such that Ω is locally
on one side of Γ, and Qis defined by Q= (0, T )×Ω. Further, functions yT∈L2(Ω)2,
yQ∈L2(Q)2, and y0∈H⊂L2(Ω)2are given. The source term fis required to belong
to L2(0, T;V0). The parameters γand νare positive real numbers. The constraints
ua, ubare required to be in L2(Q)2with ua,i(x, t)≤ub,i(x, t) a.e. on Q,i= 1,2.
3.1. Notations and preliminary results. First, we introduce some notations
and provide some results that we need later on.
To begin with, we define the solenoidal spaces
H:= {v∈L2(Ω)2: div v= 0}, V := {v∈H1
0(Ω)2: div v= 0}.
These spaces are Hilbert spaces with their norms denoted by |·|Hrespectively |·|V
and scalar products (·,·)Hrespectively (·,·)V. The dual of Vwith respect to the
scalar product of Hwe denote by V0with the duality pairing h·,·iV0,V .
We shall work in the standard space of abstract functions from [0, T] to a real Banach
space X,Lp(0, T;X), endowed with its natural norm,
kykLp(X):= kykLp(0,T ;X)= ZT
0|y(t)|p
Xdt!1/p
1≤p < ∞,
kykL∞(X):= vrai max
t∈(0,T )|y(t)|X.
In the sequel, we will identify the spaces Lp(0, T;Lp(Ω)2) and Lp(Q)2for 1 < p < ∞,
and denote their norm by kukp:= kukLp(Q)2.The usual L2(Q)2-scalar product we
denote by (·,·)Qto avoid ambiguity.
In all what follows, k · k stands for norms of abstract functions, while | · | denotes
norms of ”stationary” spaces like Hand V.
To deal with the time derivative in (3.2), we introduce the common spaces of functions
ywhose time derivatives ytexist as abstract functions,
Wα(0, T;V) := {y∈L2(0, T;V) : yt∈Lα(0, T;V0)}, W(0, T) := W2(0, T;V),
where 1 ≤α < ∞. Endowed with the norm
kykWα:= kykWα(0,T ;V)=kykL2(V)+kytkLα(V0),
these spaces are Banach spaces, respectively Hilbert spaces in the case of W(0, T).
Every function of W(0, T) is, up to changes on sets of zero measure, equivalent to
a function of C([0, T], H), and the imbedding W(0, T ),→C([0, T], H) is continuous,
cf. [1, 12]. As we saw above, there is no imbedding Wα(0, T;V) in C([0, T], H) for
α < 2.
However, the space W(0, T) enjoys the following imbedding property:
Lemma 3.1. The space W(0, T)is continuously imbedded in L4(Q)2.
8Arnd R¨
osch, Daniel Wachsmuth
Proof. For v∈V, the interpolation inequality |v|4≤c|v|1/2
H|v|1/2
Vholds, cf. [13]. Let
v∈W(0, T) be given. Then, we can readily estimate
kvk4
4≤ZT
0|v(t)|4
4dt ≤cZT
0|v|2
H|v|2
Vdt ≤ckvk2
L∞(H)kvk2
L2(V)≤ckvk4
W,
which proves the claim.
For convenience, we define the trilinear form b:V×V×V7→ Rby
b(u, v, w) = ((u·∇)v, w)2=ZΩ
2
X
i,j=1
ui
∂vj
∂xi
wjdx
together with
bQ(u, v, w) = ZT
0
b(u(t), v(t), w(t)) dt.
An important property of bis that for u∈Vand sufficient regular v, w it holds
b(u, v, w) = −b(u, w, v).(3.3)
There are several estimates of brespectively bQavailable. We mention only the
following one which we will need in the sequel. For detailed discussions consult [3, 13,
15].
Lemma 3.2. Let u, w ∈L4(Q)2and v∈L2(0, T;V)be given. Then there is a constant
c > 0independently of u, v, w such that
|bQ(u, v, w)| ≤ ckuk4kvkL2(0,T ;V)kwk4
holds.
To specify the problem setting, we introduce a linear operator A:L2(0, T;V)7→
L2(0, T;V0) by
ZT
0h(Ay)(t), v(t)iV0,V dt := ZT
0
(y(t), v(t))Vdt,
and a nonlinear operator Bby
ZT
0B(y)(t), v(t)V0,V dt := ZT
0
b(y(t), y(t), v(t))dt.
As a conclusion of Lemma 3.1, eq. (3.3), and Lemma 3.2, we find that Bis continuous
as operator from W(0, T) to L2(0, T;V0).
Testing system (3.2) by divergence-free functions, one obtains the solenoidal form of
the Navier-Stokes equations
yt+νAy +B(y) = u+f
y(0) = y0,
where the first equation has to be understood in the sense of L2(0, T;V0). It is well-
known that for all initial values y0∈Hand source terms u, f ∈L2(0, T;V0) there
exists a unique weak solution y∈W(0, T) of (3.2), cf. [3, 13].
Adjoint state regularity 9
We introduce the linearized equation by
yt+νAy +B0(¯y)y=u
y(0) = y0.(3.4)
Here, ¯yis a given state y∈W(0, T). This equation is solvable for all u∈L2(0, T;V0)
and y0∈H. Its unique solution yis in W(0, T).
3.2. Optimality condition. Now, we return to the optimization problem (3.1).
We will call a control u∈Uad locally optimal, if there exists ρ > 0 such that
J(¯y, ¯u)≤J(y, u)
for all u∈Uad with ku−¯uk2≤ρ. Here, ¯yand ydenote the states associated with ¯u
and u, respectively.
The first-order necessary condition for local optimality is stated in the next theorem.
Theorem 3.3. Let ¯ube a locally optimal control with associated state ¯y=y(¯u). Then
there exists a unique solution ¯
λ∈W4/3(0, T;V)of the adjoint equation
−¯
λt+νA¯
λ+B0(¯y)∗¯
λ=αQ(¯y−yQ) + αR~
curl curl ¯y
¯
λ(T) = αT(¯y(T)−yT).(3.5)
Moreover, the variational inequality
(γ¯u+¯
λ, u −¯u)L2(Q)2≥0∀u∈Uad (3.6)
is satisfied.
Proofs can be found in [6, 7, 14]. The regularity of ¯
λis proven in [10] for homogeneous
initial conditions, ¯
λ(T) = 0.
The adjoint state λis the solution of a linearized adjoint equation backward in time.
So it is natural, to look for its dependance of the given data. For convenience, we
denote by gthe right-hand side of (3.5), and by λTthe initial value αT(¯y(T)−yT).
Theorem 3.4. Let λT∈H,g∈L2(0, T;V0), and ¯y∈L2(0, T;V)∩L∞(0, T;H)be
given. Then there exists a unique weak solution λof (3.5) satisfying λ∈W4/3(0, T).
The mapping (g, λT)7→ λis continuous in the mentioned spaces.
Proof. At first, denote by wthe weak solution of
−wt+νAw =g
w(T) = λT.
Its existence and regularity w∈W(0, T) follows from solvability of the instationary
Stokes-equation, cf. [13]. Moreover, we get the continuity estimate
kwkW≤ckgkL2(V0)+|λT|H.(3.7)
Further, let zbe the weak solution of
−zt+νAz +B0(¯y)∗z=−B0(¯y)∗w
z(T) = 0.
10 Arnd R¨
osch, Daniel Wachsmuth
Since ¯yand ware in W(0, T), we get B0(¯y)∗w∈L4/3(0, T;V0)∩W(0, T)∗as follows.
We write for v∈W(0, T )
[B0(¯y)∗w]v=ZT
0
b(¯y, v, w) + b(v, ¯y, w)dt
≤ck¯yk4kvkL2(V)kwk4+kvk4k¯ykL2(V)kwk4.
By Lemma 3.2, we conclude
[B0(¯y)∗w]v≤ck¯ykWkwkWkvkL2(V)+kvk4.
Since kvk4≤ckvkW, we get B0(¯y)∗w∈W(0, T)∗. The space Vis continuously
imbedded in L4(Ω)2, which allows us to conclude B0(¯y)∗w∈L4/3(0, T;V0). Therefore,
we arrive at
kB0(¯y)∗wkW∗+kB0(¯y)∗wkL4/3(V0)≤ck¯ykWkwkW.(3.8)
Now, Proposition 2.2.1 in [9] respectively Proposition 2.4 in [10] imply the existence
of ztogether with the regularity z∈W4/3(0, T) and the estimate
kzkW4/3≤ckB0(¯y)∗wkL4/3(V0)+kB0(¯y)∗wkW∗≤ck¯ykWkwkW.(3.9)
We construct a solution of the inhomogeneous adjoint equation (3.5) by λ=z+w.
Using (3.7) and (3.9),
kλkW4/3≤ kzkW4/3+kwkW≤c(1 + k¯ykW)kgkL2(V0)+|λT|H
is found, and the claim is proven.
Observe, that the conditions of the previous theorem requires the initial value to be
in H, whereas the regularity λ∈W4/3(0, T) does not guarantee λ(t)→λTin Hfor
t→T.
If the data is more regular then things are much easier. If for instance f∈L2(Q)2
and y0∈Vis given together with yT∈V, then the state yand the adjoint λadmits
the same regularity: it holds that λbelongs to a space H2,1which is continuously
imbedded in C([0, T ], V ), confer [9, 13].
3.3. Example. In this last section, we will answering the question: can the
adjoint state be represented by a continuous abstract function? Clearly, if λt∈
L4/3(0, T;V0) together with λ(T)∈V0hold, then it is obvious that λis a continuous
function with values in V0. Nevertheless, we are looking for a sharper imbedding
result.
Let ε > 0 and integer k > 3/2 + 3εbe given. Set l= 1 −k/3 + εand α= 8/3k.
Notice that by the definition of kand lwe have l < 1/2. Then the function f:= fα,k,l
introduced in Section 2, fulfills:
(i) f∈Lp(0, T;V) for p < 8/3k
4/3k−ε= 2 + 2ε
4/3k−ε
(ii) d
dt f∈Lq(0, T;V0) for q < 8/3k
2k−ε=4
3+4/3ε
2k−ε
(iii) f∈C([0, T], H−s) for s > 1
3+ε
k
(iv) f6∈ C([0, T], H−s) for s < 1
3−1/2+ε
k.
Adjoint state regularity 11
Here, we observe that f∈W(2,4/3; V, V 0) for all possible kand ε. Thus, it has the
same regularity as the adjoint state λ. And we can say that the space W(2,4/3; V, V 0)
is not continuous imbedded in C([0, T ], H−s) for s < 1/3.
However, there is a positive result available.
Theorem 3.5. The space W(p, q;V, V 0)is compactly imbedded in C([0, T], H−s)for
s >
1
p+1
q−1
1 + 1
p−1
q
.
Proof. The notation is adapted to the one used in the present article. For the proof
and detailed discussions, we refer to Amann [2].
We combine these conclusions to
Corollary 3.6. The space W(2,4/3; V, V 0)is continuously imbedded in the space
C([0, T], H−s)for s > 1/3. If s < 1/3holds, then W(2,4/3; V, V 0)can not be imbedded
in C([0, T], H−s).
REFERENCES
[1] R. A. Adams. Sobolev spaces. Academic Press, San Diego, 1978.
[2] H. Amann. Compact embeddings of vector-valued Sobolev and Besov spaces. Glasnik Matem-
aticki, 35:161–177, 2000.
[3] P. Constantin and C. Foias. Navier-Stokes equations. The University of Chicago Press, Chicago,
1988.
[4] R. Dautray and J. L. Lions. Evolution problems I, volume 5 of Mathematical analysis and
numerical methods for science and technology. Springer, Berlin, 1992.
[5] H. Gajewski, K. Gr¨oger, and K. Zacharias. Nichtlineare Operatorgleichungen und Operatordif-
ferentialgleichungen. Akademie-Verlag, Berlin, 1974.
[6] M. D. Gunzburger and S. Manservisi. The velocity tracking problem for Navier-Stokes flows
with bounded distributed controls. SIAM J. Contr. Optim., 37:1913–1945, 1999.
[7] M. D. Gunzburger and S. Manservisi. Analysis and approximation of the velocity tracking
problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal., 37:1481–
1512, 2000.
[8] M. Hinterm¨uller and M. Hinze. A SQP-semi-smooth Newton-type algorithm applied to control
of the instationary Navier-Stokes system subject to control constraints. Technical Re-
port TR03-11, Department of Computational and Applied Mathematics, Rice University,
submitted.
[9] M. Hinze. Optimal and instantaneous control of the instationary Navier-Stokes equations.
Habilitation, TU Berlin, revised version, 2002.
[10] M. Hinze and K. Kunisch. Second order methods for optimal control of time-dependent fluid
flow. SIAM J. Contr. Optim., 40:925–946, 2001.
[11] M. Hinze and K. Kunisch. Second order methods for boundary control of the instationary
Navier-Stokes system. ZAMM, 84:171–187, 2004.
[12] J. L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications,
volume I. Springer, Berlin, 1972.
[13] R. Temam. Navier-Stokes equations. North Holland, Amsterdam, 1979.
[14] F. Tr¨oltzsch and D. Wachsmuth. Second-order sufficient optimality conditions for the optimal
control of Navier-Stokes equations. Preprint 30-2003, Institut f¨ur Mathematik, TU Berlin,
submitted.
[15] W. von Wahl. The equations of Navier-Stokes and abstract parabolic equations. Vieweg,
Braunschweig, 1985.
