Một số bài toán điều khiển tối ưu đối với hệ phương trình navier stokes voigt

MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION

TRAN MINH NGUYET

SOME OPTIMAL CONTROL PROBLEMS
FOR NAVIER-STOKES-VOIGT EQUATIONS
(MỘT SỐ BÀI TOÁN ĐIỀU KHIỂN TỐI ƯU
ĐỐI VỚI HỆ PHƯƠNG TRÌNH NAVIER-STOKES-VOIGT)

DOCTORAL DISSERTATION OF MATHEMATICS

Hanoi - 2019


MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION

TRAN MINH NGUYET

SOME OPTIMAL CONTROL PROBLEMS
FOR NAVIER-STOKES-VOIGT EQUATIONS

Speciality: Differential and Integral Equations
Speciality Code: 9.46.01.03

DOCTORAL DISSERTATION OF MATHEMATICS

Supervisor: PROF.DR. CUNG THE ANH

Hanoi - 2019

COMMITTAL IN THE DISSERTATION

I assure that my scientific results are new and original. To my knowledge,
before I published these results, there had been no such results in any scientific
document. I take responsibility for my research results in the dissertation. The
publications in common with other authors have been agreed by the co-authors
when put into the dissertation.
December 10, 2019
Author

Tran Minh Nguyet

i


ACKNOWLEDGEMENTS

This dissertation was carried out at the Department of Mathematics and
Informatics, Hanoi National University of Education. It was completed under
the supervision of Prof.Dr. Cung The Anh.
First and foremost, I would like to express my deep gratefulness to Prof.Dr.
Cung The Anh for his careful, patient and effective supervision. I am very lucky
to have a chance to study with him. He is an excellent researcher.
I would like to thank Assoc.Prof.Dr. Tran Dinh Ke for his help during the
time I studied at Department of Mathematics and Informatics, Hanoi National
University of Education. I would also like to thank all the lecturers and PhD
students at the seminar of Division of Mathematical Analysis for their encouragement and valuable comments.
A very special gratitude goes to Thang Long University for providing me the

funding during the time I studied in the doctoral program. Many thanks are
also due to my colleagues at Division of Mathematics, Thang Long University,
who always encourage me to overcome difficulties during my period of study.
Last but not least, I am grateful to my parents, my husband, my brother,
and my beloved daughters for their love and support.
Hanoi, December 10, 2019
Tran Minh Nguyet

ii


CONTENTS
COMMITTAL IN THE DISSERTATION

. . . . . . . . . . . . . . . . . . . . .

i

ACKNOWLEDGEMENTS .............................................................................................ii
CONTENTS ........................................................................................................iii
LIST OF SYMBOLS ..................................................................................................... 1
INTRODUCTION ......................................................................................................... 3

Chapter 1.
1.1

PRELIMINARIES AND AUXILIARY RESULTS

7

Function spaces ....................................................................................... 7
1.1.1

Regularities of boundaries .......................................................... 7

1.1.2

Lp and Sobolev spaces ................................................................ 7

1.1.3

Solenoidal function spaces ........................................................ 11

1.1.4

Spaces of abstract functions ..................................................... 12

1.1.5

Some useful inequalities ............................................................ 13

1.2

Continuous and compact imbeddings .................................................. 14

1.3

Operators ............................................................................................... 16

1.4

The nonstationary 3D Navier-Stokes-Voigt equations ........................ 20

1.5

1.4.1

Solvability of the 3D Navier-Stokes-Voigt equations with
homogeneous boundary conditions .......................................... 21

1.4.2

Some auxiliary results on linearized equations ....................... 22

Some definitions in Convex Analysis ................................................... 25

Chapter 2.

A DISTRIBUTED OPTIMAL CONTROL PROBLEM

26

2.1

Setting of the problem .......................................................................... 26

2.2

Existence of optimal solutions .............................................................. 28

2.3

First-order necessary optimality conditions ......................................... 32

2.4

Second-order sufficient optimality conditions ...................................... 41

Chapter 3.

A TIME OPTIMAL CONTROL PROBLEM

47

3.1

Setting of the problem .......................................................................... 47

3.2

Existence of optimal solutions .............................................................. 49

3.3

First-order necessary optimality conditions ......................................... 52

3.4

Second-order sufficient optimality conditions ...................................... 59
iii

Chapter 4.

AN OPTIMAL BOUNDARY CONTROL PROBLEM

67

4.1

Setting of the problem .......................................................................... 67

4.2

Solvability of the 3D Navier-Stokes-Voigt equations with nonhomogeneous boundary conditions .......................................................... 69

4.3

Existence of optimal solutions .............................................................. 75

4.4

First-order and second-order necessary optimality conditions ........... 77

4.5

4.4.1

First-order necessary optimality conditions ............................ 77

4.4.2

Second-order necessary optimality conditions......................... 81

Second-order sufficient optimality conditions ...................................... 84

CONCLUSION AND FUTURE WORK................................................................. 88
LIST OF PUBLICATIONS .................................................................................. 89

REFERENCES .............................................................................................. 90

iv


LIST OF SYMBOLS

R

the set of real numbers

R+

the set of positive real numbers

Rn

n-dimensional Euclidean vector space

A := B
A¯

A is defined by B

(., .)X

scalar product in the Hilbert space X

ǁxǁX

norm of x in the space X

X′

the dual space of the space X
duality pairing between x′ ∈ X′ and x ∈ X
X is imbedded in Y
the space of
∫ Lebesgue measurable functions f
such that Ω |f (x)| dx < +∞

⟨ x′, x⟩ X′,X
X ‹→ Y
Lp(Ω)

the closure of the set A

p

L20(Ω)

the
∫ space of functions f ∈ L2(Ω) such that

L∞(Ω)

the space of almost everywhere bounded
functions on Ω

C0∞(Ω)

the space of infinitely differentiable functions
with compact support in Ω

¯)
C(Ω

¯
the space of continuous functions on Ω

Ω f (x)dx

=0

Wm,p(Ω),
Hm(Ω),
Hm
0 (Ω),

Sobolev spaces

Hs(Ω),
H s (Γ)
H−m(Ω)

the dual space of Hm(Ω)

H−s(Γ)

the dual space of Hs(Γ)
L2(Ω) × L2(Ω) × L2(Ω) (analogously applied
for all other kinds of spaces)

L2(Ω)

0

1


(., .)
((., .))
((., .))1
|.|

the scalar product in L2(Ω)
the scalar product in H01(Ω)
the scalar product in H1(Ω)
the norm in L2(Ω)

ǁ.ǁ1

the norm in H01(Ω)
the norm in H1(Ω)

x·y

the scalar product between x, y ∈ Rn

∇

(∂x1 ,

∂x2 , · · · ,

∇y

(∂x1 ,

∂x2 , · · ·

ǁ.ǁ

∂

∂y

∂

∂y

,

∂

∂xn )
∂y

∂xn )

y·∇

y1 ∂x1 + y2

∇ · y, div y

∂x1

V

{y ∈

H, V
Lp(0, T ; X), 1 < p < ∞

the closures of V in L2(Ω) and H10(Ω)
the space of functions f : [0, T ] → X such
∫T
that 0 ǁf (t)ǁpX dt < ∞

L∞(0, T ; X)

the space of functions f : [0, T ] → X such
that ǁf (.)ǁX is almost everywhere bounded
on [0, T ]

W 1,p(0, T ; X)

{y ∈ Lp(0, T ; X) : yt ∈ Lp(0, T ; X)}

C([0, T ]; X)

the space of continuous functions from

∂

∂y1

+

∂x2 + · · · + yn ∂xn
∂

∂

∂x2 + · · · + ∂xn
C∞
0 (Ω) : div y =
∂y2

∂yn

0}

[0, T ] to X
{xk}
xk → x

sequence of vectors xk
xk converges strongly to x

xk ~ x

xk converges weakly to x

NU (u)

i.e.
a.e.

the normal cone of U at the point u
the polar cone of tangents of U at u
id est (that is)
almost every

p. 5
2D
3D

page 5
two-dimensional

three-dimensional

Q

The proof is complete

TU (u)

2


INTRODUCTION

1. Literature survey and motivation
The Navier-Stokes-Voigt (sometimes written Voight) equations was first introduced by Oskolkov in [57] as a model of motion of certain linear viscoelastic
incompressible fluids. This system was also proposed by Cao, Lunasin and
Titi in [12] as a regularization, for small values of α, of the three-dimensional
Navier-Stokes equations for the sake of direct numerical simulations. In fact, the
Navier-Stokes-Voigt system belongs to the so-called α-models in fluid mechanics
(see e.g. [38]), but it has attractive advantages over other α-models in that one
does not need to impose any additional artificial boundary conditions (except
the Dirichlet boundary conditions) to get the global well-posedness. We also
refer the interested reader to [21] for some interesting applications of NavierStokes-Voigt equations in image inpainting.
In the past years, the existence and long-time behavior of solutions to the
Navier-Stokes-Voigt equations has attracted the attention of many mathematicians. In bounded domains or unbounded domains satisfying the Poincaré inequality, there are many results on the existence and long-time behavior of
solutions in terms of existence of attractors for the Navier-Stokes-Voigt equations, see e.g. [3, 18, 19, 31, 41, 42, 60, 74]. In the whole space R3, the existence
and decay rates of solutions have been studied recently in [4, 56, 75].
The optimal control theory has been developed rapidly in the past few decades
and becomes an important and separate field of applied mathematics. The optimal control of ordinary differential equations is of interest for its applications
in many fileds such as aviation and space technology, robotics and the control

of chemical processes. However, in many situations, the processes to be optimized may not be modeled by ordinary differential equations, instead partial
differential equations are used. For example, heat conduction, diffusion, electromagnetic waves, fluid flows can be modeled by partial differential equations.
In particular, optimal control of partial differential equations in fluid mechanics was first studied in 1980s by Fursikov when he established several theorems
about the existence of solutions to some optimal control problems governed by
Navier-Stokes equations (see [25, 26, 27]).

3


One of the most important objectives of optimal control theory is to obtain
necessary (or possibly necessary and sufficient) conditions for the control to be
an extremum. Since the pioneering work [1] of Abergel and Temam in 1990,
where the first optimality conditions to the optimal control problem for fluid
flows can be found, this matter has been studied very intensively by many authors, and in various research directions such as distributed optimal control,
time optimal control, boundary optimal control and sparse optimal control. Let
us briefly review some results on optimality conditions of optimal control problems governed by Navier-Stokes equations that is one of the most important
equations in fluid mechanics. For distributed control problems, this matter was
studied in [23, 33, 36, 68]. These works are all in the case of absence of state
constraints. In the case of the presence of state constraints, the problem was
investigated by Wang [71] and Liu [52]. The time optimal control problem of
Navier-Stokes equations was investigated by Barbu in [7] and Fernandez-Cara
in [24]. Optimal boundary control problems of the Navier-Stokes equations have
been studied by many authors, see for instance, [32, 39, 40, 61] in the stationary
case, and [10, 17, 28, 29, 34, 37] in the nonstationary case. One interesting result
about Pontryagin’s principle for optimal control problem governed by 3D NavierStokes equations is introduced by B.T. Kien, A. Rösch and D. Wachsmuth in
[43]. We can see also the habilitation [35], the theses [69], [63] and references
therein, for other works on optimal control of Navier-Stokes equations.
As described above, the unique existence and long-time behavior of solutions
to the Navier-Stokes-Voigt equations, as well as the optimal control problems
for fluid flows, in particular for Navier-Stokes equations, have been considered

by many mathematicians. However, to the best of our knowledge, the optimal
control of three-dimensional Navier-Stokes-Voigt equations has not been studied
before. This is our motivation to choose the topic ”Some optimal control
problems for Navier-Stokes-Voigt equations”. Because of the physical
and practical significance, one only considers Navier-Stokes-Voigt equations in
the case of three or two dimensions. The thesis presents results on some optimal
control problems for this equations in the three-dimensional space (the most
physically meaningful case). However, all results of the thesis are still true in the
two-dimensional one (with very similar statements of results and corresponding
proofs). Namely, we will study the following problems:
(P1) The distributed optimal control problem of the nonstationary three di-

4


mensional Navier-Stokes-Voigt equations, where the objective functional is of
quadratic form and the distributed control belongs to a non-empty, closed, convex subset,
(P2) The time optimal control problem of the nonstationary three dimensional Navier-Stokes-Voigt equations, where the set of admissible controls is an
arbitrary non-empty, closed, convex subset,
(P3) The boundary optimal control problem of the nonstationary three dimensional Navier-Stokes-Voigt equations, where the objective functional is of
quadratic form and the boundary control variable has to satisfy some compatibility conditions.
2. Objectives
The objectives of this dissertation are to prove the existence of optimal solutions and to give the necessary and sufficient optimality conditions for problems
(P1), (P2), (P3), namely,
(i) to show the existence of optimal solutions and to establish the first-order
necessary and the second-order sufficient optimality conditions for problem
(P1).
(ii) to prove the existence of optimal solutions and to derive the first-order
necessary and the second-order sufficient optimality conditions for problem
(P2).

(iii) to get the existence of optimal solutions and to give the first-order and
second-order necessary, the second-order sufficient optimality conditions for
problem (P3).
3. The structure and results of the dissertation
The dissertation has four chapters and a list of references.
Chapter 1 collects several basic concepts and facts on Sobolev spaces and
partial differential equations associated with solutions of Navier-Stokes-Voigt
equations as well as some auxiliary results.
Chapter 2 presents results on the distributed optimal control problem governed by Navier-Stokes-Voigt equations.
Chapter 3 provides results on the time optimal control problem governed by
Navier-Stokes-Voigt equations.
Chapter 4 presents results on the boundary optimal control problem governed
5


by Navier-Stokes-Voigt equations.
The results obtained in Chapters 2, 3 and 4 are answers for problems (P1),
(P2), (P3), respectively.
Chapter 2 and Chapter 3 are based on the papers [CT1], [CT2] in the List of
Publications which were published in the journals Numerical Functional Analysis
and Optimization and Applied Mathematics and Optimization, respectively. The
results of Chapter 4 is the content of the work [CT3] in the List of Publications,
which has been submitted for publication.
These results have been presented at:
• Mini-workshop ”Partial Differential Equations: Analysis and Numerics”,
September 2019, Vietnam Institute for Advanced Studies in Mathematics,
Ha Noi.
• ”Vietnam-Korea Joint Conference on Selected Topics in Mathematics”, February 2017, Da Nang, Vietnam.
• The 14th Workshop on Optimization and Scientific Computation, April
2016, Ba Vi, Ha Noi.

• Seminar at Vietnam Institute for Advanced Studies in Mathematics.
• Seminar of Division of Mathematical Analysis at Hanoi National University
of Education.

6


CHAPTER

1

PRELIMINARIES AND AUXILIARY RESULTS

In this chapter, we review some basic concepts and results on function spaces,
imbeddings, operators, Navier-Stokes-Voigt equations and present some auxiliary results on linearized equations.

1.1
1.1.1

Function spaces
Regularities of boundaries

Let Ω be an open bounded set in Rn. Its boundary is denoted by Γ. The
outward normal vector of the boundary is denoted by n. We will need some
smoothness properties of Γ. In some situations, it is sufficient to assume that
Γ is locally Lipschitz,

i.e. each point x on the boundary Γ has a neighborhood Ux such that Γ∩Ux is the
graph of a Lipschitz continuous function, see e.g. [2, Subsection 4.5]. However,
this smoothness might be insufficient in some other cases. Sometimes, we will

need that
Ω is of class Cm,

where m ∈ Z+ or m = ∞. This means that the boundary Γ is a (n−1)-dimensional
manifold of class Cm and Ω is locally located on one side of Γ (see [66, p. 2] and
[51, p. 34]). It is clear that the latter property implies the first one.
1.1.2

Lp and Sobolev spaces

Let Ω be a nonempty and Lebesgue measurable subset of R n. We denote by
Lp(Ω), 1 ≤ p < ∞, the space of real-valued functions defined on Ω whose p−th
power is integrable for the Lebesgue measure dx. It is a Banach space endowed
with the norm
1/p

ǁuǁLp(Ω) =

|u(x)| p dx

∫
Ω

7

.


For p = 2, L2(Ω) is a Hilbert space with the scalar product
∫

(u, v)L2(Ω) =

u(x)v(x)dx.
Ω

We denote by L20(Ω) the space of all functions in L2(Ω) which have average of
zero on Ω
∫
L2(Ω) :=
0

f ∈ L2(Ω) :

f (x)dx = 0

.

Ω

For a multiindex α := (α1, α2, ..., αn) ∈ Nn, we set |α| =
∂|α|
Dα =
.
∂x1α1 ∂xα22 · · · ∂xαnn

Σn αi and
i=1

For m ∈ Z+, p ∈ [1, ∞), we define the Sobolev space Wm,p(Ω) to be the space of
all functions whose weak derivatives up to order m are functions in Lp(Ω). The

norm in Wm,p(Ω) is defined by
Σ
1/p
p
α

ǁD uǁLp (Ω)

ǁuǁ W m,p(Ω) :=

.

0≤|α|≤m

The following theorem gives a basic property of this space.
Theorem 1.1.1. [2, Theorem 3.2 and Theorem 3.5] Wm,p(Ω) is a separable
Banach space if 1 ≤ p < ∞.
In the case p = 2, the space Hm(Ω) := Wm,2(Ω) is a separable Hilbert space
with scalar product
Σ
(Dαu, Dαv)L2(Ω) .

(u, v)Hm(Ω) =

0≤|α|≤m

We now recall an extension theorem valid for Sobolev spaces on bounded domains.
Lemma 1.1.2. [62, Theorem 5.20] If Ω is a bounded domain of class Ck in Rn,
¯ there exists a bounded linear extension operator E
then for each open set Ω′ ⊃ Ω

such that if u ∈ Hk(Ω) then Eu ∈ H0k(Ω′) and
ǁEuǁHk (Ω′) ≤ Ck,Ω′ ǁuǁHk (Ω).

(1.1)

In fact, we have (1.1) for each Hj with 0 ≤ j ≤ k.
Let C0∞(Ω) be the space of all C∞-functions with compact support in Ω. The
closure of C0∞ (Ω) in H m (Ω) is denoted by H m
(Ω), and we use the notation H − m (Ω)
0
for the dual space of H0m(Ω).
8


Next, we introduce the space Hs(Ω) when s is a positive real number (see [2,
Section 7.42 - 7.48]). Assume that Ω is the whole space Rn or an open bounded
subset of class C1. Let s be a positive real number that is not an integer. Then
we can write s = m + σ where m ∈ N and 0 < σ < 1. We denote by Hs(Ω) the
space consisting of functions in Hm(Ω) for which the norm

ǁuǁHs(Ω) =

ǁuǁ2Hm(Ω) +

Σ ∫ ∫ |Dαu(x) − Dαu(y)|2
|α|=m

Ω

|x − y|n+2σ

Ω

1/2

dxdy

,

is finite. With this norm, Hs(Ω) is a Hilbert space.
Now, we recall the definition of integral of a function defined on the boundary
(see [20, p. 143-146]). We assume that Ω is an open bounded set with C∞
boundary Γ. Then there exists a finite family of bounded open sets {Uj}r j=1
covering Γ and a corresponding family of diffeomorphisms {φj }rj=1 mapping Uj
onto the set
B = {y ∈ Rn : y = (y′, yn), |y′| < 1, −1 < yn < 1}

with
φj (Uj ∩ Ω) = B+ := {y = (y ′ , yn ) ∈ B, yn > 0},

and so
φj (Uj ∩ Γ) = B0 := {y = (y ′ , yn ) ∈ B, yn = 0}.

If f is a function having the support in Uj, we define
∫
∫
∫
f (x)ds =

f◦φ

f (x)ds =:
Uj ∩Γ

Γ

B0

Here,
Jj (y ′ ) =

n
Σ

J kj(y ′ )

2

1

(y ′ , 0)Jj (y ′ )dy ′ .

!1/2
,

k=1

with

!
J kj (y ′ ) = det

∂(x1, ..., xk−1, xk+1, ..., xn).
∂(y1, ..., yn−1)

.

,
yn=0

1
and x = φ−
j (y). For an arbitrary function f defined on Γ , we set

∫

r ∫
Σ

f (x)ds =
Γ

f (x)ωj (x)ds,
j=1

Γ

where {ωj } is a partition of unity for Γ subordinate to {Uj }. This definition
depends neither on the mapping φj nor the partition of unity ωj considered. In

9


fact, this definition is valid in the case that Γ is locally Lipschitz (see [55, p.
120]).
For s ∈ R, s ≥ 0, we are going to define the Sobolev spaces Hs(Γ) (see [20, p.
143-146]). Assume that Γ is of class Cm (m ≥ s). Let u be a function on Γ, we
define θj u on Rn−1 by
θj u(y ′ ) =

1 ′
(ωj u)(φ−
j (y , 0))

if |y ′ | < 1,

0

otherwise.

We denote by Hs(Γ) the space of functions defined on Γ such that
θj u ∈ H s (Rn−1 ),

j = 1, . . . , r.

Endowed with the norm
ǁuǁHs(Γ) =

( r
Σ

)1/2
2
ǁθj uǁ H
s (R n−1)

,

(1.2)

j=1

the space Hs(Γ) is a Hilbert space. One can prove that the definition of Hs(Γ)
is independent of the choice of Uj , φj , ωj . The norm (1.2) depends on system
{Uj , φj , ωj } , however, one can prove that these different norms are equivalent.

We denote by H−s(Γ) the dual spaces of Hs(Γ). In the sequel, we will be
concerned with vector functions in three-dimensional space. For convenience,
we will make use of the notation
L2(Ω) := L2(Ω)3 = L2(Ω) × L2(Ω) × L2(Ω),
which is analogously applied for all other kinds of spaces. These product spaces
are equipped with the usual product norm, except the case C∞
0 (Ω) that is not a
normed space at all.
We will use the notations (., .), ((., .)), ((., .))1 and |.|, ǁ.ǁ, ǁ.ǁ1 to denote the
scalar products and corresponding norms in the spaces L2(Ω), H01(Ω), H1(Ω)
respectively, namely
∫ Σ
3
(u, v) :=

Ω

j=1

∫ Σ
3
((u, v)) :=
Ω

ujvj dx, u = (u1, u2, u3), v = (v1, v2, v3) ∈ L2(Ω),

∇uj · ∇vj dx, u = (u1, u2, u3), v = (v1, v2, v3) ∈ H01(Ω),

j=1

((u, v))1 = (u, v) + ((u, v)), u = (u1, u2, u3), v = (v1, v2, v3) ∈ H1(Ω),
10


|u| :=

√
(u, u),

ǁuǁ :=

√

((u, u)) and ǁuǁ1 :=

√
((u, u))1.

By Poincaré’s inequality, we see that two norms ǁ.ǁ and ǁ.ǁ1 are equivalent in
the space H10(Ω).
We have the following theorem, which is called a trace theorem.
Theorem 1.1.3. [30, Theorem 1.5.1.2] Let Ω be a bounded open subset with
locally Lipschitz boundary Γ and s ∈ (1/2, 1] be a real number. Then the mapping
¯ , has
T : u → u|Γ which is defined for every Lipschitz continuous function u on Ω

a unique continuous extension as an operator from Hs(Ω) onto Hs−1/2(Γ), which
is still denoted by T. This operator has a right continuous inverse.
The right inverse operator in the theorem above is called a lifting operator.
If Ω is sufficiently smooth, of class C1, for example, then the unit outer normal
n to Γ is well defined and continuous. However, we sometimes need less regularity
on Ω but still would like to have n well-defined. In this regard, we have the
following result.
Lemma 1.1.4. [55, Chapter II, Lemma 4.2] If Γ is locally Lipschitz then the
unit outer normal n exists almost everywhere on Γ.
We have the following Green’s formula on the relationship between integrals
on the domain Ω and integrals on its boundary Γ.
Theorem 1.1.5. [30, Theorem 1.5.3.1] Let Ω be a bounded open subset in Rn
with locally Lipschitz boundary Γ. Then, for every u, v ∈ H1(Ω) we have
∫ ∂u
∫ ∂v
∫
vdx +
u
dx =

T(u)T(v)nids,
Ω

∂xi

Ω

∂xi

Γ

where ni denotes the ith complement of the unit outward normal vector n to Γ
and T is the trace operator.
1.1.3

Solenoidal function spaces

We will usually work with functions that satisfy the constraint div y = 0. Let
V be the space
V = {y ∈ C∞
0 (Ω) : div y = 0}.

The closure of V in the space L2(Ω) and H10(Ω) are denoted by H and V , which
are Hilbert spaces with scalar products (., .) and ((., .)), respectively. The space
V has the following useful characterization.
11


Theorem 1.1.6. [66, Theorem I.1.6] Let Ω be an open Lipschitz domain in R3.
Then it holds

V = {u ∈ H10(Ω) : div u = 0}.

We will use notation V ′ for the dual space of V and ⟨ ., .⟩

V ′,V

for the duality

pairing between V ′ and V .
1.1.4

Spaces of abstract functions

Let T < ∞ be a given final time. We denote by Q the time-space cylinder
Ω × (0, T ). We present here some spaces of Banach-space valued functions (see

[67, p. 143] and [51, p. 6-7]).
Let X be a real Banach space. We denote by Lp(0, T ; X), 1 ≤ p ≤ ∞, the
standard Banach space of all functions from (0, T ) to X, endowed with the norm
!1/p
∫
T

ǁyǁLp(0,T ;X) :=

p

0

ǁy(t)ǁXdt

ǁyǁL∞(0,T ;X) := esssup ǁy(t)ǁX.

,

1 ≤ p < ∞,

t∈(0,T )
′

The dual space of Lp(0, T ; X), 1 < p < ∞, is Lp (0, T ; X ′) with p′ := p/(p − 1) and
X′ is the dual space of X. In particular, if X is a Hilbert space then L2(0, T ; X)

is a Hilbert space with the following inner product
∫ T
⟨ u, v⟩
=

⟨ u(t),
v(t)⟩ Xdt.
2
In the sequel, we will identify the spaces L (0, T ; L2(Ω)) and L2(Q). We denote
L2(0,T ;X)

0

by C([0, T ]; X) the space of all continuous functions from [0, T ] to X.
To deal with time derivatives in state equations, we need the following lemma.
Lemma 1.1.7. [66, Lemma 1.1] Let X be a given Banach space with dual space
X′ and let u and g be two functions belonging to L1(0, T ; X). Then, the following

three conditions are equivalent.
(i). The function u is a.e. equal to a primitive function of a function g, i.e.
∫ t
g(s)ds, ξ ∈ X, a.e. t ∈ [0, T ].
u(t) = ξ +
0

(ii). For each test function φ ∈ C0∞ ((0, T )),
∫ T
u(t)φ′(t)dt

∫

T

=−

0

g(t)φ(t)dt.
0

12


(iii). For each η ∈ X ′ ,

d
dt

⟨ η, u⟩

X ′ ,X

= ⟨ η, g⟩

X ′ ,X ,

in the scalar distribution sense on (0, T ).
Next, we introduce the common spaces of functions y whose time derivatives
yt exist as abstract functions
W 1,p(0, T ; X) := {y ∈ Lp(0, T ; X) : yt ∈ Lp(0, T ; X)}.

Endowed with the norm
ǁy ǁ2Lp(0,T ;X) + ǁyt ǁ2Lp(0,T ;X)

ǁyǁW 1,p(0,T ;X) :=

1/2

,

it is a Banach space.
1.1.5

Some useful inequalities
′

Lemma 1.1.8 (Hölder inequality). Assume that u ∈ Lp(Ω), v ∈ Lp (Ω) where
p, p′ ∈ (1, ∞), 1/p + 1/p′ = 1. Then, uv ∈ L1(Ω) and

∫

∫

u(x)v(x)dx ≤

1/p

∫

p′

p

|u(x)| dx

Ω

|v(x)| dx

1/p′

.

Ω

Ω

More general, we have the following lemma.

Lemma 1.1.9 (General Hölder inequality). Let p1, . . . . , pk ∈ (1, ∞) such that
Σk
Qk
pi
1
−1
i=1 pi = 1. If ui ∈ L (Ω) for all i ∈ {1, 2, . . . , k} then
i=1 ui L (). Moreover, one has
k
k
ăY
Y
u iă

uiLpi ().
ă
1
L ()

i=1

i=1

Lemma 1.1.10 (Poincaré’s inequality [62, Proposition 5.8]). If Ω is bounded in
some direction then there exists a constant C depending only on such that
!1/2
n

ă u ă 2
, u H01().

uL2() C
ă xi ă 2
L ()

i=1

Lemma 1.1.11. [66, Chapter III, Lemma 3.5] For any open set Ω in R3 we have
ǁvǁL4(Ω) ≤ 2

1/2

1/4

3/4

1

ǁvǁL2(Ω)ǁ∇vǁL2(Ω), ∀v ∈ H0 (Ω).

Lemma 1.1.12 (Gronwall’s inequality). Let η(.) be a nonnegative, absolutely
continuous function on [0, T ], which satisfies for a.e. t ∈ [0, T ] the differential
inequality
η′(t) ≤ φ(t)η(t) + ψ(t),
13


where φ(t) and ψ(t) are nonnegative, integrable functions on [0, T ]. Then
∫ t
∫
η(t) ≤ e

t
ϕ(s)ds
0

η(0) +

ψ(s)ds
0

for all 0 ≤ t ≤ T .
Lemma 1.1.13 (Young’s inequality with ϵ). Let 1 < p, q < ∞, 1 +
p

1

q

= 1. Then,

for every a, b, ϵ > 0 we have
ab ≤ ϵap + C(ϵ)bq,

where C(ϵ) = (ϵp)−q/pq−1.

1.2

Continuous and compact imbeddings

Theorem 1.2.1 (Rellich-Kondrachov theorem [2, Theorem 6.2]). Let Ω ⊂ Rn

be a bounded Lipschitz domain and 1 ≤ p ≤ ∞.
• If 1 ≤ p < n then W 1,p(Ω) is imbedded in Lq(Ω) ∀ 1 ≤ q ≤
imbedding is compact for 1 ≤ q < np .

np
,
n−p

and this

n−p

• If p = n then the imbedding W 1,p(Ω) ‹→ Lq(Ω), q ∈ [1, ∞), is compact.
• If p > n then the imbedding W 1,p(Ω) ‹→ C(Ω) is compact.
Let Ω be a bounded domain in R3 with locally Lipschitz boundary Γ. It
follows from the above theorem that the imbedding
H1(Ω) ‹→ L2(Ω)

is compact. As a consequence, we get the compactness of the following imbeddings:
H1(Ω) ‹→ L2(Ω),

(1.3)

V ‹→ L2(Ω),

(1.4)

V ‹→ H.

Theorem 1.2.2 (Schauder’s theorem [73, p. 282]). Let X, Y be Banach spaces.

A linear continuous operator T is compact iff its dual operator T ′ is compact.
From (1.4) and Schauder’s theorem we deduce that the imbedding
L2(Ω) ‹→ V ′
is compact.
14

(1.5)


Theorem 1.2.3. [51, Theorem 16.1 and Remark 16.1] Assume that Ω is a
bounded domain with locally Lipschitz boundary. Let s ∈ R+. Then, for every
ϵ > 0, the injection
Hs(Ω) ‹→ Hs−ϵ(Ω)

is compact.
From this theorem and the continuity property of trace operators as well as
lifting operators, we can deduce that the imbedding
H1/2(Γ) ‹→ L2(Γ)

is also compact, and so is the imbedding H1/2(Γ) ‹→ L2(Γ).
Let X be a Banach space. By the following proposition, every function in
the space W 1,p(0, T ; X) is, up to changes on sets of zero measure, equivalent
to a function of C([0, T ]; X), and the imbedding W 1,p(0, T ; X) ‹→ C([0, T ]; X) is
continuous.
Proposition 1.2.4. [62, Proposition 7.1] Suppose that u ∈ W 1,p(0, T ; X), 1 ≤
p ≤ ∞. Then
∫ t
du
u(t) = u(s) +
(τ )dτ for every 0 ≤ s ≤ t ≤ T,

s

dt

and u ∈ C([0, T ]; X) (with the usual caveat). Furthermore, we have the estimate
sup ǁu(t)ǁX ≤ CǁuǁW 1,p(0,T ;X).
0≤t≤T

We denote by D([0, T ]; X) the space of all functions f : [0, T ] → X, which
are infinitely differentiable and have compact support in (0, T ). We have the
following theorem.
Theorem 1.2.5. [51, Theorem 2.1] If X is a separable Hilbert space then
D([0, T ]; X) is dense in W 1,2(0, T ; X).

We know that H1(Ω) is separable. Hence, it follows from the continuous and
surjective property of the trace operator T : H1(Ω) → H1/2(Γ) that H1/2(Γ) is
separable, and so is the space H1/2(Γ). By the above theorem we imply that
D([0, T ]; H1/2(Γ)) is dense in W 1,2(0, T ; H1/2(Γ)).

The following theorem is very useful later.

15

(1.6)


Theorem 1.2.6. [64] Let X, Y be two Banach spaces such that X ⊂ Y with
compact imbedding. Then,
W 1,2(0, T ; X) ⊂ L2(0, T ; Y ),
W 1,2(0, T ; X) ⊂ C([0, T ]; Y ),

with compact imbeddings.
From this theorem and (1.3), (1.4), (1.5) we see that the following imbeddings
are compact:
W 1,2(0, T ; V ) ‹→ L2(Q),
W 1,2(0, T ; L2(Ω)) ‹→ L2(0, T ; V ′),
W 1,2(0, T ; L2(Ω)) ‹→ C([0, T ]; V ′),
W 1,2(0, T ; H1(Ω)) ‹→ L2(Q),
W 1,2(0, T ; H1(Ω)) ‹→ C([0, T ]; L2(Ω)),
W 1,2(0, T ; H1/2(Γ)) ‹→ C([0, T ]; L2(Γ)).

1.3

Operators

Let Ω be a bounded Lipschitz open set in R3. We introduce a trilinear form
b by
Σ
3 ∫
b(u, v, w) =

i,j=1

Ω

ui

∂v j

∂xi

wj dx,

whenever the integrals make sense. Using integration by parts, one can check
that if u, v, w ∈ H1(Ω), ∇ · u = 0 and at least one of the three functions u, v, w
belongs to the space H0 1(Ω), then
b(u, v, w) = −b(u, w, v).

(1.7)

Hence,
b(u, v, v) = 0 if u, v ∈ H1(Ω); ∇ · u = 0; and u|Γ = 0 or v|Γ = 0.

In particular, we have
b(u, v, w) = −b(u, w, v),

∀ u, v, w ∈ V,
∀ u, v ∈ V.

b(u, v, v) = 0,

16

(1.8)


Lemma 1.3.1. There exists a constant C independent of u, v, w such that
|b(u, v, w)| ≤ C|u|1/4ǁuǁ13/4|∇v||w|1/4ǁwǁ3/4
1,

∀ u, v, w ∈ H1(Ω),

(1.9)

|b(u, v, w)| ≤ Cǁuǁ1|∇v|ǁwǁ1,

∀ u, v, w ∈ H1(Ω),

(1.10)

∀ u, v ∈ H1(Ω).

(1.11)

|b(u, v, u)| ≤ C|u|1/2ǁuǁ3/2
1 |∇v|,

As a special case, the following estimates
C|u|1/4 ǁuǁ3/4 ǁvǁ|w|1/4 ǁwǁ3/4 ,

|b(u, v, w)| ≤

(1.12)
Cǁuǁǁvǁǁwǁ,

|b(u, v, u)| ≤ C|u| 1/2 ǁuǁ 3/2 ǁvǁ

hold for every u, v, w ∈ V .
Proof. We only need to prove (1.9). Using the Hölder inequality, we have
∀u, v, w ∈ H1(Ω).

|b(u, v, w)| ≤ CǁuǁL4(Ω)|∇v|ǁwǁL4(Ω),

(1.13)

We will show that
ǁzǁL4 (Ω) ≤ C|z|1/4ǁzǁ3/4
1 ,

∀z ∈ H1(Ω),

(1.14)

then from (1.13) we get (1.9) as a consequence. When z ∈ H01(Ω), inequality
(1.14) follows from Lemma 1.1.11, we now prove that this inequality holds for
every z ∈ H1(Ω). Let Ω′ ⊃ Ω be some open bounded subset in R3. From Lemma
1.1.2, there exists an extension z˜ of z such that z˜ ∈ H0 1(Ω′) and
ǁz˜ǁ Hj (Ω′ ) ≤ Cǁzǁ Hj (Ω) , for j = 0, 1.

Thus, we have

1/4

(1.15)

3/4

ǁzǁ L4 (Ω) ≤ ǁz˜ǁL4 (Ω′ ) ≤ Cǁz˜ǁL2 (Ω′ ) ǁz˜ǁH1 (Ω′ ) .

This together with (1.15) imply (1.14).

We recall that the operator grad : L2(Ω) → H−1(Ω) is defined by
∫
⟨ grad p, v⟩ H−1(Ω),H1(Ω) = −
p∇ · vdx, for v ∈ H01(Ω).
0

Ω

The following proposition shows that operator grad is an isomorphism from
L20(Ω) into H−1(Ω).

Proposition 1.3.2. [66, Chapter I, Propositions 1.1 and 1.2] If f ∈ H−1(Ω)
and ⟨ f, v⟩ = 0 ∀v ∈ V then there exists a unique function p ∈ 0L2(Ω) such that
f = grad p. Furthermore, we have
ǁpǁL2(Ω) ≤ Cǁf ǁH−1(Ω),
17


where C is a constant depending only on Ω.
We now introduce a continuous linear operator A : H1(Ω) → H−1(Ω) for
u ∈ H1(Ω), v ∈ H10(Ω) by
⟨ Au, v⟩

= (∇u, ∇v),

H−1(Ω),H1(Ω)
0

and a bilinear operator B : H1(Ω) × H1(Ω) → H−1(Ω) for u, v ∈ H1(Ω), w ∈ H01(Ω)
by

⟨ B(u, v), w⟩

H−1 (Ω),H1(Ω)
0

= b(u, v, w).

We still use the same notations to denote the restrictions of A and B on the
space V and V × V respectively, namely
A : V → V ′,

⟨ Au, v⟩

B : V × V → V ′,

V ′,V

= (∇u, ∇v), u, v ∈ V,

⟨ B(u, v), w⟩

= b(u, v, w), u, v, w ∈ V.

V ′,V

Then, we define a continuous linear operator A : L2(0, T ; H1(Ω)) → L2(0, T ; H−1(Ω))
for y ∈ L2(0, T ; H1(Ω)), v ∈ L2(0, T ; H10(Ω)) by
∫ T
⟨ Ay, v⟩
=

⟨ Ay(t), v(t)⟩

1
L2(0,T ;H−1(Ω)),L2(0,T ;H
0 (Ω))

∫

0

H−1 (Ω),H1(Ω)dt
0

T

(∇y(t), ∇v(t))dt,
0

=

and a bilinear operator B : W 1,2(0, T ; H1(Ω))×W 1,2(0, T ; H1(Ω)) → L2(0, T ; H−1(Ω))
for y, z ∈ W 1,2(0, T ; H1(Ω)), w ∈ L2(0, T ; H10(Ω)) by
∫T
⟨ B(y, z), w⟩
=

⟨ B(y(t), z(t)), w(t)⟩

1

L2(0,T ;H−1(Ω)),L2(0,T ;H
0 (Ω))

∫

H−1(Ω),H1 (Ω)dt

0

0

T

b(y(t), z(t), w(t))dt.
=

0

We also use the same notations to denote the restrictions of A and B on the
space L2(0, T ; V ) and W 1,2(0, T ; V ) × W 1,2(0, T ; V ) respectively, namely
A : L2(0, T ; V ) → L2(0, T ; V ′),

∫
⟨ Ay, v⟩
=

∫

T

⟨ Ay(t), v(t)⟩
:=

L2(0,T ;V ′),L2(0,T ;V )

V ′,V

T

(∇y(t), ∇v(t))dt,

dt
0

0

for y, v ∈ L2(0, T ; V ). (1.16)
B : W 1,2(0, T ; V ) × W 1,2(0, T ; V ) → L2(0, T ; V ′)
18


∫
⟨ B(y, z), w⟩
=

∫

L2(0,T

;V

′),L2(0,T ;V

T

⟨ B(y, z)(t), w(t)⟩

)

V ′,V

dt

0

T

b(y(t), z(t), w(t))dt, for y, z ∈ W 1,2(0, T ; V ), w ∈ L2(0, T ; V ). (1.17)

:=
0

The lemma below shows the continuity property of operator B, which is very
helpful later.
Lemma 1.3.3. (i). If yn ~ y in W 1,2(0, T ; H1(Ω)) and ∇ · y = 0, ∇ · yn = 0
for every n and for a.e. t ∈ [0, T ], then B(yn, yn) converges to B(y, y) in
L2(0, T ; H−1(Ω)) as n → ∞.

(ii). If yn ~ y in W 1,2(0, T ; V ) then B(yn, yn) converges to B(y, y) in L2(0, T ; V ′)
as n → ∞.

Proof. We only present here a proof for part (i) of the lemma since the proof for
part (ii) is very similar to the first one. For an arbitrary v ∈ L2(0, T ; H10(Ω)), we
have
⟨ B(yn, yn) − B(y, y), v⟩

∫

0

T

b(yn(s), yn(s), v(s)) − b(y(s), y(s), v(s)) ds

0

=

L2(0,T ;H−1(Ω)),L2(0,T ;H1(Ω))

∫ Tb(yn(s), yn(s), v(s)) − b(yn(s), y(s), v(s)) ds
∫ T

0

=

b(yn(s), y(s), v(s)) − b(y(s), y(s), v(s)) ds

+
0

∫

b(yn(s), yn(s) − y(s), v(s))ds +

=

T

∫

T

b(yn(s) − y(s), y(s), v(s))ds
0

0

= ⟨ B(yn, yn − y), v⟩

1
L2(0,T ;H−1(Ω)),L2(0,T ;H
0 (Ω))

+ ⟨ B(yn − y, y), v⟩

1
.
L2(0,T ;H−1(Ω)),L2(0,T ;H
0 (Ω))

Hence
ǁB(yn, yn) − B(y, y)ǁL2(0,T ;H−1(Ω))
≤ ǁB(yn, yn − y)ǁL2(0,T ;H−1(Ω)) + ǁB(yn − y, y)ǁL2(0,T ;H−1(Ω)).

(1.18)

Using (1.7) and (1.9) we obtain
∫ T
∫T
.
.
.
.
. . 0
.
0
∫ T
3/ 4
b(yn(s),
y
(s) − y(s), v(s))ds. = .
b(yn(s), v(s), yn(s) − y(s))ds.
n
≤
C
|yn(s)| ǁyn(s)ǁ1 ǁv(s)ǁ|y
ǁyn(s) − y(s)ǁ 1 ds.
n (s) − y(s)|
.

0

1/4

19

1/4

3/4