courschine .pdf



Nom original: courschine.pdf

Ce document au format PDF 1.4 a été généré par TeX / pdfeTeX-1.21a, et a été envoyé sur fichier-pdf.fr le 13/12/2011 à 20:17, depuis l'adresse IP 196.203.x.x. La présente page de téléchargement du fichier a été vue 3163 fois.
Taille du document: 633 Ko (91 pages).
Confidentialité: fichier public




Télécharger le fichier (PDF)










Aperçu du document


Fourier Analysis Methods for PDE’s
R. Danchin
November 14, 2005

2

Contents
1 An introduction to Fourier analysis
1.1 Notations and definitions . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Littlewood-Paley decomposition . . . . . . . . . . . . . . . . . .
1.2.1 Bernstein inequalities . . . . . . . . . . . . . . . . . . . . . .
1.2.2 The nonhomogeneous Littlewood-Paley decomposition . . . .
1.2.3 About the periodic case . . . . . . . . . . . . . . . . . . . . .
1.3 Littlewood-Paley decomposition and functional spaces . . . . . . . .
1.3.1 Sobolev and H¨
older spaces . . . . . . . . . . . . . . . . . . . .
1.3.2 Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 A few properties of Besov spaces . . . . . . . . . . . . . . . .
1.4 Paradifferential calculus . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Results of continuity for the paraproduct and the remainder .
1.4.3 Results of continuity for the product . . . . . . . . . . . . .
1.4.4 A result of compactness in Besov spaces . . . . . . . . . . . .
1.4.5 Results of continuity for the composition . . . . . . . . . . .
1.5 Calculus in homogeneous functional spaces . . . . . . . . . . . . . .
1.5.1 Homogeneous Littlewood-Paley decomposition . . . . . . . .
1.5.2 Homogeneous Besov spaces . . . . . . . . . . . . . . . . . . .
1.5.3 Paradifferential calculus in homogeneous spaces . . . . . . . .
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

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

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

7
7
8
8
9
10
11
11
13
15
21
21
22
23
24
27
28
29
29
35
37

. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
periodic case
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .

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

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

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

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

39
39
40
41
42
43
43
44
44
46
47
50
51

transport equation
Framework and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A priori estimates in Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . .
Solving the transport equation in Besov spaces . . . . . . . . . . . . . . . . . . .

53
53
54
58

2 The heat equation
2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 A priori estimates in Besov spaces for the heat equation . .
2.2.1 Spectral localization . . . . . . . . . . . . . . . . . .
2.2.2 Estimates for the heat equation . . . . . . . . . . . .
2.2.3 A counterexample . . . . . . . . . . . . . . . . . . .
2.2.4 Estimates in nonhomogeneous Besov spaces, and the
2.3 Optimal well-posedness results for Navier-Stokes equations .
2.3.1 The model . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 About scaling and critical spaces . . . . . . . . . . .
2.3.3 Global well-posedness for small data . . . . . . . . .
2.3.4 Further results . . . . . . . . . . . . . . . . . . . . .
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 The
3.1
3.2
3.3

3

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

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

4

CONTENTS
3.4

3.5

On the Cauchy problem for a shallow water equation .
3.4.1 About Camassa-Holm equation . . . . . . . . .
3.4.2 A well-posedness result and a blow-up criterion
3.4.3 Uniqueness . . . . . . . . . . . . . . . . . . . .
3.4.4 The proof of existence . . . . . . . . . . . . . .
3.4.5 Blow-up criterion and energy conservation . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

4 A short insight into compressible fluid
4.1 About the model . . . . . . . . . . . .
4.1.1 Physical conservation laws . . .
4.1.2 The full model . . . . . . . . .
4.1.3 Simplifying assumptions . . . .
4.1.4 Barotropic fluids . . . . . . . .
4.2 Local well-posedness in critical spaces
4.2.1 The existence proof . . . . . .
4.3 Further results . . . . . . . . . . . . .
4.4 Exercises . . . . . . . . . . . . . . . .
Bibliographie

mechanics
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

60
60
61
62
63
65
68

.
.
.
.
.
.
.
.
.

69
69
69
70
70
72
72
73
83
84
87

Introduction
Since the 80’s, Fourier analysis methods have known a growing interest in the study of
linear and nonlinear PDE’s. In particular, techniques based on Littlewood-Paley decomposition
and paradifferential calculus have proved to be very efficient. Littlewood-Paley decomposition
has been introduced more than fifty years ago in harmonic analysis but its systematic use in
the PDE’s framework is rather recent. Paradifferential calculus, as for it, has been introduced
in 1981 by J.-M. Bony for the study of the propagation of microlocal singularities in nonlinear
hyperbolic PDE’s (see [4]).
In the present notes, we aim at giving a survey of those techniques and a few examples of
how they may used to solve PDE’s. We focus on two linear models: the heat equation and
the transport equation. For each of them, an example of related nonlinear problem is given.
Although most of the results we present here belong to the mathematical folklore, we want
to point out that Fourier analysis methods are very efficient to tackle most of well-posedness
problems for evolutionary PDE’s in the whole space or in the torus.
The notes are structured as follows. The first chapter deals with Fourier analysis. We
introduce Littlewood-Paley decomposition and show how it may used to characterize functional
spaces. We also give a (non so) short insight into the theory of homogeneous spaces which
turn out to be well adapted to the study of many PDE’s and are omitted in most of textbooks
on functional analysis. Finally, we introduce some smatterings of paradifferential calculus and
prove estimates for the product of two temperate distributions, when it makes sense.
In the second chapter, we focus on the heat equation. We state a priori estimates in Besov
spaces with optimal gain of derivatives. As an application, we prove global well-posedness in
Besov spaces with critical regularity for the incompressible Navier-Stokes equations with small
data.
The third chapter is devoted to the study of transport equations associated to Lipschitz
vectorfields. We state a priori estimates in Besov spaces then apply our results to the study of
a shallow water equation.
In the last chapter, we give an example of coupling between heat equation and transport
equation, namely the compressible barotropic Navier-Stokes system. Well-posedness in Besov
spaces with critical regularity is stated.
Acknowledgments The author is grateful to J.-Y. Chemin for supplying most material for
the first chapter, to Zhouping Xin, Chiaojang Xu and Yinbin Deng for the invitation to deliver
this course at Wuhan Normal University, to Ping Zhang for his kind invitation at the Chinese
Academy of Sciences in Beijing and to Yuxin Ge for helping me to communicate between France
and China.

5

6

CONTENTS

Chapter 1

An introduction to Fourier analysis
1.1

Notations and definitions

• S stands for the Schwartz space of smooth functions over RN whose derivatives of all order
decay at infinity. The space S is endowed with the topology generated by the following
family of semi-norms:
kukM,S := sup (1 + |x|)M |∂ α u(x)| for all

u∈S

and M ∈ N.

x∈RN
|α|≤M

• The set S 0 of temperate distributions is the dual set of S for the usual pairing.
• For any u ∈ S, the Fourier transform of u denoted by u
b or Fu is defined by
Z
∀ξ ∈ RN , u
b(ξ) = Fu(ξ) :=
e−iξ·x u(x) dx.
RN

The Fourier transform maps S into and onto itself, and the inverse Fourier transform is
given by the formula F −1 = (2π)−N F.
• The Fourier transform is extended by duality to the whole S 0 by setting
<u
b, ϕ >:=< u, ϕ
b >S 0 ,S
whenever u ∈

S0

and ϕ ∈ S.

• Derivatives: for all multi-index α ∈ NN , we have
F(∂xα u) = (iξ)α Fu

and F(xα u) = (−i)|α| ∂ξα Fu.

• Algebraic properties: for (u, v) ∈ S × S 0 , we have u ∗ v ∈ S 0 and
F(u ∗ v) = Fu Fv.
The above formula also holds true for couples of distributions with compact supports.
• Multipliers: if A is a smooth function
with polynomial growth at infinity, and u ∈ S 0 (RN )

then we set A(D)u := F −1 A Fu .
• The open (resp. closed) ball with radius R centered at x0 ∈ RN is denoted by B(0, R)
(resp. B(0, R)).
• The shell {ξ ∈ RN | R1 ≤ |ξ| ≤ R2 } is denoted by C(0, R1 , R2 ).
• The notation A . B means that A ≤ CB for some “irrelevant” constant C (which may
change from line to line but whose meaning is clear from the context). Likewise, A ≈ B
means that A . B and B . A.
7

8

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS

1.2

The Littlewood-Paley decomposition

1.2.1

Bernstein inequalities

As recalled in the previous section, in Fourier variables differentiating with respect to xj amounts
to multiplying by the function ξ 7→ iξj .
As far as one is concerned with estimates in Lebesgue spaces and whenever the distribution
we consider is well localized in Fourier variables, Bernstein lemma states that differentiating
amounts to multiplying by an appropriate constant
Lemma (Bernstein). Let k be in N. Let (R1 , R2 ) satisfy 0 < R1 < R2 . There exists a constant
C depending only on r1 , r2 , N, such that for all 1 ≤ a ≤ b ≤ ∞ and u ∈ La , we have
1

1

(1.1)

Supp u
b ⊂ B(0, R1 λ) ⇒ sup|α|=k k∂ α ukLb ≤ C k+1 λk+N ( a − b ) kukLa ,

(1.2)

Supp u
b ⊂ C(0, R1 λ, R2 λ) ⇒ C −k−1 λk kukLa ≤ sup|α|=k k∂ α ukLa ≤ C k+1 λk kukLa .

Proof: Arguing by rescaling, one can assume with no loss of generality that λ = 1.
Now, fix a smooth function φ compactly supported and such that φ ≡ 1 in a neighborhood
of the ball B(0, R1 ). We notice that u
b = φb
u. Hence, denoting g := F −1 φ, we get for all
multi-index α,
Z
∂ α u(x) =

∂ α g(y)u(x − y)dy.

Taking advantage of Young inequality, we thus get
k∂ α ukLb ≤ k∂ α gkLc kukLa

with

1
1 1
=1+ − ·
c
b a

Because
k∂ α gkLc ≤ k∂ α gkL∞ + k∂ α gkL1 ≤ C k+1 ,
the proof of the first inequality is complete.
For proving (1.2), we first notice that the inequality on the right is a particular case of
(1.1). Next, we introduce a smooth function ϕ
e with compact support in RN \ {0} and
such that ϕ ≡ 1 in a neighborhood of the shell C(0, R1 , R2 ).
P
As |α|=k (iξ)α (−iξ)α = |ξ|2k , we have
X
(1.3)
u=
gα ? ∂ α u with gbα (ξ) := (iξ)α |ξ|−2k ϕ(ξ).
e
|α|=k

Making use of Young inequality, one can now conclude to the left inequality in (1.2).
Remark. In other words, if u
b is supported in a ball of radius λ then differentiating once is not
worse that multiplying by λ. If u
b is supported in the shell {ξ ∈ RN | R1 λ ≤ |ξ| ≤ R2 λ} then,
up to an irrelevant constant, differentiating once amounts to multiplying by λ.
In most applications, the functions we deal with are not spectrally supported in a shell or
in a ball. Hence, if one wants to take advantage of the nice properties exhibited in Bernstein
lemma, one first has to split the function into pieces which are spectrally supported in a shell
or in a ball. This may be done by introducing a dyadic partition of unity in Fourier variables.
There are two main ways to proceed. Either the decomposition is made indistinctly over the
whole space RN (and we say that the decomposition is homogeneous), or the low frequencies
are treated separately (and the decomposition is said to be nonhomogeneous).
Both decompositions have advantages and drawbacks. The nonhomogeneous one is more
suitable for characterizing the usual functional spaces whereas the properties of invariance by
dilation of the homogeneous decomposition may be more adapted for studying certain PDE’s or
stating optimal functional inequalities having some scaling invariance.

1.2. THE LITTLEWOOD-PALEY DECOMPOSITION

1.2.2

9

The nonhomogeneous Littlewood-Paley decomposition

Let α > 1 and (ϕ, χ) be a couple of smooth functions valued in [0, 1], such that ϕ is supported
in the shell {ξ ∈ RN | α−1 ≤ |ξ| ≤ 2α}, χ is supported in the ball {ξ ∈ RN | |ξ| ≤ α} and
∀ξ ∈ RN , χ(ξ) +

X

ϕ(2−q ξ) = 1.

q∈N

For u ∈ S 0 , one can define nonhomogeneous dyadic blocks as follows. Let
∆q u := 0 if q ≤ −2,
∆−1 u := χ(D)u = e
h ? u with e
h := F −1 χ,
Z
∆q u := ϕ(2−q D)u = 2qN h(2q y)u(x − y)dy with h = F −1 ϕ,

if q ≥ 0.

One can prove that we have
u=

X

in S 0 (RN )

∆q u

q∈Z

for all temperate distribution u (see exercise 1.2). The right-hand side is called nonhomogeneous
Littlewood-Paley decomposition of u.
It is also convenient to introduce the following low frequency cut-off:
X
Sq u :=
∆p u.
p≤q−1

Of course, S0 u = ∆−1 u. Because ϕ(ξ) = χ(ξ/2) − χ(ξ) for all ξ ∈ RN , one can prove that,
more generally, we have
Z
−q
Sq u = χ(2 D)u = e
h(2q y)u(x − y)dy for all q ∈ N.
The Littlewood-Paley decomposition is “almost” orthogonal in L2 . Assuming for instance that
α = 4/3, we have the following result1 :
Proposition 1.2.1. For any u ∈ S 0 (RN ) and v ∈ S 0 (RN ), the following properties hold:
∆p ∆q u ≡ 0

if

∆q (Sp−1 u ∆p v) ≡ 0

|p − q| ≥ 2,
if

|p − q| ≥ 5.

Remark. At this point, one can wonder why it is so important to choose smooth cut-off functions
χ and ϕ for defining a Littlewood-Paley decomposition. Obviously, setting ∆0−1 u := 1|ξ|≤1 (D)u
and ∆0q u := 12q ≤|ξ|≤2q+1 (D)u would define a dyadic spectral decomposition which, in addition,
would be orthogonal in L2 .
In most applications however, it is crucial that we have
k∆q ukLp ≤ C kukLp
for some constant C independent of q.
Alas, unless p = 2, the above inequality fails to be true with ∆0q u instead of ∆q u (see
exercise 1.5).
1

Of course, similar properties may be proved for any α > 1.

10

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS

1.2.3

About the periodic case

Throughout, a1 , · · · , aN denote N positive reals. We denote by TN
a the periodic box with
period 2πai in the i-th direction, and QN
:=
(0,
2πa
)
×
·
·
·
×
(0,
2πa
1
N ). We also introduce
a
N
N
e
Za := Z/a1 × · · · × Z/aN the dual lattice associated to Ta .
We claim that the analysis of the previous section for temperate distributions defined on the
whole space RN may be carried out to S 0 (TN
a ) with very few changes.
0
N
Indeed, decompose u ∈ S (Ta ) into Fourier series:
Z
X
1
u(x) =
u
bβ eiβ·x with u
bβ := N
e−iβ·y u(y) dy.
|Ta | TN
a
eN
β∈Z
a

Denoting
hq (x) =

X

ϕ(2−q β)eiβ·x ,

eN
β∈Z
a

one can now define the periodic dyadic blocks as follows:
Z
X
1
−q
iβ·x
hq (y)u(x − y) dy
∆per
u(x)
:=
ϕ(2
β)b
u
e
=
β
q
|TN
a | TN
a

for all

q∈Z

eN
β∈Z
a

and the low frequency cut-off:
Sqper u(x) := u
b0 +

X

∆per
p u(x) =

p≤q−1

X

χ(2−q β)b
uβ eiβ·x .

eN
β∈Z
a

It is obvious that ∆per
p u = 0 for negative enough p (depending on a) and that
X
0
N
u=u
b0 +
∆per
q u in S (Ta ).
q

Now, to any temperate distribution u over RN , one can associate the periodic distribution uper
defined by
X
uper (x) :=
u(x + α) where ZN
2πa := 2πa1 Z × · · · × 2πaN Z.
α∈ZN
2πa

For uper , both periodic and nonhomogeneous Littlewood-Paley decompositions are available. It
turns out that periodic blocks and nonhomogeneous blocks coincide in the following sense:
Proposition 1.2.2. For all temperate distribution u over RN , one has
per
per
∀q ∈ Z, ϕ(2−q D)u
= ∆per
.
q u
Proof: This is actually an easy consequence of the following Poisson formula for θ ∈ S 0 (RN ):
X
X
1
b
θ(2πα),
θ(β)
=
(2π)N
N
N
β∈Z

α∈Z

which, after a change of variables yields
(1.4)

∀u ∈ S 0 (RN ), ∀x ∈ RN ,

X
1 X iβ·x
e
u
b
(β)
=
u(x + α).
|TN
a |
N
eN
β∈Z
a

α∈Z2πa

1.3. LITTLEWOOD-PALEY DECOMPOSITION AND FUNCTIONAL SPACES

11

Now, on one hand, for all x ∈ RN , we have by definition of ϕ(2−q D),
per
R
P
(x) = 2qN α∈ZN RN h(2q y)u(x+α−y) dy,
ϕ(2−q D)u
Z
2πa
= 2qN
h(2q y) uper (x − y) dz.
RN

On the other hand, by taking advantage of (1.4), we get
X

1 X iβ·(x−y)
h 2q (x + α − y) ,
e
ϕ(2−q β) = 2qN
N
|Ta |
N
α∈Z2πa

eN
β∈Z
a

whence
per (x)
∆per
q u

Z
1 X
=
eiβ(x−y) ϕ(2−q β)uper (y) dy,
|
|TN
N
a
e N Ta
β∈Z
a
X Z
= 2qN
h(2q (x+α−y)) uper (y) dy,
N

QN
a

α∈Z2πa
Z
= 2qN
h(2q (x−z)) uper (z) dz.
RN

The proof is complete.
In what follows, we shall focus on distributions defined on RN . We want to point out that all
the properties described in the next sections remain true in the periodic setting provided the
dyadic blocks have been defined as indicated above.

1.3

Littlewood-Paley decomposition and functional spaces

Many functional spaces over RN (or TN
older or Sobolev spaces may be characterized
a ) such as H¨
in terms of Littlewood-Paley decomposition.

1.3.1

Sobolev and H¨
older spaces

Let us first recall how nonhomogeneous Sobolev spaces H s are defined.
Definition. Let s ∈ R. A temperate distribution u belongs to H s (RN ) if u
b ∈ L2loc (RN ) and
Z
kukH s :=

2 s

2

(1 + |ξ| ) |b
u(ξ)| dξ

1

2

< ∞.

RN

It is classical that H s endowed with the norm k · kH s is a Banach space2 . Now, from the
definition of (χ, ϕ), one easily infers that
X
1
(1.5)
∀ξ ∈ RN , ≤ χ2 (ξ) +
ϕ2 (2−q ξ) ≤ 1,
3
q∈N

whence the following result:
Proposition 1.3.1. There exists a constant C such that for all s ∈ R, we have
X
1
2
kuk
22qs k∆q uk2L2 ≤ C |s|+1 kuk2H s .
s ≤
H
C |s|+1
q
Hence Littlewood-Paley decomposition supplies a simple characterization of Sobolev spaces:
u belongs to H s if and only if the sequence (2qs k∆q ukL2 )q∈Z belongs to `2 (Z).
2

Actually it is even an Hilbert space.

12

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
Let us now focus on H¨
older spaces.

Definition. Let r ∈ (0, 1). We denote by C r the set of bounded functions u : RN → R such
that there exists C ≥ 0 with
∀(x, y) ∈ RN × RN , |u(x) − u(y)| ≤ C|x − y|r .

(1.6)

More generally, if r > 0 is not an integer, we denote by C r the set of [r] times3 differentiable
functions u such that ∂ α u ∈ C r−[r] for all |α| ≤ r.
One can easily prove that the set C r endowed with the norm
kukC r :=

X

k∂ α ukL∞

|α|≤[r]

|∂ α u(x) − ∂ α u(y)|
+ sup
|x − y|r−[r]
x6=y

!

is a Banach space.
In the case r ∈ R+ \ N, the Littlewood-Paley decomposition supplies a very simple characterization of C r :
Proposition 1.3.2. There exists a constant C such that for all r ∈ R+ \ N and u ∈ C r we
have
C r+1
sup 2qr k∆q ukL∞ ≤
kukC r .
[r]!
q
Conversely, if the sequence (2qr k∆q ukL∞ )q≥−1 is bounded then
kukC r ≤ C

r+1



1
1
+
r − [r] [r] + 1 − r



sup 2qr k∆q ukL∞ .
q

Proof: Let us just sketch the proof (for more details see [10]).
R
Let us first notice that, owing to xα h(x)dx = 0 for all multi-index α, one can write
∀q ∈ N, ∆q u(x) = 2qN

(1.7)

Z

[r]


X
1 k
h(2q (x − y)) u(y) −
D u(x)(y − x)(k) dy.
k!
k=1

Applying the [r]-th order Taylor formula for bounding the right-hand side of (1.7) and
using the fact that k∆−1 ukL∞ ≤ C kukL∞ yields the first inequality.
For
the converse inequality, we notice that since for all multi-index we have ∂ α u =
P proving
α
q ∂ ∆q u, Bernstein lemma insures that
k∂ α ukL∞ ≤

C 1+r
sup 2qr k∆q ukL∞
r − [r] q

whenever |α| ≤ [r].
Next, for all multi-index α of size [r], all (x, y) ∈ RN × RN such that |x − y| ≤ 1 and all
Q ∈ N, we have
α

α

|∂ u(x) − ∂ u(y)| ≤

Q−1
X

α

α

|∂ ∆q u(x) − ∂ ∆q u(y)| +

q=−1
3

From now on, the notation [r] stands for the integer part of r.

+∞
X
q=Q

|∂ α ∆q u(x) − ∂ α ∆q u(y)|.

1.3. LITTLEWOOD-PALEY DECOMPOSITION AND FUNCTIONAL SPACES

13

Making use of Bernstein lemma, we end up with
|∂ α u(x) − ∂ α u(y)| ≤ C r+1 sup 2qr k∆q ukL∞

Q
X

q≥0

2−q(r−[r]−1) |x − y| +

q=−1

X


2−q(r−[r]) .

q≥Q+1

Choosing Q = [− log2 |x − y|] + 1 yields the wanted inequality.
Remark. The above characterization of H¨older spaces is false if r is an integer (see exercise 1.9).

1.3.2

Besov spaces

From now on, we make the convention that for all non-negative sequence (aq )q∈Z , the notation
P
1
r r stands for sup a in the case r = ∞.
a
q q
q
q
The characterizations of Sobolev and H¨older spaces given in the previous part naturally lead
to the following definition:
Definition. Let 1 ≤ p, r ≤ ∞ and s ∈ R. For u ∈ S 0 (RN ), we set
kuk

s
Bp,r

:=

X

qs

2 k∆q ukLp

r

1
r

.

q
s is the set of temperate distributions u such that kuk s < ∞.
The Besov space Bp,r
Bp,r

Before going further into the study of Besov spaces, let us state two important lemmas. The
first one reads:
Lemma 1.3.3. Let (uq )q∈N be a sequence of bounded functions such that the Fourier transform
of uq is supported in dyadic shells. Let us assume that, for some M ≥ 0, we have
kuq kL∞ ≤ C2qM .
Then the series

P

q

uq is convergent in S 0 .

Proof: After rescaling, relation (1.3) rewrites
X
(1.8)
uq = 2−qk
2qN gα (2q ·) ? ∂ α uq .
|α|=k

Therefore, for any test function φ in S, we have
X
(1.9)
huq , φi = (−1)k 2−qk
huq , 2qN gˇα (2q ·) ? ∂ α φi with gˇ(z) = g(−z).
|α|=k

Hence
X

|huq , φi| ≤ C2−qk

2qM k∂ α φkL1 .

|α|=k

P

Let us choose k > M . Then
q huq , φi is a convergent series, the sum of which is less
than CkφkP,S for some large enough integer P. Thus the formula
X
hu, φi := lim
h∆q0 u, φi
q→∞

defines a temperate distribution.

q 0 ≤q

14

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS

The second important lemma is the following one:
Lemma 1.3.4. Let s ∈ R and 1 ≤ p, r ≤ ∞. Let (uq )q≥−1 be a sequence of functions such that
X

qs

2 kuq kLp

r

1
r

< ∞.

q≥−1

(i) If Supp
b−1 ⊂ B(0, R2 ) and Supp u
bq ⊂ C(0, 2q R1 , 2q R2 ) for some 0 < R1 < R2 then
P u
s
u := q≥−1 uq belongs to Bp,r and there exists a universal constant C such that
s
kukBp,r
≤C

1+|s|

X

qs

2 kuq kLp

r

1
r

.

q≥−1

(ii) If s is positive and Supp u
bq ⊂ B(0, 2q R) for some R > 0 then u :=
s
Bp,r and there exists a universal constant C such that
s
kukBp,r

C 1+s

s

X

qs

2 kuq kLp

r

P

q≥−1 uq

belongs to

1
r

.

q≥−1

Proof: Under the “hypothesis
of the first assertion and according to Bernstein lemma, we have

P
−s
q N
p
. Lemma 1.3.3 thus implies that q uq is a convergent series in S 0 .
kuq kL∞ ≤ C2
Next, we notice that there exists an integer N0 so that
|q 0 − q| ≥ N0 =⇒ ∆q0 uq = 0.
Therefore, with the convention that uq = 0 if q ≤ −2, we can write that
k∆q0 ukLp

X

= k

∆q0 uq kLp

|q−q 0 |<N0

X

≤ C

kuq kLp .

|q−q 0 |<N

0

So, we obtain that
0

2q s k∆q0 ukLp

≤ C

0

X

2q s kuq kLp

|q−q 0 |≤N0

X

≤ C 1+|s|

2qs kuq kLp ,

|q−q 0 |≤N0

and we deduce from convolution inequalities that
s
kukBp,r
≤C

1+|s|

X

rqs

2

kuq krLp

1
r

,

q∈N

which is exactly the first result.
For proving the second result, we first notice that for any q, we have
kuq kLp ≤ C2−qs .

1.3. LITTLEWOOD-PALEY DECOMPOSITION AND FUNCTIONAL SPACES

15

P
As s is positive, this implies that q uq is a convergent series in Lp . Next, we notice that
there exists some N1 ∈ N such that
q 0 ≥ q + N1 =⇒ ∆q0 uq = 0.
Now, we write that
k∆q0 ukLp


X
=


q>q 0 −N1



∆q 0 u q
p,
L

X

≤ C

kuq kLp .

q>q 0 −N1

So, we get that
0

X

2q s k∆q0 ukLp ≤ C

0

2(q −q)s 2qs kuq kLp .

q≥q 0 −N1

In other words, we have
0

2q s k∆q0 ukLp ≤ C(ck ) ? (d` )

with ck = 1[−N1 ,+∞[ (k)2−ks

and d` = 2`s ku` kLp .

Applying convolution inequalities for series completes the proof of the second property.
s does not depend on the choice of the couple (χ, ϕ)
Corollary. The definition of the space Bp,r
defining the Littlewood-Paley decomposition.

Remark. The Besov spaces have been obtained by taking first the Lp norm on each dyadic
block, then taking a weighted `r norm. It turns out that taking first the weighted `r norm
and next the Lp norm over RN is also relevant. This yields new Banach spaces called Triebels . Using such spaces may be appropriate for studying certain
Lizorkin spaces and denoted by Fp,r
0 coincides with the Lebesgue space Lp
problems. In particular, if 1 < p < ∞, the space Fp,2
s
and, more generally, Fp,2 coincide with the potential space Hps of temperate distributions u
s
such that (I − ∆) 2 belongs to Lp .
The reader is referred to [36] or [38] for a more complete study of Triebel-Lizorkin spaces.

1.3.3

A few properties of Besov spaces

In the following proposition, we give a first list of important properties of Besov spaces.
Proposition 1.3.5. Let s ∈ R and 1 ≤ p, r ≤ ∞.
s is a Banach space which is continuously embedded in S 0 .
(i) Topological properties: Bp,r
s if and
(ii) Density: the space Cc∞ of smooth functions with compact support is dense in Bp,r
only if p and r are finite.
s
(iii) Duality: for all s ∈ R and 1 ≤ p, r < ∞, the space Bp−s
0 ,r 0 is the dual space of Bp,r .
s
s
If 1 ≤ p < ∞, the completion Bp,∞
of Cc∞ for the norm k · kBp,∞
is the predual of Bp−s
0 ,1 .
s is separable. The same holds for B s .
(iv) Separability: If 1 ≤ p, r < ∞ then the space Bp,r
p,∞

(v) Embeddings: we have
s ,→ B se whenever s
(a) Bp,r
e < s or se = s and re ≥ r,
p,e
r
s−N ( p1 − p1e )

s ,→ B
(b) Bp,r
pe,r

whenever pe ≥ p,

16

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
N
p
0
is continuously embedded in
(c) we have B∞,1
,→ C ∩ L∞ . If p < ∞ then the space Bp,1
the space C0 of continuous bounded functions which decay at infinity.

s
(vi) Fatou property: if (un )n∈N is a bounded sequence of Bp,r
which tends to u in S 0 then
s and
u ∈ Bp,r
s
s .
kukBp,r
≤ lim inf kun kBp,r
θs+(1−θ)e
s

s ∩ B se and θ ∈ [0, 1] then u ∈ B
(vii) Complex interpolation: if u ∈ Bp,r
p,r
p,r

and

1−θ
kukB θs+(1−θ)es ≤ kukθBp,r
.
s kuk s
Be
p,r

p,r

θs+(1−θ)e
s

s
se
(viii) Real interpolation: if u ∈ Bp,∞
∩ Bp,∞
and s < se then u belongs to Bp,1
θ ∈ (0, 1) and there exists a universal constant C such that

kukB θs+(1−θ)es ≤
p,1

for all

C
kuk1−θ
.
kukθBp,∞
s
s
e
Bp,∞
θ(1−θ)(e
s −s)

s is continuously embedded in S 0 . By definition, B s is a
Proof: Let us first prove that Bp,r
p,r
0
subspace of S . Thus it suffices to prove that there exist a constant C and an integer M
such that for any φ ∈ S we have
s kφkM,S .
|hu, φi| ≤ CkukBp,r

(1.10)

Taking advantage of (1.9) with ∆q u instead of uq , we see that for all k ∈ N, there exists
an integer Mk and a constant Ck such that

|h∆q u, φi| ≤ Ck 2−q 2q(1−k) k∆q ukL∞ kφkMk ,S .
According to Bernstein lemma, we have k∆q ukL∞ ≤ C2
chosen so large as to satisfy s − N/p ≥ 1 − k, we have

qN
p

k∆q ukLp . Hence, if k has been

s kφkM ,S
|h∆q u, φi| ≤ Ck 2−q kukBp,r
k

which, after summation on q, yields inequality(1.10).
s . Inequality (1.10) implies that for
Next, consider (u(n) )n∈N a Cauchy sequence in Bp,r
any test function φ in S , sequence (hu(n) , φi)n∈N is a Cauchy sequence in R. Thus the
formula
hu, φi := lim hu(n) , φi
n→∞

s , sequence (∆ u(n) )
defines a temperate distribution. By definition of the norm of Bp,r
q
n∈N
is a Cauchy sequence in Lp for any q . Thus an element uq of Lp exists such that
(∆q u(n) )n∈N converges to uq in Lp . As the sequence (u(n) )n∈N converges to u in S 0 we
actually have ∆q u = uq .

Fix a Q ∈ N and a positive ε. Since for all q ≥ −1, ∆q u(n) tends to ∆q u in Lp , we have
for all n ∈ N,
X
q≤Q

qs

2 k∆q (u

(n)

− u)kLp

r

1
r

= lim

X

m→∞

q≤Q

qs

2 k∆q (u

(n)

−u

(m)

)kLp

r

1
r

.

1.3. LITTLEWOOD-PALEY DECOMPOSITION AND FUNCTIONAL SPACES

17

s
Because the argument of the limit in the right-hand side is bounded by ku(n) − u(m) kBp,r
(n)
s
and (u )n∈N is a Cauchy sequence in Bp,r , one can now conclude that there exists a n0
(independent of Q) such that for all n ≥ n0 , we have

X

qs

2 k∆q (u

(n)

− u)kLp

r

1
r

≤ ε.

q≤Q
s .
Letting Q go to infinity insures that (u(n) )n∈N tends to u in Bp,r
s
Let us tackle the proof of (ii). Consider first the case where r is finite. Let u be in Bp,r
and ε > 0. Since r is finite, there exists an integer q such that

ε
s
ku − Sq ukBp,r
< ·
2

(1.11)

Now let φ be in Cc∞ . For any q 0 ∈ N, Bernstein lemma insures that we have,
0

2q s k∆q0 (φSq u − Sq u)kLp

0

0

≤ 2−q 2q ([s]+2) k∆q0 (φSq u − Sq u)kLp
≤ Cs 2−q

0

k∂ α (φSq u − Sq u)kLp .

sup
|α|=[s]+2

From the above inequality, we get that
(1.12)


s
kφSq u − Sq ukBp,r
≤ Cs k(1 − φ)Sq ukLp +

sup


k∂ α ((1−φ)Sq u)kLp .

|α|=[s]+2

Let us consider a sequence (φn )n∈N such that all the derivatives of φn of order less than
or equal to [s] + 2 are uniformly bounded with respect to n and such that φn ≡ 1 in a
neighborhood of the ball B(0, n). If p is finite, combining Leibniz formula and Lebesgue
theorem, we discover that
lim k(1−φn )Sq ukLp +

n→∞

sup

k∂ α ((1−φn )Sq u)kLp = 0·

|α|=[s]+2

Thus a function φ in Cc∞ exists such that

Cs k(1 − φ)Sq ukLp +

ε
k∂ α ((1−φ)Sq u)kLp < · .
2
|α|=[s]+2
sup

Combining (1.11) and (1.12), we end up with
s
kφSq u − ukBp,r
< ε.

As Sq u is a smooth function, this completes the proof in the case p, r < ∞.
Now, it is obvious that the set Cb∞ of smooth functions with bounded derivatives at all
s . Therefore C ∞ cannot be a dense subset of B s .
orders is embedded in any space B∞,r
c
∞,r
s
Finally, the closure of Cc∞ for the Besov norm Bp,∞
is the space of temperate distributions
u such that
lim 2qs k∆q ukLp = 0,
q→∞

s . This completes the proof of (ii).
which is a strict subspace of Bp,∞

18

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
In order to prove the properties of duality, we use the fact that the map
u 7−→ (∆q u)q≥−1
s and4 `r (Lp ).
is a (bi)continuous isomorphism between Bp,r
s

Assume that 1 ≤ p < ∞. On one hand, if in addition 1 ≤ r < ∞, it is obvious that
0
0
0
`r−s (Lp ) is the dual of `rs (Lp ). On the other hand, `1−s (Lp ) is the dual space of the set of
Lp valued sequences (zq )q≥−1 such that
lim 2qs kzq kLp = 0.

q→+∞

s
Now, one can prove that Bp,∞
is the space of temperate distributions u such that

lim 2qs k∆q ukLp = 0,

q→+∞

whence the desired result.
The proof of (iv) also relies on the use of the map u 7−→ (∆q u)q≥−1 . The details are left
to the reader.
Let us now prove (v). Considering that `r (Z) ⊂ `re(Z) for r ≤ re, the first embedding is
straightforward. In order to prove the second embedding, we apply Bernstein lemma and
get


kS0 ukLpe ≤ CkS0 ukLp

qN

and k∆q ukLpe ≤ C2

1
− p1e
p

k∆q ukLp

if q ∈ N,

whence the result.
N
p
,→ Cb we use again Bernstein lemma and get that
For proving that Bp,1

k∆q ukL∞ ≤ 2

qN
p

k∆q ukLp .

P
This insures that the series
∆q u of continuous bounded functions converges uniformly on
N
R . Hence u is a bounded continuous function. Besides, it is obvious that the embedding
N

p
is continuous. If p is finite, one can use in addition that Cc∞ is dense in Bp,1
and conclude
that u decays at infinity.

s which
Let us now focus on the proof of (vi). Let (un )n∈N be a bounded sequence of Bp,r
0
tends to some u in S . This insures that for all q ∈ Z, sequence (∆q un )n∈N tends to ∆q u
in S 0 . Since (∆q un )n∈N is a bounded sequence in Lp ∩ Cb∞ , one can conclude that ∆q u
belongs to Lp ∩ Cb∞ and that

k∆q ukLp ≤ lim inf k∆q un kLp .
Now, for all Q ∈ N, we have


PQ

q=−1

2qs k∆

q ukLp

r

1



r



PQ

q=−1


≤ lim inf

2qs lim inf

PQ

q=−1

2qs k∆q un kLp

s .
≤ lim inf kun kBp,r

Letting Q go to infinity completes the proof of (vi).
4

k∆q un kLp

Here `rs stands for the set of sequences (zq )q≥−1 such that k(2qs zq )k`r < ∞.

r

r

1
r

,

1
r

,

1.3. LITTLEWOOD-PALEY DECOMPOSITION AND FUNCTIONAL SPACES

19

Property (vii) is a straightforward consequence of H¨older inequality.
For proving (viii), we write
X
X
kukB θs+(1−θ)es =
2q(θs+(1−θ)es) k∆q ukLp +
2q(θs+(1−θ)es) k∆q ukLp
p,1

q≤Q

q>Q

for some Q to be chosen hereafter.
Now, by definition of the Besov norms, we have
2q(θs+(1−θ)es) k∆q ukLp

s
≤ 2q(1−θ)(es−s) kukBp,∞
,

2q(θs+(1−θ)es) k∆q ukLp

≤ 2−qθ(es−s) kukBp,∞
.
s
e

Thus we infer that
s
kukB θs+(1−θ)es ≤ kukBp,∞
p,1

X

2q(1−θ)(es−s) + kukBp,∞
s
e

q≤Q
s
≤ kukBp,∞

X

2−qθ(es−s)

q>Q

2(Q+1)(1−θ)(es−s)
2(1−θ)(es−s) − 1

+ kukBp,∞
s
e

2−Qθ(es−s)
·
1 − 2−θ(es−s)

In order to complete the proof of (viii), it suffices to choose Q such that
kukBp,∞
s
e
s
kukBp,∞

≤ 2Q(es−s) < 2

kukBp,∞
s
e
s
kukBp,∞

.

s
s , or in other words, what happens if one lets θ tends
One can wonder how far is Bp,∞
from Bp,1
to 1 in proposition 1.3.5.(viii). Of course, one already knows that
s
s
s
Bp,1
,→ Bp,r
,→ Bp,∞
.

A more precise answer is given by the following logarithmic interpolation result:
Proposition 1.3.6. There exists a constant C such that for all s ∈ R, ε > 0 and 1 ≤ p ≤ ∞,
we have


kukBp,∞
s+ε
1+ε
s
s
kukBp,1
≤C
kukBp,∞
1 + log
.
s
ε
kukBp,∞
Proof: Let Q be a positive integer to be fixed hereafter. We have
X
X
s
kukBp,1
=
2qs k∆q ukLp +
2qs k∆q ukLp ,
q<Q

q≥Q

whence,
s
s
+
kukBp,1
≤ (Q + 1)kukBp,∞

Choosing for Q the closest positive integer to

2−Qε
kukBp,∞
s+ε .
1 − 2−ε

kukBp,∞
s+ε
1
log2
yields the result.
s
ε
kukBp,∞

We now want to study how multipliers operate on Besov spaces. Before stating our result, we
need to define the multipliers we are going to consider.

20

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS

Definition. A smooth function f : RN → R is said to be a S m -multiplier if for all multi-index
α, there exists a constant Cα such that
∀ξ ∈ RN , |∂ α f (ξ)| ≤ Cα (1 + |ξ|)m−|α| .
Proposition 1.3.7. Let m ∈ R and f be a S m -multiplier. Then for all s ∈ R and 1 ≤ p, r ≤ ∞
s to B s−m .
the operator f (D) is continuous from Bp,r
p,r
Proof: Let ϕ
e be a smooth function supported in a shell and such that ϕ
e ≡ 1 on Supp ϕ. It
is clear that we have
∆q f (D)u = ϕ(2
e −q D)f (D) ∆q u

for all

q ∈ N.

Hence, by virtue of convolution inequalities, we have
k∆q f (D)ukLp ≤ kkq kL1 k∆q ukLp
with

1
kq (x) :=
(2π)N

Z

eix·ξ f (ξ)ϕ(2
e −q ξ) dξ.

By performing an easy change of variables, we notice that kkq kL1 = k`q kL1 with
Z
1
`q (y) :=
eiy·η ϕ(η)f
e
(2q η) dη.
(2π)N
Now, for all M ∈ N, we have
(1 +

|y|2 )M `q (y)

Z


(1 − ∆η )M eiy·η ϕ(η)f
e
(2q η) dη,

Z


eiy·η (1 − ∆η )M ϕ(η)f
e
(2q η) dη,
Z

X
q|β|
cα,β 2
eiy·η ∂ α ϕ(η)
e ∂ β f (2q η) dη,

=
=
=

|α|+|β|≤2M

for some integers cα,β (whose exact value does not matter).
Hence, using the fact that integration may be restricted to Supp ϕ
e and that |∂ β f (2q η)| ≤

m−|β|
Cβ 1 + 2q |η|
, we get
(1 + |y|2 )M |`q (y)| ≤ CM 2qm .
Choosing M > N/2, we thus conclude that
kkq kL1 = k`q kL1 ≤ C2qm
whence
∀q ∈ N, 2q(s−m) kf (D)∆q ukLp ≤ C2qs k∆q ukLp .
Stating a similar inequality for q = −1 is left to the reader. This yields the proposition.
Proposition 1.3.8. A constant C exists which satisfies the following properties. Let s < 0,
s
if and only if (2qs kSq ukLp )q∈N ∈ `r (N).
(p, r) ∈ [1, ∞]2 and u ∈ S 0 . Then u belongs to Bp,r
Moreover, we have
1
s
kukBp,r

2

X
q

qs

2 kSq ukLp

r

1
r


1
s .
≤C 1+
kukBp,r
|s|

1.4. PARADIFFERENTIAL CALCULUS

21

Proof: On one hand, we have
2qs k∆q ukLp

≤ 2qs (kSq+1 ukLp + kSq ukLp )
≤ 2−s 2(q+1)s kSq+1 ukLp + 2qs kSq ukLp ,

which proves the inequality on the left. On the other hand, we can write that
X
2qs kSq ukLp ≤ 2qs
k∆q0 ukLp
q 0 ≤q−1

X



0

0

2(q−q )s 2q s k∆q0 ukLp .

q 0 ≤q−1

As s is negative, we get the result.

1.4

Paradifferential calculus

When dealing with nonlinear problems, one often has to study the functional properties of
products of two temperate distributions u and v.
Characterizing distributions such that the product uv makes sense is an intricate question
which is intimately related to the notion of wavefront (see e.g [1] for an elementary introduction).
In this section, we shall see that very simple arguments based on the use of Littlewood-Paley
decomposition yield sufficient conditions for uv to be defined, and continuity results for the map
(u, v) 7→ uv.

1.4.1

Definitions

For u and v two temperate distributions, we have the following formal decomposition:
X
uv =
∆p u∆q v.
p,q

The fundamental idea of paradifferential calculus is to split uv into three parts, both of them
being always defined. The first part, denoted by Tu v and called paraproduct of v by u corresponds to terms ∆p u ∆q v where p is small in comparison with q. The second term, Tv u is
the symmetric counterpart of Tv u (i.e. we keep only the terms corresponding to large frequencies of u multiplied by small frequencies of v ). The third and last term (the remainder term)
corresponds to the dyadic blocks of u and v with comparable frequencies.
This very simple splitting device goes back to the pioneering work by J.-M. Bony in [4]. In
what follows, we shall adopt the following definition for paraproduct and remainder:
Definition. Let u and v be two temperate distributions. We denote
X
X
Tu v =
∆p u ∆q v =
Sq−1 u ∆q v
p≤q−2

q

and
R(u, v) =

X

e qv
∆q u ∆

fq := ∆q−1 + ∆q + ∆q+1 .
with ∆

q

At least formally, we have the following Bony decomposition:
(1.13)

uv = Tu v + Tv u + R(u, v).

Of course, it may happen that the product uv is not defined. However, the reader may retain
the following principles:

22

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
• The paraproduct of two temperate distributions u and v is always defined. This is due to
the fact that the general term of the paraproduct is spectrally localized in dyadic shells.
Besides, the regularity of Tu v is mainly determined by the regularity of v. In particular,
Tu v cannot be more regular than v.
• The remainder may not be defined. Roughly, it is defined as soon as u and v belong to
functional spaces whose sum of regularity index is positive. In that case, the regularity
exponent of R(u, v) is the sum of the regularity exponents of u and v .

1.4.2

Results of continuity for the paraproduct and the remainder

The bilinear paraproduct and remainder operators benefit from continuity properties in most
usual functional spaces. In the present chapter, we focus on Besov spaces. The reader is referred
to [36] and [37] for a more complete study.
As regards the paraproduct, we have the following results:
Proposition 1.4.1. Let 1 ≤ p, r ≤ ∞ and s ∈ R.
s
s
(i) The paraproduct T is a bilinear continuous operator from L∞ × Bp,r
to Bp,r
and there
exists a constant C such that
|s|+1
s ;B s ) ≤ C
kT kL(L∞ ×Bp,r
.
p,r

(ii) If σ > 0 and 1 ≤ r, r1 , r2 ≤ ∞ are such that 1/r = 1/r1 + 1/r2 then T is bilinear
−σ × B s
s−σ
continuous from B∞,r
p,r2 to Bp,r and there exists a constant C such that
1
kT kL(B∞,r
−σ

s−σ
s
×Bp,r
;Bp,r
)
1
2



C |s−σ|+1
·
σ


Proof: According to proposition 1.2.1, the sequence F(Sq−1 u∆q v) q∈Z is supported in dyadic
shells. Hence, in view of proposition 1.3.5, it suffices to prove that
X
1
r r
qs
s .
2 kSq−1 u∆q vkLp
. kukL∞ kvkBp,r
q

hkL1 kukL∞ , this is actually straightforward. This yields the first
Since kSq−1 ukL∞ ≤ ke
result.
For proving the second result, we use that, because σ > 0, we have

X σ(q0 −q) −q0 σ
∆q0 u ∞ .
2q(s−σ) kSq−1 u∆q vkLp ≤ 2qs k∆q vkLp
2
2
L
q 0 ≤q−2

Therefore, combining H¨
older and convolution inequalities for series, we get
X
1 X

r r
q(s−σ)

2−kσ kukB∞,r
,
2
kSq−1 u∆q vkLp
−σ kvkB s
p,r
q∈Z

k≥2

1

2

whence the desired inequality.
Remark. By combining the above results of continuity for the paraproduct with the embeddings
stated in proposition 1.3.5, one can get a score of other results of continuity. For instance,
N
p

−ε

−ε , we discover that T is continuous from
by using that for all ε > 0, we have Bp11,r1 ,→ B∞,r
1
N
p

−ε

s
s−ε for all 1 ≤ p, p , r, r , r ≤ ∞ such that 1/r = 1/r + 1/r .
Bp11,r1 × Bp,r
to Bp,r
1
1 2
1
2
2

1.4. PARADIFFERENTIAL CALCULUS

23

Proposition 1.4.2. Let (s1 , s2 ) ∈ R2 and 1 ≤ p, p1 , p2 , r, r1 , r2 ≤ ∞. Assume that
(1.14)

1
1
1

+
≤ 1,
p
p1 p2
Bps11,r1

× Bps22,r2

s1 +s2 +N

1
1
1
p − p1 − p2

Then the remainder R maps
C such that
kR(u, v)k

1
1
1

+
r
r1 r2

Bp,r

in

and s1 + s2 > 0.

s1 +s2 +N
Bp,r

1
− p1 − p1
p
1
2


and there exists a constant

|s1 +s2 |+1
≤C
kukBps1 ,r kvkBps2 ,r .
1 1
2 2
s1 + s2

Proof: It suffices to treat the case where 1/r = 1/r1 + 1/r2 and 1/p = 1/p1 + 1/p2 . The
general case then follows from the embeddings of proposition 1.3.5.(v).
Now, by definition of the remainder operator, we have
X
e q v.
R(u, v) =
∆q u ∆
q


e q , the support of F ∆q u∆
fq v is included in
On one hand, by definition of ∆q and ∆
B(0, 3 · 2q+3 ). On the other hand, by virtue of H¨older inequality for functions, we have


e q vkLp ≤ 2qs1 k∆q ukLp1 2qs2 k∆
e q vkLp2 .
2q(s1 +s2 ) k∆q u ∆
Applying H¨
older inequality for series, we thus have
q(s +s )

e q vkLp r ≤ Ckuk s1 kvk s2 .
2 1 2 k∆q u ∆
Bp ,r
Bp ,r
`
1

1

2

2

As s1 + s2 > 0 has been assumed, lemma 1.3.4 yields the desired result.
Remark. By combining proposition 1.3.5.(v) with the above proposition, one can get other
results of continuity for the remainder. Moreover, condition (1.14) may be somewhat relaxed
(see exercise 1.10).

1.4.3

Results of continuity for the product

We do not aim at giving an exhaustive list of the mapping properties of (u, v) 7→ uv in Besov
spaces. As a matter of fact, memorizing such a list would be quite useless: it is actually
far wiser to appeal to results of continuity for the paraproduct and remainder, and to Bony’s
decomposition.
For example, by combining propositions 1.4.1 and 1.4.2, and the continuous embeddings
stated in proposition 1.3.5, one gets the following important results:
s ∩ L∞ is an algebra and we have
Proposition 1.4.3. Let s > 0 and 1 ≤ p, r ≤ ∞. Then Bp,r
s
s
s .
kuvkBp,r
. kukL∞ kvkBp,r
+ kvkL∞ kukBp,r

Proof: According to propositions 1.4.2 and 1.4.1, we have
s
kR(u, v)kBp,r
s
kTu vkBp,r
s
kTv ukBp,r

s kvkB 0
. kukBp,r
,
∞,∞
s ,
. kukL∞ kvkBp,r
s .
. kvkL∞ kukBp,r

0
One can easily check that L∞ ,→ B∞,∞
, hence applying Bony’s decomposition yields the
proposition.

24

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS

Proposition 1.4.4. Let 1 ≤ p1 , p2 , p3 , p4 , r ≤ ∞ and (s1 , s2 , s3 , s4 ) ∈ R4 be such that
s1 + s2 −

N
N
= s3 + s4 − ,
p1
p4

1
1
+
≤ 1,
p1 p2

Then the product is continuous from Bps11,∞ ∩ Bps22,r
kuvk

N
s1 +s2 − p
1

Bp2 ,r

2

s1 + s2 > 0

and

s1 +s2 − pN

1

to Bp2 ,r

s1 , s3 <

N
.
p1

and we have

. kukBps1 ,∞ kvkBps2 ,r + kvkBps3 ,∞ kukBps4 ,r .
1

2

3

4

Proof: By virtue of propositions 1.4.1 and 1.4.2 and embeddings, we have
kTu vk

N
s1 +s2 − p
1

. kuk

Bp2 ,r

kR(u, v)k

N
s1 +s2 − p
1

Bp2 ,r

kTv uk

N
s3 +s4 − p
3

. kuk

Bp4 ,r

kvkBps2 ,r . kukBps1 ,∞ kvkBps2 ,r ,

N
s1 − p

2

B∞,∞ 1

1

2

. kukBps1 ,∞ kvkBps2 ,r ,

N
s3 − p

B∞,∞ 3

1

2

kvkBps4 ,r . kukBps3 ,∞ kvkBps4 ,r .
4

3

4

Taking advantage of Bony’s decomposition completes the proof.
Proposition 1.4.5. For all φ ∈ S, 1 ≤ p, r ≤ ∞ and s ∈ R, the map Mφ : u → φu is
s .
continuous in Bp,r
Proof: The proof relies on Bony decomposition. Indeed, according to proposition 1.4.1 and
Besov embeddings, we have
s
kTφ ukBp,r

s
kTu φkBp,r

s ,
. kφkL∞ kukBp,r

s kuk ∞
s kukB s
. kφkBp,r
 kφkBp,r
L
p,r
.
s
kφk
kuk
.
kφk
kuk
N
N
N

Bp,r
s− p −1
p +1
p +1

Bp,r

B∞,∞

if s >
if s ≤

Bp,r

N
p,
N
p,

and, by virtue of proposition 1.4.2,
(
s
kR(u, φ)kBp,r
.

s
kφkL∞ kukBp,r

if s > 0,

kφkB∞,∞
1−s kukB s
p,r

if s ≤ 0,

which completes the proof.

1.4.4

A result of compactness in Besov spaces

One can now state a result of compactness for Besov spaces which will prove to be very useful
for solving nonlinear PDE’s.
Proposition 1.4.6. Let 1 ≤ p, r ≤ ∞, s ∈ R and ε > 0. For all φ ∈ Cc∞ , the map u 7→ φu is
s+ε to B s .
compact from Bp,r
p,r
As a preliminary step, we need to state the following result :
Lemma 1.4.7. Let a1 , · · · , aN be positive and δ ∈ (0, (mini ai )/4). There exists a constant C
s (TN ) supported in a cube of size
such that for all 1 ≤ p, r ≤ ∞ and s ∈ R, and u ∈ Bp,r
a
2πa1 − 2δ × · · · × 2πaN − 2δ, we have
X
per
per
C −1 kukBp,r
kBp,r
with
u
:=
u( · + α).
s (RN ) ≤ ku
s (TN ) ≤ CkukB s (RN )
a
p,r
α∈ZN
2πa

1.4. PARADIFFERENTIAL CALCULUS

25

Proof: With no loss of generality, one can assume that
Supp u ⊂ QN
a,δ := [δ, 2πa1 − δ] × · · · × [δ, 2πaN − δ].
Let θ be a smooth function supported in QN
a,δ/2 and equals to one on a neighborhood of
N
qN
q
Qa,δ . Denoting hq := 2 h(2 ·), we have for all q ∈ N, M ∈ N and x ∈ RN ,
∆q u(x) =
hu, θhq (x − ·)i,

= (Id−∆)−M u, (Id−∆)M θhq (x − ·) .
By virtue of Besov embeddings, one can choose M so large as to satisfy


s
Bp,r
(RN ) ,→ v ∈ S 0 (RN ) | (Id−∆)−M v ∈ L∞ (RN ) .
By taking advantage of Leibniz formula, we thus get for some large enough C,
Z
X
|∂α θ(y)||∂β hq (x − y)| dy.
|∆q u(x)| ≤ CkukBp,r
s (RN )
|α|+|β|≤2M

RN

5
If one assumes that x 6∈ QN
a then one has |x − y| ≥ |x − πa| − π|a| + δ/2 whenever y
belongs to Supp θ. Therefore one has for all K ∈ N,
X Z
C
|x − y|K |∂α θ(y)||∂β hq (x − y)| dy
|∆q u(x)| ≤
s (RN )
kukBp,r
δ K
N
|x−πa|−π|a|+ 2
|α|+|β|≤2M R

and it is easy to conclude that there exists a constant CK such that
(1.15) |∆q u(x)| ≤ CK |x−πa| − π|a| +

δ −K (2M−K)q
2
kukBp,r
for all x ∈ RN \ QN
s (RN )
a .
2

By choosing K = 2M + 1 with M large enough, one can now easily conclude that
(1.16)

k∆q ukLp (RN \QN
≤ C2−q kukBp,r
s (RN )
a )

for some constant C depending only on δ and on a.
Next, we notice that, by virtue of proposition 1.2.2, we have
X
per
∀x ∈ RN , ∆per
(x) − ∆q u(x) =
∆q u(x + α).
q u
α∈ZN
2πa \{0}

Therefore, taking M large enough and using (1.15) with K = 2M + 1, one gathers that
for all x ∈ QN
a , we have
per
|∆per
(x) − ∆q u(x)| ≤ C2−q kukBp,r
s (RN ) ,
q u

whence
(1.17)

per
k∆per
− ∆q ukLp (QN
≤ C2−q kukBp,r
s (RN ) .
q u
a )

Note that if one replaces the function hq by the function 2qN F −1 χ(2q ·) in the above
computations then one gets


per per
−q
(1.18)
max kSq ukLp (RN \QN
,
kS
u
−S
uk
p
N
s (RN ) .
q
L (Qa ) ≤ C2 kukBp,r
q
a )
5

We take the max norm in RN to simplify the computations.

26

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
Let q0 ∈ N be such that 8C2−q0 ≤ 1. From (1.16), (1.17) and (1.18), we get

1

X
r r

per per r
qs
per per
+
2 k∆q u kLp (QN
kSq0 u kLp (QN
a )
a )

q≥q0
1 kuk

X
s (RN )
r

Bp,r
r

.
− kSq0 ukrLp (RN ) +
2qs k∆q ukLp (RN )


2
q≥q0

One can easily show that


kSqper
uper krLp (QN )
0
a

+

X

2

qs

r
per
k∆per
kLp (QN
q u
a )

1
r

≈ kuper kBp,r
s (TN )
a

q≥q0

and that


kSq0 ukrLp (RN )

+

X

qs

2 k∆q ukLp (RN )

r

1
r

≈ kukBp,r
s (RN )

q≥q0

whence the desired result.
One can now prove proposition 1.4.6. According to proposition 1.4.5, we already know that
s+ε (RN ) in B s (RN ). We shall prove the compactness by decomposing T
Tφ : u 7→ φu maps Bp,r
φ
p,r
into a product of continuous maps, one of them being compact.
With no loss of generality, one can assume that φ is supported in some cube QN
a,δ which
satisfies the assumptions of proposition 1.4.7. Now, one can use the decomposition
T φ = J ◦ I ◦ Π ◦ Mφ
where
• J stands for the extension map by 0 outside QN
a from the subset of those distributions
N
N
over Ta whose restriction to Qa is supported in QN
a,δ , to the set of temperate distributions
N
N
over R supported in Qa,δ ,
s+ε (TN ) to B s (TN ),
• I is the canonical embedding from Bp,r
a
p,r
a

• Π is the map u 7→ uper introduced in proposition 1.2.2,
s+ε (RN ) with values in B s+ε (RN ).
• Mφ is the map u 7→ φu over Bp,r
p,r

According to proposition 1.4.5, the map Mφ is continuous. Besides, according to lemma 1.4.7,
s (TN ) whose
the map J (resp. Π ) is continuous from the subspace of those distributions of Bp,r
a
N , to B s (RN ) (resp. from the subspace of distributions
restriction to QN
is
supported
in
Q
a
p,r
a,δ
s+ε (RN ) supported in QN , to B s+ε (TN )).
of Bp,r
p,r
a
a,δ
We claim that I may be approximated by a sequence of operators with finite rank. This
will yield compactness.
Indeed, introduce the finite rank operator I n defined by
X
I n (v) :=
∆per
q v.
q≤n

Using proposition 1.2.1, we discover that for all q ∈ N, we have
X
n
per
∆per
∆per
q (I − I )(v) =
q ∆ v.
j>n
|j−q|≤1

1.4. PARADIFFERENTIAL CALCULUS

27

We thus have for some constant C independent of q,

n
2qs k∆per
≤ C2−nε 2q(s+ε) k∆per
,
q (I − I )(v)kLp (TN
q vkLp (TN
a )
a )
whence
−nε
.
kI − In )(v)kBp,r
kvkBp,r
s+ε
s (TN ) ≤ C2
(TN
a
a )

s+ε (TN ); B s (TN ) . The proof of proposition 1.4.6 is
This insures that In tends to I in L Bp,r
a
p,r
a
complete.

1.4.5

Results of continuity for the composition

Let us state the main result of this section:
Proposition 1.4.8. Let I be an open interval of R. Let s > 0 and σ be the smallest integer
s
such that σ ≥ s. Let F : I → R satisfy F (0) = 0 and F 0 ∈ W σ,∞ (I; R). Assume that v ∈ Bp,r
s
has values in J ⊂⊂ I . Then F (v) ∈ Bp,r and there exists a constant C depending only on s,
I, J, and N, and such that
σ
s
s .
kF (v)kBp,r
≤ C 1+kvkL∞ kF 0 kWσ,∞ (I) kvkBp,r
Proof: The proof is based on Meyer’s first linearization method (see e.g. [1], chapter 2).
Of course, one can change F for a function Fe ∈ Wσ+1,∞ (R) compactly supported in I
and such that Fe ≡ F on a neighborhood of J . So let us assume that F belongs to
Wσ+1,∞ (R) and has compact support in I.
The starting point of the proof is the following formal decomposition6
X
(1.19)
F (v) =
F (Sq0 +1 v) − F (Sq0 v).
q 0 ≥−1

According to first order Taylor’s formula, we have for all q 0 ≥ −1,
Z
F (S

q 0 +1

v) − F (S v) = m ∆ v
q0

q0

q0

with m :=
q0

1

F 0 (Sq0 v + τ ∆q0 v) dτ.

0

One can easily prove that the mp ’s are Meyer multipliers, namely


0


(1.20)
∀k ∈ {0, · · · , σ}, Dk mq0 ∞ ≤ Ck 2q k (1 + kvkL∞ )k kF 0 kWk,∞ .
L

In particular, inequality (1.20) with k = 0 implies that


F (Sq0 +1 v) − F (Sq0 v) p ≤ C2−q0 s sup 2qs k∆q vk p .
L
L
q

Since s is positive, we conclude that (1.19) holds true in Lp .
s . For all q ≥ −1, we have
We now have to prove that F (v) belongs to Bp,r

X

∆q F (v) =


X
∆ q mq 0 ∆ q 0 v .
∆ q mq 0 ∆ q 0 v +
q 0 ≥q

−1≤q 0 ≤q−1

|
6

Remind that S−1 v ≡ 0.

{z

∆1q

}

|

{z

∆2q

}

28

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
Taking advantage of Bernstein lemma, we get for all (q 0 , q) such that −1 ≤ q 0 ≤ q − 1,




∆q mq0 ∆q0 v p . 2−qσ Dσ ∆q mq0 ∆q0 v p ,
L
L
whence, combining Leibniz formula and (1.20),



0
0
2qs ∆q mq0 ∆q0 v Lp . kF 0 kWσ,∞ (I) (1 + kvkL∞ )σ 2(q −q)(σ−s) 2q σ ∆q0 v Lp .
Since σ > s, convolution inequalities enable us to conclude that
X

r
2 ∆1q Lp
qs

1
r

s
. kF 0 kWσ,∞ (I) (1 + kvkL∞ )σ kvkBp,r

q

and proposition 1.3.4 insures that

P

s .
∆1q belongs to Bp,r

Bounding the term pertaining to ∆2q is easy. Indeed, we have according to (1.20),



0
0
2qs ∆q (mq0 ∆q0 v) Lp . kF 0 kL∞ (I) 2(q−q )s 2q s ∆q0 v Lp ,
so that, since s > 0,
X

r
2 ∆2q Lp
qs

1
r

s .
. kF 0 kL∞ (I) kvkBp,r

q

Applying once again proposition 1.3.4 completes the proof.
Finally, combining propositions 1.4.3 and 1.4.8 with the following equality:
Z
F (v) − F (u) = (v − u)

1

F 0 (u + τ (v − u)) dτ,

0

we readily get the following
Corollary 1.4.9. Let I be an open interval of R and F : I → R. Let s > 0 and σ be the
smallest integer such that σ ≥ s. Assume that F 0 (0) = 0 and that F 00 belongs to Wσ,∞ (I; R).
s have values in J ⊂⊂ I. There exists a constant C = C
Let u, v ∈ Bp,r
s,I,J,N such that
σ
s
kF (v) − F (u)kBp,r
≤ C 1+kvkL∞ kF 00 kWσ,∞ (I)


s
s
× kv − ukBp,r
sup ku + τ (v − u)kL∞ + kv − ukL∞ sup ku + τ (v − u)kBp,r
.
τ ∈[0,1]

1.5

τ ∈[0,1]

Calculus in homogeneous functional spaces

Nonhomogeneous functional spaces are not the most natural spaces for studying mathematical
problems which have a property of invariance by dilation. For instance, it is well known that
the Sobolev embedding H 1 (R3 ) ,→ L6 (R3 ) involves only the L2 norm of the gradient and not
the whole H 1 norm. We shall also see in the next chapters that many interesting PDE’s have
some properties of scaling invariance.
This is a good motivation for introducing a homogeneous Littlewood-Paley decomposition
where the low frequencies are treated exactly as the high frequencies, and to define homogeneous
functional spaces.

1.5. CALCULUS IN HOMOGENEOUS FUNCTIONAL SPACES

1.5.1

29

Homogeneous Littlewood-Paley decomposition

Let (χ, ϕ) be as in section 1.2.2. The homogeneous dyadic blocks are defined by
˙ q u := ϕ(2−q D)u


for all

q ∈ Z.

We also introduce the following low frequency cut-off:
S˙ q u := χ(2−q D)

for all

q ∈ Z.

The above definition deserves two important preliminary remarks:
P
˙ q u = u modulo a polynomial only.
• For u ∈ S 0 , we have q∈Z ∆
P
˙ p u.
• In contrast with the nonhomogeneous case, we do not have S˙ q u = p≤q−1 ∆
In light of the first remark, working with distributions modulo polynomials seems to be the most
appropriate choice. As a matter of fact, this is the viewpoint of most authors of textbooks on
abstract functional analysis (see e.g. [38] or [36]).
Since in the next chapters, we aim at applying homogeneous Littlewood-Paley decomposition
for solving nonlinear PDE’s however, it is not suitable to work with distributions defined modulo
polynomials. This motivates the following definition (after J.-Y. Chemin in [12]):
Definition. We denote by Sh0 the space of temperate distributions u such that
lim S˙ j u = 0

j→−∞

in S 0 .

Remarks. (i) A polynomial u does not belong to Sh0 unless it is identically 0. Indeed, if u is
a polynomial then we have S˙ j u = u for all j in Z.
(ii) It is obvious that Sh0 is the space of temperate distributions u which satisfy
X
˙ j u in S 0 .
u=

j

(iii) The space Sh0 is not a closed subspace of S 0 for the topology of weak convergence. Indeed,
consider a sequence (fn )n∈N with fn (x) = f (x/n) and f ∈ S such that f (0) = 1. Then
(fn )n∈N tends weakly to the constant function 1 which does not belong to Sh0 .
Examples. (i) If a temperate distribution u is such that its Fourier transform u
b is locally
0
integrable near 0, then u belongs to Sh .
(ii) If u is a temperate distribution such that for some function θ in Cc∞ (RN ) with value 1
near the origin, we have θ(D)u in Lp for some p ∈ [1, +∞[, then u belongs to Sh0 . In
s is included in S 0 .
particular, when p if finite, any nonhomogeneous Besov space Bp,r
h

1.5.2

Homogeneous Besov spaces

Definition 1.5.1. Let u be a temperate distribution, s a real number, and 1 ≤ p, r ≤ ∞. Then
we set
X
1
r
rqs ˙
r
2 k∆q ukLp
kukB˙ s :=
p,r

q∈Z

with the usual change if r = ∞.
For the semi-norms we have defined, we can prove the following inequalities

30

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS

Theorem 1.5.2. A constant C exists such that for all s ∈ R,
r1 ≤ r2 ⇒ kukB˙ s

≤ kukB˙ s ,

p,r2

p1 ≤ p2 ⇒ kuk

p,r1

1 − 1
p2

(p
B˙ p2 ,r 1
s−N

) ≤ CkukB˙ ps1 ,r .

Moreover we have the following interpolation inequalities for any θ in ]0, 1[ and s < se:
kukB˙ θs+(1−θ)es ≤ kukθB˙ s kuk1−θ
,
B˙ se
p,r

kukB˙ θs+(1−θ)es
p,1

p,r

p,r

C
kukθB˙ s kuk1−θ
.

s
e
B˙ p,∞
p,∞
θ(1 − θ)(e
s − s)

The proof is the same as in the nonhomogeneous framework, and thus omitted. Finally, the
following logarithmic interpolation inequalities are available:
Proposition 1.5.3. There exists a constant C such that for all s ∈ R, ε > 0 and 1 ≤ p ≤ ∞,
we have

kuk s−ε + kuk s+ε
1+ε
B˙ p,∞
B˙ p,∞
≤ C
kukB˙ s
(1.21)
1 + log
,
kukB˙ s
p,∞
p,1
ε
kukB˙ s
p,∞

(1.22)

kukB˙ s

p,1

≤ C

1+ε
ε

1 + kukB˙ s

p,∞

log e + kukB˙ p,∞
s−ε + kuk ˙ s+ε
Bp,∞



.

Proof: The proof is almost the same as in the nonhomogeneous framework. We write for
some positive integer Q to be fixed hereafter:
X
X
X
2qs k∆q ukLp +
2qs k∆q ukLp +
2qs k∆q ukLp ,
kukB˙ s =
p,1

q≤−Q

q≥Q

|q|<Q

whence, from elementary computations,

2−Qε
kuk
s−ε + kuk ˙ s+ε .
˙
B
B
p,∞
p,∞
p,∞
p,1
1 − 2−ε
kuk s−ε + kuk s+ε
1
B˙ p,∞
B˙ p,∞
yields (1.21).
Choosing for Q the closest positive integer to log2
ε
kukB˙ s
kukB˙ s ≤ (2Q − 1)kukB˙ s

+

p,∞

The second inequality may be easily deduced from (1.21).
Homogeneous Besov spaces have some invariance properties by dilation. More precisely, if for
any temperate distribution u and λ > 0, we introduce the temperate distribution uλ defined
for all x ∈ RN by uλ (x) := u(λx), then we have
Proposition 1.5.4. If kukB˙ s

p,r

is finite, so is kuλ kB˙ s

p,r

kuλ kB˙ s ≈ λ

s− N
p

and we have

kukB˙ s .
p,r

p,r

Besides, equality is true if λ = 2m for some m ∈ Z.
Proof: Let α be a positive real. We have
ϕ(α

−1

D)uλ (x) = α

N

Z
h(α(x − y))u(λy) dy.

1.5. CALCULUS IN HOMOGENEOUS FUNCTIONAL SPACES

31

By the change of variables z = λy , we get that
Z
ϕ(α−1 D)uλ (x) = αN λ−N h(αx − αλ−1 z)u(z) dz,


= ϕ(λα−1 D)u (αx).
For q ∈ Z, let us denote vq := ϕ(2[log2 λ]−log2 λ−q D)uλ . Taking α = 2q−[log2 λ] λ in the above
equality, we get
s− N
˙ q−[log λ] ukLp
2qs kvq kLp ≈ λ p 2(q−[log2 λ])s k∆
2
with an equality if log2 λ is an integer.
Thus we deduce that
X

qs

2 kvq kLp



1
r

≈λ

s− N
p

kukB˙ s .
p,r

q
q
Since Supp vbq ⊂ C(0, 43 2q , 16
3 2 ), we have

˙ q uλ =


X

˙ p uq .


|p−q|≤2

Now, straightforward computations yield the desired result.
Next, we notice that k · kB˙ s

is actually only a semi-norm in the sense that, if u is a polynomial
˙ q u = 0, (because the support of u
then, for any integer q we have ∆
b is the origin), and so
s
˙
kukB˙ s = 0. Therefore, if we define the homogeneous Besov space Bp,r as the set of temperate
p,r
distributions u such that kukB˙ s is finite, we may run into troubles later when studying non
p,r
linear problems because we will not be able to tell a polynomial from a null function !
In the present lecture notes, we adopt the following definition:
p,r

s
Definition. Let s be a real number and (p, r) be in [1, ∞]2 . The space B˙ p,r
is the set of
0
distributions u in Sh such that kukB˙ s is finite.
p,r

In the homogeneous framework, the results corresponding to proposition 1.3.4 read:
Proposition 1.5.5. Let s ∈ R and 1 ≤ p, r ≤ ∞ satisfy
(1.23)

s<

N
,
p

or

s=

N
p

and

r = 1.

(i) Let (uq )q∈Z be a sequence of functions such that
X

qs

2 kuq kLp

r

1
r

< ∞.

q

If Supp u
bq ⊂ C(0, 2q R1 , 2q R2 ) for some 0 < R1 < R2 then u :=
and there exists a constant C such that
X
1
r r
1+|s|
qs
(1.24)
kukB˙ s ≤ C
2 kuq kLp
.
p,r

q

Let (uq )q∈Z be a sequence of functions such that
X
q

qs

2 kuq kLp

r

1
r

< ∞.

P

q∈Z uq

s
belongs to B˙ p,r

32

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS
If Supp u
bq ⊂ B(0, 2q R) for some positive R and if in addition s is positive then u :=
P
˙s
q∈Z uq belongs to Bp,r and there exists a constant C such that
kukB˙ s

(1.25)

p,r

1

r r
C 1+s X qs
2 kuq kLp
.

s
q

Proof: Let us prove (i).
P
On one hand, using Bernstein lemma, it is easy to see that the series q≤0 uq is convergent
P
0
in L∞ and that u− :=
q≤0 uq belongs to Sh . On the other hand, since kuq kL∞ ≤
P
q( N −s)
0
, lemma 1.3.3 insures that the series
C2 p
q>0 uq is convergent in S . Besides, the
P
Fourier transform of u+ := q>0 uq is supported away from the origin, hence u+ belongs
to Sh0 . Since u = u− + u+ , one can conclude that u belongs to Sh0 . Now, the proof of
(1.24) may be done exactly as in lemma 1.3.4.
Proving (ii ) is similar. Indeed, we have for any q ∈ Z,
kuq kLp ≤ C2−qs

q( N
−s)
p

kuq kL∞ ≤ C2

and

.

P
P
∞ and belongs to S 0 whereas
p
Hence
q≤0 uq converges in L
q>0 uq converges in L
h
0
(and also belongs to Sh since its Fourier transform is bounded away from 0). Next, the
proof of (1.25) goes along the lines of lemma 1.3.4.
As an important corollary of proposition 1.5.5, we get that, under condition (1.23), the definition
s does not depend on ϕ.
of B˙ p,r
The following proposition, the proof of which is left to the reader, describes the relations
between homogeneous and nonhomogeneous spaces.
s ,→ B s .
Proposition 1.5.6. Let s be a negative number (or s = 0 and r = 1). Then B˙ p,r
p,r
s ,
Besides, if s < 0, a constant C (independent of s) exists so that, for any u belonging to B˙ p,r
we have
C
s
kukBp,r

kukB˙ s .
p,r
−s
s ,→ B
s
˙ p,r
when p is finite, and
Let s be a positive number (or s = 0 and r = ∞). Then Bp,r
s
0
s
B∞,r ∩ Sh is a subset of B˙ ∞,r . If s > 0, a constant C exists (independent of s) so that, for any
s , we have
u belonging to Bp,r
C
s .
kukB˙ s ≤ kukBp,r
p,r
s
Let us notice that there is no monotonicity property with respect to s for homogeneous
spaces. The reason why is that homogeneous Besov spaces carry on informations about both
low and high frequencies.
In homogeneous spaces, the counterpart of proposition 1.3.8 reads:

Proposition 1.5.7. There exists a constant C which satisfies the following properties. Let
s if and only if
s < 0, (p, r) ∈ [1, ∞]2 and u a distribution in Sh0 . Then u belongs to B˙ p,r
(2qs kS˙ q ukLp )q∈Z ∈ `r (Z).
Moreover, we have
1
kukB˙ s ≤
p,r
2

X
q

k(2 kS˙ q ukLp )q
qs

r

1
r


1
≤C 1+
kukB˙ s .
p,r
|s|

1.5. CALCULUS IN HOMOGENEOUS FUNCTIONAL SPACES

33

Proof: The proof goes along the lines of proposition 1.3.8.
Proposition 1.5.8. Let f be a smooth function on RN \ {0} which is homogeneous of degree
m. Let 1 ≤ p, r ≤ ∞. Assume that
s−m<

N
,
p

or

r=1

and

s−m≤

N
·
p

s to B
s−m .
˙ p,r
Then f (D) is continuous from B˙ p,r

Proof: Assume that r > 1. Introduce a smooth function ϕ
e supported in a shell and such
that ϕ
e ≡ 1 on Supp ϕ. As f is homogeneous of degree m, we have
˙ q f (D)u = 2qm [ϕf
˙ q u.

e ](2−q D)∆
Since F −1 (ϕf
e ) belongs to S, we readily get




˙

˙
∆q f (D)u p ≤ C2qm ∆
q u

Lp

L

whence
X

˙ q f (D)ukLp
2q(s−m) k∆

r

,

1
r

. kukB˙ s .
p,r

q

Now, we have
S˙ q f (D)u =

X

˙ q0 u,
S˙ q f (D)∆

q 0 ≤q

whence according to Bernstein lemma,



X q0 ( N +m−s) 0
˙


qs ˙
p
0
S
f
(D)u
.
2
2

u
q

q
L

Lp

q 0 ≤q

.2

+m−s)
q( N
p

kukB˙ s .
p,∞

Since N/p + m − s > 0, it is now clear that S˙ q f (D)u tends to 0 when q goes to −∞.
s−m . The proof in the case r = 1 is left to the reader.
Hence f (D)u belongs to B˙ p,r
s .
Let us now focus on the topological properties of the spaces B˙ p,r
s ,k · k
Proposition 1.5.9. For all s ∈ R and 1 ≤ p, r ≤ ∞, the couple (B˙ p,r
s ) is a normed
B˙ p,r
s is densely embedded in B
s .
˙ p,r
space. If besides r is finite then Cc∞ ∩ B˙ p,r

Proof: It is obvious that k · kB˙ s

is a semi-norm. Let us assume that kukB˙ s = 0 for some
p,r
u in
This implies that Supp u
b ⊂ {0} and thus that for any j ∈ Z we have S˙ j u = u.
As u belongs to Sh0 , we must have lim S˙ j u = 0 so that we can conclude that u = 0.
p,r

Sh0 .

j→−∞

s , it is obvious that the sequence of general term
Now, if r is finite and u ∈ B˙ p,r
X
∆q u
|q|≤n
s and tends to u in B
s . Arguing like in 1.3.5.(ii), it is then easy to
˙ p,r
belongs to C ∞ ∩ B˙ p,r
s which tends to u in B
s .
˙ p,r
exhibit a sequence of functions of Cc∞ ∩ B˙ p,r

Theorem 1.5.10. If s < Np or s =
is continuously embedded in S 0 .

N
p

s ,k·k
and r = 1 then (B˙ p,r
B˙ s ) is a Banach space which
p,r

34

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS

s
Proof: Let us first prove that B˙ p,r
is continuously embedded in S 0 . The case r = 1 and
P ˙
P ˙
s = N/p is easy because the series
∆j u is convergent in L∞ . As u = j ∆
j u, this

implies that u belongs to L . Besides, we have
N

p
0
,→ B˙ ∞,1
,→ L∞ ,→ S 0 .
B˙ p,1

(1.26)

N

s− p
s ,→ B
˙ ∞,∞
and arguing like for proving
Let us now assume that s < N/p. Using that B˙ p,r
(1.10), one can find a large integer M such that for all nonnegative j, we have

˙ j u, φi| ≤ 2−j kuk
|h∆

s− N

p
B˙ ∞,∞

kφkM,S .

For negative j , one can write that for large enough M , we have
˙ j u, φi| . 2j
|h∆



j



. 2

Because u belongs to Sh0 , we have hu, φi =

N
p

N
p

−s

−s



kuk

s− N

p
B˙ ∞,∞

kφkL1



kukB˙ s kφkM,S .
p,r

˙

P

j h∆j u, φi.

Therefore, for large enough M ,

|hu, φi| ≤ Cs kukB˙ s kφkM,S

(1.27)

p,r

s ,→ S 0 .
and we can conclude that B˙ p,r

We still have to prove that for all triplet (s, p, r) satisfying the hypothesis of the theorem,
s
s .
the set B˙ p,r
is a Banach space. So let us consider a Cauchy sequence (un )n∈N in B˙ p,r
Using (1.26) or (1.27), this implies that a temperate distribution u exists such that the
sequence (un )n∈N converges to u in S 0 . We now have to state that u belongs to Sh0 . Let
us first assume that s < N/p. Since un belongs to Sh0 , we have, thanks to (1.27),
∀j ∈ Z , ∀n ∈ N , |hS˙ j un , φi| . 2

j



N
p

−s



kun kB˙ s kφkM,S .
p,r

As the sequence (un )n∈N tends to u in S 0 , we have
j
∀j ∈ Z , |hS˙ j u, φi| ≤ Cs 2



N
p

−s



kφkM,S sup kun kB˙ s .
n

p,r

Thus u belongs to Sh0 .
N

p
The case when u belongs to B˙ p,1
is a little bit different. Let ε > 0. As (un )n∈N is a
N

p
0 , there exists an integer n such that
Cauchy sequence in B˙ p,1
,→ B˙ ∞,1
0

∀j ∈ Z , ∀n ≥ n0 ,

X
k≤j

Let us choose j0 small enough so that
X
k≤j0

˙ k un kL∞ ≤
k∆

ε X ˙
+
k∆k un0 kL∞ .
2
k≤j

˙ k un kL∞ ≤ ε ·
k∆
0
2

As un belongs to Sh0 , we have
∀j ≤ j0 , ∀n ≥ n0 , kS˙ j un kL∞ ≤ ε.

1.5. CALCULUS IN HOMOGENEOUS FUNCTIONAL SPACES

35

As sequence (un )n∈N tends to u in L∞ , this implies that
∀j ≤ j0 , kS˙ j ukL∞ ≤ ε.
This proves that u belongs to Sh0 . Next, arguing like in the nonhomogeneous case completes the proof.
s is no longer a
Remark. It turns out that when s > N/p (or s = N/p and r > 1) the space B˙ p,r
Banach space. In the one dimensional case for instance, it is can be easily seen that the sequence
(fn )n∈N defined by

fbn (ξ) =

χ(ξ)
ξ log |ξ|

if |ξ| ≥ 2−n ,

and 0

1

elsewhere,

1

2
2
is a Cauchy sequence in B˙ 2,∞
but cannot have a limit in B˙ 2,∞
since the function ξ 7→ ξ χ(ξ)
log |ξ|
does not belong to S 0 !
This defect of convergence due to low frequencies is typical to homogeneous functional spaces
with high regularity index. It is sometimes called infrared divergence.
There is a way to modify the definition of homogeneous Besov spaces so as to get a Banach
space regardless of the regularity index. This is called realizing homogeneous Besov spaces. It
turns out that realizations coincide with definition 1.5.2 when s < N/p or s = N/p and r = 1.
In the other cases however, realizations are not functional spaces but spaces defined up to a
polynomial whose degree depends on s − N/p and on r (see e.g [5] or [32]). It goes without
saying that solving PDE’s in such spaces may be quite unpleasant.

1.5.3

Paradifferential calculus in homogeneous spaces

We designate homogeneous paraproduct of v by u and denote by T˙u v the bilinear operator:
X
˙ q v.
S˙ q−1 u ∆
T˙u v :=
q

˙
We designate homogeneous remainder of u and v and denote by R(u,
v) the bilinear operator:
X
˙
˙ pu ∆
˙ q v.
R(u,
v) =

|p−q|≤1

It is clear that, formally, we have the following homogeneous Bony decomposition:
(1.28)

˙
uv = T˙u v + T˙v u + R(u,
v).

The properties of continuity of homogeneous paraproduct and remainder on homogeneous Besov
spaces are described in the following propositions.
Proposition 1.5.11. There exists a constant C such that for any couple of real numbers (s, σ)
with σ positive and for any (p, r, r1 , r2 ) in [1, +∞]4 with 1/r = 1/r1 + 1/r2 , we have
kT˙ kL(L∞ ×B˙ s

˙s
p,r ;Bp,r )

≤ C |s|+1

if condition (1.23) is satisfied, and
kT˙ kL(B˙ ∞,r
−σ

s−σ
s
×B˙ p,r
;B˙ p,r
)
1
2

if s − σ < N/p, or s − σ = N/p and r = 1.



C |s−σ|+1
·
σ

36

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS

˙ q v. Arguing like in proposition 1.4.1, one can easily prove that
Proof: Let wq := S˙ q−1 u ∆
X

qs

2 kwq kLp

r

1
r

. kukL∞ kvkB˙ s

p,r

and

X

2

q(s−σ)

kwq kLp

r

1
r

.
. kukB˙ ∞,r
−σ kvk ˙ s
B
1

q

q

p,r2

Since the sequence (Fwq )q∈Z is supported in dyadic shells, applying proposition 1.5.5
completes the proof.
Proposition 1.5.12. There exists a constant C which satisfies the following inequalities. For
any (s1 , s2 ), any 1 ≤ p1 , p2 , p ≤ ∞ and any 1 ≤ r, r1 , r2 ≤ ∞ such that
1
1
1

+
≤1
p
p1 p2

s1 + s2 > 0,

and

1
1
1

+
≤ 1,
r
r1 r2

we have
˙
s
kRk
L(B˙ p1 ,r
1

s
σ12
×B˙ p22 ,r2 ;B˙ p,r
)
1



C |s1 +s2 |+1
s1 + s2


with

σ12 := s1 + s2 − N

1
1
1
+

p1 p 2 p


,

provided that σ12 < N/p, or σ12 = N/p and r = 1.
fq v and arguing like in proposition 1.4.2, we easily get
Proof: Denoting wq := ∆q u ∆
X

2

qσ12

kwq kLp

r

1
r

. kukB˙ ps1 ,r kvkB˙ ps2 ,r .
1

q

1

2

2

Since the sequence the sequence (Fwq )q∈Z is supported in dyadic balls, applying proposition 1.5.5 completes the proof.
Proposition 1.5.13. Let I be an open interval of R. Let s > 0 and σ be the smallest integer
s
such that σ ≥ s. Let F : I → R satisfy F (0) = 0 and F 0 ∈ Wσ,∞ (I; R). Assume that v ∈ B˙ p,r
s
has values in J ⊂⊂ I. Assume that condition (1.23) is satisfied. Then F (v) ∈ B˙ p,r
and there
exists a constant C = Cs,I,J,N such that
σ
kF (v)kB˙ s ≤ C 1+kvkL∞ kF 0 kWσ,∞ (I) kvkB˙ s .
p,r

p,r

Proof: Arguing like in the proof of proposition 1.4.8 and using Bernstein lemma, we get for
all j ∈ Z,



˙

F (Sj+1 v) − F (S˙ j v) p ≤ C2−js 2js k∆j vkLp ,

L

˙

j( N −s)
2js k∆j vkLp .
F (Sj+1 v) − F (S˙ j v) ∞ ≤ C2 p
L

Since s is positive, F (0) = 0 and condition (1.23) is satisfied, this insures that the series
X
F (S˙ j+1 v) − F (S˙ j v)
j∈Z

converges to F (v) in Lp + L∞ , thus in S 0 .
˙ q F (v) into
Next, for all q ∈ Z, we split ∆
X
X
˙ q F (v) =
˙ j v) +
˙ q (mj ∆
˙ j v).

∆q (mj ∆

j<q

j≥q

Following the lines of the proof of proposition 1.4.8 and applying proposition 1.5.5 leads
to the desired result.

1.6. EXERCISES

1.6

37

Exercises

Exercise 1.1. Prove that for any α > 1, there exists two smooth functions ϕ and χ such
that ϕ is supported in the shell {ξ ∈ RN | α−1 ≤ |ξ| ≤ 2α} and χ is supported in the ball
{ξ ∈ RN | |ξ| ≤ α}, and
X
∀ξ ∈ RN , χ(ξ) +
ϕ(2−q ξ) = 1.
q∈N

Exercise 1.2. Prove that for all temperate distribution u, the equality u =
true in S 0 (RN ).

P

q∈Z ∆q u

holds

Exercise 1.3. Prove proposition 1.2.1.
Exercise 1.4. Prove inequality (1.5).
Exercise 1.5. For q ∈ Z, denote ∆0q u := 12q ≤|ξ|≤2q+1 (D)u. Prove that the inequality
0
∆q u p ≤ C kuk p
L
L
for some constant C independent of q is false in the case p 6= 2.
Hint: Try with the function u = χ.
0 with the Lebesgue space Lp .
Exercise 1.6. Compare the Besov space Bp,r
s
s
s
Exercise 1.7. Let Bp,∞
be the completion of Cc∞ for the k · kBp,∞
norm. Prove that u ∈ Bp,∞
if and only if limq→+∞ 2qs k∆q ukLp = 0.
r−1 (RN ) is the set of temperate distributions u
Exercise 1.8. Let r ∈ (0, 1). Prove that B∞,∞
such that there exist N + 1 functions u0 , · · · , uN of C r verifying

u = u0 +

N
X

∂j uj .

j=1

Exercise 1.9. Let C?1 be the Zygmund space of bounded functions u such that
∀(x, y) ∈ RN × RN , |u(x + y) + u(x − y) − 2u(x)| ≤ C|y|
for some constant C.
1
Prove that C?1 = B∞,∞
.
Exercise 1.10. Let 1 ≤ p, p1 , p2 , r ≤ ∞ and s ∈ R. Let r0 := r/(r − 1). Assume that
1/p ≤ 1/p1 + 1/p2 .

−N

Prove that the remainder R maps Bps1 ,r × Bp−s
0 in Bp,∞
2 ,r

1
− p1 − p1
p
1
2

.

Exercise 1.11. Prove (1.20).
Exercise 1.12. Let I be an open interval of R and F : I → R. Assume that F (0) = 0 and
that F 0 is bounded.
0 with values in J ⊂⊂ I , the function F (v) belongs to B 0 and
Prove that for all v ∈ Bp,1
p,1
that
kF (v)kBp,1
. kF 0 kL∞ (I) kvkBp,1
0
0 .
Exercise 1.13. Prove that u belongs to Sh0 if and only if for any θ in Cc∞ (RN ) with value 1
near the origin, we have lim θ(λD)u = 0 in S 0 .
λ→∞

38

CHAPTER 1. AN INTRODUCTION TO FOURIER ANALYSIS

Exercise 1.14. Prove proposition 1.5.6.
Exercise 1.15. We say that a temperate distribution u tends weakly to 0 at infinity if u(λ ·)
tends to 0 in S 0 when λ goes to infinity.
Prove that u ∈ S 0 belongs to Sh0 if and only if u tends weakly to 0 at infinity.
Exercise 1.16. Let s be in ]0, N [. Prove that for any p in [1, ∞], we have
N
−s
1
p
˙ p,∞

B
.
s
|·|

s when s > N/p or s = N/p
Exercise 1.17. Find divergent Cauchy sequences for the space B˙ p,r
and r > 1.

Exercise 1.18. Let 1 ≤ p1 , r1 , p2 , r2 ≤ ∞ and (s1 , s2 ) ∈ R2 . Assume that
s1 <

N
p1

or s1 =

N
and r1 = 1.
p1

Prove that Lp1 ∩ B˙ ps22,r2 and B˙ ps11,r1 ∩ B˙ ps22,r2 are Banach spaces.
Exercise 1.19. Let 1 ≤ p, r ≤ +∞, s ∈ R and ε > 0. Let φ ∈ Cc∞ (RN ).
s to B
s−ε + B
s .
˙ p,r
˙ p,r
Prove that the map u 7→ φu is compact from B˙ p,r
Exercise 1.20. Let s be a positive real number and (p, r) ∈ [1, ∞]2 . Prove the existence of a
constant C such that for all u in Sh0 , we have


s t∆

C −1 kukB˙ p,r
ukLp r + dt ≤ CkukB˙ p,r
−2s ≤ kt e
−2s .
L (R ,

t

)

Hint: use that for any positive s and c, we have
X
2j
sup
ts 22js e−ct2 < ∞.
t>0

j∈Z

Exercise 1.21. Let s be in ]0, 1[ and (p, r) ∈ [1, ∞]2 . Prove that there exists a constant C
such that for any u in Sh0 , we have
kτ u − uk p
−z
L
C −1 kukB˙ s ≤
r N dz ≤ CkukB˙ p,r
s
p,r
|z|s
L (R ; N )
|z|
Exercise 1.22.

1) Let (s1 , s2 ), (p1 , p2 , p) and (r1 , r2 ) be such that
1
1
1
1
1

+
≤ 1 and
+
= 1.
p
p1 p2
r1 r2


:= s1 + s2 − N p11 + p12 − p1 satisfies σ12 < N/p, or σ12 = N/p and
s1 + s2 ≥ 0,

Assume that σ12
r = 1.

σ12 and that
Prove that the remainder is continuous from B˙ ps11,r1 × B˙ ps22,r2 to B˙ p,∞

˙
s
kRk
L(B˙ p1 ,r
1

1

s

σ

12 )
×B˙ p22 ,r2 ;B˙ p,∞

≤ C |s1 +s2 |+1 .

2) Adapt exercise 1.12 to the homogeneous framework.

Chapter 2

The heat equation
In this chapter, we state estimates in Besov spaces for the heat equation. Such estimates are
fundamental for solving certain nonlinear PDE’s of parabolic type. As an example, we show
that incompressible Navier-Stokes equations are locally well-posed in Besov spaces with critical
index of regularity.

2.1

Generalities

The basic heat equation reads

(H)

∂t u − µ∆u = f,
u|t=0 = u0 .

Above, the external source term f = f (t, x) and the initial data u0 = u0 (x) are given. The
diffusion µ is a positive constant. We restrict ourselves to the evolution for positive times t,
and, for the sake of simplicity, we always assume that x belongs to the whole space RN . Similar
result would hold in the torus TN
a , though.
Giving an exhaustive list of properties of the heat equation is not our goal here. We however
have to recall a few important facts for (H) that will be needed for stating estimates in Besov
spaces.
Let u0 ∈ S(RN ) and f ∈ C(R+ ; S(RN )). Applying the partial Fourier transform with respect
to the space variable (still denoted by b ), we get the following linear ordinary differential
equation for all ξ ∈ RN :

(H)

∂t u
b(t, ξ) + µ|ξ|2 u
b(t, ξ) = fb(t, ξ),
u
b|t=0 (ξ) = u
b0 (ξ),

whence
(2.1)

−µ|ξ|2 t

u
b(t, ξ) = e

Z
u
b0 (ξ) +

t

e−µ|ξ|

2 (t−τ )

fb(τ, ξ) dτ.

0

Performing the inverse Fourier transform, we end up with the following well known representation
formula for all t ∈ R+ and x ∈ RN :
Z

Z
Z t
|x−y|2
|x−y|2
1
1
− 4µ(t−τ )
− 4µt
(2.2) u(t, x) =
e
u
(y)
dy
+
e
f
(τ,
y)
dy
dτ.
0
N
N
0 (4πµ(t−τ )) 2
RN
(4πµt) 2 RN
By arguing by duality, formulae (2.1) and (2.2) may be extended to temperate distributions.
39

40

CHAPTER 2. THE HEAT EQUATION

Introducing the heat semi-group es∆
rewritten in a more concise way:
µt∆

(2.3)

u(t) = e


s≥0

for the Laplacian operator, equality (2.2) may be
Z

t

u0 +

eµ(t−τ )∆ f (τ ) dτ.

0

Although formulae (2.1) and (2.2) are explicit, they are not so convenient for getting a priori
estimates in functional spaces. In fact, even for stating the basic energy equality, it is far more
efficient to multiply (H) by u, integrate over RN and perform an integration by parts in the
term with the Laplacian. At least formally, we end up with
Z
1d
kuk2L2 + µ kDuk2L2 = f u dx,
2 dt
thus, integrating over the time interval [0, T ], we get
Z T
Z
2
2
2
ku(T )kL2 + 2µ
kDu(t)kL2 dt = ku0 kL2 + 2

0

0

TZ

f (t, x)u(t, x) dt dx.
RN

We notice that starting from u0 ∈ L2 and f ∈ L2 (0, T ; H˙ −1 ), the above equality provides an
estimate for Du in L2 (0, T × RN ). In other words, it gives a gain of one derivative for u with
respect to u0 and of two derivatives with respect to f. One can wonder if it is possible to gain
more than one derivative with respect to u0 by considering Lρ norms in time (ρ < 2) with
values in Sobolev spaces.
In the next section, we shall see that very simple arguments based on Littlewood-Paley
decomposition enable us to gain two derivatives when taking a L1 norm in time. Besides, the
method we are going to present apply indistinctly to the Lp framework for all p ∈ [1, +∞].

2.2

A priori estimates in Besov spaces for the heat equation

The fundamental idea is to localize the heat equation through a Littlewood-Paley decomposition.
˙ q u0
It is then easy to prove Lρ (0, T ; Lp ) estimates for each dyadic block in term of norms of ∆
˙ q f. If one assumes that u0 and f belong to some Besov spaces, performing a (weighted)
and ∆
r
` summation is the most natural next step. In doing so however, one does not obtain an
s ) since the time integration has been performed before
estimate in a space of type Lρ (0, T ; B˙ p,r
the summation.
This leads to the definition of the following spaces first introduced by J.-Y. Chemin and N.
Lerner in [13] then extended in [9].
Definition. For T > 0, s ∈ R, 1 ≤ r, ρ ≤ ∞, we set (with the usual convention if r = ∞):
X
1
r r
qs ˙
.
kukLeρ (B˙ s ) :=
2 k∆q ukLρ (Lp )
T

p,r

T

q

s ) as the set of temperate distributions u over (0, T ) × RN
e ρ (B˙ p,r
We then define the space L
T
0
such that lim S˙ q u = 0 in S (0, T × RN ) and kukLeρ (B˙ s ) < ∞.
q→−∞

T

p,r

In a similar way, we set
kukLeρ (B˙ s

p,r )

:=

X

˙ q ukLρ (R+ ;Lp )
2 k∆
qs

r

1
r

,

q

s ) as the set of temperate distributions u over R+ × RN such that
e ρ (B˙ p,r
and define the space L
lim S˙ q u = 0 in S 0 (R+ × RN ) and kukLeρ (B˙ s ) < ∞.
q→−∞

p,r

2.2. A PRIORI ESTIMATES IN BESOV SPACES FOR THE HEAT EQUATION
Remark 2.2.1.

(i) According to Minkowski inequality, we have

kukLeρ (B˙ s
T

41

p,r )

≤ kukLρ (B˙ s

p,r )

T

if

r ≥ ρ,

kukLeρ (B˙ s
T

p,r )

≥ kukLρ (B˙ s
T

if

p,r )

r ≤ ρ.

(ii) All the properties of continuity for the product, composition, remainder and paraproduct
s ). The general principle
e ρ (B˙ p,r
stated in chapter 1 may be easily generalized to the spaces L
T
is that the time exponent ρ behaves according to H¨
older inequality.
For instance, we have the following tame estimate :
kuvkLeρ (B˙ s
T

p,r )

. kukLρ1 (L∞ ) kvkLeρ2 (B˙ s
T

T

p,r )

+ kvkLρ3 (L∞ ) kukLeρ4 (B˙ s
T

T

p,r )

whenever s > 0, 1 ≤ p ≤ ∞, 1 ≤ ρ, ρ1 , ρ2 , ρ3 , ρ4 ≤ ∞ and
1
1
1
1
1
=
+
=
+ ·
ρ
ρ1 ρ2
ρ 3 ρ4
Remark. Of course similar definitions may be given in the nonhomogeneous framework, leading
s ). The details are left to the reader.
e ρ (Bp,r
to some functional spaces denoted by L
T

2.2.1

Spectral localization

In this section, we prove estimates for the semi-group of the heat equation restricted to functions
with compact supports away from the origin in Fourier variables. These estimates are based on
the following result.
Lemma 2.2.2. Let φ be a smooth function supported in the shell C(0, R1 , R2 ) with 0 < R1 < R2 .
There exist two positive constants κ and C depending only on φ and such that for all 1 ≤ p ≤ ∞,
τ ≥ 0 and λ > 0, we have




φ(λ−1 D)eτ ∆ u p ≤ Ce−κτ λ2 φ(λ−1 D)u p .
L
L
Proof: Performing a change of variable, one can assume with no loss of generality that λ = 1.
Now, let φe be a smooth function supported in C(0, R10 , R20 ) for some R10 < R1 and R20 > R2
and such that φe ≡ 1 in a neighborhood of C(0, R1 , R2 ). We have




−τ |ξ|2
e
F φ(D)eτ ∆ u (ξ) = φ(ξ)e
F(φ(D)u)(ξ).
Thus, φ(D)eτ ∆ u = kτ ? φ(D)u with
−N

Z

kτ (z) := (2π)

RN

2

e dξ.
e−τ |ξ| eiz·ξ φ(ξ)

According to convolution inequalities, we have


φ(D)eτ ∆ u p ≤ kkτ k
L

L1

kφ(D)ukLp .

Therefore it only remains to prove that there exist two positive constants κ and C such
that
(2.4)

∀τ ∈ R+ , kkτ kL1 ≤ Ce−κτ .

For that, we use the fact that for all m ∈ N, we have
Z

2
2
m
−N
e (Id − ∆ξ )m eiz·ξ dξ,
(1 + |z| ) kτ (z) = (2π)
e−τ |ξ| φ(ξ)
Z

2
−N
e
= (2π)
eiz·ξ (Id − ∆ξ )m e−τ |ξ| φ(ξ)
dξ.

42

CHAPTER 2. THE HEAT EQUATION
From the last equality and the fact that the integration may be restricted to the shell
C(0, R10 , R20 ), we easily conclude that there exists a constant Cm such that
∀z ∈ RN , (1 + |z|2 )m |kτ (z)| ≤ Cm e−κτ ,
whence inequality (2.4).

2.2.2

Estimates for the heat equation

Let us now state our main result for the heat equation.
s
Theorem 2.2.3. Let T > 0, s ∈ R and 1 ≤ ρ, p, r ≤ ∞. Assume that u0 ∈ B˙ p,r
and
2

2

s−2+ ρ
s+ ρ
s ) and there exists
e ρ (B˙ p,r
e ρ (B˙ p,r
e ∞ (B˙ p,r
f ∈L
). Then (H) has a unique solution u in L
)∩L
T
T
T
a constant C depending only on N and such that for all ρ1 ∈ [ρ, +∞], we have


1
1
−1
µ ρ1 kuk ρ s+ ρ2 ≤ C ku0 kB˙ s + µ ρ kf k ρ s−2+ ρ2 .
e 1 (B˙ p,r
L
T

1

e (B˙ p,r
L
T

p,r

)

)

s ).
If in addition r is finite then u belongs to C([0, T ]; B˙ p,r

Proof: Since u0 and f are temperate distributions, equation (H) has a unique solution u in
S 0 (0, T × RN ), which satisfies
Z t
2
u
b(t, ξ) = e−µt|ξ| u
b0 (ξ) +
e−µ(t−τ ) fb(τ, ξ) dτ.
0

As F(S˙ q u0 ) (resp. F(S˙ q f )) tends to 0 in S 0 (RN ) (resp. S 0 (0, T × RN )) when q goes to
−∞, we easily gather that also S˙ q u goes to zero in S 0 (0, T × RN ) when q goes to −∞.
˙ q to (H) and using formula (2.3) yields
Next, we notice that, applying ∆
˙ q u(t) = eµt∆ ∆
˙ q u0 +


t

Z

˙ q f (τ ) dτ.
eµ(t−τ ) ∆

0

Therefore,


˙


u(t)
q


Lp




˙ q u0
≤ eµt∆ ∆


Lp

+

Z t

µ(t−τ )∆ ˙

e

f

)


q

Lp

0

dτ.

By virtue of lemma 2.2.2, we thus have for some κ > 0,
Z t




2q
˙

˙

−κµ22q t
k∆q u0 kLp +
e−κµ2 (t−τ ) ∆
∆q u(t) p . e
q f (τ )
L

Lp

0

dτ.

Applying convolution inequalities, we get
˙ q uk ρ1 p .
k∆
L (L )



T

1 − e−κµT ρ1 2
κµρ1 22q

2q



1
ρ1

˙ q u0 kLp +
k∆



1 − e−κµT ρ2 2
κµρ2 22q

2q



1
ρ2

˙ qf k ρ p
k∆
L (L )
T

with 1/ρ2 = 1 + 1/ρ1 − 1/ρ.
Finally, taking the `r (Z) norm, we conclude that (with the usual convention if r = +∞)
(2.5)

kuk

s+ 2
e ρ1 (B˙ p,r ρ1
L
T

.
)

X
2q r
ρ1
1 − e−κµT ρ1 2
q

κµρ1 22q

˙ q u0 kLp
2 k∆
qs

r

1
r

X
1
2q 1
ρ2
r r
1 − e−κµT ρ2 2
q(s−2+ ρ2 ) ˙
ρ
+
2
k∆q f kL (Lp )
,
T
κµρ2 22q
q

2.2. A PRIORI ESTIMATES IN BESOV SPACES FOR THE HEAT EQUATION

43

2

s+ ρ
s ) and yields the desired inequality.
e ∞ (B˙ p,r
e ρ (Bp,r
)∩L
which insures that u ∈ L
T
T
s ) in the case where r is finite may be easily deduced from
That u belongs to C([0, T ]; B˙ p,r
s
s
the density of S ∩ B˙ p,r in B˙ p,r (see proposition 1.5.9).
s ) spaces enables us to gain two
e ρ (B˙ p,r
Remark. In the case u0 ≡ 0 (resp. f ≡ 0) the use of L
T
derivatives for u compare to the regularity of f (resp. u0 ). Remind that the classical result
of maximal regularity for the heat equation which states that D2 u ∈ Lp (0, T × RN ) whenever
2− 2

u0 ∈ Bp,p p and f ∈ Lp (0, T × RN ) breaks down for p = 1.

2.2.3

A counterexample

In this section, we aim at convincing the reader that the results stated in the previous section
s ) spaces is not a technical artifact.
e ρ (Bp,r
are optimal and that the appearance of L
T
Let us consider the simple case of the free heat equation ∂t u − ∆u = 0 with initial data u0
in L2 . From (2.2), it is easy to see that the function (t, x) 7→ tD2 u(t, x) belongs to L1 (R+ ; L2 ).
Can we expect u to be in L1loc (R+ ; H 2 ) ? The answer is negative. One can even state a more
accurate result:
Proposition 2.2.4. Let u0 be in Sh0 . The solution to (H) with f ≡ 0 belongs to L1 (R+ ; H˙ 2 )
0 .
if and only if u0 is in B˙ 2,1
0 then theorem 2.2.3 states that u belongs to L
e 1 (B˙ 2 ). As L
e 1 (B˙ 2 ) =
Proof: If u0 is in B˙ 2,1
2,1
2,1
2 ) and B
˙ 2 ,→ H˙ 2 , we thus have u ∈ L1 (R+ ; H˙ 2 ) as expected.
L1 (R+ ; B˙ 2,1
2,1

Conversely, let us assume that u ∈ L1 (R+ ; H˙ 2 ). According to Parseval formula and to
(2.1), we thus have
Z

+∞ Z

4 −2t|ξ|2

|ξ| e

I :=

2

1

2

dt < ∞.

|b
u0 (ξ)| dξ

RN

0

Z

2

1

2

|b
u0 (ξ)| dξ

Now, denoting cq :=

, easy computations yield

2q ≤|ξ|≤2q+1

I≥

XZ
p∈Z

41−p X

4−p

2(q−p) 2
24q e−2
cq

1

2

dt.

q∈Z

Keeping only the term q = p in the second sum, we end up with
X
I≥α
cp
p∈Z

for some positive constant α.
P
0 .
Hence
cq has to be finite. In other words, u0 has to belong to B2,1

2.2.4

Estimates in nonhomogeneous Besov spaces, and the periodic case

One can wonder if the estimates stated in theorem 2.2.3 remain true in nonhomogeneous Besov
spaces. On one hand, the block ∆−1 u corresponding to the low frequencies of u cannot be
handled by mean of lemma 2.2.2. On the other hand, by using the representation formula (2.2),
we easily get
Z
T

k∆−1 u(T )kLp ≤ k∆−1 u0 kLp +

0

k∆−1 f kLp dt,

44

CHAPTER 2. THE HEAT EQUATION

whence, if 1 ≤ ρ ≤ ρ1 ≤ ∞,
1

k∆−1 ukLρ1 (Lp ) ≤ T ρ1 k∆−1 u0 kLp + T

1+ ρ1 − ρ1

T

1

k∆−1 f kLρ (Lp ) .
T

Of course the other dyadic blocks may be treated as in the homogeneous case. We end up with
the following statement:
2

s−2+ ρ
s and f ∈ L
e ρ (Bp,r
Theorem 2.2.5. Let s ∈ R and 1 ≤ ρ, p, r ≤ ∞. Let T > 0, u0 ∈ Bp,r
).
T
2

s+ ρ
s ) and there exists a constant C
e ρ (Bp,r
e ∞ (Bp,r
Then (H) has a unique solution u in L
)∩L
T
T
depending only on N and such that for all ρ1 ∈ [ρ, +∞], we have


1
1
1
1+ 1 − 1
−1
s
+ (1 + T ρ1 ρ )µ ρ kf k ρ s−2+ ρ2 .
µ ρ1 kuk ρ s+ ρ2 ≤ C (1 + T ρ1 )ku0 kBp,r
e 1 (Bp,r
L
T

1

)

e (Bp,r
L
T

)

s ).
If in addition r is finite then u belongs to C([0, T ]; Bp,r

Remark. Compare to the homogeneous case, the constant appearing in the above inequality now
depends on T. When dealing with nonlinear parabolic PDE’s, this may preclude from proving
global in time existence results (even for small data). Avoiding the time dependence is thus a
good motivation for using homogeneous spaces when dealing with such PDE’s.
Let us finally make a short remark concerning the periodic case. In the case where u0 and
f are periodic and have zero average, it is clear that the unique solution u to (H) has also
zero average. The above analysis may be carried out and leads again to the estimates stated in
theorem 2.2.3.
In the general
not have zero average, the above estimates hold
R
R case where u0 and f need
−1
for u − |TN
|
u
dx
and
the
study
of
u
dx
has to be done separately.
a
TN
T
a

2.3

Optimal well-posedness results for Navier-Stokes equations

In this section, we give an example of application of theorem 2.2.3 for solving a nonlinear PDE’s
related to the heat equation. We focus on the incompressible Navier-Stokes equations which
have been extensively studied recently. It goes without saying that the same method works for
a great deal of semi-linear heat equations.

2.3.1

The model

The incompressible Navier-Stokes equations write

 ∂t u + u · ∇u − µ∆u + ∇Π = f,
div u = 0,
(N S)

u|t=0 = u0 .
Above u = u(t, x) stands for the unknown velocity field which is a time dependent vectorfield
and the scalar function Π = Π(t, x) stands for the pressure. From a mathematical viewpoint,
∇Π is the Lagrange multiplier associated to the divergence free constraint. The initial velocity
u0 and the external force f are given functions. It isPunderstood that the convective term
N
j
i
u · ∇u stands for the vector field whose i-th entry is
j=1 u ∂j u . Finally, the parameter µ
1
(the viscosity) has to be positive .

By performing the change of unknown function u(t, x) = v(t, x/ µ), one can restrict the study to the case
µ = 1, an assumption which is made in most of papers devoted to Navier-Stokes equations. Since we aim at
keeping track of the dependence with respect to µ , we will not perform this change of function.
1

2.3. OPTIMAL WELL-POSEDNESS RESULTS FOR NAVIER-STOKES EQUATIONS

45

Note that, strictly speaking, (NS) does not enter in the class of semi-linear parabolic equations. Since we will consider only the case where the space variable x belongs to the whole space
however, we shall see in a moment that (NS) may be reduced to a (system of) semilinear heat
equations with nonlocal nonlinearity.
We notice that for suitably smooth divergence-free vectorfields, the convective term rewrites
div u ⊗ u. This leads to the following definition of weak solution:
Definition. A distribution u ∈ S 0 (0, T × RN ; RN ) is called a weak solution of (N S) if div u = 0
in S 0 (0, T × RN ; R) and if 2
Z TZ
RN

0

Z TZ
u·∂t ϕ dxdt+
0

Z TZ

i j

u u ∂j ϕi dxdt−µ

RN

0

Z TZ
∇u·∇ϕ dxdt+

RN

0

Z
f ·ϕ dxdt =

RN

u0 ϕ(0) dx
RN


for all divergence free ϕ in Cc∞ [0, T ); S(RN ; RN ) .
If u is a smooth enough weak solution of (NS), taking ϕ = u in the above relation yields
the following energy equality:
(2.6)

ku(t)k2L2

Z
+ 2µ
0

t

kDu(τ )k2L2

dτ =

ku0 k2L2

Z tZ
f · u dx dt.

+2
0

Taking advantage of (2.6) and of compactness arguments, J. Leray in 1934 proved the existence
of global weak solutions with bounded energy in the case where f ≡ 0. Let us state his main
result (see [33] for more details):
Theorem (Leray). Let N ≥ 2. For all divergence free vectorfield u0 with coefficients in L2 ,
system (N S) with f ≡ 0 has a global weak solution u in L∞ (R+ ; L2 ) with Du ∈ L2 (R+ ; L2 ),
which satisfies
(2.7)

ku(t)k2L2

Z
+ 2µ
0

t

kDu(τ )k2L2 dτ ≤ ku0 k2L2 .

If N = 2, then (2.6) is true. Besides the solution u belongs to C(R+ ; L2 ) and is unique in the
set of divergence-free vector fields with coefficients in L∞ (R+ ; L2 ) and gradient in L2 (R+ ; L2 ).
Since the work by J. Leray in 1934, the problem of uniqueness in the energy space when
N ≥ 3 has remained unsolved. Considering smoother data and restricting the set of admissible
solutions is the usual way to get existence and uniqueness results. Again, this has been first
noticed by J. Leray in [33]:
Theorem (Leray). Let N = 3. There exists a positive constant c such that for all divergencefree vector field u0 with coefficients in H 1 which satisfies
ku0 kL2 k∇u0 kL2 ≤ cµ2

or

ku0 k2L2 ku0 kL∞ ≤ cµ3 ,

system (N S) with no external force has a unique global solution u ∈ C(R+ ; H 1 ) which also
satisfies D2 u ∈ L2 (R+ ; L2 ).
One can alternately consider large data in H 1 . One still obtain existence and uniqueness of
a solution but for small time only.
Whether global existence and uniqueness holds true is an outstanding open problem.
2

From now on, we adopt the summation convention over repeated indices.

46

2.3.2

CHAPTER 2. THE HEAT EQUATION

About scaling and critical spaces

In this section, we aim at finding functional spaces E as large as possible for which any vectorfield
u0 in E generates a unique solution on a small time interval. If one restricts to the Sobolev
spaces framework, the optimal result is due to H. Fujita and T. Kato in [28]. Their original
statement pertains to the three-dimensional case in bounded domains. The statement below in
dimension N = 2, 3 has been proved by J.-Y. Chemin in [9].
N
Theorem (Fujita-Kato). Let u0 be a solenoidal vector-field with coefficients in H˙ 2 −1 . Let f
N
have coefficients in L2 (0, T ; H˙ 2 −2 ). There exists a positive time T such that (N S) has a unique
N
N
solution in C([0, T ]; H˙ 2 −1 ) with gradient in L2 (0, T ; H˙ 2 −1 ).
Moreover, there exists a constant c depending only on N and such that (N S) has a unique
N
N
global solution in Cb (R+ ; H˙ 2 −1 ) with gradient in L2 (R+ ; H˙ 2 −1 ) whenever

ku0 k ˙

(2.8)

H

1

N −1
2

+ µ 2 kf k

L2 (H˙

N −2
2
)

≤ cµ.

Compare to Leray’s results, there are two important advances in Fujita and Kato’s approach.
The first one is that suitably smooth solutions to (NS) may be interpreted as a fixed point of
some functional over Banach spaces.
Indeed, let P denote the orthogonal projector of L2 (RN ; RN ) over solenoidal vectorfields.
One can easily show that P is the 0 order multiplier defined by P := Id − ∇(−∆)−1 div . In
other words, in Fourier variables, we have for any u with coefficients in L2 ,
ξ·u
b(ξ)
c
ξ·
Pu(ξ)
=u
b(ξ) −
|ξ|2
Note that the above definition may be extended to distributions of Sh0 .
Now, we remark that Navier-Stokes equations rewrite
g
(N
S)

∂t u − P div(u ⊗ u) − µ∆u = Pf,

u|t=0 = u0 .

Hence, by virtue of (2.3), we have at least formally,
Z t
Z t

µt∆
µ(t−τ )∆
(2.9)
u(t) = e u0 +
e
Pf (τ ) dτ −
eµ(t−τ )∆ P div u(τ ) ⊗ u(τ ) dτ.
0

0

Any solution of (NS) which satisfies (2.9) is called (after the theory of analytic semi-groups) a
mild solution of Navier-Stokes equations. One can prove that any suitably smooth mild solution
is also a weak solution to (NS) (see [32] for a detailed study).
Thus solving (NS) amounts to finding a fixed point for the functional v 7→ Φ(v) defined by
Z t

µt∆
Φ(v)(t) := e u0 +
eµ(t−τ )∆ P (f − div(v ⊗ v))(τ ) dτ.
0

The second advance in Fujita-Kato’s approach has to do with the choice of the functional
framework for the solution u and for the data.
g
The idea is that when appealing to contracting mapping arguments for solving (N
S), it is
suitable that u belongs to a functional space X such that the linear term ∂t u − µ∆u and the
nonlinear term P div(u ⊗ u) have the same regularity.
g
This may be interpreted in terms of scaling invariance for (N
S). Indeed, we notice that if
g
u solves (N S) with data u0 and f, so does for any λ > 0 the vectorfield
uλ : (t, x) 7→ λu(λ2 t, λx)

2.3. OPTIMAL WELL-POSEDNESS RESULTS FOR NAVIER-STOKES EQUATIONS

47

with data
u0,λ : x 7→ λu(λx)

and

fλ : (t, x) 7→ λ3 f (λ2 t, λx).

It is also clear that a space where the heat and convective terms have the same regularity must
have a norm invariant for all λ by the change u 7→ uλ . This leads to the following definition:
Definition. A critical space for initial data is any Banach space E ⊂ S 0 (RN ) whose norm is
invariant for all λ by u0 7→ u0,λ .
A critical space for external forces is any Banach space F ⊂ S 0 (R+ × RN ) whose norm is
invariant for all λ by f 7→ fλ .
A critical space for solutions to (NS) is any Banach space X ⊂ S 0 (R+ × RN ) whose norm is
invariant for all λ by u 7→ uλ .
N
One can easily check that Fujita-Kato’s theorem enters in this framework. Indeed, H˙ 2 −1
N
is invariant by u0 7→ u0,λ , the space L2 (R+ ; H˙ 2 −2 ) is invariant by f 7→ fλ and the space of
N
N
divergence-free vector-fields with coefficients in C(R+ ; H˙ 2 −1 ) and gradient in L2 (R+ ; H˙ 2 −1 ) is
invariant by u 7→ uλ .

2.3.3

Global well-posedness for small data

In the present section, we investigate the problem of existence of global (mild) solutions to
Navier-Stokes equations for small data which belong to functional spaces which are invariant by
the scaling exhibited in the previous section.
Our main result is the following:
Theorem 2.3.1. Let 1 ≤ r ≤ ∞ and 1 ≤ p < ∞. There exists a constant c > 0 independent of
N
−1
p
µ such that for all divergence free vector-field u0 with coefficients in B˙ p,r
and external force
N

−1
p
e 1 (B˙ p,r
f with coefficients in L
) such that

ku0 k

(2.10)

N −1

p
B˙ p,r

+ kPf k

N −1

p
e 1 (B˙ p,r
L

< cµ,
)
N

N

+1
−1
p
p
e ∞ (B˙ p,r
e 1 (B˙ p,r
system (N S) has a unique solution u in L
)∩L
) which satisfies

kuk

(2.11)

N −1

p
e ∞ (B˙ p,r
L

+ µkuk
)

N +1

p
e 1 (B˙ p,r
L

< 2cµ.
)

N

N

−1
+1
p
p
e 1 (B˙ p,r
Besides, if r is finite then u belongs to C(R+ ; B˙ p,r
) and uniqueness holds true in L
)∩
N

−1
p
e ∞ (B˙ p,r
L
) with no smallness condition.
N

−1
p
According to proposition 1.5.4, all the spaces B˙ p,r
are scaling invariant for the NavierN
N
−1
−1
2
= B˙ 2,2 . Hence theorem 2.3.1 is a natural generalStokes equations. Besides, we have H˙ 2
ization of Fujita and Kato’s theorem.
Well-posedness in critical Besov spaces has been first proved by M. Cannone in [7] for 3 <
s ) spaces for
e ρ (B˙ p,r
p ≤ 6 and r = ∞, then extended in [8] and [35]. The idea of using L
T
solving Navier-Stokes equations is due to J.-Y. Chemin in [11]. Related results in more general
functional spaces have been proved in [31]. Critical spaces in which global well-posedness for
small data may be proved with the method below have been characterized in [2].

The proof of theorem 2.3.1 is based on the following lemma.

48

CHAPTER 2. THE HEAT EQUATION

Lemma 2.3.2. Let (X, k · kX ) be a Banach space and B : X × X → X be a bilinear continuous
operator with norm K. Then for all y ∈ X such that 4KkykX < 1, equation x = y + B(x, x)
1
). Besides x satisfies kxkX ≤ 2kykX .
has a unique solution x in the ball B(0, 2K
Proof: We rule out the case K = 0 which is obvious. Now, the result is a mere consequence
of the contracting mapping theorem.
R
) into
Indeed, let R := 4KkykX and F : x 7→ y + B(x, x). On one hand, F maps B(0, 2K
R 2
0
itself provided that R ≤ 1. On the other hand, for all (x, x ) ∈ B(0, 2K ) , we have

kF (x0 ) − F (x)kX ≤ Rkx0 − xkX .


R
whenever R < 1, which insures the
Hence F is a contracting mapping on B 0, 2K
existence of x.
1
Routine computations then lead to kxkX ≤ 2kykX and to uniqueness in B(0, 2K
).

Let us now prove theorem 2.3.1:
First step: Existence
The existence of a solution for (NS) with data u0 and f will be obtained by applying lemma
2.3.2 for convenient y, B and (X, k · kX ).
For X, we shall take the space of divergence free distributions over R+ ×RN with coefficients
N
N
+1
−1
p
p
e 1 (B˙ p,r
e ∞ (B˙ p,r
in L
)∩L
) endowed with the norm
kvkX := kvk
We then set y : t 7→ eµt∆ u0 +
formula

Rt
0

N −1

p
e ∞ (B˙ p,r
L

+ µkvk
)

N +1

p
e 1 (B˙ p,r
L

.
)

eµ(t−τ )∆ Pf dτ and define the bilinear functional B by the
Z

t

B(v, w)(t) = −

eµ(t−τ )∆ P div(v(τ ) ⊗ w(τ )) dτ.

0

We claim that y belongs to X, that B maps X × X in X and that there exists some constant
C such that

(2.12)
,
kykX ≤ C ku0 k Np −1 + kPf k
N −1
p
e 1 (B˙ p,r
L

B˙ p,r

(2.13)

)

∀(v, w) ∈ X 2 , kB(v, w)kX ≤ Cµ−1 kvkX kwkX .
N

N

−1
−1
p
p
e 1 (B˙ p,r
Indeed, as u0 is divergence free and belongs to B˙ p,r
, and as f is in L
), theorem 2.2.3
insures that y belongs to X and satisfies (2.12).
Next, using Bony’s decomposition and div v = div w = 0, one can write

˙
div(v ⊗ w) = T˙∂j v wj + T˙wj ∂j v + ∂j R(v,
wj )
with the summation convention over repeated indices.
N

1


p 2
e ρ (B˙ p,r
Hence, combining propositions 1.4.1 and 1.4.2, remark 2.2.1 and the embedding L
) ,→
− 12
ρ
e (B˙ ∞,∞ ) for ρ = 4/3 or ρ = 4, we get
L

kdiv(v ⊗ w)k

N −1

p
e 1 (B˙ p,r
L

≤ Ckvk
)

N +1
2

p
e 34 (B˙ p,r
L

kwk
)

N −1
2

p
e 4 (B˙ p,r
L

.
)

2.3. OPTIMAL WELL-POSEDNESS RESULTS FOR NAVIER-STOKES EQUATIONS

49

Putting forward complex interpolation, we have
kvk

N +1
p
2
e 43 (B˙ p,r
)
L

kwk

3

1

≤ kvk 4

kvk 4

≤ µ− 4 kvk,

≤ kwk

kwk

≤ µ− 4 kwk,

N +1
p
e 1 (B˙ p,r
L
)
1
4
N +1
p
e 1 (B˙ p,r
L
)

N −1
p
2
e 4 (B˙ p,r
)
L

3

N −1
p
e ∞ (B˙ p,r
L
)
3
4
N −1
p
e ∞ (B˙ p,r
L
)

1

whence
kdiv(v ⊗ w)k

N −1

p
e 1 (B˙ p,r
L

≤ Cµ−1 kvkX kwkX .
)

Finally, by using the fact that P is an homogeneous multiplier of degree 0, and by applying
theorem 2.2.3, be conclude that B(v, w) belongs to X and that (2.13) is satisfied.
Now, lemma 2.3.2 may be applied provided that 4CkykX < µ. According to (2.12) this
condition will be satisfied if
ku0 k

N −1

p
B˙ p,r

+ kPf k

N −1

p
e 1 (B˙ p,r
L

< cµ
)

for some small enough constant c.
This achieves the proof of existence of a global solution in X for (NS), and of uniqueness
under condition (2.11).
Second step: Uniqueness in the case where r is finite
Let u1 and u2 be two solutions of (NS) in X. Denoting δu := u2 − u1 , we have for all t ∈ R+ ,
Z

t

eµ(t−τ )∆ P div(δu ⊗ u2 + u1 ⊗ δu) dτ.

δu(t) = −
0

By going along the lines of the proof of (2.13), one can easily check that for all positive T and
N
N
4
+ 21
− 12
p
p
e 3 (B˙ p,r
e 4 (B˙ p,r
divergence free (v, w) in L
)×L
), we have
T

T

kP div(v ⊗ w)k

N −1

p
e 1 (B˙ p,r
L
T

≤ Ckvk

4

N +1
2

p
e 3 (B˙ p,r
L
T

)

kwk

N −1
2

p
e 4 (B˙ p,r
L
T

)

.
)

Hence, denoting XT the space of functions of X restricted to [0, T ]×RN , and applying theorem
2.2.3, we get with obvious notation,


kδukXT ≤ C kδuk 4 Np + 1 ku2 k
.
N − 1 + ku1 k 4
N + 1 kδuk
N −1
p
p
p
e 3 (B˙ p,r
L
T

2

)

e 4 (B˙ p,r
L
T

2

e 3 (B˙ p,r
L
T

)

2

)

e 4 (B˙ p,r
L
T

2

)

Using complex interpolation, we conclude that
(2.14)
with

kδukXT ≤ Z(T )kδukXT
3
Z(T ) := C µ− 4 ku2 k

1

N −1
p
2
e 4 (B˙ p,r
L
)
T

+ µ− 4 ku1 k

N +1
4
p
2
e 3 (B˙ p,r
L
)
T



.

Now, Lebesgue dominated convergence theorem insures that Z is a continuous nondecreasing
function which vanishes at zero. Hence δu ≡ 0 in XT for small enough T.
Finally, because the function t 7→ kδuk
is also continuous, a standard connectivity
N −1
p
e 1 (B˙ p,r
L
t

)

argument enable us to conclude that δu ≡ 0 on R+ × RN .



Documents similaires


courschine
article14 sghir aissa
maxwell quaternion 2
article4 sghir aissa
article10 sghir aissa
vladimir voevodsky last interview translated from russian


Sur le même sujet..