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Lecture notes for 8th summer school on nonlinear PDE analysis

Pseudo-Differential Operators

And Their Applications

Chao-Jiang XU, Wuhan University

July 2010, Xi’an, China

Contents

Chapter 1. Sobolev spaces 11.1. Fourier transformation 11.2. Fractional derivative formulas 21.3. Definition of Sobolev space 4* Logarithmic Sobolev space 61.4. Sobolev embedding Theorem 81.5. Compactness and interpolation inequalities 121.6. Sobolev space on an open domain and trace Theorem 17

Chapter 2. Pseudo-differential operators 192.1. Symbols class 192.2. Asymptotic expansion 202.3. Definition of pseudo-differential operators 222.4. Algebra of pseudo-differential operators 232.5. Continuity in Sobolev spaces 25

Chapter 3. Non linear Fourier analysis on ℝn 293.1. Littlewood-Palay theory 293.2. Characterization of Sobolev spaces 323.3. Characterization of Holder spaces 353.4. Sobolev embedding Theorem 383.5. Besov spaces 40* Around function space B1

∞,∞ 41* Logarithmic Sobolev spaces 43

Chapter 4. Analysis of non linear partial differential equations 494.1. Paramultipication 494.2. Paradifferential operators 524.3. Paradifferential equations 584.4. Regularity of solution of non-linear equations 614.5. Paradifferential calculus in Besov space 63

Chapter 5. Transport equations and Euler’s equation 675.1. Ordinary differential equations 675.2. Commutation lemmas 685.3. Propagation of regularity for transport equations 695.4. Euler’s equations 705.5. Existence and uniqueness of solution of a model equation 73

I

II CONTENTS

Chapter 6. Existence of solution for shallow water equations 776.1. Shallow water equations 776.2. The local existence of solution 796.3. The global existence for small initial data 836.4. Losing energy estimates 87

Chapter 7. Regularity of solutions 937.1. Hypo-ellipticity of Hormander’s operators 937.2. Analytic smoothing effect of kinetic equations 987.3. Gevrey effect for Kac’s equations 106

Bibliography 127

CHAPTER 1

Sobolev spaces

This is a preparation chapter, we recall without detail some basic properties ofSobolev spaces and the important inequalities which we will use to the analysis ofpartial differential equation.

1.1. Fourier transformation

We use the following notations in this lectures notes : For � = (�1, ⋅ ⋅ ⋅ , �d) ∈ℕd, x = (x1, ⋅ ⋅ ⋅ , xd) ∈ ℝd, setting

∣�∣ = �1 + ⋅ ⋅ ⋅+ �d, ∣x∣ = (∣x1∣2 + ⋅ ⋅ ⋅+ ∣xd∣2)1/2,

and

x� = x�11 ⋅ ⋅ ⋅x�1

1 , ∂�x = ∂�1x1⋅ ⋅ ⋅ ∂�dxd , D�

x =(1

i∂x1

)�1 ⋅ ⋅ ⋅(1

i∂xd)�d .

The Fourier transformation is defined on the functions of L1(ℝd) by

ℱ(f)(�) = f(�) =

∫ℝde−ix ⋅ �f(x)dx,

then ℱ : L1(ℝd) → L∞(ℝd) is a continuous maps. We have also

∂�� f(�) = (−ix)�f(�), ℱ(D�xf)(�) = ��f(�),

and

(1 + ∣�∣2)k∂�f(�) =

∫ℝde−ix ⋅ �(I −△x)

k(−ix)�f(x)dx,

where we use the identity

(I −△x)ke−ix ⋅ � = (1 + ∣�∣2)ke−ix ⋅ �.

The Fourier transformation is symmetric in the following sense. Let f, g ∈ L1(ℝd), ifwe apply Fubini theorem, we get the fundamental relation

⟨f, ℱg⟩ =

∫∫ℝ2d

e−ix ⋅ �f(x)g(�)dxd� = ⟨ℱf, g⟩.

We define also the inverse Fourier transformation by

ℱ(f)(�) =

∫ℝdeix ⋅ �f(x)dx.

Theorem 1.1.1. .

(1) If we have f, f ∈ L1(ℝd), then

f = (2�)−dℱ(f).

1

2 1. SOBOLEV SPACES

(2) The operator (2�)−d2ℱ can be extended from L1 ∩ L2(ℝd) to an onto unitary

operator on L2(ℝd), the inverse of which is (2�)−d2 ℱ , and we have the following

Blanchrel Formula :

(2�)−d2∥ℱ(u)∥L2(ℝd) = ∥u∥L2(ℝd).

The Fourier transformation is an isomorphism from S(ℝd) to S(ℝd), where S(ℝd) ⊂C∞(ℝd) is the Schwartz function space with semi-norm

∥f∥k,S = sup∣�∣≤k

(1 + ∣x∣)k∣∂�f(x)∣.

Then the Fourier transformation can be extended to an isomorphism from tempereddistribution space S ′(ℝd) to S ′(ℝd).

Let us give some examples of explicit Fourier transformation.

(1) For a ∈ ℝd, we have ℱ(�a)(�) = e−ia ⋅ �.(2) ℱ(1)(�) = (2�)d�0(�).(3) Let z ∈ ℂ such that ℜe z > 0 then

ℱ(e−z∣ ⋅ ∣

2)

(�) =(�z

)d/2e−∣�∣2

4z ,

where z1/2 = ∣z∣1/2e−i�/2 if z = ∣z∣e−i� with � ∈]− �/2, �/2[.(4) If 0 < � < d, then

ℱ(

1

∣ ⋅ ∣�

)(�) = cd∣�∣�−d,

with

cd =

∫Sd−1

∫ ∞0

e−er!1rd−1−�drd!.

(5)

ℱ(

1

1 + x2

)(�) = �e−∣�∣.

Finally, if A is a linear automorophism of ℝd, we have that

ℱ(u ∘ A) =1

∣ detA∣

((ℱu)∘(tA−1

)).

And

ℱ(u ★ v) = ℱ(u)ℱ(v)

when the convolution u ★ v is well-defined.

1.2. Fractional derivative formulas

For 0 < � < 1 and g ∈ S(ℝd), we define the fractional derivative by

∣Dx∣� g(x) = ℱ−1(∣ ⋅ ∣�g( ⋅ )

)(x).

Then, we have the following formula.

1.2. FRACTIONAL DERIVATIVE FORMULAS 3

Lemma 1.2.1. Let 0 < � < 1, then for any g ∈ S(ℝd), we have

(1.2.1) ∣Dx∣� g(x) = C�

∫ℝd

g(x)− g(x+ x′)

∣x′∣n+�dx′

with C� ∕= 0 being a complex constant depending only on � and the dimension d.

In fact, using Fourier inverse formula∫ℝd

g(x)− g(x+ x′)

∣x′∣d+�dx′ = (2�)−d

∫ℝdℱ(g)(�) ei x⋅�

(∫ℝn

1− e−i x′⋅�

∣x′∣d+�dx′)d�

On the other hand, it is clear that∫ℝd

1− e−i x′⋅�

∣x′∣d+�dx′ = ∣�∣�

∫ℝd

1− e−i u⋅�∣�∣

∣u∣d+�du.

Observing∫ℝd

1−ei u⋅ �∣�∣

∣u∣d+� du ∕= 0 is a complex constant depending only on � and the

dimension d, but independent of � ∈ ℝn, then the above two equalities give (1.2.1).

In the study of infinitely degenerate elliptic equations, we need also the logarithmictype derivative. For s > 0 and g ∈ S(ℝd), we define

(log ⟨Dx⟩)s g(x) = ℱ−1((log ⟨∣ ⋅ ∣⟩)sg( ⋅ )

)(x),

where for � ∈ ℝd,

(log ⟨∣ � ∣⟩)s =(1

2log(e+ ∣ � ∣2)

)s.

We will use the notation

(1.2.2) log � =

{log �, if 0 < � < 1 ;0, if 1 ≤ � .

We have the following similar formula.

Lemma 1.2.2. Let s > 0, then for any g ∈ S(ℝd), we have

(1.2.3) (log ⟨Dx⟩)s g(x) = c0

∫ℝd

(g(x)− g(x+ x′)

)∣log ∣x′∣∣s−1

∣x′∣ddx′ +

(g ★ cs

)(x)

where c0 > 0 and∣ℱ(cs)(�)∣ ≤ C0(log ⟨∣�∣⟩)s−1, ∀ � ∈ ℝn,

the constants c0, C0 depending only on s and the dimension d.

In fact, same calculus as for Lemma 1.2.1,∫ℝn

(g(x)− g(x+ x′)

)∣log ∣x′∣∣s−1

∣x′∣ddx′ =

((log ⟨Dx⟩)s g ★ c

)(x),

where, the function c is, for � ∈ ℝd,

ℱ(c )(�) =(

log ⟨ ∣�∣ ⟩)−s ∫

{∣z∣≤1}

(1− e−iz⋅�

)∣ log ∣z∣ ∣s−1

∣z∣ndz.

The estimate of this function is exactly the same as the proof of Proposition 1.3.5.

4 1. SOBOLEV SPACES

1.3. Definition of Sobolev space

Definition 1.3.1. Let s ∈ ℝ, the Sobolev space Hs(ℝd) is the set

Hs(ℝd) = {f ∈ S ′(ℝd); f ∈ L2loc(ℝd); (1 + ∣�∣2)s/2f(�) ∈ L2(ℝd)},

with the norm

∥f∥Hs(ℝd) = ∥(1 + ∣ ⋅ ∣2)s/2f∥L2(ℝd).

The homogeneous Sobolev space is the set

Hs(ℝd) = {f ∈ S ′(ℝd); f ∈ L1loc(ℝd); ∣�∣sf(�) ∈ L2(ℝd)}.

Then the Fourier transformation is an isomorphism from Hs(ℝd) to L2(ℝd; (1 +∣�∣2)sd�), and Hs(ℝd) is a Hilbert space with inner product

(u, v)Hs =

∫ℝd

(1 + ∣�∣2)su(�)¯v(�)d�.

But for the homogeneous Sobolev spaces, Hs(ℝd) is a Hilbert space if and only ifs < d/2, with the norm

∥f∥Hs(ℝd) = ∥∣ ⋅ ∣sf∥L2(ℝd).

In the case of s ≥ d/2, this define only a semi-norm.

Proposition 1.3.1. S(ℝd) is dense in Hs(ℝd) for any real s. Moreover, S(ℝd) isdense in Hs(ℝd) for any ∣s∣ < d/2.

Therefore, D(ℝd) is also dense in Hs(ℝd) for any real s. By Fourier transformation,this Proposition can be deduced by the density of S(ℝd) in L2(ℝd; (1 + ∣�∣2)sd�).

We study now the equivalent definition of Sobolev spaces.

Proposition 1.3.2. Let m ∈ ℕ, then

Hm(ℝd) = {f ∈ L2(ℝd); ∂�f ∈ L2(ℝd),∀∣�∣ ≤ m},

and the norm is equivalent to ( ∑∣�∣≤m

∥∂�f∥2L2(ℝd)

)1/2

.

Here, we use the simple fact,

ℱ(∂�u) = (−i�)�u(�)

and there exist Cm > 0 such that for any � ∈ ℝd,

C−1m

(1 +

∑0<∣�∣≤m

∣��∣2)≤ (1 + ∣�∣2)m ≤ Cm

(1 +

∑0<∣�∣≤m

∣��∣2).

Proposition 1.3.3. .

(1) Let 0 < s < 1, then

Hs(ℝd) ={f ∈ L2(ℝd);

∫∫ℝd×ℝd

∣f(x+ y)− f(x)∣2

∣y∣d+2sdxdy < +∞

},

1.3. DEFINITION OF SOBOLEV SPACE 5

and the norm is equivalent to(∥f∥2

L2(ℝd) +

∫∫ℝd×ℝd

∣f(x+ y)− f(x)∣2

∣y∣d+2sdxdy

)1/2

.

(2) Let s = m+ �,m ∈ ℕ, 0 < � < 1, then

Hs(ℝd) ={∂�f ∈ L2(ℝd),∀∣�∣ ≤ m;∫∫

ℝd×ℝd

∣∂�f(x+ y)− ∂�f(x)∣2

∣y∣d+2�dxdy < +∞,∀∣�∣ = m

},

and the norm is equivalent to( ∑∣�∣≤m

∥∂�f∥2L2(ℝd) +

∑∣�∣=m

∫∫ℝr×ℝd

∣∂�f(x+ y)− ∂�f(x)∣2

∣y∣d+2�dxdy

)1/2

.

In fact, using the Planchrel formula, we have:∫∫ℝd×ℝd

∣f(x+ y)− f(x)∣2

∣y∣d+2sdxdy =

∫ℝd∣f(�)∣2

∫ℝd

∣eiy ⋅ � − 1∣2

∣y∣d+2sdyd�,

and ∫ℝd

∣eiy ⋅ � − 1∣2

∣y∣d+2sdy = ∣�∣2s

∫ℝd

∣eiy ⋅�∣�∣ − 1∣2

∣y∣d+2sdy = cs∣�∣2s.

It is clary that

cs =

∫ℝd

∣eiy ⋅�∣�∣ − 1∣2

∣y∣d+2sdy

is a positive constant depending only on s and the dimension d, but not on the �∣�∣ ∈

Sd−1.By using the Propositions 1.3.2 and 1.3.3, the definition of Sobolev space is invariant

by diffeomorphism of ℝd. Let Θ : ℝd → ℝd be a global diffeomorphism, i. e. thereexists C > 0 such that

C−1∣x− y∣ ≤ ∣Θ(x)−Θ(y)∣ ≤ C∣x− y∣, ∀ x, y ∈ ℝd,

C−1 ≤ ∣ det(∇Θ(x))∣ ≤ C, ∀ x ∈ ℝd.

And the same for the inverse Θ−1. Then

u ∈ Hs(ℝd) ⇐⇒ u ∘Θ ∈ Hs(ℝd),

with equivalent norm. Using this way and partition of unity, we can define the Sobolevspace on the manifolds.

For the Sobolev space of negative index, we have


(1) For any s > 0, H−s(ℝd) is the dual of Hs(ℝd).(2) Let m ∈ ℕ, u ∈ H−m(ℝd), then there exist u0, u� ∈ L2(ℝd), ∣�∣ = m such that

u = u0 +∑∣�∣=m

∂�u�.

6 1. SOBOLEV SPACES

In fact, we can use the following algebraic identity

∣�∣2m =∑

1≤j1,⋅⋅⋅ ,jm≤d

�2j1⋅ ⋅ ⋅ �2

jm =∑∣�∣=m

(i�)�(−i�)�.

Let � ∈ C∞0 (B(0, 2)) with value 1 near the unit ball, and setting

u(�) = �(�)u(�) +∑∣�∣=m

(−i�)�v�(�)

with

v�(�) = (1− �(�))(i�)�

∣�∣2mu(�).

Then

u0 = ℱ−1(�u), u� = ℱ−1v� ∈ L2(ℝd),

and

u = u0 +∑∣�∣=m

∂�u�.

We give here some basic properties of Sobolev spaces deduced immediately fromdefinition :

∙ �0 ∈ Hs(ℝd) for any s < −d/2.∙ L1(ℝd) ∈ Hs(ℝd) for any s < −d/2.∙ ℰ ′(ℝd) ⊂ H−∞(ℝd) =

∪s∈ℝH

s(ℝd), that means if u is a distribution withcompact support, then there exists s ∈ ℝ such that u ∈ Hs(ℝd).∙ 1 /∈ H−∞(ℝd), because

ℱ(1)(�) = (2�)d�0(�) /∈ L2loc(ℝd).

* Logarithmic Sobolev space

In the study of infinite degenerate elliptic operator, and also for the Debye-Yukawatype potential collision operators, we use a logarithmic type Sobolev space.

Definition 1.3.2. Let s > 0, we define the following logarithmic Sobolev’s space.

Hslog(ℝd) = {u ∈ L2(ℝd);

(log ⟨�⟩

)s ∣u(�)∣ ∈ L2(ℝd)},

where ⟨�⟩ = (e+ ∣�∣2)1/2.

If X = (X1, ⋅ ⋅ ⋅ , Xm) is an infinitely degenerate elliptic system of vector fieldsdefined on an open domain Ω ⊂ ℝd. Under some conditions, we can get the logarithmicregularity estimate (see [83]),

(1.3.1) ∥(log ⟨D⟩)su∥2L2 ≤ C

{ m∑j=1

∥Xju∥2L2 + ∥u∥2

L2

}, ∀u ∈ C∞0 (Ω),

where ⟨D⟩ = (e + ∣D∣2)1/2. The simplest example is the system in ℝ3 such as X1 =∂x1 , X2 = ∂x2 , X3 = exp(−∣x1∣−1/s)∂x3 with s > 0.

We have similary the logarithmic derivative norm in L2.

* LOGARITHMIC SOBOLEV SPACE 7

Proposition 1.3.5. Let s > 1/2, then for any u ∈ S(ℝd)

(1.3.2) ∣∣(log ⟨D⟩)su∣∣2L2(ℝd) = c0

∫∣u(x)− u(y)∣2∣log ∣x− y∣∣2s−1

∣x− y∣ddxdy

+O(∣∣u∣∣2L2(ℝd))

where the constants c0, C0 depending only on s and the dimension d.

Proof: It follows from the Planchrel formula that∫∣u(x)− u(y)∣2∣log ∣x− y∣∣2s−1

∣x− y∣ddxdy =

∫∣u(x)− u(x+ z)∣2∣log ∣z∣∣2s−1

∣z∣ddxdz

=

∫∣ log ∣z∣∣2s−1

∣z∣d

(∫∣eiz⋅� − 1∣2∣u(�)∣2d�

)dz =

∫I(�)∣u(�)∣2d�,

where

I(�) =

∫{∣z∣≤1}

∣eiz⋅� − 1∣2∣ log ∣z∣ ∣2s−1

∣z∣ddz.

Hence, it suffices to show that there exists a constant C > 0 such that

C−1I(�) ≤ ∣ log < � > ∣2s ≤ C(I(�) +O(1)).

When ∣�∣ ≤ 1, we have by the Taylor’s expansion that ∣eiz⋅� − 1∣2 ≤ ∣z∣2. Thus,

∣I(�)∣ ≤∫{∣z∣≤1}

∣ log ∣z∣ ∣2s−1

∣z∣d−2dz = cd−1

∫ 1

0

�(∣log �∣)2s−1 d� = O(1),

where cd−1 is the area of the unit sphere in ℝd. When ∣�∣ ≥ 1, by the change of variablesz∣�∣ = y, by denoting ! = �/∣�∣, we have

I(�) =

∫{∣y∣≤∣�∣}

∣∣∣eiy⋅! − 1∣2∣ log(∣y∣∣�∣

) ∣∣∣2s−1

∣y∣ddy

=

∫ ∣�∣0

∣∣∣ log(�∣�∣

) ∣∣∣2s−1

�

(∫Sd−1

∣ei�(#⋅!) − 1∣2d#)d�

=

∫ 1

0

⋅ ⋅ ⋅ d�+

∫ ∣�∣1

⋅ ⋅ ⋅ d� := I1(�) + I2(�),

where we have used the polar coordinate (�, #). Since I(�) is rotationally invariant, wecan simply take ! = (1, 0, ⋅ ⋅ ⋅ , 0) so that∫

Sd−1

∣ei�(#⋅!) − 1∣2d# = cd−2

∫ 2�

0

∣ei�#1 − 1∣2d#1 =cd−2

�

∫ 2��

0

∣eit − 1∣2dt,

where cd−2 denotes the area of the unit sphere in ℝd−1. Set

c1 =

∫ 2�

0

∣eit − 1∣2dt > 0.

If � ≥ 1, then we have

c1cd−2/2 ≤ c1cd−2[�]

�≤∫Sd−1

∣ei�(#⋅!) − 1∣2d# ≤ c1cd−2[�+ 1]

�≤ 2c1cd−2.

8 1. SOBOLEV SPACES

Therefore, I2(�) is bounded above and below by some uniform constants times

∫ ∣�∣1

∣∣∣ log(�∣�∣

) ∣∣∣2s−1

�d� =

∫ log ∣�∣

0

� 2s−1d� = (log ∣�∣)2s/(2s),

where we have used the change of variables � = log( ∣�∣�

). If � ≤ 1, then the Taylor

expansion implies ∣∣∣∣∫Sd−1

∣ei�(#⋅!) − 1∣2d#∣∣∣∣ ≤ cd−1�

2.

Since ∣ log( ∣�∣�

)∣ = ∣ log �∣+ ∣ log ∣�∣∣ when � ≤ 1 and ∣�∣ ≥ 1, we have

∣I1(�)∣ ≤ cd−1

∫ 1

0

�(∣ log �∣+ ∣ log ∣�∣∣)2s−1d� = O((log < � >)2s−1

).

Combining all the above estimates completes the proof of the proposition 1.3.5.

1.4. Sobolev embedding Theorem

In this section, we study the Sobolev embedding theorem. We define the Holderfunction space, for k ∈ ℕ, � ∈]0, 1[,

Ck,�(ℝd) ={∂�u ∈ L∞(ℝd), ∣�∣ ≤ k; sup

∣x−y∣∕=0

∣∂�u(x)− ∂�u(y)∣∣x− y∣�

< +∞, ∣�∣ = k},

with the norm

∥u∥Ck,� =∑∣�∣≤k

∥∂�u∥L∞(ℝd) +∑∣�∣=k

sup∣x−y∣∕=0

∣∂�u(x)− ∂�u(y)∣∣x− y∣�

.

It is Banach space.

Theorem 1.4.1. (a) For any 0 < s < d/2, we have continuous embedding Hs(ℝd) ⊂Lps(ℝd) where ps = 2d

d−2s;

(b) We have continuous embedding Hd/2(ℝd) ⊂ BMO.(c) For any 0 < s − d/2 = k + �, k ∈ ℕ, � ∈]0, 1[, we have continuous embedding

Hs(ℝd) ⊂ Ck,�(ℝd);

Remark : The space Hd/2(ℝd) is not included in L∞(ℝd) as it is indicated by thefollowing example. Let us define u by

u(�) =�(�)∣�∣−d

1 + log(2 + ∣�∣),

where � ∈ D(ℝd ∖{0}) is a non-negative function. It is obvious that u ∈ Hd/2(ℝd). Onthe other hand u does not belongs to L1(ℝd), u ∈ L1

loc(ℝd), u ≥ 0 so that u /∈ L∞(ℝd) .

1.4. SOBOLEV EMBEDDING THEOREM 9

Proof of Theorem 1.4.1 : (a) By density of S(ℝd) in Hs(ℝd), we prove the followingSobolev inequality

(1.4.1) ∥u∥Lps (ℝd) ≤ Cs∥u∥Hs(ℝd)

for u ∈ S(ℝd) . We have for p = ps ∈ [0,+∞[, thanks to Fubini Theorem

∥u∥pLp(ℝd)

=

∫ℝd∣u(x)∣pdx

= p

∫ℝd

∫ ∣u(x)∣

0

�p−1d�dx = p

∫ ∞0

�p−1ℳ(∣u∣ > �

)d�,

whereℳ( ⋅ ) is the Lesbesgue measure. We decompose new the function u in low andhigh frequencies, more precisely, take a positive constant A to determiner, we set

u = u1,A + u2,A

withu1,A = ℱ−1

(1B(0,A)u

), u2,A = ℱ−1

(1Bc(0,A)u

).

Since Supp ℱ(u1,A is compact, u1,A is bounded and

∥u1,A∥L∞ ≤ (2�)−d∥ℱ(u1,A)∥L1 ≤ (2�)−d∫B(0,A)

(1 + ∣�∣2)−s/2(1 + ∣�∣2)s/2∣u(�)∣d�

≤ (2�)−d∥u∥Hs

(∫B(0,A)

(1 + ∣�∣2)−sd�

)1/2

≤ Cs∥u∥HsAd2−s.

The triangle inequality implies that for any positive A,

{x ∈ ℝd; ∣u(x)∣ > �} ⊂ {x ∈ ℝd; ∣u1,A(x)∣ > �}∪{x ∈ ℝd; ∣u2,A(x)∣ > �}.

We choose

A = A� =

(�

4Cs∥u∥Hs

) pd

,

thenℳ{x ∈ ℝd; ∣u1,A�(x)∣ > �} = 0.

And we have

∥u∥pLp(ℝd)

= p

∫ ∞0

�p−1ℳ(∣u2,A�∣ > �

)d�.

It is well known that

ℳ(∣u2,A� ∣ > �

)=

∫(∣u2,A�

∣>�) dx

≤∫(∣u2,A�

∣>�) 4∣u2,A�(x)∣2

�2dx ≤

4∥u2,A�∥2L2

�2.

So that

∥u∥pLp(ℝd)

≤ 4p

∫ ∞0

�p−3∥u2,A�∥2L2d�

≤ 4p(2�)−d∫ ∞

0

�p−3

∫∣�∣≥A�

∣u(�)∣2d�d�.

10 1. SOBOLEV SPACES

By the definition of A�, we have

∣�∣ ≥ A� ⇐⇒ � ≤ C� = 4Cs∥u∥Hs ∣�∣dp .

Fubini Theorem implies that

∥u∥pLp(ℝd)

≤ 4p(2�)−d∫ℝd

(∫ C�

0

�p−3d�

)∣u(�)∣2d�

≤ 4p(2�)−d

p− 2

(4Cs∥u∥Hs

)p−2∫ℝd∣�∣

d(p−2)p ∣u(�)∣2d�.

As s = d(12− 1

p), (a) is proved by the following (first) Sobolev inequality.

(1.4.2) ∥u∥Lps (ℝd) ≤ Cs∥u∥Hs

with ps = 2dd−2s

, the constant Cs is also called Sobolev constant.

(c) We consider only the case 0 < s− d/2 < 1, we have firstly

(1.4.3) ∥u∥L∞ ≤ (2�)−d∥u∥L1 ≤ Cs∥u∥Hs

where ∫(1 + ∣�∣2)−sd� = C2

s <∞.

Using same decomposition u = u1,A + u2,A, u1,A is smooth and

∣u1,A(x)− u1,A(y)∣ ≤ ∥∇u1,A∥L∞∣x− y∣.Fourier inverse formula give

∥∇u1,A∥L∞ ≤∫ℝd∣�∣∣u1,A(�)∣d� ≤ ∥u∥Hs

(∫∣�∣≤A

∣�∣2−2sd�

)1/2

≤ Cs∥u∥HsA1−s+ d2 .

On the other hand

∥u2,A∥L∞ ≤∫ℝd∣u2,A(�)∣d� ≤ ∥u∥Hs

(∫∣�∣≥A

∣�∣−2sd�

)1/2

≤ Cs∥u∥HsA−s+d2 .

We get finally

∣u(x)− u(y)∣ ≤ ∥∇u1,A∥L∞∣x− y∣+ 2∥u2,A∥L∞

≤ Cs∥u∥Hs

(∣x− y∣A1−s+ d

2 + A1−s+ d2

),

choose A = ∣x− y∣−1, for 0 < ∣x− y∣ < 1, we have proved

∣u(x)− u(y)∣ ≤ Cs∥u∥Hs∣x− y∣s−d2 .

For ∣x− y∣ ≥ 1,(1.4.3) implies

∣u(x)− u(y)∣ ≤ 2∥u∥L∞ ≤ Cs∥u∥Hs∣x− y∣s−d2 .

We get finally,

(1.4.4) ∥u∥Cs−

d2≤ Cs∥u∥Hs .

It is also called Sobolev inequality . We have proved (c).


The limit case (b) for s = d2

is some, recall the norm of BMO, for any euclidian ballB of radius 0 < R small enough,∫

B

∣u− uB∣dx

∣B∣≤ ∥u1,A − (u1,B)B∥L2(B, ∣B∣−1dx)

+2

∣B∣1/2∥u2, A∥L2(ℝd).

We have∥u1,A − (u1,B)B∥L2(B, ∣B∣−1dx)

≤ R∥∇u1, A∥L∞

≤ R∥u∥Hd/2(ℝd)

(∫∣�∣≤A

∣�∣2−dd�)1/2

≤ CRA∥u∥Hd/2(ℝd).

1

∣B∣1/2∥u2,A∥L2(ℝd) ≤ C(AR)−d/2

(∫∣�∣≥A

∣�∣d∣u(�)∣2d�)1/2

.

We infer that ∫B

∣u− uB∣dx

∣B∣≤ C∥u∥Hd/2

(RA+ (RA)−d/2

).

So that take A = R−1, we have proved,

(1.4.5) ∥u∥BMO

= supB⊂ℝd

∫B

∣u− uB∣dx

∣B∣≤ C∥u∥Hd/2 .

It is also called Sobolev inequality . We have proved (b).

To understand the non smoothness of functions in Hs with s ≤ d2. We have the

following density Theorem.

Theorem 1.4.2. If s ≤ d2, then D(ℝd ∖ { 0 }) is dense in Hs(ℝd).

Proof : Let u ∈ D(ℝd ∖ { 0 })⊥ the orthogonal in Hs, then

us = ℱ−1((1 + ∣�∣2)su(�)

)∈ H−s(ℝd),

and for any ' ∈ D(ℝd ∖ { 0 }),

0 = (u, ')Hs =

∫ℝdus(�) ¯'(�)d� = ⟨us, '⟩.

This implies that Supp us ⊂ {0}, i. e.

us =∑∣�∣≤N

a�∂��0 ∈ H−s(ℝd).

Now −s ≥ −d/2 implies that us = 0 which proved Theorem 1.4.2.

We have the following Hardy’s inequality : For any s ∈ [0, d2[, there exists Cs > 0

such that

(1.4.6)

∫ℝd

∣u(x)∣2

∣x∣2sdx ≤ Cs∥u∥2

Hs(ℝd),

for any u ∈ Hs(ℝd). Again the classical Hardy’s inequality is with s = 1 (so thatd ≥ 3, and in form

(1.4.7)

∫ℝd

∣u(x)∣2

∣x∣2dx ≤ C∥∇u∥2

L2(ℝd),


for any u belong to the homogeneous Sobolev space H1(ℝd).

For the logarithmic Sobolev space given in the Definition 1.3.2, we have the followinglogarithmic Sobolev inequality. If s > 1/2, there exists Cs > 0 such that

(1.4.8)

∫ℝd∣u(x)∣2

∣∣∣∣log

(e+∣u(x)∣2

∥u∥2L2

)∣∣∣∣2s−1

dx ≤ Cs∥u∥2Hs

log,

for any u ∈ Hslog(ℝd). This implies the following continuous embedding

Hslog(ℝd) ⊂ L2

(logL)s−1/2(ℝd).

We have also the following stability of Sobolev spaces by nonlinear compositionresults (see [102]).

Lemma 1.4.1. Let F ∈ C∞(ℝ), F (0) = 0, s ≥ 0, if u ∈ Hs(ℝd) ∩ L∞(ℝd), thenF (u) ∈ Hs(ℝd) with

∥F (u)∥Hs ≤ CM,s∥u∥Hs

where the constant CM,s depends only on ∥F (j)∥L∞([−M,M ]),M[s]+1 with ∥u∥L∞ = M .

The same result is true for u ∈ L∞(]0, T [;Hs(ℝN)) ∩ L∞(]0, T [×ℝd) .

For the logarithmic Sobolev space, if u ∈ Hslog(ℝd)∩L∞(ℝd), then F (u) ∈ Hs−1/2

log (ℝd).

This means that Hs(ℝd) ∩ L∞(ℝd) is an algebra for any s ≥ 0. Since Hs(ℝd) ⊂L∞(ℝd) if s > d/2, then Hs(ℝd) is an algebra if s > d/2.

1.5. Compactness and interpolation inequalities

We have the following interpolation results. If s1 < s2, s = (1− �)s1 + �s2, � ∈]0, 1[and u ∈ Hs2(ℝd), then we have the convexity inequality

(1.5.1) ∥u∥Hs ≤ ∥u∥1−�Hs1∥u∥�Hs2

which derives easily from Holder inequality. We have also interpolation inequality, for" > 0,

(1.5.2) ∥u∥Hs ≤ "∥u∥Hs2 + "−�

1−� ∥u∥Hs1 .

The following compactness theorem is known by Rellich theorem. It is a key Theo-rem in the proof of existence of weak solutions for many non linear partial differentialequations.

Theorem 1.5.1. Let K be a compact subset of ℝd and s < s′. Denote

HsK(ℝd) = {f ∈ Hs(ℝd); Suppu ⊂ K}

Then, the embedding of Hs′K(ℝd) into Hs

K(ℝd) is a compact linear operator.

Remark : More precisely, if {un} ⊂ Hs′K(ℝd) such that

supn∥un∥Hs′ ≤ C,

then there exists u ∈ Hs′(ℝd), and there exists a subsequence {unk} such that

unk → u in Hs(ℝd),

1.5. COMPACTNESS AND INTERPOLATION INEQUALITIES 13

for any s < s′.We can prove this Theorem by the interpolation inequality (7.3.40). But we present

here the following more general compactness theorem.

Theorem 1.5.2. Let s < s′, then the multiplication by a function of S(ℝd) is acompact operator from Hs′(ℝd) into Hs(ℝd).

Proof : Let � ∈ S(ℝd) and {un} ⊂ Hs′(ℝd) such that ∥un∥Hs′ ≤ 1, we have toprove that we can extract a subsequence and {unk} such that {�unk} converge inHs(ℝd). Since Hs′(ℝd) is an Hilbert space, the weak compactness theorem ensures thatwe can extract a subsequence {unk} such that {unk} converge weakly to an elementu ∈ Hs′(ℝd) with ∥u∥Hs′ ≤ 1. Denote by vk = unk − u, we have to prove

supk∥vk∥Hs′ ≤ C =⇒ �vk → 0 in Hs(ℝd).

Now for any R > 0,∫(1 + ∣�∣2)s∣ℱ(�vk)(�)∣2d� ≤

∫∣�∣≤R

(1 + ∣�∣2)s∣ℱ(�vk)(�)∣2d�∫∣�∣≥R

(1 + ∣�∣2)s−s′(1 + ∣�∣2)s

′ ∣ℱ(�vk)(�)∣2d�

≤∫∣�∣≤R

(1 + ∣�∣2)s∣ℱ(�vk)(�)∣2d� +1

(1 +R2)s′−s∥�vk∥2

Hs′ .

For any small " > 0, take R big enough, we have

1

(1 +R2)s′−s∥�vk∥2

Hs′ ≤"

2.

On the other hand,

ℱ(�vk)(�) = (2�)−d∫�(� − �)vk(�)d� =

∫(1 + ∣�∣2)s

′ �(�)vk(�)d�

where for any � ∈ ℝd,

�(�) = (2�)−d(1 + ∣�∣2)−s′�(� − �) ∈ S(ℝd).

As vk → 0 weakly in Hs′(ℝd), it turns out that for any � ∈ ℝd,

limk→∞ℱ(�vk)(�) = 0.

If we have the estimate

(1.5.3) sup∣�∣≤R, k∈ℕ

∣ℱ(�vk)(�)∣ ≤M < +∞,

then Lebesgue dominant theorem implies that

limk→∞

∫∣�∣≤R

(1 + ∣�∣2)s∣ℱ(�vk)(�)∣2d� = 0

which leads to the convergence of �vk to 0 in Hs(ℝd).


We prove now (1.5.3), it is clear that∣∣ℱ(�vk)(�)∣∣ ≤ (2�)−d

∫ ∣∣�(�−�)∣∣ ∣∣vk(�)

∣∣d� ≤ ∥vk∥Hs′

(∫(1+ ∣�∣2)−s

′ ∣�(�−�)∣2d�)1/2

.

Since � ∈ S(ℝd), we have

∣�(� − �)∣2 ≤ CN0(1 + ∣� − �∣2)−N0

with N0 = d/2 + ∣s′∣+ 1. Thus, for any ∣�∣ ≤ R,∫(1 + ∣�∣2)−s

′∣�(� − �)∣2d� ≤ CR + 2N0CN0

∫∣�∣≥2R

(1 + ∣�∣2)−d/2−1d� ≤MR <∞

where we have used

�, � ∈ ℝd, ∣�∣ ≤ R, ∣�∣ ≥ 2R =⇒ ∣� − �∣ ≥ ∣�∣/2.We have finish the proof of Theorem 1.5.2.

For the function space HsK(ℝd), we have the following theorem.

Theorem 1.5.3. Let s > 0 and K a compact subset of ℝd, then there exists CK,s > 0such that

(1.5.4) C−1∥u∥2Hs ≤

∫∣�∣2s∣u(�)∣2d� ≤ C∥u∥2

Hs .

for any u ∈ HsK(ℝd). In the other words, for any s > 0 and K a compact subset of ℝd,

we have thatHsK(ℝd) = Hs

K(ℝd)

with equivalent norms.

The inequality (1.5.4) is known by Poincare inequality. The classical form is fors = 1 and present as

(1.5.5) ∥u∥L2 ≤ CK∥∇u∥L2 ,

for all u ∈ H1K(ℝd).

Proof : The right inequality is obvious. We prove the left inequality of (1.5.4) byabsolve , we suppose that there exists a sequence {un} of Hs

K(ℝd) such that

∥un∥Hs = 1, limn→∞

∫∣�∣2s∣un(�)∣2d� = 0.

By Rellich theorem, there exists a subsequence {unk} which converges to an elementu ∈ L2

K(ℝd) with ∥u∥L2 = 1. By Cauchy-Schwarz inequality,

∣unk(�)− u(�)∣ ≤ (2�)−d∫K

∣unk(x)− u(x)∣dx ≤ CK∥unk − u∥L2 .

Then unk converges uniformly to u. So Lebesgue dominant theorem implies that forany R > 0, ∫

∣�∣≤R∣�∣2s∣u(�)∣2d� = 0,

this implies that u = 0 and thus u = 0, we have proved (1.5.4) by contradiction.

1.5. COMPACTNESS AND INTERPOLATION INEQUALITIES 15

We have also the following Poincare inequality for functions supported in smallballs.

Proposition 1.5.1. Let 0 ≤ t ≤ s, there exists a constant C such that for any � > 0

(1.5.6) ∥u∥Ht ≤ C �s−t ∥u∥Hs

for any u ∈ HsB�

(ℝd), where B� = {x ∈ ℝd; ∣x− x0∣ ≤ �}.

Remark : The classical Poincare inequality (1.5.5) is holds for the function u ∈ H1(ℝd)such that Supp u ⊂ M = {x = (x1, ⋅ ⋅ ⋅ , xd) ∈ ℝd; ∣x1∣ ≤ �} (so that no necessarycompact). And the constant CK in (1.5.5) can be as Cd �.

From (1.5.4), we can also take the homogeneous norms in (1.5.6).

Proof of Proposition 1.5.1: By translation, we can take x0 = 0. Setting v(x) =u(�x), then v is supported in the unity ball. On the other hand, we have the followingtrivial inequality,

∥v∥Ht ≤ C∥v∥Hs , ∀ t ≤ s,

Using now the fact that

v(�) = �−du

(�

�

).

We have finish the proof of Proposition 1.5.1 by using (1.5.4) and By dilatation.

In the study of kinetic equations, we also need to consider the Sobolev spaces withweighted.

Definition 1.5.1. (1) For p ∈ [1,+∞[ and r > 0, we define the function space

Lpr(ℝd) = {f ∈ Lp(ℝd); ∥f∥pLpr(ℝd)

=

∫ℝd∣f(x)∣p < x >p r dx < +∞}

where < x >= (1 + ∣x∣2)1/2.(2) For p ∈ [1,+∞[ and r > 0, we define the function space

Lp(

logL)r

(ℝd) = {f ∈ Lp(ℝd);

∫ℝ3

∣f(x)∣p(

log(e+ ∣f(x)∣2))p r

dx < +∞}.

(3) For s, r ∈ ℝ, we also define the weighted Sobolev space Hsr (ℝd) by its norm:

∥f∥2Hsr (ℝd) =

∫ℝd∣Λsf(x)∣2 < x >2r dx,

where

Λsf = ℱ−1(

1 + ∣�∣2)s/2f(�)).

∙ Lpr(ℝd) is Banach space, but Lp(

logL)r

(ℝd) is not, since

∥f∥Lp(logL)r =

(∫ℝ3

∣f(x)∣p(

log(e+ ∣f(x)∣2))p r

dx

)1/p

is only a semi-norm.


∙ If s = k is a positive integer, then we can use the equivalent norm

∥f∥2Hkr (ℝd) =

∑∣�∣≤k

∫ℝd∣∂�f(x)∣2 < x >2r dx.

∙ If s ∈ ℝ+, k ∈ ℕ, then we can use the equivalent norm

∥f∥2Hsk(ℝd) =

∑∣�∣≤k

∫ℝd

(1 + ∣�∣2)s∣ℱ(x�f)(�)∣2d�.

We have the following interpolation inequality for the weighted in Sobolev space(see [50]).

Lemma 1.5.1. For any s ∈ ℝ+, r ∈ ℕ, " > 0. Then, there exits a constant Ks,r,",d >0 such that for any f ∈ S(ℝd),

(1.5.7) ∥f∥2Hsr (ℝd) ≤ Ks,r,",d∥f∥Hs−"

2r (ℝd)∥f∥Hs+"0 (ℝd).

Remark : By density, (1.5.7) is holds for any f ∈ Hs−"2r (ℝd)

∩Hs+"

0 (ℝd).

Proof : Write

∥f∥2Hsr (ℝd) =

∑∣�∣≤r

∫ℝd

(1 + ∣�∣2)s∣ℱ(x�f)(�)∣2d�

≤∑∣�∣≤r

∣∣∣∣∫ℝd

(1 + ∣�∣2)s(∂�f

)(�)(∂�

¯f)(�)d�

∣∣∣∣≤

∑∣�∣≤r

∣∣∣∣∫ℝd∂�(

(1 + ∣�∣2)s(∂�f

)(�))

¯f(�)d�

∣∣∣∣ .Notice that

∂�(

(1 + ∣�∣2)sg(�))

=∑�≤�

P�−�(�)((1 + ∣�∣2)s−(∣�−�∣))∂�g(�)

where P�−�(�) are polynomials of degree ∣�−�∣. Then there exits constants K�,� suchthat ∣∣∣P�−�(�)

(1 + ∣�∣2

)−∣�−�∣ ≤ K�,�.

1.6. SOBOLEV SPACE ON AN OPEN DOMAIN AND TRACE THEOREM 17

We obtain the estimate

∥f∥2Hsr (ℝd) ≤

∑∣�∣≤r

∑�≤�

∣∣∣∣∫ℝdP�−�(�)(1 + ∣�∣2)s−(∣�−�∣)(∂�+� f

)(�)

¯f(�)d�

∣∣∣∣≤

∑∣�∣≤r

∑�≤�

∫ℝd

∣∣∣P�−�(�)(1 + ∣�∣2)s−(∣�−�∣)−"∣∣∣1/2∣∣∂�+� f(�)

∣∣×∣∣∣P�−�(�)(1 + ∣�∣2)s−(∣�−�∣)+"

∣∣∣1/2∣∣f(�)∣∣d�

≤∑∣�∣≤r

∑�≤�

K�,�

(∫ℝd

(1 + ∣�∣2)s+"∣f(�)∣2d�)1/2

×(∫

ℝd(1 + ∣�∣2)s−"∣ℱ

(x�+�f(x)

)(�)∣2d�

)1/2

.

The lemma is proven.

We have also

Lemma 1.5.2. Let r > 0, " > 0, then there exists a constant C" > 0 such that forany f ∈ S(ℝd),

(1.5.8) ∥f∥2L2r(ℝd) ≤ C"∥f∥L1

2r(ℝd)∥f∥Hd/2+"(ℝd).

This is a direct application of Sobolev embedding (1.4.3),∫ℝd

(1 + ∣x∣2)r∣f(x)∣2dx ≤ ∥f∥L12r∥f∥L∞ ≤ C"∥f∥L1

2r∥f∥Hd/2+" .

1.6. Sobolev space on an open domain and trace Theorem

Definition 1.6.1. Let Ω be a regular open domain of ℝd, s ∈ ℝ. We denote byHs(Ω) the space of distributions u ∈ D′(Ω) which is the restriction on Ω of someelement u ∈ Hs(ℝd). We equip the quotient norm

(1.6.1) ∥u∥Hs(Ω) = inf ∥u∥Hs(ℝd),

where u describes the family of extension of u in Hs(ℝd). Hs0(Ω) is the closed of C∞0 (Ω)

in Hs(Ω).

Then Hs(Ω) and Hs0(Ω) are Hilbert spaces. For an integer index k ∈ ℕ and any

open domain Ω of ℝd, we set

ℋk(Ω) ={u ∈ L2(Ω); ∂�u ∈ L2(Ω), ∣�∣ ≤ k

}.

This is also a Hilbert space. We have

Proposition 1.6.1. If Ω is a regular open domain of ℝd, k ∈ ℕ, then

(1.6.2) Hk(Ω) = ℋk(Ω).


Since Ω is regular, we can suppose without loss of generality that Ω = ℝd+. Then

Seeley’s extension operators is a continuous maps from ℋk(ℝd+) to ℋk(ℝd) = Hk(ℝd).

See [27]. Remark that this Proposition is not true if Ω is not regular, and ℋk0(Ω) is

complicate in this case.

Let x = (x1, x′) ∈ ℝ×ℝd−1, we study now the trace of a function of Hs(ℝd) on the

hyperplane {x ∈ ℝd;x1 = 0}.

Theorem 1.6.1. Let s ∈ ℝ, s > 12. The trace operator : C∞0 (ℝd) → C∞0 (ℝd−1

defined by (u)(x′) = u(0, x′)

can be extended into a continuous maps from Hs(ℝd) into Hs− 12 (ℝd−1). Moreover, it

is an onto maps.

The trace Theorem is true for the restriction maps into a regular hyper-surface.But in this case, we need to define the Sobolev space on an hyper-surface (manifolds).We does not continue in this direction since we don’t study the boundary problem.

Remark that the limitation s > 12

is necessary, in fact, there exists the elements in

H12 (ℝd) which does not admit trace in L2(ℝd−1).

In the case of d = 1, this deduced from �0 /∈ H−1/2(ℝ). If d ≥ 2, take g ∈ S(ℝd−1),we consider the sequence {uk}k∈ℕ ⊂ S ′(ℝd) defined by

uk(�) = 12≤∣�1∣≤k

∣�1∣−1(

log ∣�1∣)− 3

4g(�′).

Then∥uk∥2

H1/2(ℝd) ≤ 4 log 2 ∥g∥2H1/2(ℝd−1).

But

uk(0, x′) = (2�)−1 g(x′)

∫2≤∣�1∣≤k

∣�1∣−1(

log ∣�1∣)− 3

4d�1

= (�)−1 g(x′)

∫ k

2

r−1(

log r)− 3

4dr = Ck g(x′)

where Ck = 4(�)−1((log k)1/4 − (log 2)1/4

). Thus

∥uk(0, ⋅ )∥2L2(ℝd−1) = C2

k∥g∥2L2(ℝd−1) → +∞.

This implies that the trace operator can’t be extended into a continuous maps fromH1/2(ℝd) into L2(ℝd−1).

CHAPTER 2

Pseudo-differential operators

In this chapter, we recall some basic properties of pseudo-differential calculus whichwe will use to the analysis of kinetic equations. For the more detail of this theory, thereexists many standard reference, see for examples [13, 67, 69, 70, 91, 92].

2.1. Symbols class

Recall that ifP (x,D) =

∑∣�∣≤m

a�(x)D�x

is a differential operator, where the coefficients a� ∈ C∞(ℝd), then for any u ∈ S(ℝd),we have

P (x,D)u(x) = (2�)−d∫ℝdeix ⋅ �P (x, �)u(�)d�,

whereP (x, �) =

∑∣�∣≤m

a�(x)��x

is a polynomials of � ∈ ℝd. We extend now this formula to more general functionsP (x, �).

We consider now symbol class.

Definition 2.1.1. Let m ∈ ℝ, then Sm is the set of all a ∈ C∞(ℝd ×ℝd) such thatfor all �, � ∈ ℕ, we have, for all x, � ∈ ℝd,

(2.1.1) ∣∂�� ∂�xa(x, �)∣ ≤ C�,�(1 + ∣�∣)m−∣�∣.Sm is called the space of symbols of order m. We write S−∞ =

∩Sm, S∞ =

∪Sm.

It is clair that

a ∈ Sm ⇒ ∂�� ∂�xa ∈ Sm−∣�∣; a ∈ Sm1 , b ∈ Sm2 ⇒ ab ∈ Sm1+m2 .

Some examples :

(1) P (x, �) =∑∣�∣≤m a�(x)�� ∈ Sm if a� ∈ C∞b (ℝd), where

C∞b (ℝd) = {f ∈ C∞(ℝd); ∂�f ∈ L∞(ℝd), ∀� ∈ ℕd}.We say that P (x, �) is a differential symbol of order m.

(2) Let a(�) ∈ C∞(ℝd ∖ {0}) a (positive) homogeneous function of order m ∈ ℝ inthe sense : ∀� > 0, a(��) = �ma(�), then a(�) = (1− �(�))a(�) ∈ Sm where

� ∈ C∞0 (ℝd); �(�) = 1, ∣�∣ ≤ 1

2and �(�) = 0, ∣�∣ ≥ 1.

We will use serval time this cutoff function in this chapter.

19

20 2. PSEUDO-DIFFERENTIAL OPERATORS

(3) If a ∈ S(ℝd), then a(�) ∈ S−∞.(4) The function eix ⋅ � is not a symbol.

The following Lemma is very easy but very utile .

Lemma 2.1.1. If a1, ⋅ ⋅ ⋅ , ak ∈ S0 and F ∈ C∞(ℂk), then F (a1, ⋅ ⋅ ⋅ , ak) ∈ S0.

2.2. Asymptotic expansion

We consider now a sequence of symbols aj ∈ Smj , j ∈ ℕ with the index decreasingand mj ↘ −∞. We give the following definition.

Definition 2.2.1. We say that∑aj is an asymptotic expansion of a symbol a ∈

Sm0, and write

a(x, �) ∼∑

aj(x, �).

If for any k ≥ 0,

a(x, �)−k∑j=0

aj(x, �) ∈ Smk+1 .

A symbol a ∈ Sm is called a classical symbol if a(x, �) ∼∑aj(x, �) and aj(x, �) is

(positive) homogeneous of order m − j for ∣�∣ ≥ 1 and any j ∈ ℕ. In this case, am iscalled the principal symbol of a.

The following Theorem give a sense for this asymptotic expansion.

Theorem 2.2.1. Let aj ∈ Smj , j ∈ ℕ with the index decreasing and mj ↘ −∞.Then there exists a symbol a ∈ Sm0 such that

a(x, �) ∼∑

aj(x, �).

We need the classical Borel Lemma.

Lemma 2.2.1. Let {bj}j∈ℕ ⊂ ℂ. There exists f ∈ C∞(ℝ) such that f (j)(0) = bjfor all j ∈ ℕ.

In this sense, we say also that we have asymptotic expansion

f(x) ∼∑

bjxj

j!when x → 0.

Proof : Setting

f(x) =∞∑j=0

bj�(�jx)xj

j!

with �j ↗ +∞ to choose, where � is cutoff function near to0. For j > k, we have∣∣∣∣ dkdxk(bj�(�jx)

xj

j!

)∣∣∣∣ =

∣∣∣∣∣k∑l=0

C lkbj�

k−l(�jx)�k−lj

xj−l

(j − l)!

∣∣∣∣∣ ≤ C∣bj∣�k−jj

1

(j − k)!.

We choose 1 + ∣bj∣ ≤ �j ↗ +∞, then f ∈ C∞(ℝ) and f (j)(0) = bj for all j ∈ ℕ.

2.2. ASYMPTOTIC EXPANSION 21

Proof of Theorem 2.2.1: Now it is similar as Borel Lemma for 1/∣�∣ → 0. We setnow

a(x, �) =∑

aj(x, �) =∑

(1− �("j�))aj(x, �),

with "j ↘ 0 to choose. Since

(1− �("j�)) = 0, if ∣�∣ ≤ 1

2"j.

Then for any R > 0 fixe, there exists j0 ∈ ℕ such that for all j ≥ j0

R ≤ 1

2"j,

then for any ∣�∣ ≤ R and for all j ≥ j0

aj(x, �) = (1− �("j�))aj(x, �) = 0.

So that the series is locally finite and the a ∈ C∞(ℝd × ℝd).Now for j ∈ ℕ (big enough), we claim now "j > 0 small enough such that

(2.2.1) ∣∂�� ∂�x aj(x, �)∣ ≤ 2−j(1 + ∣�∣)1+mj−∣�∣,

for any x, � ∈ ℝd and any ∣�∣+ ∣�∣ ≤ j. In fact, by Leibnitz formula,

∂�� ∂�x aj(x, �) = (1− �("j�))∂

�� ∂

�xaj(x, �)−

∑0< ≤�

C �"∣ ∣j (∂ �)("j�))∂

�− � ∂�xaj(x, �),

then∣∂�� ∂�x aj(x, �)∣ ≤ C�,�,j"j(1 + ∣�∣)1+mj−∣�∣

with

C�,�,j = supx,�∈ℝd

{ ∣1− �("j�)∣"j∣�∣

∣∂�� ∂�xaj(x, �)∣(1 + ∣�∣)−(mj−∣�∣)

+∑

0< ≤�

C �∣"j�∣∣ ∣(∂ �)("j�))∣∂�− � ∂�xaj(x, �)∣(1 + ∣�∣)−(mj−∣�− ∣)

}.

Since aj ∈ Smj , we choose 0 < "j such that

"j sup∣�∣+∣�∣≤j

C�,�,j ≤ 2−j.

We have proved (2.2.1). We have also proved that

a ∈ Sm0 .

Because there exists N0 such that mN0

+ 1 ≤ m0 and

a =∑j<N0

aj +∞∑

j=N0

aj.

It is clary that ∑j<N0

aj ∈ Sm0 ,

and (2.2.1) implies that∞∑

j=N0

aj ∈ SmN0+1 ⊂ Sm0 .


Now for the fixed �, � ∈ ℕd, k ∈ ℕ, we have to prove

(2.2.2)∣∣∣∂�� ∂�x(a(x, �)−

k∑j=0

aj(x, �))∣∣∣ ≤ C�,�,j(1 + ∣�∣)mk+1−∣�∣.

We choose N ≥ ∣�∣+ ∣�∣ big such that mN + 1 ≤ mk+1, then (2.2.1) implies that∣∣∣∂�� ∂�x(a(x, �)−N−1∑j=0

aj(x, �))∣∣∣ ≤ (1 + ∣�∣)mk+1−∣�∣.

On the other hand, we have

a(x, �)−k∑j=0

aj(x, �) =(a(x, �)−

N−1∑j=0

aj(x, �))

+N−1∑j=k+1

aj(x, �)

+k∑j=0

(aj(x, �)− aj(x, �)

).

Then∑N−1

j=k+1 aj(x, �) ∈ Smk+1 and aj − aj ∈ S−∞ deduce then (2.2.2) which prove theTheorem 2.2.1.

2.3. Definition of pseudo-differential operators

Definition 2.3.1. Let a ∈ S(ℝd × ℝd) and u ∈ S(ℝd), we define,

(2.3.1) a(x,D)u(x) = (2�)−d∫ℝdeix ⋅ �a(x, �)u(�)d�.

a(x,D) is called the pseudo-differential operator.

The formula (2.3.1) can be extended immediately to the differential symbol of orderk ∈ ℕ,

a(x, �) =k∑

∣�∣≤m

a�(x)��,

where a� ∈ C∞b (ℝd), ∣�∣ ≤ k. And a(x,D)u ∈ S(ℝd) for any u ∈ S(ℝd). Moregenerally, we have

Theorem 2.3.1. Let m ∈ ℝ, then the pseudo-differential operator defined in (2.3.1)can be extended to a symbol a ∈ Sm and

a(x,D) : S(ℝd) −→ S(ℝd)

is a continuous maps. The commutators with derivative Dj and multiplication by xjare

[a(x,D), Dj] = i(∂xja

)(x,D), [a(x,D), xj] = −i

(∂�ja

)(x,D).

One calls a(x,D) a pseudo-differential operator of order m of symbol a. We denote bya(x,D) ∈ Op(Sm).

2.4. ALGEBRA OF PSEUDO-DIFFERENTIAL OPERATORS 23

Proof : Since u ∈ S, it is clear that (2.3.1) defines a continuous function with

∣a(x,D)u(x)∣ ≤ (2�)−d∫ℝd

(1 + ∣�∣)m∣u(�)∣d� supx,�∈ℝd

(∣a(x, �)∣(1 + ∣�∣)−m

).

So that for prove the Theorem, it is enough to get the commutators formula. A directcalculus give

Dja(x,D)u(x) = a(x,D)Dju(x)− i(∂xja

)(x,D)u(x).

Since ℱ(xju) = −Dju(�), we have that

a(x,D)(xju) = xja(x,D)u− i(∂�ja

)(x,D)u(x).

By iteration, we have that x�D�a(x,D)u is a linear combination of term(∂�′

� ∂�′

x a)(x,D)x�

′′D�′′u; �′ + �′′ = �, �′ + �′′ = �.

Hence x�D�a(x,D)u is bounded. The proof is complete.

2.4. Algebra of pseudo-differential operators

If we introduce the definition of Fourier transformation of u in (2.3.1), it followsthat the Schwartz kernel of a(x,D) is give by

(2.4.1) K(x, x− y) = (2�)−d∫ℝdei(x−y) ⋅ �a(x, �)d�,

a(x,D)u(x) =

∫K(x, x− y)u(y)dy

which exists as an oscillatory integral, and we can interpret as (2�)−da(x, x−y) where ais the Fourier transformation of a(x, �) with respect to the � variable. Then by Fourierinverse formula,

a(x, �) =

∫K(x, x− y)e−iy ⋅ �dy.

Remark : Here the Fourier transformation and the inverse formula is in temperatedistribution space S ′. Schwartz kernel theorem is also with kernel in S ′(ℝ2d), and itdefine a continuous maps from S to S ′. But for the symbol class a ∈ Sm, we have acontinuous maps from S to S.

We study now the adjoint of a(x,D) with respect to the scalar product in L2,

(u, v) =

∫ℝdu(x)v(x)dx; u, v ∈ S(ℝd).

We define the adjoint operator a∗(x,D) by

(a(x,D)u, v) = (u, a∗(x,D)v), , ∀u, v ∈ S(ℝd).

Theorem 2.4.1. If a ∈ Sm, then a∗(x,D) is also a pseudo-differential operator withsymbol a∗(x, �) ∈ Sm and

a∗(x, �) ∼∑�∣

1

�!∂��D

�x a(x, �).


We have therefore a(x,D) can be extended to a continuous map from S ′ to S ′, as theadjoint of a∗(x,D).

The symbol of adjoint operator a∗(x,D) is given by oscillatory integral

a∗(x, �) = (2�)−d∫e−iy ⋅ �a(x− y, � − �)dyd�,

and the asymptotic expansion deduced from Taylor formula. We omit the detail ofproof, and send to [13, 67, 92].

Now we study the composition of operators.

Theorem 2.4.2. If a1 ∈ Sm1 and a2 ∈ Sm2, then as operators in S or in S ′,a1(x,D)a2(x,D) =

(a1♯a2

)(x,D)

is also a pseudo-differential operator of order m1 + m2 and the following asymptoticexpansion for the symbol.

(2.4.2)(a1♯a2

)(x, �) ∼

∑�

1

�!

(∂�� a1

)(x, �)

(D�x a2

)(x, �).

We have again the oscillatory integral(a1♯a2

)(x, �) = (2�)−d

∫e−i(x−y) ⋅ �−�)a1(x, �)a2(y, �)dyd�.

From this composition results, we have immediately the following commutatorsresults.

Theorem 2.4.3. If a1 ∈ Sm1 and a2 ∈ Sm2, then

[a1(x,D), a2(x,D)] = a1(x,D)a2(x,D)− a2(x,D)a1(x,D) = b(x,D)

is a pseudo-differential operator of order m1 +m2 − 1 such that

(2.4.3) b(x, �) =1

i

{a1, a2

}(x, �) + rm1+m2−2(x, �),

where rm1+m2−2 ∈ Sm1+m2−2, and{f, g

}(x, �) =

d∑j=1

((∂�jf)(x, �)(∂xj g)(x, �)− (∂xjf)(x, �)(∂�j g)(x, �)

)is called “‘Poisson bracket” of functions f(x, �) and g(x, �).

For a ∈ Sm a classical symbol, we say that a(x,D) is a elliptic pseudo differentialoperator of order m if for some positive constants c, the principal symbol satisfies

∣am(x, �)∣ ≥ c∣�∣m, ∀ ∣�∣ ≥ 1.

We have now the inversion of elliptic operators.

Theorem 2.4.4. Let a(x,D) be an elliptic pseudo differential operator of order m,then there exists b ∈ S−m such that

(2.4.4) (a ♯ b)(x, �)− 1 ∈ S−∞ and (b ♯ a)(x, �)− 1 ∈ S−∞.

2.5. CONTINUITY IN SOBOLEV SPACES 25

Proof : In fact, if we take

b0(x, �) =(1− �(�))

am(x, �)∈ S−m

where � is the cutoff function near to 0. Then Theorem 2.4.2 implies

(a ♯ b0)(x, �) = 1− r(x, �),

with r ∈ S−1. Now for k ≥ 0 setting

bk(x,D) = b0(x,D) ∘ r(x,D)k ∈ Op (S−m−k).

Theorem 2.2.1 implies that there exists b ∈ S−m such that for any k ∈ ℕ,

a(x,D)b(x,D)− Id = a(x,D)(b(x,D)−

∑j<k

bj(x,D))− r(x,D)k ∈ Op (S−k).

We have proved Theorem 2.4.4.

Exercice 2.4.1. Assume that �, ∈ C∞b such that Supp�∩Supp = ∅. Prove thatfor any pseudo-differential operators a(x,D), we have

� a(x,D) ∈ Op (S−∞).

2.5. Continuity in Sobolev spaces

Theorem 2.5.1. Let a ∈ S0, then a(x,D) is bounded in L2(ℝd).

For the proof we need a classical lemma of Schur :

Lemma 2.5.1. Let K ∈ C0(ℝd × ℝd) and

supy

∫∣K(x, y)∣dx ≤ C, sup

x

∫∣K(x, y)∣dy ≤ C,

then the integral operators with kernel K has norm ≤ C in L2(ℝd).

Recall the integral operators of kernel K(x, y) is defined by

Ku(x) =

∫ℝdK(x, y)u(y)dy.

Then, by Cauchy-Schwarz’s inequality and Fubini Theorem

∥Ku∥2L2 =

∫ ∣∣∣∣∫ K(x, y)u(y)dy

∣∣∣∣2 dx ≤ ∫ (∫ ∣K(x, y)∣∣u(y)∣2dy∫∣K(x, y)∣dy

)dx

≤ C

∫ (∫∣K(x, y)∣dx

)∣u(y)∣2dy ≤ C2∥u∥2

L2 .


Proof of Theorem 2.5.1 : Assume first that a ∈ S−d−1, then the kernel K of theoperator a(x,D) is continuous and

∣K(x, x− y)∣ ≤ (2�)−d∫∣a(x, �)∣d� ≤ C1

∫(1 + ∣�∣)−d−1d� ≤ C2.

Now (x− y)�K(x, x− y) is the kernel of the commutators

[xj1 , [xj2 ⋅ ⋅ ⋅ , [xjd , a(x,D)]]] = i∣�∣(∂�� a)(x,D),

it is a pseudo differential operators of order −d− 1− ∣�∣, so that

(1 + ∣x− y∣)d+1∣K(x, x− y)∣ ≤ Cd.

Therefore, the kernel K satisfies the condition of Lemma 2.5.1, and a(x,D) is boundedin L2.

Next we prove by induction that a(x,D) is L2 continuous if a ∈ Sk and k ≤ −1.We have

∥a(x,D)u∥2L2 = (a(x,D)u, a(x,D)u) = (b(x,D)u, u)

where b(x,D) = a∗(x,D)a(x,D) is order 2k. The continuity of a(x,D) is therefore aconsequence of that of b(x,D), and

∥a(x,D)u∥2L2 ≤ ∥b(x,D)u∥L2 ∥u∥L2 ≤ C∥u∥2

L2 .

So that, from a ∈ S−d−1, we get the continuity for a ∈ S −d−12 , a ∈ S −d−1

4 , ⋅ ⋅ ⋅ hance fora ∈ S−1.

Assume now a ∈ S0 and choose

M > 2 sup ∣a(x, �)∣2.Then, by Lemma 2.1.1,

c(x, �) = (M − ∣a(x, �)∣2)1/2 ∈ S0,

since M − ∣a(x, �)∣2 ≥ M/2 and we can choose F ∈ C∞(ℝ) with F (t) = t1/2 whent ≥M/2.

Now Theorem 2.4.1 and 2.4.2 show that

c∗(x,D)c(x,D) = M − a∗(x,D)a(x,D) + r(x,D),

where r ∈ S−1. Then

0 ≤ ∥c(x,D)u∥2L2 = M∥u∥2

L2 − ∥a(x,D)u∥2L2 +

(r(x,D)u, u

)L2 .

Since r(x,D) is already known to be L2 continuous, We have that

∥a(x,D)u∥2L2 ≤M∥u∥2

L2 +(r(x,D)u, u

)L2 ≤ C∥u∥2

L2 .

It follows from the proof that the norm of a(x,D) can be estimate by a semi-normof a in S0. There is a very simple proof of L2 continuity which requires no smoothnessat all in � but instead some decay as x→∞.

Theorem 2.5.2. Let a(x, �) be a measurable function which d+1 times continuouslydifferentiable with respect tox for fixed �, if

sup�∈ℝd

∑∣�∣≤d+1

∫∣∂�xa(x, �)∣dx ≤M < +∞.

2.5. CONTINUITY IN SOBOLEV SPACES 27

Then a(x,D) is bounded in L2(ℝd) with norm ≤ CM .

In fact, we have

ℱ(a(x,D)u(x)

)(�) =

∫A(� − �, �)u(�)d�,

where

A(�, �) = (2�)−d∫e−ix ⋅ �a(x, �)dx.

By hypothesis(1 + ∣�∣)d+1∣A(�, �)∣ ≤ CM.

which implies that∫∣A(� − �, �)∣d� ≤ CM,

∫∣A(� − �, �)∣d� ≤ CM.

Then Lemma 2.5.1 implies that

∥ℱ(a(x,D)u(x)

)( ⋅ )∥2

L2 ≤ CM∥u∥2L2

which completes the proof.

Theorem 2.5.3. Let a ∈ Sm, then a(x,D) is a continuous operator from Hs(ℝd)to Hs−m(ℝd) for every s.

Moreover, if a(x,D) is elliptic of order m, then

a(x,D) : Hs(ℝd) −→, Hs−m(ℝd)

is an isomorphism.

Let Λsx = (1 + ∣Dx∣2)s/2, then

∥a(x,D)u∥Hs−m ≤ C∥u∥Hs , ∀ u ∈ S(ℝd)

is equivalent to

∥Λs−mx a(x,D)Λ−sx (Λs

xu)∥L2 ≤ C∥Λsxu∥L2 ∀ Λs

xu ∈ S(ℝd).

Since Λs−mx a(x,D)Λ−sx = a(x,D) is a pseudo-differential operator of order 0, it is

continuous in L2, so that

∥a(x,D)v∥L2 ≤ C∥v∥L2 ∀ v ∈ S(ℝd).

and u ∈ Hs is equivalent to Λsxu ∈ L2. In the elliptic case, we use the continuity of

inverse operator of order −m. We complete the proof by the density of S(ℝd) in L2.

Garding inequality

If a(D) is a pseudo-differential operator with constant coefficients and a(�) ≥ 0,then a(D) is “positive” in the sense(

a(D)u, u)≥ 0, ∀ u ∈ S.

The Garding inequality is the following : For m ∈ ℝ, if a ∈ S2m+1 and ℜe a(x, �) ≥0, then

ℜe(a(x,D)u, u

)≥ −C∥u∥2

Hm , ∀ u ∈ S.We have now the following weak Garding inequality.


Proposition 2.5.1. Let a ∈ S2m a classical symbol, suppose that there exists c > 0such that

ℜe a2m(x, �) ≥ c∣�∣2m, ∀ ∣�∣ ≥ 1.

Then, for any N ∈ ℕ, there exists CN > 0 such that

(2.5.1) ℜe(a(x,D)u, u

)≥ c

2∥u∥2

Hm − CN∥u∥2H−N , ∀ u ∈ S.

Proof : We have first

ℜe(a(x,D)u, u

)=

(a(x,D) + a∗(x,D)

2u, u

)=(b(x,D)u, u

).

Using Theorem 2.4.1

b(x, �) =1

2(a(x, �) + a∗(x, �) = ℜe a2m(x, �) + r(x, �)

with r ∈ S2m−1, then there exists C > 0 such that

b(x, �)− 3

4c(1 + ∣�∣2)m ≥ � > 0, ∀ ∣�∣ ≥ C.

Then

d(x, �) =

(b(x, �)− 3

4c(1 + ∣�∣2)m

)1/2 (1− �(C−1�)

)∈ Sm.

Using the composition formula of Theorem 2.4.2,

(d∗ ♯ d)(x, �) = b(x, �)− 3

4c(1 + ∣�∣2)m + r(x, �)

with r ∈ S2m−1. Then

0 ≤ ∥d(x,D)u∥2L2 =

((b(x,D)− 3

4c(1 + ∣D∣2)m + r(x,D)

)u, u

).

By the continuity of r(x,D) ∈ Op(S2m−1, we get

ℜe(a(x,D)u, u

)≥ 3c

4∥u∥2

Hm − C0∥u∥Hm∥u∥Hm−1 .

Now interpolation inequality (7.3.40) deduced

∥u∥Hm−1 ≤ "∥u∥Hm + C",N∥u∥H−N .Choose

C0" =c

4.

We have proved the Proposition 2.5.1.

CHAPTER 3

Non linear Fourier analysis on ℝn

In this chapter, we study the functions spaces which we need in the theory ofpartial differential equations. We will focus to Fourier analysis of function space, it isa fundamental theory in modern analysis of partial differential equations.

3.1. Littlewood-Palay theory

In this section, we study the proprieties of a functions who’s Fourier transformationis supported in a ball or a ring. For a constant K > 1, we define the ball and ring asfollowing

ℬ = {� ∈ ℝn; ∣�∣ ≤ K}, C = {� ∈ ℝn;K−1 ≤ ∣�∣ ≤ 2K}.

3.1.1. Bernstein inequalities.

Lemma 3.1.1. For 1 ≤ a ≤ b ≤ +∞, a constant C > 0 depends only on n,K, a, bsuch that for any � > 0 :

1) If u ∈ La(ℝn), Supp u ⊂ �ℬ, then for any k ∈ ℕ, we have

sup∣�∣=k∥D�u∥Lb ≤ Ck+1�k+n( 1

a− 1b)∥u∥La .

2) if u ∈ La(ℝn), Supp u ⊂ � C, then for any k ∈ ℕ, we have

C−k−1�k∥u∥La ≤ sup∣�∣=k∥D�u∥La ≤ Ck+1�k∥u∥La ,

and for any smooth homogeneous function � of degree m ∈ ℝ,

∥�(D)u∥Lb ≤ C�,m�k+n( 1

a− 1b)∥u∥La .

Proof: 1) : Choose ∈ C∞0 (ℝn), Supp ⊂ 2ℬ, (�) = 1 for � ∈ ℬ, and set �(�) = (�−1�), g(x) = ℱ−1( )(x), then the hypothesis Supp u ⊂ �ℬ implies u(�) = �(�)u�),

u(x) = (�ng�−1 ∗ u(x) ∈ C∞(ℝn)

and

D�u(x) = �n+∣�∣((D�g)�−1 ∗ u(x).

Since ∥�n(D�g)�−1∥L1 = C(n, �) < +∞, which give

∥D�u∥L∞ ≤ C(n, �)�∣�∣∥u∥L∞ .

We have proved 1) for a = b = +∞. The general case is deduced from Younginequality

∥f ∗ g∥Lb ≤ ∥f∥La∥g∥Lc29

30 3. NON LINEAR FOURIER ANALYSIS ON ℝn

where 1b

= 1a

+ 1c− 1, and the fact, for any 1 ≤ c ≤ +∞

∥�n(D�g)�−1∥Lc = ∥(D�g)∥Lc ≤ ∥(D�g)∥L+∞ + ∥(D�g)∥L1

≤ 2∥(1 + ∣ ⋅ ∣2)n(D�g)∥L+∞

≤ ∥(1−Δ)n(( ⋅ )� )∥L1 = C(n, �) < +∞

2) Set C = {� ∈ ℝn; (2K)−1 ≤ ∣�∣ ≤ 2K}, and choose ∈ C∞0 (C), (�) = 1

for � ∈ C, set �(�) = (�−1�), g(x) = ℱ−1( )(x), then the hypothesis Supp u ⊂ � Cimplies u(�) = �(�)u�), For � ∈ ℕd, setting

g�,� = ℱ−1(

(i�)�∣�∣−2k �(�)).

Then ∑∣�∣=k

(−i�)�g�,� = �(�),

which implies

u =∑∣�∣=k

g�,� ∗ ∂�u.

By Young inequality,

∥u∥La ≤∑∣�∣=k

∥g�,�∥L1∥D�u∥La ,

and an elementary calculus∥g�,�∥L1 ≤ C�−k,

For the second part of 2), only difference with 1) is for negative m, but it is same asabove.

Exercise : Let 1 ≤ a > +∞. A constant C exists such that for any � > 0 andu ∈ L2a with Supp u ⊂ � C, we have

∥ua∥L2 ≤ C�−1∥∇ua∥L2 .

Indiction : This results is some sense surprising, for example, Supp u2 is not in� C. We can use the identity

u2a =d∑j=1

u2a−1∂juj

with

uj = ℱ−1(− i�j∣�∣−2u(�)

).

3.1.2. Dyadic partition of unity. We consider now a dyadic decomposition ofℝn, for K > 1 a fixed constant (we can choose K = 2), and p ∈ ℕ+ we set

(3.1.1) Cp = 2pC = {� ∈ ℝn;K−12p ≤ ∣�∣ ≤ K2p+1},and C−1 = ℬ, then {Cp}+∞

−1 is a recover of ℝn.

Lemma 3.1.2. There exist N1 which depends only on K such that for any p

#{q; Cq∩Cp ∕= ∅} ≤ N1.

3.1. LITTLEWOOD-PALAY THEORY 31

That means {Cp}+∞−1 is a uniformly finite recover of ℝn.

Proof: Fixed p, and suppose that Cq∩Cp ∕= ∅. The case q ≥ p and q ≤ p is similar,

we consider only q ≤ p and p ∕= −1. If � ∈ Cq∩Cp, q ∕= −1, then

K−12p ≤ ∣�∣ ≤ K2q+1,

which give 2p−q ≤ 2K2, and p− q ≤ [1 + 2 log2K]. Note N1 = 2[1 + 2 log2K] + 2, wehave proved Lemma.

Lemma 3.1.3. There exist ', ∈ C∞0 (ℝn), with Supp ⊂ C−1, Supp' ⊂ C, suchthat for any � ∈ ℝn and any N0, we have

(�) +∞∑p=0

'(2−p�) = 1,

and

(�) +

N0−1∑p=0

'(2−p�) = (2−N0�).

Proof: Take � ∈ C∞0 (ℝn) with 0 ≤ � ≤ 1, Supp � ⊂ C, and �(�) = 1 for K−1 ≤∣�∣ ≤ 2K. Set

s(�) =∞∑

p=−∞

�(2−p�), � ∈ ℝn ∖ 0,

then from Lemma 3.1.2, s ∈ C∞(ℝn ∖ 0). We define now

'(�) = �(�)/s(�).

If ∣�∣ ≥ K, p ≤ −1, we have 2−p∣�∣ = 2∣p∣∣�∣ ≥ 2∣p∣K ≥ 2K, and 2−p� /∈ C, then�(2−p�) = 0. Therefore, if ∣�∣ ≥ K,

∞∑p=0

'(2−p�) =∞∑

p=−∞

�(2−p�)

s(2−p�)=

∑∞p=−∞ �(2

−p�)∑∞p=−∞ �(2

−p�)= 1.

here we have used the fact s(2−p�) =∑∞

q=−∞ �(2−(q+p)�) =

∑∞p1=−∞ �(2

−p1�) = s(�).

Take now (�) = 1−∑∞

p=0 '(2−p�), then ∈ C∞0 (ℝn), Supp ⊂ ℬ. Now for any N0,

(2−N0�) +∞∑p=0

'(2−p−N0�) = 1

= (�) +

N0−1∑p=0

'(2−p�) +∞∑

p=N0

'(2−p�).

Since∑∞

p=0 '(2−p−N0�) =∑∞

p1=N0'(2−p1�), we have proved Lemma.

For , ' ∈ C∞0 (ℝn), we can define the pseudo-differential operators (D) as

(D)u(x) = △−1u = ℱ−1( (�)u(�)

), '(2−pD)u = △pu = ℱ−1

('(2−p�)u(�)

),

and setting

Sq(u) =

q−1∑p=−1

△pu = (2−qD)u.


Definition 3.1.1. For u ∈ S ′(ℝn), we define their Littlewood-Paley decomposition(or dyadic decomposition) {up}∞p=−1 as u−1 = (D)u, up = '(2−pD)u.

It is evident that up ∈ S ′ for any u ∈ S ′, and Suppup ⊂ Cp, we have

Theorem 3.1.1. For u ∈ S ′, we have u =∑∞

p=−1 up, in sense of S ′.Proof: For any f ∈ S, we have

f(�) = limq→∞

Sq(f) =∞∑

p=−1

fp

in the sense of S, then for any u ∈ S ′, using Parseval formula, we have

(u, f) = (2�)−n(u, f) = (2�)−n∞∑

p=−1

(u, fp)

= (2�)−n∞∑

p=−1

(up, f) = (2�)−n(∞∑

p=−1

up, f)

= (∞∑

p=−1

up, f).

We have proved Theorem.

3.2. Characterization of Sobolev spaces

In this section, we will give a characterization of Sobolev spaces Hs by Littlewood-Paley decomposition.

Theorem 3.2.1. For s > 0, the following properties are equivalent.(a) u ∈ Hs(ℝn);(b) u =

∑∞p=−1 up, with Suppup ⊂ Cp and

∥up∥L2 ≤ cp2−ps, {cp} ∈ l2;

(c) u =∑∞

p=−1 up, with Suppup ⊂ 2p ℬ and

∥up∥L2 ≤ cp2−ps, {cp} ∈ l2;

(d) u =∑∞

p=−1 up, with up ∈ C∞ and for any � ∈ ℕn,

∥D�up∥L2 ≤ cp,�2−p(s−∣�∣), {cp,�} ∈ l2.Remark : The equivalence of (a) and (b) is true for all s ∈ ℝ.

Proof: Suppose that u ∈ Hs(ℝn) ⊂ S ′, and Littlewood- Paley decomposition of uas∑∞

p=−1 up, then Suppup ⊂ Cp, and

∥up∥2Hs =

∫(1 + ∣�∣2)s∣up(�)∣2d�

=

∫(1 + ∣�∣2)s∣'(2−p�)u(�)∣2d�

=

∫Cp

(1 + ∣�∣2)s∣'(2−p�)u(�)∣2d�,

3.2. CHARACTERIZATION OF SOBOLEV SPACES 33

since for any s ∈ ℝ and � ∈ Cp, we have K122ps ≤ (1 + ∣�∣2)s ≤ K222ps, we obtain

(3.2.1) K1∥up∥L2 ≤ 2−ps∥up∥Hs ≤ K2∥up∥L2 .

Set now cp = ∥up∥Hs , we want to prove {cp} ∈ l2, since for ∣p − q∣ ≥ N1, Cp∩Cq = ∅,

then series

Sq,N1u =∞∑k=0

uq+kN1

is orthogonal, and

∥Sq,N1u∥2Hs =

∫(1 + ∣�∣2)s∣Sq,N1u∣2d�

=∞∑k=0

∫(1 + ∣�∣2)s∣uq+kN1(�)∣2d�

=∞∑k=0

∥uq+kN1∥2Hs ,

but∞∑

p=−1

∥up∥2Hs ≤ N1

∞∑k=0

∥uq+kN1∥2Hs = N1∥Sq,N1u∥2

Hs ≤ ∥u∥2Hs ≤ N1∥u∥2

Hs .

We have proved (a) → (b) for any s ∈ ℝ. In inverse, if we have (b), then (3.2.1) give∥up∥Hs ≤ K2cp, and

∥u∥2Hs ≤

N1−1∑q=0

∥Sq,N1u∥2Hs =

N1−1∑q=0

(∞∑k=0

∥uq+kN1∥2Hs

)

≤ K22

∞∑p=0

c2p < +∞.

that means u ∈ Hs(ℝn), then (b) ⇔ (a) also for any s ∈ ℝ.(b) → (c) is evident, since Cp ⊂ B(0, 2K2p).Suppose that we have (c), and u =

∑∞p=−1 up, with Suppup ⊂ 2p ℬ, then up ∈ C∞,

and for all ∀� ∈ ℕn, we have

∥D�up∥L2 = ∥D�up∥L2 = ∥��up(�)∥L2

≤ K∣�∣1 2p∣�∣∥up∥L2 = K

∣�∣1 2p∣�∣∥up∥L2

≤ K∣�∣1 cp2

−ps+p∣�∣ = cp,�2−ps+p∣�∣,

we have proved (c) → (d).We prove now (c)→ (a), it is now necessary s > 0. From (c), we have immediately

u =∑up ∈ L2, and the properties Suppup ⊂ 2p ℬ implies

vk = '(2−kD)u = '(2−kD)∑

∣p−k∣≤N1

up.


Then, we have

∥vk∥2L2 =

∥∥∥∥∥∞∑

p=k−N1

'(2−pD)up

∥∥∥∥∥2

L2

=

∫ ∣∣∣∣∣∞∑

p=k−N1

'(2−kD)up(x)

∣∣∣∣∣2

dx

≤∫ ( ∞∑

p=k−N1

22ps∣'(2−kD)up(x)∣2)(

∞∑p=k−N1

2−2ps

)dx

≤ C2−2ks

∞∑p=−1

22ps∥'(2−kD)up∥2L2 .

Here we have used the fact s > 0 to get∑∞

p=k−N12−2ps ≤ C2−2ks. As in the proof of

(a) → (b), we have∞∑

k=−1

∥'(2−kD)up∥2L2 ≤ C∥up∥2

L2 .

We set c2k =

∑∞p=−1 22ps∥'(2−kD)up∥2

L2 , then {ck} ∈ l2, and u =∑vk verifies the

condition of (b), we have proved (c) → (b) ⇔ (a).Now it is enough to prove (d) → (a). Under assumption of (d), we have firstly

u =∑up ∈ L2. Take � ∈ ℕn, ∣�∣ = s0 > s > 0, and k(�) = (2−k�) ∈ C∞0 (ℝn) with

Supp k ⊂ B(0, C22k+1), k(�) = 1, ∣�∣ ≤ C12k, then

Supp k(1− k) ⊂ {� ∈ ℝn;C12k ≤ ∣�∣ ≤ C22k+1}.

Set uk(�) = k(�)uk(�) + (1− k(�))uk(�) = u(1)k (�) + u

(2)k (�), we have

∥uk∥2L2 = ∥uk∥2

L2 =

∫∣u(1)k (�) + u

(2)k (�)∣2d�

=

∫∣u(1)k (�)∣2d� + 2

∫ k(�)(1− k(�))∣uk(�)∣2d� +

∫∣u(2)k ∣

2d�.

Since 0 ≤ k(�)(1− k(�)) ≤ 1, we have

∥u(1)k ∥

2L2 + ∥u(2)

k ∥2L2 ≤ ∥uk∥2

L2 ≤ c2k2−2ks.

Similarly,

∥u(1)k ∥

2Hs0 + ∥u(2)

k ∥2Hs0 ≤ ∥uk∥2

Hs0 ≤ c2k2−2k(s−s0).

Set u(1) =∑u

(1)k , u(2) =

∑u

(2)k , then u = u(1) + u(2), and from (c), u(1) ∈ Hs, for u(2),

we have

∥'(2−pD)u(2)∥2L2 =

∫∣∑

k≤p+N0

'(2−pD)u(2)k ∣

2dx

≤

( ∑k≤p+N0

2−2k(s−s0)

)(∫ ∑k≤p+N0

22k(s−s0)∣'(2−pD)u(2)k ∣

2dx

)

≤ 1− 2−2(p+N0+1)(s−s0)

1− 2−(s−s0)2−2ps0

∑k≤p+N0

22k(s−s0)∥'(2−pD)u(2)k ∥

2Hs0 .

3.3. CHARACTERIZATION OF HOLDER SPACES 35

But for s0 > s > 0, we have

1− 2−2(p+N0+1)(s−s0)

1− 2−(s−s0)2−2ps0 ≤ C2−2ps

with C independent on p. Set now c2p =

∑k≤p+N0

22k(s−s0)∥'(2−pD)u(2)k ∥2

Hs0 , then∑p

c2p ≤

∑k

22k(s−s0)∥u(2)k ∥

2Hs0 < +∞.

We have proved u(2) =∑

p '(2−pD)u(2) ∈ Hs.

3.3. Characterization of Holder spaces

For 0 < � < 1, we define the Holder space by

C�(ℝn) = {u ∈ L∞(ℝn); [u]� = supx,y

∣u(x)− u(y)∣∣x− y∣�

< +∞},

with the norm ∥u∥C� = ∥u∥L∞ + [u]�. And for � ∈ ℝ+ ∖ℕ, � = [�] + �, 0 < � < 1, wedefine

C�(ℝn) = {u ∈ C [�](ℝn);D�u ∈ C�(ℝn), ∣�∣ ≤ [�]},with the norm

∥u∥C� =∑∣�∣≤[�]

∥D�u∥C� .

Then C� are Banach space, and it is evident that they norm are equivalent to

∥u∥′C� =∑∣�∣≤[�]

∥D�u∥L∞ +∑∣�∣=[�]

[D�u]�.

For � = 1, Zygmund space C1∗ (module the affine functions) is defined by C1

∗ ={u ∈ C0(ℝn); [u]∗1 < +∞}, where [u]∗1 = supℎ,x∈ℝn ∣u(x+ℎ)+u(x−ℎ)−2u(x)∣/∣ℎ∣, then

C1∗ is also Banach space with norm [u]∗1. For m ∈ ℕ, we define Cm

∗ = {u ∈ C1∗ ;D

�u ∈C1∗ , ∣�∣ ≤ m− 1}. From now for � ∈ ℝ+, without confusion we denote C�, if � ∈ ℕ it

is Zygmund space, if � ∈ ℝ+ ∖ ℕ it is Holder space.For Holder space we have a similar results as Theorem 3.2.1.

Theorem 3.3.1. For any � > 0, and � = l + �, l ∈ ℕ, 0 < � ≤ 1, the followingproperties are equivalents:

(a) u ∈ C�;(b) u =

∑∞p=−1 up with Suppup ⊂ Cp, and ∥up∥L∞ ≤ C2−p�;

(c) u =∑∞

p=−1 up with Suppup ⊂ B(0, K12p), and ∥up∥L∞ ≤ C2−p�;

(d) u =∑∞

p=−1 up with up ∈ C l+1, and for all � ∈ ℕn, ∣�∣ ≤ l + 1, ∥D�up∥L∞ ≤C�2

−p(�+∣�∣).

Proof: Suppose that u ∈ C�, and u =∑∞

p=−1 up his Littlewood-Paley decomposi-

tion. Since Suppu−1 ⊂ B(0, K1), we have

∥u−1∥L∞ ≤ C(n)K1∥u∥L∞ .


For p > −1, denote by ' the transformation of ', then 'p(x) = 2np'(2px), and for any� ∈ ℕn, we have

(3.3.1)

∫x�'(x)dx = D�

�

∫e−ix�'(x)dx∣�=0 = D�

�'(0) = 0.

Now for any f ∈ C�(ℝn), � = l + �, the Taylor formula give

f(x) =∑∣�∣<l

1

�!∂�f(y)(x− y)�

+l

�!(x− y)�

∫ ∑∣�∣=l

∂�f(y + t(x− y))(1− t)l−1dt

=∑∣�∣≤l

1

�!∂�f(y)(x− y)�

+l

�!(x− y)�

∫ ∑∣�∣=l

[∂�f(y + t(x− y))− ∂�f(y)](1− t)l−1dt,

then we have

∣f(x)−∑∣�∣≤l

1

�!∂�f(y)(x− y)�∣ ≤ Cl∣x− y∣�∥f∥C� .

And for up = 'p ∗ u, using (3.3.1), we have

up(x) =

∫'p(x− y)u(y)dy

=

∫'p(x− y)[u(y)−

∑∣�∣≤l

1

�(x− y)�∂�u(x)]dy,

hence

∥up∥L∞ ≤ C∥u∥C�∫∣'p(x− y)∣∣x− y∣�dy

= C∥u∥C�2−p�∫∣'(x)∣∣x∣�dx

≤ C�∥u∥C�2−p�.

We have proved (a) → (b).(b) → (c) is evident.(c) → (d) is reduced by Lemma 3.1.1 .We prove now (d) → (a), take sum we have immediately u ∈ C l(ℝn), and for all

∣�∣ = l, p ∈ ℕ+, x, y ∈ ℝn,

∣∂�up(x)− ∂�up(y)∣ ≤M2p(1−�)∣x− y∣.

3.3. CHARACTERIZATION OF HOLDER SPACES 37

Fixed ∣x− y∣ > 0 small, take p0 such that 2p0 ≤ ∣x− y∣−1 ≤ 2p0+1, then

∂�u(x)− ∂�u(y) =∑p≤p0

[∂�up(x)− ∂�up(y)]

+∑p>p0

[∂�up(x)− ∂�up(y)]

= I + II.

For the first term, we have

I ≤M ∣x− y∣∑p≤p0

2p(1−�) ≤ 2M ∣x− y∣2p0(1−�) ≤ 2M ∣x− y∣�.

For the second term, we have

II ≤∑p>p0

∣∂�up(x)∣+∑p>p0

∣∂�up(y)∣

≤ 2M∑p>p0

2−p� ≤ 4M2−p0� ≤ 8M ∣x− y∣�,

we obtain for ∣�∣ = l,

∣∂�u(x)− ∂�u(y)∣ ≤ C∣x− y∣�.Then u ∈ C�(ℝn).

Remark : Since the Littlewood-Paley decomposition is well-defined in the dis-tribution space S ′. We large the definition of Holder space for index � < 0 as theequivalence of (a) and (b) in Theorem 3.2.1 (There exists a general looking in Besovspace).

Definition 3.3.1. For � ∈ ℝ, and u ∈ S ′, if his Littlewood-Paley decompositionu =

∑∞p=−1 up satisfy ∥up∥L∞ ≤ C2−p�, then we say u ∈ C�.

This definition must justifier by the following inverse results.

Exercise Let {uq}q∈ℕ be a sequence in L∞ such that, for some constant C > 0 ands ∈ ℝ

supp uq ⊂ Cq, and ∥uq∥L∞ ≤ C2qs, ∀q ∈ ℕthen u =

∑uq converge in S ′.

Using this definition, we have in Theorem 3.3.1 the equivalence of (a) and (b) forall � ∈ ℝ, and as for Sobolev space, we have

Theorem 3.3.2. Suppose that P (D) ∈ Sm1,0, then P : C� → C�−m is continuous forall � ∈ ℝ.

Proof: Without loss general, we can suppose that P (�) is homogeneous of degree mfor ∣�∣ ≥ A. Choose N0 big enough such that K−12N0 ≥ A. For u ∈ C� and u =

∑up

his Littlewood-Paley decomposition, set vp = P (D)up. Then vp(�) = P (�)up(�), andSuppvp ⊂ Cp, choose now Φ ∈ C∞0 (ℝn) with Φ(�) = 1 on C0, and SuppΦ ⊂ C ′0 = {� ∈ℝn;K ′−1 ≤ ∣�∣ ≤ 2K ′}, then up(�) = Φ(2−p�)up(�). Set Ψ(�) = P (�)Φ(�), and it is


Fourier transformation of ℎ(x), then Ψp(�) = P (�)Φ(2−p�) = 2mpP (2−p�)Φ(2−p�), andit is the transformation of 2(m+n)pℎ(2px), and

vp(x) = 2(m+n)p(ℎ(2p⋅) ∗ up)(x) = 2mp∫ℎ(t)up(x− 2−pt)dt.

Since ℎ ∈ L1(ℝn), Hausdorff-Young inequality give

∥vp∥L∞ ≤ C2mp∥up∥L∞ ≤ C2−p(�−m).

We have proved v = P (D)u ∈ C�−m.

3.4. Sobolev embedding Theorem

Using the Littlewood-Paley decomposition, we have a very simple proof of Sobolevembedding theorem.

Theorem 3.4.1. (a) For any s ∈ ℝ, we have continuous embedding Hs ⊂ Cs−n/2;(b) For any 0 < s < n/2, we have continuous embedding Hs ⊂ Lp where p = 2n

n−2s;

(c) We have continuous embedding Hn/2 ⊂ VMO.

Proof: (a) Suppose that u ∈ Hs, u =∑∞

p=−1 up the Littlewood-Paley decomposi-

tion. Take Φ(�) as in the proof of Theorem 3.3.2, set ℎ = Φ, then

up(x) = 2np[ℎ(2p⋅) ∗ up](x) = 2np∫up(t)ℎ[2p(x− t)]dt,

and Schwartz inequality give

∥up∥L∞ ≤ ∥up∥L2∥2npℎ(2px)∥L2

= ∥up∥L2

(∫∣2npℎ(2px)∣2dx

)1/2

= 212np∥up∥L2∥ℎ∥L2 .

We have proved for u ∈ Hs,

∥up∥L∞ ≤ 2−p(s−n2

)cp∥ℎ∥L2 .

Then u ∈ C�−n2 .

(b) We give here a direct proof by Fourier transformation, similar proof can beeasily given by use Littlewood-Paley decomposition. For f ∈ Lp, we have

∥f∥pLp = p

∫ ∞0

�p−1m{∣f ∣ > �}d�,

where m{∣f ∣ > �} is the Lebesgue measure of set {x ∈ ℝn; ∣f ∣ > �}. For A > 0, set

f1,A

= ℱ−1(1B(0,A)f), f = f1,A

+ f2,A

, we have

∥f1,A∥L∞ ≤ ∥f

1,A∥L1 ≤ ∥f∥Hs(

∫B(0,A)

⟨�⟩−2sd�)1/2

≤ CAn/2−s∥f∥Hs .


For any A > 0, we have {∣f ∣ > �} ⊂ {2∣f1,A∣ > �}

∪{2∣f

2,A∣ > �}. Choose A� =

(�/(4C∥f∥Hs))p/n, then m{∣f1,A�∣ > �/2} = 0, and

∥f∥pLp ≤ p

∫ ∞0

�p−1m{2∣f2,A�∣ > �}d�.

but

m{∣f2,A�∣ > �/2} =

∫{∣f

2,A�∣>�/2}

dx

≤∫{∣f

2,A�∣>�/2}

4∣f2,A�

(x)∣2

�2dx

≤ 4

�2∥f

2,A�∥2L2 .

We obtain

∥f∥pLp ≤ p

∫ ∞0

�p−3∥f2,A�∥2L2d�

= p(2�)n∫ℝ+×ℝn

�p−31{(�,�);∣�∣≥A�}∣f(�)∣2d�d�.

By the definition of A�, we have ∣�∣ ≥ A� → � ≤ C� = 4C∥f∥Hs⟨�⟩n/p. Then Fubinitheorem implies that

∥f∥pLp ≤ p(2�)n∫ℝn

(

∫ C�

0

�p−3d�)∣f(�)∣2d�

≤ Cp∥f∥p−2Hs

∫ℝn⟨�⟩n(p−2)/p∣f(�)∣2d�,

since n(p− 2)/p = 2s, we have proved (b).(c) By the definition, u ∈ VMO if and only if u ∈ L1

loc(ℝn), and

supB∣B∣−1

∫B

∣u− uB∣dx < +∞;(3.4.1)

limdiamB→0

∣B∣−1

∫B

∣u− uB∣dx = 0,(3.4.2)

where B = B(x0, R), the norm is give by (3.4.1). Take now u ∈ Hn/2(ℝn), since S isdense in Hn/2(ℝn), we need only to prove (3.4.1). For any balls B(x0, R) with R > 0small enough, there exist N0 such that 2−N0−1 ≤ R ≤ 2−N0 , we cup the Littlewood-Paley decomposition of u as u =

∑∞p=−1 up =

∑N0−1p=−1 up +

∑∞p=N0

up = u(1) + u(2),then (

∣B∣−1

∫B

∣u− uB∣dx)2

≤ ∣B∣−1

∫B

∣u− uB∣2dx

≤ 2∣B∣−1

∫B

∣u(1) − u(1)B ∣

2dx+ 2∣B∣−1

∫B

∣u(2) − u(2)B ∣

2dx.


For the first term, using Poincare inequality and results of (a), we have

∣B∣−1

∫B

∣u(1) − u(1)B ∣

2dx ≤ CR2∣B∣−1

∫B

∣Du(1)∣2dx

≤ CR2∥Du(1)∥2L∞ ≤ CR2

(N0−1∑p=−1

C2p

)2

≤ CR222N0 ≤ C

where C = C0∥u∥2Hn/2 independent on x0, R. For the second term,

∣B∣−1

∫B

∣u(2) − u(2)B ∣

2dx ≤ 2∣B∣−1

∫B

∣u(2)∣2dx

≤ CR−n∫ℝn∣u(2)(�)∣2d� ≤ CR−n

∫∣�∣≥K−12N0

∣u(�)∣2d�

≤ C

∫∣�∣≥K−12N0

(1 + ∣�∣)n∣u(�)∣2d� ≤ C∥u∥2Hn/2 .

We have proved (3.4.1) for small R, since u ∈ Hn/2(ℝn) ⊂ L2(ℝn), (3.4.1) is evidentfor R > R0 with constant C = C(R0).

3.5. Besov spaces

Using Littlewood-Paley decomposition, we can also define Besov space.

Definition 3.5.1. For s ∈ ℝ, p, r ∈ [1,+∞], we say u ∈ Bsp,r, if u ∈ S ′ and

his Littlewood-Paley decomposition {uq} satisfies the condition: ∥uq∥Lp ≤ cq2−qs, with

{cq} ∈ lr. The norm is defined by ∥u∥Bsp,r

= ∥2qs∥uq∥Lp∥lr .

By the definition, we have Bs2,2 = Hs, and B�

∞,∞ = C�. We have immediately thefollowing embedding results.

Proposition 3.5.1. We have the continuous embedding B0p,1 ⊂ Lp and Lp ⊂ B0

p,∞.

The proof is trivial. The first inclusion come from the fact that {△qu}q∈ℕ is con-vergent in Lp. The second one come from the fact that for any q ∈ ℕ, we have∥△qu∥Lp ≤ C∥u∥Lp .

Since the definition of Besov space use a Littlewood-Paley decomposition, we haveto prove that the space Bs

p,r does not depends on the choice of the functions , '. It is the following Lemma.

Lemma 3.5.1. Let C ′ be a ring in ℝn, s ∈ ℝ, 1 ≤ p, r ≤ +∞. Let {uq}q∈ℕ be asequence of smooth functions such that

Supp uq ⊂ 2qC ′ and ∥{2qs∥uq∥Lp}q∈ℕ∥ℓr < +∞.

Then we have

u =∑q∈ℕ

uq ∈ BsP,r, and ∥u∥Bsp,r ≤ C∥{2qs∥uq∥Lp}q∈ℕ∥ℓr .

* AROUND FUNCTION SPACE B1∞,∞ 41

Proof : From Bernstein inequality, we have

∥uq∥L∞ ≤ C2q(np−s).

Then (see exercise) u =∑uq converge in S ′. We study now the Littlewood-Paley

decomposition of u, i. e. {△q′u}. There exists N0 such that

∣q − q′∣ ≥ N0 ⇒ 2q′C ∩ 2qC ′ = ∅ ⇒ △q′uq = 0.

So that

∥△q′u∥Lp = ∥∑

∣q−q′∣<N0

△q′uq∥Lp ≤ C∑

∣q−q′∣<N0

∥uq∥Lp ,

and

2q′s∥△q′u∥Lp ≤ C

∑q′≥−1, ∣q−q′∣<N0

2q′s∥uq∥Lp ≤ C

∑q′≥−1, ∣q−q′∣<N0

2(q′−q)s(

2qs∥uq∥Lp).

We deduce

aq′ = 2q′s∥△q′u∥Lp ≤ {ck}k∈ℤ ∗ {dm}m∈ℤ(q′),

with

ck = 1[−N0,N0](k)2ks and dm = 1ℕ(m)2ms∥um∥Lp .The classical Young inequality between {ck}k∈ℤ ∈ ℓ1, and {dm}m∈ℤ ∈ ℓr gives that

∥u∥Bsp,r ≤ C∥{dm}m∈ℤ∥ℓr ,

which proves the lemma.

* Around function space B1∞,∞

We study some fine proprieties of Zygmund space B1∞,∞ = C1

∗ . There are similar

results for Bk∞,∞, k ∈ ℕ. Those function spaces take a very important role in the

studied of fluid mechanics equations.

Definition 3.5.2. A function � from [0, a] to ℝ+ is a modulus of continuity if � isan increasing continuous function such that �(0) = 0. We say that � is admissible ifthe function Γ defined by

(3.5.1) Γ(y) = y�(1

y)

is non decreasing and satisfies∫ ∞x

1

y2Γ(y)dy ≤ C

Γ(x)

x, ∀x ≥ 1

a.

The basic examples is for � ∈]0, 1] the functions

�(s) = s�, �(s) = s(− log s)�, �(s) = s(− log s)(− log(− log s)

)�are admissible modulus of continuity.

Definition 3.5.3. .


∙ Let � be a modulus of continuity. We define the function space

C�(Ω,ℝn) = {f ∈ L∞(Ω,ℝn); ∣f(x)− f(y)∣ ≤ C�(∣x− y∣)∀x, y ∈ Ω, ∣x− y∣ ≤ a}with norm

∥f∥C� = ∥f∥L∞ + supx,y∈Ω,0<∣x−y∣≤a

∣f(x)− f(y)∣�(∣x− y∣)

.

∙ Let Γ be an increasing function on [b,+∞[. The space BΓ is the space ofbounded continuous functions u on ℝd such that

∥u∥BΓ= ∥u∥L∞ + sup

q≥0

∥∇Squ∥L∞Γ(2q)

< +∞.

Remark : When Γ(s) = s1−�, we have BΓ = B�∞,∞. When �(s) = s(− log s) the space

C� is also called LL the logarithmic Lipschitz space.

Proposition 3.5.2. When � is an admissible modulus of continuity, and Γ is definedby (3.5.1), we have C� = BΓ.

Remark : We have immediately B1∞,∞ ⊂ LL.

Sobolev embedding theorem in Besov space. We have now a general versionof the embedding theorem in Besov space

Theorem 3.5.1. Let 1 ≤ p1 ≤ p2 ≤ ∞ and 1 ≤ r1 ≤ r2 ≤ ∞ . Then Bsp1,r1

is

continuously embedded in Bs−n( 1

p1− 1p2

)

p2,r2 .

We use Bernstein inequality and the fact ℓr1 ⊂ ℓr2 . We give now some importantproperties of Besov space

Proposition 3.5.3. 1) The space Bsp,r is a Banach space.

2) The space D(ℝn) is dense in Bsp,r if and only if 1 ≤ r < +∞.

3) Let � be a smooth function on ℝn which is homogeneous of degree m ∈ ℝ outsideℬ. Then for any (s, p, r) ∈ ℝ × [1,+∞]2, the operator �(D) maps continuously Bs

p,r

into Bs−mp,r .

Tangential Sobolev spaces. .Since the problems for nonlinear equation is often with boundary value, so we have

to study the functions space defined on a smooth domain of ℝn. After localization nearto boundary and a changing of variables, we study only demi-space ℝn

+ = {(x′, xn) ∈ℝn;xn > 0}, where x′ = {x1, ⋅ ⋅ ⋅ , xn−1}. We define the tangential Sobolev space(s, s′ ∈ ℝ),

Hs,s′(ℝn) = {u ∈ S ′; (1 + ∣�∣2)s(1 + ∣�′∣2)s′ ∣u∣2 ∈ L1}.

Then Hs,s′(ℝn) is an algebra, if s + s′ > n/2, s > 1/2, s + 2s′ > 1/2. Denote D =(D′, Dn),△p = '(2−pD),△′p = '(2−pD′, 0),△p,p′ = △p ∘ △p′ . Then for any u ∈ S ′,

u =

N1∑p′=−1

△−1,p′u+∑p≥0

△p,−1u+∑p,p′≥0

△p,p′u.

* LOGARITHMIC SOBOLEV SPACES 43

This is double dyadic decomposition. We can also give a characterization of tangentialspace Hs,s′ as for usual Sobolev space.

Theorem 3.5.2. u ∈ Hs,s′(ℝn), if and only if

N1∑p′=−1

4p′s′∥△−1,p′u∥2

L2 +∑p≥0

4ps∥△p,−1u∥2L2

+∑p,p′≥0

4ps+p′s′∥△p,p′u∥2

L2 < +∞.

We have also the similar results as that of Theorem 3.2.1 and for Besov space.

* Logarithmic Sobolev spaces

In the infinite degenerate case, we use also a logarithmic type Sobolev space. Letℓ > 0, we define the following logarithmic Sobolev’s space.

H(logℓ⟨�⟩) = {u ∈ L2(ℝn); logℓ⟨�⟩u(�) ∈ L2(ℝn)},

where ⟨�⟩ = (e + ∣�∣2)1/2. We study now the Littlewood-Paley decompositon for thisfunction space .

Let C0 = {� ∈ ℝn; e < ⟨�⟩ < e3}, Ck = ekC0, k ∈ ℕ, C−1 = {� ∈ ℝn; ⟨�⟩ < e2}, thereexist ∈ C∞0 (]0, e2[), ' ∈ C∞0 (]e, e3[) such that

(⟨�⟩) +∞∑j=0

'(e−j⟨�⟩) = 1, ∀� ∈ ℝn.

For f ∈ L2(ℝn), we set Λ = (e−Δ)1/2 and

Δ−1f = (Λ)f, Δjf = '(e−jΛ)f, j ∈ ℕ.

Then f =∑

Δjf in L2, and

Lemma 3.5.2. Let ℓ > 0, we have that1) if u ∈ H(logℓ⟨�⟩), then

∥Δju∥L2 ≤ cjj−ℓ, ∥{cj}∥ℓ2 ≤ ∥u∥H(logℓ⟨�⟩).

2) if u ∈ L2(ℝn), and

∥Δju∥L2 ≤ cjj−ℓ, {cj} ∈ ℓ2,

then u ∈ H(logℓ⟨�⟩), and for any S ≥ 1

S2ℓ∥ logℓ Λu∥2L2(ℝn) ≤ C1ℓ

2ℓ∥u∥2L2 + CS

2 S2ℓ∥{cj}∥2

ℓ2 ,

with C1, C2 independent of S, ℓ and u.

Proof : 1) For u ∈ H(logℓ⟨�⟩), we have

∥Δju∥2L2 =

∫'(e−j⟨�⟩)2∣u(�)∣2d� ≤ j−2ℓ

∫Cj

log2ℓ⟨�⟩'(e−j⟨�⟩)2∣u(�)∣2d�.


We set

c2j =

∫Cj

log2ℓ⟨�⟩'(e−j⟨�⟩)2∣u(�)∣2d�.

Then the fact 2(�) +∑∞

j=0 '(e−j⟨�⟩)2 ≤ 1 implies that

∞∑j=−1

c2j ≤

∫ℝn

log2ℓ⟨�⟩∞∑

j=−1

'(e−j⟨�⟩)2∣u(�)∣2d� ≤ ∥u∥2H(logℓ⟨�⟩).

2) For S > 0, we have

S2ℓ∥ logℓ Λu∥2L2(ℝn) ≤ 3

∑S(j+3)≤ℓ

(S(j + 3))2ℓ

∫'(e−j⟨�⟩)2∣u(�)∣2d�

+ 3∑

S(j+3)>ℓ

(S(j + 3))2ℓ∥Δju∥2L2

≤ 3ℓ2ℓ∥u∥2L2 + 3S2ℓ

∑S(j+3)>ℓ

(j + 3)2ℓj−2ℓc2j

≤ 3ℓ2ℓ∥u∥2L2 + 3S2ℓ

∑j

(1 + 3/j)2S(j+3)c2j

≤ 3ℓ2ℓ∥u∥2L2 + 3(e626)SS2ℓ∥{cj}∥2

L2 .

As in the classical case, for the second part in the preceding lemma, we have moregeneral results

Lemma 3.5.3. Suppose that {uk}k∈ℕ a sequence of L2, with Supp uk ⊂ B(0, Kek)and for ℓ > 1/2

∥uk∥L2 ≤ ckk−ℓ, {ck} ∈ ℓ2.

Then u =∑

k uk ∈ H(logℓ−1/2⟨�⟩) and for any S ≥ 1,

S2ℓ−1∥ logℓ−1/2 Λu∥2L2(ℝn) ≤ C1(ℓ− 1/2)2ℓ−1∥u∥2

L2 + CS2 S

2ℓ−1(2ℓ− 1)∥{ck}∥2ℓ2 ,

with C1, C2 independent on S, ℓ and u.

Remark : We have a loss of 1/2 for the index.

Proof : Since ℓ > 1/2, we have that u =∑

k uk converge in L2, in fact,

∥u∥L2 ≤∑k

∥uk∥L2 ≤∑k

ckk−ℓ ≤ ∥{ck}∥ℓ2

(∑k

k−2ℓ

)1/2

.

We suppose now S = 1, since the general case of S is similar as lemma 3.5.2. We set

u =∞∑

j=−1

Δju =∞∑

j=−1

vj =∞∑

j=−1

∑k

Δjuk.


Then

∥u∥2H(logℓ−1/2⟨�⟩) ≤ 2∥

∑j+3≤ℓ−1/2

Δju∥2H(logℓ−1/2⟨�⟩) + 2∥

∑j+3>ℓ−1/2

Δju∥2H(logℓ−1/2⟨�⟩)

≤ 2(ℓ− 1/2)2ℓ−1∥u∥2L2 + 2

∑j+3>ℓ−1/2

(j + 3)2ℓ−1∥Δju∥2L2 .

On the other hand, there exists N1 > 0 (depends only on K) such that for any j >k +N1, Cj

∩B(0, Kek) = ∅, then Δjuk = 0. We have vj =

∑k≥j−N1

Δjuk, and

∥Δju∥2L2 =

∫∣∑

k≥j−N1

Δjuk∣2dx ≤

( ∑k≥j−N1

k−2ℓ

)( ∑k≥j−N1

∫k2ℓ∣Δjuk∣2dx

)≤ (2ℓ− 1)(j −N1)−2ℓ+1

∑k≥j−N1

k2ℓ∥Δjuk∥2L2 .

Set now c2j =

∑k≥j−N1

k2ℓ∥Δjuk∥2L2 , we have∑

j

c2j ≤

∑k

k2ℓ∥uk∥2L2 ≤

∑k

c2k.

Finally, for j + 3 > ℓ− 1/2,(j + 3

j −N1

)2ℓ−1

≤(j + 3

j −N1

)2(j+3)

≤ e2(N1+3)(N1 + 4)2(N1+3) ≤ C2.

We have proved the lemma.

Lemma 3.5.4. Suppose that {uk} a sequence in C∞(ℝn) and for ℓ > 1/2 thereexists a function v ∈ H(logℓ⟨�⟩) such that for any � ∈ ℕn, there exist B∣�∣ > 0

∥D�uk∥L2 ≤ B∣�∣ek∣�∣∥Δkv∥L2 .

Then u =∑

k uk ∈ H(logℓ−1/2⟨�⟩) and for any S ≥ 1,

S2ℓ−1∥u∥2H(logℓ−1/2⟨�⟩) ≤ CS

((ℓ− 1/2)2ℓ−1∥v∥2

L2 + S2ℓ−1(2ℓ− 1)∥v∥2H(logℓ⟨�⟩)

),

with CS depending only on B0, B[S]+2 and C1, C2 the constants in lemmas 3.5.2 andlemma 3.5.3.

Proof : As in the lemma 3.5.3, we have u =∑

k uk ∈ L2. We decompose,

uk = u1k + u2

k = (e−k−1Λ)uk + (1− (e−k−1Λ))uk.

Then u1 =∑u1k satisfies the hypothesis of lemma 3.5.3, we have for S ≥ 1,

S2ℓ−1∥u1∥2H(logℓ−1/2⟨�⟩) ≤ C1(ℓ− 1/2)2ℓ−1B2

0∥v∥2L2 + CS

2 B20S

2ℓ−1(2ℓ− 1)∥v∥2H(logℓ⟨�⟩).

We study now u2 =∑u2k, with the conditions

Suppu2k ⊂ {� ∈ ℝn; ⟨�⟩ ≥ ek}, ∥D�u2

k∥L2 ≤ B�ek∣�∣∥Δkv∥L2 .

For k ≥ p+ 3, Cp∩{� ∈ ℝn; ⟨�⟩ ≥ ek} = ∅, we have

Δpu2 = vp =

∑k≤p+2

Δpu2k.


Then

∥vp∥2L2 ≤

( ∑k≤p+2

e2k

)( ∑k≤p+2

e−2k∥Δpu2k∥2

L2

)≤ 2e2(p+2)

∑k≤p+2

e−2k∥Δpu2k∥2

L2 ≤ 2e4p−2ℓ+1∑k≤p+2

e−2k∥⟨D⟩ logℓ−1/2 ΛΔpu2k∥2

L2 .

Set now c2p =

∑k≤p+2 e

−2k∥⟨D⟩ logℓ−1/2 ΛΔpu2k∥2

L2 , we have

∞∑p=−1

c2p ≤

∑k

e−2k∥⟨D⟩ logℓ−1/2 Λu2k∥2

L2 .

By lemma 3.5.2, we have

S2ℓ−1∥ logℓ−1/2 Λ(u2)∥2L2(ℝn) ≤ C1(ℓ− 1/2)2ℓ−1∥u∥2

L2 + CS2 S

2ℓ−1∥{cp}∥2ℓ2 .

We study now ∥{cp}∥ℓ2 . For simplicity the notation, we replace ℓ − 1/2 by ℓ in whatfollows,

∥⟨D⟩ logℓ Λu2k∥2

L2 =

∫⟨�⟩−2([S]+1)⟨�⟩2[S]+4 log2ℓ⟨�⟩(1− (e−k−1⟨�⟩))2∣uk(�)∣2d�,

and if ([S] + 1)(k + 2) ≥ ℓ,

⟨�⟩−2([S]+1) log2ℓ⟨�⟩(1− (e−k−1⟨�⟩))2 ≤ e−2([S]+1)(k+2)(k + 2)2ℓ(1− (e−k−1⟨�⟩))2;

if ([S] + 1)(k + 2) < ℓ,

⟨�⟩−2([S]+1) log2ℓ⟨�⟩(1− (e−k−1⟨�⟩))2 ≤ e−2([S]+1)(k+2)

(ℓ

[S] + 1

)2ℓ

(1− (e−k−1⟨�⟩))2.

Consequently

∞∑p=−1

c2p ≤

∑([S]+1)(k+2)<ℓ

e−2ke−2([S]+1)(k+2)

(ℓ

[S] + 1

)2ℓ

∥uk∥2H[S]+2(ℝn)

+∑

([S]+1)(k+2)≥ℓ

e−4([S]+1)e−2k([S]+2)

(1 +

2

k

)2ℓ

k2ℓ∥uk∥2H[S]+2(ℝn).

From the hypothesis of lemma,

∥uk∥H[S]+2(ℝn) ≤ B[S]+2ek([S]+2)∥Δkv∥L2 ,

we have∞∑

p=−1

c2p ≤ B[S]+2(S−2ℓℓ2ℓ∥v∥2

L2 + ∥v∥2H(logℓ⟨�⟩)).

We have proved the lemma with the constant CS depends on B0, B[S]+2 and C1, C2.


To study the regularity up to the boundary, we follows [89, 104, 107] and introducethe following tangential logarithmic Sobolev spaces : For ℓ > 0, we set

H(logℓ⟨(�′, 0)⟩,ℝn) = {u ∈ L2(ℝn); logℓ⟨(�′, 0)⟩u(�) ∈ L2(ℝn)},and

H(logℓ⟨(�′, 0)⟩,ℝn+) = {u ∈ L2(ℝn

+); logℓ⟨(�′, 0)⟩ℱx′u(�′, xn) ∈ L2(ℝn+)},

where � = (�′, �n) ∈ ℝn−1 × ℝ,ℝn+ = {(x′, xn);x′ ∈ ℝn−1, xn > 0}. We have

H(logℓ⟨(�′, 0)⟩,ℝn)∣ℝn+ = H(logℓ⟨(�′, 0)⟩,ℝn+).

We use now the tangential Littlewood-Paley decomposition :

Δ′−1f = (Λ′)f, Δ′jf = '(e−jΛ′)f, j ∈ ℕ,

where ℱ('(Λ′)f) = '(⟨(�′, 0)⟩)f , and the function spaces H(logℓ⟨(�′, 0)⟩,ℝn+) is char-

acterized by ∑j2ℓ∥Δ′ju∥2

L2(ℝn+) < +∞.We have the similar results as lemmas 3.5.2–3.5.4 for the tangential function spaces.

CHAPTER 4

Analysis of non linear partial differential equations

There are two essentials problems in the theory of nonlinear partial differentialequations, the first one is nonlinearity, it restrict to consider weak solution in somealgebra. The second one is that we must work for operators with no regularitiescoefficients. We will use Littlewood-Paley decomposition of first chapter to studythose problems.

4.1. Paramultipication

Theorem 4.1.1. If 0 ≤ s, then Hs ∩ L∞ is an algebra.

Remark : If s > n/2, by Sobolev embedding theorem, we have proved that then Hs

is an algebra.Proof: If s = 0 it is trivial. So that we consider the case s > 0. Take u, v ∈ Hs ∩ L∞,denote by u =

∑up, v =

∑vq their Littlewood-Paley decompositions, then we have

uv =∑p,q

upvq.

For some N0 big enough, B(0, 4K2−N0) is a very small balls, set

C ′ = C +B(0, 4K2−N0),

then {C ′p} possess same properties of {Cp}. We define

(4.1.1) Squ =∑

−1≤p≤q−N0

up,

and

Tuv =∑q

(Squ)vq,(4.1.2)

R(u, v) =∑

∣p−q∣<N0

upvq.(4.1.3)

Then, we have

(4.1.4) uv = Tuv + Tvu+R(u, v).

Since SuppSqu ⊂ B(0, K2q−N0+1), we have Supp ˆ(Squ)vq ⊂ C ′q and

∥(Squ)vq∥L2 ≤ ∥Squ∥L∞∥vq∥L2 ≤ ∥u∥L∞c′q2−ps

49

50 4. NON LINEAR PDE

with {c′q} ∈ l2, using (b) of Theorem 3.2.1, we have proved Tuv, Tvu ∈ Hs. For R(u, v),we have

R(u, v) =∑

−N0<j<N0

Rj(u, v) =∑

−N0<j<N0

∞∑q=j

uq−jvq.

SinceSuppuq−jvq ⊂ Suppuq−j + Suppvq ⊂ B(0, K ′2q),

and∥uq−jvq∥L2 ≤ ∥uq−j∥L∞∥vq∥L2 ≤ C2−(q−j)(s−n/2)cq2

−qs = c′q2−q(2s−n/2).

Using (c) of Theorem 3.2.1, we have proved Rj(u, v) ∈ Hs, s > 0, which prove Theorem.

Definition 4.1.1. For a ∈ L∞ with compact support, we define paramultiplicationoperators Ta : S ′ → S ′ by

(4.1.5) Tau =∑

(Sqa)uq,

where u ∈ S ′, {aq}, {uq} the Littlewood-Paley decomposition, and Sqa =∑

p≤q−N0ap.

Remark : 1). We can define the paramultiplication operator Ta for a ∈ C� with� < 0.

2). This definition depends on the constant K of Littlewood -Paley decomposition,the partition of unity (defined by function '), and the constant N0. then the definitionisn’t canonic. We will analyze the relation of Ta and (K,',N0) in each case. We have

Theorem 4.1.2. If a ∈ L∞ with compact support, then for any s, � ∈ ℝ, we havethat Ta : Hs → Hs, Ta : C� → C� is continuous and the norms of operators satisfies

(4.1.6) ∥Ta∥ℒ(Hs,Hs) ≤ Cs∥a∥L∞ ; ∥Ta∥ℒ(C�,C�) ≤ C�∥a∥L∞ .

Proof: We prove Theorem only for C�, since

(2q−N0�) = (�) +

N0−q−1∑p=0

'(2−p�),

thenSqa(�) = (2q−N0�)a(�),

and for any u ∈ C�, we have

Supp ˆ(Sqa)uq ⊂ SuppSqa+ Suppuq ⊂ C ′q.

Set ℎ(�) = (�), then

Sqa(x) = 2n(q−N0)[ℎ(2q−N0⋅) ∗ a](x),

and∥Sqa∥L∞ ≤ ∥a∥L∞∥ℎ∥L1 .

Hence we obtain

∥(Sqa)uq∥L∞ ≤ ∥ℎ∥L1∥a∥L∞∥uq∥L∞ ≤ C∥u∥C�∥ℎ∥L1∥a∥L∞2−q�.

Using Theorem 3.3.1, we have proved (4.1.6).We study now the dependence of Ta on (K,',N0).

4.1. PARAMULTIPICATION 51

Theorem 4.1.3. Let � > 0, a ∈ C�, and Ta defined by (K,',N0). Suppose that(K ′, '′) define anther Littlewood-Paley decomposition, and {Aq} a sequence in C∞

verifies ∥a − Aq∥L∞ ≤ C2−q�, SuppAq ⊂ B(0, C2q). For u ∈ C�, denote by u =∑up

decomposition associated with (K,'), and u =∑vp decomposition associated with

(K ′, '′). Then for any N ′0, we have

(4.1.7) Tau−∑q

Aq−N ′0vq ∈ C�+�.

Proof: by definition of Tau, we have

Tau−∑q

Aq−N ′0vq =∑

p≤q−N0

apuq −∑

p≤q−N0

apvq

+∑

p≤q−N0

apvq −∑q

Aq−N ′0vq

=∑p

ap[∑

q≥p+N0

(uq − vq)]

+∑q

[∑

p≤q−N0

ap − Aq−N ′0 ]vq

=∑p

apvp +∑q

fq.

Without loss generality, we can suppose that K ′ ≥ K, then

Supp ˆvp ⊂ C ′p+N0, ∥vp∥L∞ ≤ C2−p�,

andSupp apvp ⊂ C

′′

p+N0, ∥apvp∥L∞ ≤ C2−p(�+�).

For fq, it is evident that Supp fq ⊂ C ′q, and

∥fq∥L∞ ≤ (∥a−∑

p≤q−N0

ap∥L∞ + ∥a− Aq−N ′0∥L∞)∥vq∥L∞ ≤ C2−q(�+�).

We have proved Theorem.

Corollary 4.1.1. Suppose that a ∈ C�, � > 0, and Ta, T′a the para-multiplication de-

fined by dyadic decomposition (K,',N0), and (K ′, '′, N ′0). Then Ta−T ′a ∈ ℒ(C�, C�+�)and Ta − T ′a ∈ ℒ(Hs, Hs+�), and

∥Ta − T ′a∥ℒ(C�,C�+�) ≤ C�∥a∥�;(4.1.8)

∥Ta − T ′a∥ℒ(Hs,Hs+�) ≤ Cs∥a∥�.(4.1.9)

The proof of Corollary is directly by use Theorem 4.1.3 with Aq =∑

p≤q−N ′0a′p,

where {a′p} is the Littlewood-Paley decomposition of a with respect to (K ′, '′).We have now for � > 0, a ∈ C�,

Ta ≡ T ′a(modℒ(C�, C�+�), and, modℒ(Hs, Hs+�)).

If an operators in ℒ(C�, C�+�), or ℒ(Hs, Hs+�), we called an �-regularization operators,and denote by S−�. Then the paramultiplication Ta is well-defined by function a andmodule S−�. For paramultiplication, we have also the calculus of operators.


Theorem 4.1.4. Let a, b ∈ C�, � > 0, with compact support, then

Ta ∘ Tb − Tab ∈ S−�;(4.1.10)

T ∗a − Ta ∈ S−�,(4.1.11)

their norm can be estimate by C∥a∥�∥b∥� and C∥a∥�.

The proof of (4.1.10) is also directly by use Theorem 4.1.3 with

Aq =∑

p2≤q−N0

ap2(Sq−N0b)uq =∑

vq.

For (4.1.11), using the formula

(T ∗au, v) = (u, Tav) =∑q

∑p≤r−N0

∫uqapvrdx,

and

(Tau, v) =∑r

∑p≤q−N0

∫apuqvrdx.

we can get

∣((T ∗a − Ta)u, v)∣ ≤ C∥a∥�∥u∥s∥v∥−s−�.which prove (4.1.11).

4.2. Paradifferential operators

Operators with non smooth coefficients and regularization. We have studyparamultiplication defined by a non smooth function, in fact this is a regularization ofmultiplier. We study now the regularization of pseudo-differential operators with nonsmooth coefficients.

Definition 4.2.1. (a) For � > 0,m ∈ R, we denote by lm� = {l(x, �); l is homo-geneous of degree m, and belong to C∞(ℝn ∖ 0) for variables �, for x it is belong toC�, � > 0 with compact support (uniformly respect to �)}.

(b) For any l ∈ lm� , denote by Sq(l(x, �)) = (2−qDx)l(x, �), then Sq(l(x, �)) ∈ Sm1,0.For any u ∈ S ′ we define

Tlu(x) =∑q

Sq−N0(l(x,D))uq(x).

If l(x, �) =∑

j lj(x, �) is a finite sum, we denote by Tl =∑

j Tlj .

Remark that if l(x, �) = a(x)ℎ(�), then Tl = Ta ∘ ℎ(D), for general l(x, �) we canuse sphere harmonic decomposition

l(x, �) =∑�

a�(x)ℎ�(�).

Theorem 4.2.1. For l ∈ lm� , we have that Tl : Hs → Hs−m ( or C� → C�−m) iscontinuous for any �, s ∈ ℝ.

4.2. PARADIFFERENTIAL OPERATORS 53

Proof: We prove only Theorem for l(x, �) = a(x)ℎ(�), with ℎ(�) ∈ C∞(ℝn ∖0) andhomogeneous of degree m, and a ∈ C� with compact support. Theorem 3.3.2 give thatℎ(D) : Hs → Hs−m is continuous, and Theorem 4.1.2 give that Ta : Hs−m → Hs−m isalso continuous. We have proved Theorem.

As in the Corollary 4.1.1, we have Tl − T ′l ∈ Sm−�, then Tl is well-defined byl(x, �) modulo Sm−�. We study now symbolic calculus of operators Tl, it is similar topseudo-differential operators.

Theorem 4.2.2. Let a ∈ C�, � > 0, � /∈ ℕ, and with compact support, ℎ ∈ C∞(ℝn ∖0) homogeneous of degree m. Then

R = ℎ(D) ∘ Ta −∑∣�∣≤[�]

1

�!TD�a ∘ ℎ�(D) ∈ Sm−�,

where ℎ�(�) = ∂�� (�).

Proof: Since Suppℱ(Sq−N0(a)uq) ⊂ C ′q, choose C ′q ⊂ C′′q and '0 ∈ C∞0 (C ′′0 ) with

'0(�) = 1 on C ′0, set ℎ(�) = ℎ(�)'(�), then for � ∈ C ′q,

ℎ(�) = 2mqℎ(2−q�).

If r(�) = ℎ(�), then r ∈ S and for M > n+ �/2,

∥(1 + ∣x∣�)r(x)∥L1 ≤ C∥ℎ∥C2M (Sn−1).

Now for u ∈ Hs, since for � ∈ C ′q, ℎ�(�) = ℎ�(�) and it is Fourier transformationof (−ix)�r(x), we have

R(u) =∑q

2mq[ℎ(2−qD)Sq−N0(a)−∑∣�∣≤[�]

1

�!Sq−N0(D�a)ℎ�(2−qD)]uq

=∑q

2mq∫r(t)[Sq−N0(a)(x− 2−qt)

−∑∣�∣≤[�]

1

�!Sq−N0(D�a)(x)(−i2−qt)�]uq(x− 2−qt)dt =

∑q

fq.

It is easy to see that Suppfq ⊂ C ′q, we study now estimation of ∥fq∥L2 . Since a ∈ C�,we have

∣fq(x)∣ ≤ 2mq∫∣r(t)∣∥Sq−N0(a)∥C�2−q�∣t∣�∣uq(x− 2−qt)∣dt

≤ C2q(m−�)∥a∥C�∫∣t∣�∣r(t)∣∣uq(x− 2−qt)∣dt

≤ Ccq2q(m−�−s)∥a∥C�∥ℎ∥C2M (Sn−1)∥u∥Hs

= c′q2q(m−�−s).

Which prove that R(u) ∈ Hs+�−m, and

∥R∥Sm−� ≤ C∥a∥C�∥ℎ∥C2M (Sn−1).

We study now the composition of two operators.


Theorem 4.2.3. Let lj(x, �) ∈ lmj� , j = 1, 2, set

(4.2.1) (l1#l2)(x, �) =∑∣�∣≤[�]

1

�!∂�� l1(x, �)D�

x l2(x, �).

Then Tl1 ∘ Tl2 − Tl1#l2 ∈ Sm1+m2−�.

Proof: Let l1(x, �) = a(x)ℎ(�), l2(x, �) = b(x)g(�), then

Tl1 ∘ Tl2 = Ta ∘ ℎ(D) ∘ Tb ∘ g(D)

= Ta ∘∑∣�∣≤[�]

1

�!TD�b ∘ ℎ�(D) ∘ g(D) + Ta ∘R ∘ g(D)

=∑∣�∣≤[�]

Ta#D�b ∘ ℎ�(D) ∘ g(D) + Ta ∘R ∘ g(D) +R� ∘ ℎ�(D) ∘ g(D).

using Theorem 4.2.2, we have R ∈ Sm1−�, and Theorem 4.1.4 implies R� ∈ S−(�−∣�∣).We have proved Theorem.

Theorem 4.2.4. Let l ∈ lm� , set

(4.2.2) l∗(x, �) =∑∣�∣≤[�]

1

�!∂��D

�x l(x, �),

then T ∗l ≡ Tl∗ mod(Sm−�).

From those theorems, we sew that the operators Tl is very similar to pseudo-differential operators, but here the symbolic calculus is only to the order [�], andthe rest terms is in S−�. On the other hand, for non smooth symbol l ∈ lm� , we canalso define the associated pseudo-differential operators l(x,D).

Theorem 4.2.5. Let l ∈ lm� , � > m, then for all s > m− �,

l(x,D)− Tl ∈ ℒ(Hs, Hs′),

where s′ < min{�, s+ �−m}.Proof: Let l(x, �) = a(x)ℎ(�), for u ∈ Hs, we have v = ℎ(D)u ∈ Hs−m, Tlu =

Ta ∘ ℎ(D)u = Tav, and

Tlu− l(x,D)u = Tav − av = Tva+R(a, v).

Since s+ �−m > 0, we have R(a, v) ∈ Hs+�−m, for

Tva =∑q

Sq−N0(v)aq =∑q

fq,

we have Suppfq ⊂ C ′q, and

∥fq∥L2 ≤ ∥aq∥L∞∥Sq−N0(v)∥L2 ≤ C∥a∥C�∥v∥Hs−m2−q�∑

p≤q−N0

2−p(s−m)Cp,

if s−m > 0, then∑

p 2−p(s−m)Cp ≤ C < +∞, and we have for any � > 0, Tva ∈ H�−�;if s−m < 0, then ∑

p≤q−N0

2−p(s−m)Cp ≤ C2−q(s−m),


and for any � > 0, Tva ∈ Hs+�−m−�; if s−m = 0, we have ∥Sq−N0(v)∥L2 ≤ C∥v∥L2 , andTva ∈ H�−�. We have proved Theorem.

From this Theorem, we have immediately following results.

Corollary 4.2.1. Let m > 0, l ∈ lmm, then l(x,D)− Tl ∈ ℒ(H�, L2), for any � > 0.

Theorem 4.2.6. Let l ∈ lm� , � > 0, � /∈ ℕ, then Tl ∈ Lm1,1 with symbol

(4.2.3) �(Tl)(x, �) =∑q

Sq−N0(l(x, �))'(2−q�).

Proof: It is evident Tl = �(Tl)(x,D), we need only to prove �(Tl) ∈ Sm1,1. Takel(x, �) = a(x)ℎ(�), then

�(Tl)(x, �) =∑q

Sq−N0(a)(x)ℎ(�)'(2−q�),

and for all �, � ∈ ℕn,

∂�� ∂�x�(Tl)(x, �) =

∑q

∂�xSq−N0(a)(x)∂�� (ℎ(�)'(2−q�)).

Since a ∈ C�0 ⊂ L∞, we have

∥∂�xSq−N0(a)∥L∞ ≤ C(n�)2q∣�∣∥a∥L∞ .Choose now ' ∈ C∞0 (ℝn) such that Supp' ⊂ C ′0, and '(�) = 1 on Supp', 0 ≤ ' ≤ 1,then

∣∂�� ℎ(�)'(2−q�)∣ ≤ C�∑

�1+�2=�

∣ℎ�1(�)2−q∣�2∣'(2−q�)∣

≤ C�∥ℎ∥C2M (Sn−1)2−q∣�2∣∣�∣m−∣�2∣'(2−q�)

≤ C�2q(m−∣�∣)'(2−q�).

Since on Supp'(2−q�), we have K ′−12q ≤ ∣�∣ ≤ K ′2q+1, we obtain

∣∂�� ∂�x�(Tl)∣ ≤ C�,�∣�∣m−∣�∣−∣�∣∑q≥N0

'(2−q�) ≤ C�,�∣�∣m−∣�∣−∣�∣.

which prove �(Tl) ∈ Sm1,1.We know Sm1,1 is bad class for symbolic calculus, there isn’t asymptotic symbolic

calculus in this class. But for his subclass Tl, we have a convenable symbolic calculusas that for pseudo-differential operators.

Paradifferential operators. We define now symbolic class, and paradifferentialoperators.

Definition 4.2.2. (a) Let Ω ⊂ ℝn be an open domain, for m ∈ ℝ, � > 0, we defineΣm� (Ω) the function class defined on Ω× (ℝn ∖ 0) of form

l(x, �) = lm(x, �) + lm−1(x, �) + ⋅ ⋅ ⋅+ lm−[�](x, �),

where lm−k(x, �) belong to C∞(ℝn ∖ 0) and homogeneous of degree m− k for variables

�; for variables x, lm−k ∈ C�−kloc (Ω).


(b) Let lj ∈ Σmj� (Ω), j = 1, 2, we define

l1#l2 =∑

∣�∣+k1+k2≤[�]

1

�!∂�� l

1m1−k1

D�x l

2m2−k2

,

then l1#l2 ∈ Σm1+m2� (Ω).

(c) Let l ∈ Σm� (Ω), we define

l∗ =∑

∣�∣+k≤[�]

1

�!∂��D

�x lm−k,

then l∗ ∈ Σm� (Ω).

For symbol class Σm� (Ω), we define an operators class.

Definition 4.2.3. Let Ω ⊂ ℝn be an open domain, and L : D′(Ω)→ D′(Ω) a propersupported operators. L is called a para-differential operators of order m, if there existl ∈ Σm

� (Ω), such that for any K ⊂⊂ Ω, and � ∈ C∞0 (Ω), �(x) = 1 on K, we have that

L− �T�l : Hscomp(K)→ Hs−m+�

comp ,

is continuous. We denote the class of operators by Op(Σm� (Ω)), and l = �(L) the

symbol of operators L.

It is evident for L ∈ Op(Σm� (Ω)) and any s ∈ ℝ,

L : Hsloc(Ω)→ Hs−m

loc (Ω).

The following Theorem is essential results for paradifferential operators.

Theorem 4.2.7. Let Ω ⊂ ℝn be an open domain, m ∈ ℝ, � > 0, then(a) For L ∈ Op(Σm

� (Ω)), there exist unique symbol �(L) ∈ Σm� (Ω), and

� : Op(Σm� (Ω))→ Σm

� (Ω)

is a surjection, and ker� is a continuous maps from Hs

loc(Ω) to Hs−m+�

loc(Ω).

(b) For Lj ∈ Op(Σmj� (Ω)), j = 1, 2, then

L1 ∘ L2 ∈ Op(Σm1+m2� (Ω)); �(L1 ∘ L2) = �(L1)#�(L2).

(c) For L ∈ Op(Σm� (Ω)), then L∗ ∈ Op(Σm

� (Ω)), and

�(L∗) = (�(L))∗.

(d) For L ∈ Lm1,0, a proper supported pseudo-differential operators, and∑

j lm−j(x, �)his symbol, then for any � > 0 we have

L ∈ Op(Σm� (Ω)); �(L) =

∑0≤[�]

lm−j.

Using definition of paradifferential operators and the properties of operators T�l,we need only to prove � surjection. For l ∈ Σm

� (Ω), we construct L ∈ Op(Σm� (Ω))

verifies �(L) = l. Choose {Ωj} a local finite recover of Ω with Ωj ⊂⊂ Ω, and {'j} the


partition of unity associated with {Ωj}. Take �j ∈ C∞0 (Ωj), �j(x) = 1 on Supp'j, wedefine for u ∈ D′,

Lu =∑j

�jT�j l('ju).

Then L is necessary paradifferential operators. The uniqueness of maps � give byfollowing Theorem.

Theorem 4.2.8. Let L ∈ Op(Σm� (Ω)), l = lm + ⋅ ⋅ ⋅+ lm−[�] one of his symbol, if for

some s ∈ ℝ and � > 0, L : Hs → Hs−m+� is continuous, then lm = 0.

Using this theorem, if L ∈ Op(Σm� (Ω)), and L : Hs → Hs−m+�, then �(L) = 0. The

proof of Theorem 4.2.8 give by following symbolic inverse calculus.

Theorem 4.2.9. Let l ∈ Σm� (Ω), k ∈ Σm′

� (Ω), and l(x, �) ∕= 0 on Suppk, then there

exist ℎ, ℎ′ ∈ Σm′−m� (Ω) such that

l#ℎ = ℎ′#l = k.

We collect some results for paradifferential operators in following corollary.

Corollary 4.2.2. (a) For any d > 0, we have Op(Σm� (Ω)) ⊂ Op(Σm+d

�+d (Ω)).

(b) For L ∈ Op(Σm� (Ω)), � > 1, if �m(L) = 0, then L ∈ Op(Σm−1

�−1 (Ω)).

(c) Let Lj ∈ Op(Σm−j� (Ω)), j = 1, 2, � > 1, then [L1, L2] ∈ Op(Σm1+m2−1

�−1 (Ω)), and

�m1+m2−1([L1, L2]) =1

i{�m1(L1), �m2(L2)}.

The paradifferential operators defined on function space C� and Besov space Bsp,r

is similarly.

Tangential paradifferential operators. By using the double dyadic decomposi-tion and tangential Sobolev spaceHs,s′ given at the section 1.5, we define now tangentialparamultiplication

T ′au =∑q

(S ′qa)△′qu,

where S ′qa =∑q−N0

p=−1△′pu. We will use also double index paramultiplication

Πau =∑q,q′

Sq,q′a△q,q′u,

where Sq,q′a =∑q−N0

p=−1

∑q′−N0

p′=−1△p,p′u. We have

Theorem 4.2.10. (a) If a ∈ H t,t′(ℝn), t > 1/2, t+ t′ > n/2, then Πa : Hs,s′ → Hs,s′

is continuous for any s, s′ ∈ ℝ.(b) If a ∈ H t,t′(ℝn), t > 1/2, t + t′ > n/2, then T ′a : Hs,s′ → Hs,s′ is continuous

for any s′ ∈ ℝ and −t < s ≤ t. And Πa − T ′a : Hs,s′ → Hs,s′+�(t,t′), where �(t, t′) =min(s+ s′ − n/2, s− 1/2), if s′ ∕= (n− 1)/2; = s− 1/2− �, � > 0, if s′ = (n− 1)/2.

(c) If a ∈ H t,t′(ℝn), t > 1/2, t + t′ > n/2, then Ta : Hs,s′ → Hs,s′ is continuous forany s ∈ ℝ and (t+ t′−1/2 < s′ ≤ (t+ t′−1/2). And Πa−Ta : Hs,s′ → Hs,s′+(t+t′−n/2),for any s ∈ ℝ, and −(t+ t′ − 1/2) < s′ ≤ (n− 1)/2.


the operators T ′a is well-defined by modulo ℒ(Hs,s′ , Hs,s′+t+t′−n/2). We have alsosimilar symbolic calculus, and tangential paralinearization.

Theorem 4.2.11. Let F ∈ C∞(ℝ), F (0) = 0, u ∈ Hs,s′(ℝn), s > 1/2, s + s′ > n/2,then

F (u)− T ′F ′(u)

u ∈ Hs,s′+�(s,s′).

The proof of F (u)−ΠF ′(u)

u ∈ Hs,s′+�(s,s′) is similar to Theorem 4.3.3, then Theorem4.2.11 reduced by Theorem 4.2.10.

Remark that for the operators T ′a, variables xn is only a parameter, then T ′a∣ℝn+ is

well-defined, so we can use tangential paramultiplication T ′a to study boundary valueproblems.

4.3. Paradifferential equations

Paralinearization. We study now the theory of linearization of nonlinear partialdifferential equations, the simplest cas is composition of nonlinear functions.

Theorem 4.3.1. Let F ∈ C∞(ℝ1), F (0) = 0. If f ∈ Hs(ℝn) ∩ L∞(ℝn), s > 0is a real function (or f ∈ C�(ℝn), � > 0), then the composition F (f) ∈ Hs(ℝn) (orF (f) ∈ C�(ℝn)).

Proof: For simplifier notations, we consider the case � = s − n/2 > 0, f =∑∞p=−1 fp, the Littlewood-Paley decomposition, then Theorem 3.2.1 and 3.3.1 give

∥f − Sp(f)∥L∞ ≤ C2−p�∥f∥C� ,that means Sp(f) converge uniformly to f on ℝn, and we have

F (f) =∞∑

p=−1

[F (Sp(f))− F (Sp−1(f))],

here we note S−2(f) = 0. Then

F (Sp(f))− F (Sp−1(f)) = fp

∫ 1

0

F ′(Sp−1(f) + tfp)dt = mpfp,

where mp(x) depends on f , we call first linearization formula

F (f) =∞∑

p=−1

mpfp

=∞∑

p=−1

mp(x)'(2−pD)f = Lf.

We will prove in Theorem 4.3.2, that L =∑∞

p=−1mp(x)'(2−pD) ∈ L01,1, then L : Hs →

Hs is continuous for s > 0.

Theorem 4.3.2. Let {mp} ∈ C∞(ℝn) be a functions family verifies

(4.3.1) ∥∂�mp∥L∞ ≤ C�2p∣�∣.

Then the operators L(x,D) =∑∞

p=−1mp(x)'(2−pD) is continuous from Hs to Hs, andC� to C� for any s > 0, � > 0.

4.3. PARADIFFERENTIAL EQUATIONS 59

Using theorems 3.2.1 and 3.3.1, the proof of theorem 4.3.2 is evident. Then the restproof of theorem 4.3.1 is to prove (4.3.1) for mp(x) =

∫ 1

0F ′(Sp−1(f) + tfp)dt. Since

∥∂�(Sp−1(f) + tfp)∥L∞ ≤ C�2p∣�∣ for all � ∈ ℕn, then it is easy to get

(4.3.2) ∥∂�F ′(Sp−1(f) + tfp)∥L∞ ≤ C�2p∣�∣.

which finish the proof of theorem 4.3.1.

Theorem 4.3.3. Let F ∈ C∞(ℝ1), F (0) = 0, and f be a real functions, then wehave:

(i) If f ∈ Hs(ℝn), s > n/2, then there exist g ∈ H2s−n/2(ℝn) such that

F (f) = TF ′(f)f + g.

(ii) If f ∈ C�(ℝn), � > 0, then there exist g ∈ C2�(ℝn) such that

F (f) = TF ′(f)f + g.

Proof: Using the first linearization formula, we will prove that there exist R ∈L−�1,1 , (� > 0, � = s− n/2 > 0), such that

(4.3.3) L− TF ′(f) = R.

By definition, we have

�(R) =∑p≥N0

[mp(x)− Sp−N0(F ′(f))(x)]'(2−p�)

+∑p<N0

mp(x)'(2−p�).

It is evident∑

p<N0mp(x)'(2−p�) ∈ S−∞. Using theorems 3.2.1 and 3.3.1, for the first

terms we need only to prove

∥∂�x (mp(x)− Sp−N0(F ′(f))(x))∥L∞ ≤ C�2p(∣�∣−�).

This implies by Taylor formula and (4.3.2).The conclusion of Theorem 4.3.3 is also true for vector, if F (x, y) ∈ C∞(ℝn ×

ℝN), F (x, 0) = 0, and real functions u1, ⋅ ⋅ ⋅ , uN ∈ C�(ℝn), � > 0 or in Hs(ℝn), s > n/2,then there exist g ∈ C2�(ℝn), (H2s−n/2(ℝn)), such that

(4.3.4) F (x, u1, ⋅ ⋅ ⋅ , uN)−N∑j=1

TFjuj = g,

where Fj = ∂F∂yj

(x, u1(x), ⋅ ⋅ ⋅ , uN(x)).

In the logarithmic case, we have :

Theorem 4.3.4. Suppose that F ∈ C∞(ℝ), F (0) = 0, and u ∈ H(logℓ⟨�⟩)∩L∞(ℝn)

a real function for ℓ > 1/2. Then F (u) ∈ H(logℓ−1/2⟨�⟩)∩L∞(ℝn) and for any S ≥ 1

S2ℓ−1∥F (u)∥2H(logℓ−1/2⟨�⟩) ≤ CS

((ℓ− 1

2

)2ℓ−1∥u∥2L2 + S2ℓ−1(2ℓ− 1)∥u∥2

H(logℓ⟨�⟩)

),

with CS depending only on Sup∣t∣≤∥u∥L∞ ∣F(j)(t)∣ and ∥u∥jL∞ for j = 0, 1, ⋅ ⋅ ⋅ , [S] + 2 .


Proof : We have firstly

∥F (u)∥L2 = ∥F (u)− F (0)∥L2 ≤(sup∣t∣≤∥u∥L∞

)∣F ′(t)∣∥u∥L2 .

We denote, for k ≥ 1, Sku =∑k−2

j=−1 Δju, then for u ∈ H(logℓ⟨�⟩)∩L∞(ℝn), we have

F (u) = limk→+∞ F (Sku) in L2, so that

F (u) = F (S1u) +∞∑k=2

(F (Sku)− F (Sk−1u)) =∞∑k=1

fk

with f1 = F (S1u) and for k > 1

fk =

∫ 1

0

F ′(Sk−1u+ tΔku)dtΔku.

Since for any � ∈ ℕn,

∥D�(Sk−1u+ tΔku)∥L∞ ≤ C∣�∣ek∣�∣∥u∥L∞ , ∥D�Δku∥L2 ≤ e(k+3)∣�∣∥Δku∥L2 ,

the Faa-di-Bruno formula implies that

∥D�fk∥L2 ≤ B∣�∣ek∣�∣∥Δku∥L2

with B∣�∣ depending only on Sup∣t∣≤∥u∥L∞ ∣F(j)(t)∣ and ∥u∥jL∞ for j = 0, 1, ⋅ ⋅ ⋅ , ∣�∣+ 2.

Then∑

k fk satisfies the hypothesis of lemma 3.5.4, we have proved the theorem.

Paralinearization of non linear equations. We consider now nonlinear partialdifferential equation of order m,

(4.3.5) F [u] = F (x, u, ⋅ ⋅ ⋅ , ∂�u, ⋅ ⋅ ⋅ )∣�∣≤m = 0,

where F ∈ C∞(Ω × ℝN) is real function, Ω ⊂ ℝn an open domain, N = #{� ∈ℕn; ∣�∣ ≤ m}. Eventually, F is linear for some derivation of u, we can rewrite F asfollowing

F [u] =∑

k0<k≤m

∑∣�∣=k

A�(x, u, ⋅ ⋅ ⋅ , ∂�u, ⋅ ⋅ ⋅ )∣�∣≤p(k)∂�u

+ Ak0(x, u, ⋅ ⋅ ⋅ , ∂�u, ⋅ ⋅ ⋅ )∣�∣≤k0 .

where p(k) < k, if A� depends only on x, we put p(∣�∣) = −∞. Set

d = max

(k0,

k + p(k)

2

).

then if F is full nonlinear, d = m. If F is quasilinear, k0 = m− 1, p(m) = m− 1, k =m, d = m − 1/2. If F is semilinear, k0 = m − 1, k = m, p(m) = 0, d = m − 1. If F isonly nonlinear for u, k0 = 0, p(k) = −∞, d = 0. If F is linear, then d = −∞. We cannow define the symbol of linearized operators of F on function u ∈ C�.

Theorem 4.3.5. Let u ∈ C�

loc(Ω), � > max(k0, p(k)). Set

(4.3.6) p(x, �) =∑

∣�∣>2d−�

F�(x, u(x), ⋅ ⋅ ⋅ )(i�)�,

where F� = ∂u�F . Then p(x, �) ∈ Σm�+m−2d(Ω).

4.4. REGULARITY OF SOLUTION OF NON-LINEAR EQUATIONS 61

Using the symbol (4.3.6), we can give the paralinearization of nonlinear equation(4.3.5).


loc(Ω), � > max(k0, p(k)), and P paradifferential opera-

tor with symbol p(x, �) defined by (4.3.6), and u is a solution of equation (4.3.5).

(a) If � > d, then Pu ∈ C2�−2d

loc.

(b) If s ≥ n/2 + �, � > d− n/4, u ∈ Hs

loc(Ω), then Pu ∈ Hs+�−2d

loc.

The proof of this theorem is just as that of Theorem 4.3.3. We have proved that ifu ∈ C�(Ω)

∩Hs(Ω), � > max(k0, p(k)), s > 0 is a solution of equation (4.3.5), then

(4.3.7) Pu ∈ C2�−2d(Ω)∩

Hs+�−2d(Ω).

From a nonlinear equation F [u] = 0, we have get a linear (paradifferential) equationPu = f , with f more regular than the solution u. Since for the operators P , we havethe symbolic calculus as that for pseudo-differential operators, then microlocal analysisfor nonlinear equations is carry out.

4.4. Regularity of solution of non-linear equations

Microlocal elliptic regularity.

Definition 4.4.1. For (x0, �0) ∈ T ∗ℝn ∖ 0, we say u ∈ Hsx0,�0

(or u ∈ C�x0,�0

),if there exists a neighborhood Vx0 of x0 in ℝn

x, and a conic neighborhood Γ of �0 inℝn� ∖ 0, such that for any ' ∈ C∞0 (Vx0), and ∈ C∞(Γ) homogeneous of degree 0 in �,

con supp ⊂ Γ, we have

(D)('u) ∈ Hs, (or C�).

We study now microlocal regularity for nonlinear elliptic equations.


loc(Ω), � > d, a solution of equation (4.3.5), and

(x0, �0) ∈ Ω× ℝn ∖ 0, pm(x0, �0) ∕= 0. Then u ∈ C2�+m−2dx0,�0

.

Proof: Using Theorem 4.3.6, there exists f ∈ C2�−2d

loc(Ω) such that Pu = f , where

paradifferential operators P ∈ Op(Σm�+m−2d(Ω)). Since pm(x0, �0) ∕= 0, there exists a

conic neighborhood Γ of (x0, �0), and a classic pseudo-differential operator K of degree0, with the symbol no vanish on Γ′ ⊂⊂ Γ, and con supp�(K) ⊂ Γ. Then the symboliccalculus of paradifferential operators (Theorem 4.2.9) give q ∈ Σ−m�+m−2d(Ω) such thatq#p = �(K). If Q is the paradifferential operator of symbol q, then Q ∘ P = K + Rwith R ∈ S−(�+m−2d), we have

Ku = Qf −Ru ∈ C2�+m−2d

loc(Ω),

which prove Theorem.

Hypoellipticity for nonlinear equations. For nonlinear hypoellipticity, we con-sider only second order equation

(4.4.1) F (x, u,∇u,∇2u) = 0.


For u ∈ C�

loc(Ω), � > 2, we define the linearized operator of F associated with u by

(4.4.2) L(x,D) =n∑

j,k=1

ajk(x)∂j∂k +n∑j=1

bj(x)∂j + c(x),

where

ajk(x) = akj(x) = ∂ujkF (x, u(x),∇u(x),∇2u(x)),

bj(x) = ∂ujF (x, u(x),∇u(x),∇2u(x)),

and

c(x) = ∂uF (x, u(x),∇u(x),∇2u(x)) ∈ C�−2

loc(Ω).

Definition 4.4.2. We say that the linearized operators L is subelliptic, if for anyx ∈ Ω, we have (ajk(x)) ≥ 0, and for any K ⊂⊂ Ω, there exists � > 0, C > 0 such thatfor all ' ∈ C∞0 (K) we have

(4.4.3) ∥'∥2H� ≤ C{∣⟨L', '⟩∣+ ∥'∥2

L2}.

Remark that if L is elliptic, then it is subelliptic with � = 1. There is many suf-ficient condition for subellipticity, we have study the Hormander’s condition, Oleinik-Radkevic’s condition, and Fefferman-Phong’s condition.

We prove now nonlinear hypoellipticity.


loc(Ω), � ≥ 4 be a real solution of equation (4.4.1). If

the linearized operator L is subelliptic, then u ∈ C∞(Ω).

The proof of this theorem is using paralinearization theorem to transform the non-linear equation into a linear paradifferential equation, then paradifferential symboliccalculus give the results as in the classic cas.

Let P ∈ Op(Σ2�−2) be the paradifferential operators of symbol of L, then

P =n∑k=1

∂kGk +G0 + P0,

where

�(Gk) = gk(x, �) =n∑j=1

ajk(x)(i�j), k = 1, ⋅ ⋅ ⋅ , n,

�(G0) = g0(x, �) =n∑j=1

(bj(x)−n∑k=1

∂xkajk(x))(i�j), �(P0) = c(x).

Since

u ∈ C�

loc(Ω) = C�

loc(Ω)

∩H4

loc(Ω),

from (4.3.7), we have

(4.4.4) Pu = f ∈ C�

loc(Ω)

∩H4

loc(Ω).

This is a linear equation, we have the following a priori estimates.

4.5. PARADIFFERENTIAL CALCULUS IN BESOV SPACE 63

Theorem 4.4.3. Assume that (ajk(x)) ≥ 0, then for any K ⊂⊂ Ω, s ∈ ℝ, thereexist C > 0, such that for all v ∈ C∞0 (K), and any � > 0, we have

2n∑j=1

∥Gjv∥2Hs + ∥G0v∥2

Hs−1/2 ≤ C{∥Pv∥2Hs + ∥v∥2

Hs+�},

where �(Gn+l) =∑n

j,k=1 ∣�∣−1∂xlajk(x)�j�k, l = 1, ⋅ ⋅ ⋅ , n.

Using the positivity of (ajk(x)) and the commutators of P with pseudo-differentialoperators (1−△)s/2, the proof of this theorem is evident. Now subellipticity of oper-ators L, and Corollary 4.2.1 give immediately subelliptic estimate for paradifferentialoperators

∥v∥2Hs+� ≤ C{∥Pv∥2

Hs + ∥v∥2Hs}.

By regularization processes, we obtain for any ' ∈ C∞0 (K), there exist '1, '2 ∈C∞0 (Ω), � > 0, C > 0, such that for u ∈ Hs

loc solution of equation (4.4.1), we have

∥'u∥2Hs+� ≤ C{∥'1Pu∥2

Hs + ∥'2u∥2Hs}.

From this estimates, we prove immediately theorem 4.4.2. Since from u ∈ C�

loc(Ω) ⊂

H4

loc(Ω), we get u ∈ H4+�

loc(Ω), then by iteration u ∈ H4+N�

loc(Ω) for any N , which prove

u ∈ C∞(Ω).

4.5. Paradifferential calculus in Besov space

We study now the paramultiplication operators Ta on Besov space.

Theorem 4.5.1. There exists a constant C > 0 such that for any 1 ≤ p, r, p1, p2, r1, r2 ≤+∞, we have

∥Tuv∥Bsp,r ≤ C ∣s∣+1∥u∥L∞∥v∥Bsp,r , ∀s ∈ ℝ;

∥Tuv∥Bs+tp,r1,2

≤ C ∣s∣+∣t∣+1

∣t∣∥u∥

Bt∞,r1∥v∥

Bsp,r2, ∀s ∈ ℝ, t < 0;

∥R(u, v)∥Bs1+s2p1,2,r1,2

≤ C ∣s1∣+∣s2∣+1

s1 + s2

∥u∥Bs1p1,r1

∥v∥Bs2p2,r2

, ∀s1, s2 ∈ ℝ, s1 + s2 > 0,

where 1p1,2

= 1p1

+ 1p2≤ 1, 1

r1,2= min(1, 1

r1+ 1

r2). For

s1 + s2 ≥ 0,1

p1,2

≤ 1

p1

+1

p2

, frac1r1 +1

r2

= 1

we have

∥R(u, v)∥B�1,2p1,2,∞

≤ C ∣s∣+∣t∣+1∥u∥Bs1p1,r1

∥v∥Bs2p2,r2

,

where �1,2 = s1 + s2 − n(

1p1

+ 1p2− 1

p1,2

).

The proof of this theorem is similar to that of Theorem 4.1.1 and 4.1.2. We haveto estimate

∥Sq−N0uΔqv∥Lp ≤ C∥u∥L∞ ,


and for t < 0

∥Sq−N0uΔqv∥Lp ≤C

∣t∣cq,r12−qt∥u∥

Bt∞,r1

where {cq,r1}q∈ℕ is an element of the unit sphere of ℓr1,2(ℕ).We have immediately the following corollary,

Corollary 4.5.1. There exists C > 0 such that1) For any s ∈ ℝ,

∥uv∥Bsp,r ≤C ∣s∣+1

∣s∣

(∥u∥L∞∥v∥Bsp,r + ∥u∥Bsp,r∥v∥L∞

).

2) For s1 + s2 > 0 and s1 < n/p1

∥uv∥Bsp2,r2 ≤C ∣s1∣+∣s2∣+1

s1 + s2

(∥u∥Bs1p1,∞∥v∥Bs2p2,r2 + ∥u∥Bs2p2,r2∥v∥Bs1p1,∞

),

with s = s1 + s2 − n/p1.3) For s1 + s2 ≥ 0, s1 < n/p1 and 1 = 1/r1 + 1/r2,

∥uv∥Bsp,∞ ≤ C ∣s1∣+∣s2∣+1(∥u∥Bs1p1,r1∥v∥Bs2p2,r2 + ∥u∥Bs2p2,r2∥v∥Bs1p1,r1

),

with s = s1 + s2 − n/p1.

We have also the following paralinearization theorem in Besov space

Theorem 4.5.2. Let F ∈ C∞(ℝ), F (0) = 0, s > 0, u ∈ Bsp,r∩L∞ with 1 ≤ p, r ≤ ∞,

then F (u) ∈ Bsp,r and

∥F (u)∥Bsp,r ≤ C(s, F, ∥u∥L∞)∥u∥Bsp,rIn the Besov case, we have the following more general version results.

Theorem 4.5.3. Let F ∈ C∞b (ℝ) the space of smooth bounded functions F (0) = 0,s > 0, u ∈ Bs

p,r with 1 ≤ p, r ≤ ∞. Assume that ∇u ∈ B−1∞,∞ then F (u) ∈ Bs

p,r and

∥F (u)∥Bsp,r ≤ C(s, F, ∥∇u∥B−1∞,∞

)∥u∥Bsp,r .

Remark : If u ∈ Bn/pp,r then ∇u ∈ B−1

∞,∞. Thus the space Bn/pp,r is stable under the non

linear composition of (bounded) function of C∞b (ℝ). This is in particular the case for

Sobolev space Bn/22,2 = Hn/2.

Semilinear elliptic equations. We give here a very simple application of Theo-rem 4.5.1. We consider the following semilinear elliptic equation

Δu = a(u)∣∇u∣2,where a( ⋅ ) is a smooth function.

Theorem 4.5.4. Let u ∈ H1loc ∩ Cs

loc, 1/2 < s < 1 be a weak solution of aboveequation, then u ∈ C∞

This Theorem is true for system and general second order elliptic operators, so thatwe can use this Theorem to prove the regularity of weak harmonic maps.

From Theorem 4.5.1, we prove firstly the following estimate for multiplication of 3functions.

4.5. PARADIFFERENTIAL CALCULUS IN BESOV SPACE 65

Lemma 4.5.1. Let s ≥ 2, 1/2 < � < 1, assume that

u ∈ Bs1,∞ ∩ C�, v, w ∈ Bs−1

1,∞ ∩ C�−1.

Then∥uvw∥Bs−2+a

1,∞≤ C∥u∥Bs1,∞∩C�∥v∥Bs−1

1,∞∩C�−1∥w∥Bs−11,∞∩C�−1 .

Setting A = uv and Aw = TAw + TwA+R(A,w), for s ≥ 2, 1/2 < � < 1, we have

∥A∥C�−1 = ∥uv∥B�−1∞,∞≤ ∥u∥Bs1,∞∩C�∥v∥Bs−1

1,∞∩C�−1 ,

and∥TAw∥Bs−2+�

1,∞≤ C∥A∥C�−1∥w∥Bs−1

1,∞,

∥TwA+ (R(A,w)∥Bs−2+�1,∞

≤ C∥A∥Bs−11,∞∥w∥C�−1 .

We have proved the Lemma.Now we can finish the proof of Theorem 4.5.4. After cutt-off, we suppose that

u ∈ H1 ∩ C�, thenF (u) = a(u)∣∇u∣2 ∈ L1 ⊂ B0

1,∞.

Using Proposition 3.5.3, we get u ∈ B21,∞ ∩ C�. Now Lemma 4.5.1 implies that

F (u) = a(u)∣∇u∣2 ∈ L1 ⊂ B�1,∞,

use again Proposition 3.5.3, we get u ∈ B2+�1,∞ ∩ C�. So that we prove the following

induction results

u ∈ B2+k�1,∞ ∩ C� =⇒ u ∈ B2+(k+1)�

1,∞ , ∀ k ∈ ℕ.The u ∈ ∩s≥0B

s1,∞ ⊂ C∞. We have proved Theorem 4.5.4.

CHAPTER 5

Transport equations and Euler’s equation

In the end of precedent chapter, we have given some application of Littlewood-Paleytheory, studied the regularity of weak solutions. We study now, in this chapter andnext chapter, the existence of solutions for some nonlinear equation come from fluidmechanics, so it is another type of application of Littlewood-Paley theory,

5.1. Ordinary differential equations

The Cauchy-Lipschitz theorem revisited. Let Ω be an open subset of ℝn (itis also true for a Banach space E), and I an open interval of ℝ, F is a continuousfunction defined on I × Ω. We consider the Cauchy problem for ordinary differentialequations

(5.1.1)

{x = F (t, x)x∣t=t0 = x0

If F is locally uniformly Lipschitz with respect to variable x ∈ Ω, the Cauchy-LipschitzTheorem give the existence and uniqueness of solution for above Cauchy problem. If itis only continuous or Holderien with index strictly small then 1, we have the existenceof solution(Arzela-Peano Theorem), but loss the uniqueness. We consider now thegeneralized version of classical Cauchy-Lipschitz theorem, The Osgood theorem.

Definition 5.1.1. Let � be a modulus of continuity. We say that � is an Osgoodmodulus of continuity if and only if, for any 0 < a < 1,∫ a

0

ds

�(s)= +∞.

Some examples of Osgood modulus of continuity :

�(s) = s, �(s) = s(− log s)�, �(s) = s(− log s)(− log(− log s)

)�, � < 1.

But they are not Osgood if � ≥ 1, neither the function �(s) = s�, � < 1.

Theorem 5.1.1. Let us assume that � is a Osgood modulus of continuity, F ∈L1loc(I;C�(Ω,ℝn)) and (t0, x0) ∈ I × Ω. Then an open sub-interval J of I exists such

that t0 ∈ J ⊂ I and that the Cauchy problem (5.1.1) has a unique continuous solutionsdefined on J .

This theorem is a local one, we state now some condition for blow up .

Theorem 5.1.2. Under the hypothesis of Theorem 5.1.1 with I = ℝ and Ω = ℝn,let us also assume that there exists a locally bounded function M and a locally integrablefunction � from ℝ+ into ℝ+ such that

∥F (t, u)∥ ≤ �(∣t∣)M(∥u∥).67

68 5. EULER’S EQUATION

Then if ]T∗, T∗[ is the maximal interval of existence of a solution u, and if T ∗ is finite,

we have

lim supt→T ∗∥u(t)∥ =∞.similar results for T∗.

There is many applications of this two theorem to incompressible viscous fluidestudied.

5.2. Commutation lemmas

We study now the estimate of commutator of operators Δj = '(2−jD) with multi-plication :

[Δj, a]b = Δj(a b)− aΔj(b).

We have

Lemma 5.2.1. There exists C > 0 such that for any p ∈ [1,+∞], any a ∈ Lip andb ∈ Lp, we have

∥[Δj, a]b∥Lp ≤ C2−j∥∇a∥L∞∥b∥Lp , ∀j ∈ ℕ.

Let us think Δj as convolution, we have([Δj, a]b

)(x) = 2jd

∫ℝdℎ(2d(x− y))(a(y)− a(x))b(y)dy.

As the function a is Lipschtiz, we have

∣a(y)− a(x)∣ ≤ ∥∇a∥L∞∣y − x∣.

It turns out that∣∣([Δj, a]b)(x)∣∣ ≤ 2jd∥∇a∥L∞

∫ℝd∣ℎ(2d(x− y))∣∣y − x∣ ∣b(y)∣dy.

Then Young inequality implies that

∥[Δj, a]b∥Lp ≤ 2−j∥ℎ( ⋅ )∣ ⋅ ∣∥L1∥∇a∥L∞∥b∥Lp .

This concludes the proof of the lemma.Remark : This lemma can be interpreted as a gain of one derivative by commutationbetween the operator Δj and the multiplication by a lipschitz function.

Lemma 5.2.2. Let a = (a1, ⋅ ⋅ ⋅ , ad) be a vector field on ℝd.

∙ If s ∈]0, 1[, we have, for any p, r ∈ [1,+∞]

∥[a ⋅ ∇, Δj]f∥Lp ≤ Cs2−js∥f∥Bsp,r∥∇a∥L∞ .

∙ If a ∈ Bsp,r with s ∈]1, +∞[ and p, r ∈ [1,+∞], we have

∥[a ⋅ ∇, Δj]f∥Lp ≤ Cs2−js(∥f∥Bsp,r∥∇a∥L∞ + ∥∇f∥L∞∥∇a∥Bs−1

p,r

).

∙ If s ∈]− 1, 1[, div a = 0, we have, for any p, r ∈ [1,+∞]

∥[a ⋅ ∇, Δj]f∥Lp ≤ Cs2−js∥f∥Bsp,r∥∇a∥L∞ .

5.3. PROPAGATION OF REGULARITY FOR TRANSPORT EQUATIONS 69

This is again paramultiplication calculus, We only point out some important step.Let us write again Bony’s decomposition.

a ⋅ ∇f =d∑

k=1

(Tak∂kf + T∂kfa

k + ∂kR(ak, f))−R(div a, f).

So we study the commutation with each of the above operator. We get

[Δj, Tak∂k] =∑

∣j−j′∣≤N0

[Δj, Sj′−1(ak)]Δj′∂k.

Applying lemma 5.2.1 with b = Δj′∂kf and a as Sj′−1(ak), we have

∥[Δj, Sj′−1(ak)]Δj′∂kf∥Lp ≤ Cs2−jscj∥f∥Bsp,r∥∇a∥L∞ ,

where {cj} ∈ ℓr with norm 1. Then we have

∥[Δj, Tak∂k]f∥Lp ≤ Cs2−jscj∥f∥Bsp,r∥∇a∥L∞ .

A direct calculus give

∥ΔjT∂kfak∥Lp ≤ Cs2

−js∥∇f∥L∞∥∇a∥Bs−1p,r,

and if s < 1,

∥ΔjT∂kfak∥Lp ≤ Cs2

−js∥∇f∥Bs−1p,r∥∇a∥B0

∞,∞ .

We get finally, if s > 1,

(5.2.1) ∥[Δj, T∂kak ]f∥Lp ≤ Cs2−js∥∇f∥L∞∥∇a∥Bs−1

p,r.

And if s < 1

(5.2.2) ∥[Δj, T∂kak ]f∥Lp ≤ Cs2−js∥f∥Bsp,r∥∇a∥B0

∞,∞ .

The term R(div a, f) is very easy to estimate, for any s > 0

(5.2.3) ∥[Δj, R(div a, ⋅)]f∥Lp ≤ Cscj2−js∥f∥Bsp,r∥∇a∥B0

∞,∞ .

If the vector field a is divergence free, this term disappears.Now for s > −1, we have

(5.2.4) ∥[Δj, ∂kR(ak, ⋅)]f∥Lp ≤ Cscj2−js∥f∥Bsp,r∥a∥B1

∞,∞ .

We concludes the proof of lemma.

5.3. Propagation of regularity for transport equations

We study now the following transport equation

(5.3.1) ∂tf + a ⋅ ∇f = g, f ∣t=0 = f0,

where a(t, x) = (a1(t, x), ⋅ ⋅ ⋅ , ad(t, x)) is a C∞ vector field on [0, T ]× ℝd. For r > −1,set

V (t) = max

{∥∇v(t)∥L∞ ,

∥∇v(t)∥Cr−1

r − 1

}.

We have the following “a priori estimate”.


Lemma 5.3.1. Let (f, g) ∈ L∞([0, T ];Cr)× L1([0, T ];Cr), r > 0 satisfies the equa-tion (5.3.1) such that

g = g1 + g2 with ∥g2(t)∥Cr ≤ C(r)V (t)∥f(t)∥Cr .

If −1 < r < 1, we suppose that diva(t) = 0. Then we have the following estimate

∥f(t)∥Cr ≤ ∥f0∥CreC(r)∫ t0 V (�)d� +

∫ t

0

∥g1(�)∥CreC(r)∫ t� V (� ′)d� ′d�.

Proof : Let be the integral path of vector field a, we have

f(t, x) = f0( −1(t, x)) +

∫ t

0

g(s, (s, −1(t, x)))ds.

So that we have trivially

∥f(t)∥L∞ ≤ ∥f0∥L∞ +

∫ t

0

∥g(t′)∥L∞dt′.

Now we try to apply this trivially estimate to Δjf . From equation (5.3.1) we have

∂tΔjf + a ⋅ ∇Δjf = Δjg + [a ⋅ ∇, Δj]f, Δjf ∣t=0 = Δjf0.

Using now Lemma 5.2.2, we get

∥Δjf(t)∥L∞ ≤ ∥Δjf0∥L∞ +

∫ t

0

∥Δjg1(t′)∥L∞dt′ + C(r)2−jr∫ t

0

V (t′)∥f(t′)∥Crdt′.

Multiplier by 2jr we have

∥f(t)∥Cr ≤ ∥f0∥Cr +

∫ t

0

∥g1(t′)∥Crdt′ + C(r)

∫ t

0

V (t′)∥f(t′)∥Crdt′.

The Gronwall lemma give Lemma 5.3.1

5.4. Euler’s equations

We consider now the following Euler’s equation :

(5.4.1)

⎧⎨⎩ ∂tv + div v ⊗ v = −∇pdiv v = 0v∣t=0 = v0

where

(div v ⊗ v)i =d∑j=1

∂j(vivj), i = 1, ⋅ ⋅ ⋅ , d.

For more detail about Euler’s equation and Navier-Stokes equations, see [24, 28, 73,74].

By differentiation the equation, we have

∂t∂kvi +

d∑j=1

∂k∂j(vivj) = −∂k∂ip.

5.4. EULER’S EQUATIONS 71

Using

div v =d∑j=1

∂jvj = 0,

we have

Δp = −d∑

j,k=1

∂j∂k(vjvk)

and “formally” we have

(5.4.2) p = −Δ−1

(d∑

j,k=1

∂j∂k(vjvk)

).

Lemma 5.4.1. Let d = 2, then the operators

Tjk = Δ−1∂j∂k, j, k = 1, 2

are continuous from Cr ∩ Lp to Cr ∩ Lp for all r ∈ ℝ, p ∈]1,+∞[

For the proof of this lemma see J.-Y. Chemin [28] Th. 1.31 and Th. 2.5.1.We define a bilinear operators

B(v, w) = B1(v, w) +B2(v, w),

with

B1(v, w) =d∑j,k

∇Δ−1(T∂jwk∂kv

j + T∂kvj∂jwk)

and

B2(v, w) =d∑j,k

∇TjkR(vj, wk).

We have now a representation of pression p.

Proposition 5.4.1. If div v = 0 and it is enough regular, we have

B(v, v) = −∇p =d∑

j,k=1

∇Tjk(vjvk).

Proof : In fact, by paraproduct decomposition

d∑j,k=1

∇Tjk(vjvk) =d∑

j,k=1

∇Tjk(Tvjv

k + Tvkvj +R(vj, vk)

)=

d∑j,k=1

∇Tjk(Tvjv

k + Tvkvj)

+B2(v, v)

=d∑

j,k=1

∇Δ−1∂j∂k(Tvjv

k + Tvkvj)

+B2(v, v).


Since div v =∑d

j=1 ∂jvj = 0, we have also

∑dj=1 T∂jvj = T∑d

j=1 ∂jvj = 0 as operators.

We haved∑

j,k=1

∇Tjk(vjvk) ==d∑

j,k=1

∇Δ−1∂j(T∂kvjv

k + Tvk∂kvj)

+B2(v, v)

=d∑j,k

∇Δ−1(T∂jvk∂kv

j + T∂kvj∂jvk)

+B2(v, v).

For the operators B, we have the following proprieties.


(1) For r > 1, there exists C(r) > 0 such that

∥B(v, w)∥Cr ≤ C(r)(∥v∥Lip∥w∥Cr + ∥w∥Lip∥v∥Cr

).

(2) For −1 < r < 1, there exists C(r) > 0 such that

∥B(v, w)∥Cr ≤ C(r) min{∥v∥Lip∥w∥Cr , ∥w∥Lip∥v∥Cr

}.

(3) For r > 1, there exists C(r) > 0 such that

∥divB(v, v)− Tr((Dv)2

)∥L∞ ≤ C(r)∥v∥Cr∥div v∥L∞

where Tr((Dv)2

)=∑

jk ∂jvk∂kv

j.

In fact, by the continuity of operator Tjk, we have

∥B2(v, w)∥Cr ≤ Cd∑

j,k=1

∥R(vj, wk)∥Cr+1 ≤ C∥v∥C1∗∥w∥Cr

where we need the condition 1 + r > 0.We have also

∥B1(v, w)∥Cr ≤d∑

j,k=1

(∥T∂jwk∂kv

j∥Cr−1 + ∥T∂kvj∂jwk∥Cr−1

).

If r > 1, we have

∥T∂jwk∂kvj∥Cr−1 ≤ ∥∂jwk∥L∞∥∂kvj∥Cr−1 ≤ ∥w∥Lip∥v∥Cr ,

and

∥T∂kvj∂jwk∥Cr−1 ≤ ∥v∥Lip∥w∥Cr .

Similar calculus for −1 < r < 1 and laster estimation.Now the Cauchy problem (5.4.1) for Euler’s equation is equivalent to find a regular

solution of following problem :

(5.4.3)

{∂tv + v ⋅ ∇v = B(v, v)v∣t=0 = v0

with propriety div v = 0.We prove firstly the propriety of divergence free.

5.5. EXISTENCE AND UNIQUENESS OF SOLUTION OF A MODEL EQUATION 73

Proposition 5.4.3. Let v0 ∈ Cr(ℝd,ℝd), r > 1 and div v0 = 0. Suppose thatv ∈ L∞([0, T ]; Cr(ℝd,ℝd)) is a solution of Cauchy problem (5.4.3), then we have

div v(t) = 0, ∀t ∈ [0, T ].

Proof : We have{∂t(div v) + v ⋅ ∇(div v) = divB(v, v)− Tr

((Dv)2

)div v∣t=0 = div v0 = 0

Using L∞ estimate of transport equation and last estimate of Proposition 5.4.2 , wehave

∥div v(t)∥L∞ ≤ C

∫ t

0

∥v(s)∥Cr∥div v(s)∥L∞ds,

then Gronwell inequality implies ∥div v(t)∥L∞ = 0 for all t ∈ [0, T ].

5.5. Existence and uniqueness of solution of a model equation

We study now the existence of solution for Cauchy problem (5.4.3).

Theorem 5.5.1. Suppose that v0 ∈ Cr(ℝd,ℝd) and r > 1. Then there exists T ∗ andv ∈ L∞loc([0, T ∗[; Cr(ℝd,ℝd)) an unique solution of Cauchy problem (5.4.3). Moreover,if T ∗ < +∞, we have ∫ T∗

0

∥v(s)∥C1ds = +∞.

We prove the existence of solution by iteration : Setting v1 = S2v0, and

(5.5.1)

{∂tvn+1 + vn ⋅ ∇vn+1 = B(vn, vn),vn+1∣t=0 = Sn+2(v0),

n ∈ ℕ.

Each equation is a transport equation. We have that v1 = S2(v0) ∈ B∞∞,∞ andSn+2(v0) ∈ B∞∞,∞. So that, by induction, if n(t, x) is integral path of vector fieldvn, we have that

(5.5.2) vn+1(t, x) = Sn+2(v0)( n(−t, x)) +

∫ t

0

B(vn, vn)(s, n(s− t, x))ds.

Therefore, we can construct the sequence {vn}n∈ℕ which is C∞. We study now theconvergence of this sequence by two steps :

Proposition 5.5.1. We consider the case r > 1.

(1) There exists T1 > 0 such that {vn} is a bounded sequence in L∞([0, T1]; Cr).(2) There exists 0 < T2 ≤ T1 such that {vn} is a Cauchy sequence in L∞([0, T1]; Cr−1).

After this two steps, and using the following interpolation estimate

∥vn − vm∥Cr′ ≤ ∥vn − vm∥�Cr∥vn − vm∥1−�Cr−1 ,

where r′ = �r + (1 − �)(r − 1), 0 < � < 1, we proved that {vn} is also a Cauchysequence in L∞([0, T1]; Cr′), and the limit v ∈ L∞([0, T1]; Cr). Now the operatorB(⋅, ⋅) is continuous from L∞([0, T1]; Cr′) × L∞([0, T1]; Cr′) to himself. The product(v ⋅ ∇v) is also continuous on L∞([0, T1]; Cr) × L∞([0, T1]; Cr−1). We have provedthat the limit v is a solution of Cauchy problem (5.4.3).

We prove now the Proposition 5.5.1, and the uniqueness of solution of Cauchyproblem (5.4.3).


Bounded estimation for big norm. We consider r > 1, setting C0 = ∥ℱ−1( )∥L1 ,we want to prove

(5.5.3) ∥vn∥L∞([0,T1];Cr) ≤ 8C0∥v0∥Cr , ∀n ∈ ℕ.It is trivial for n = 1, suppose that (5.5.3) is true for any j ≤ n, prove now (5.5.3) forn+ 1. We apply now Lemma 5.3.1 with g2 = 0 and

V (t) = ∥∇vn(t)∥Cr−1 ≤ 8C0∥v0∥Cr .The estimate of

∥B(vn, vn)∥Cr ≤ C∥vn∥Cr∥vn∥Lip ≤ C(8C0∥v0∥Cr

)2.

Lemma 5.3.1 give

∥vn+1(t)∥Cr ≤ C0∥v0∥Cr eCt8C0∥v0∥Cr + tC(8C0∥v0∥Cr

)2eCt8C0∥v0∥Cr .

So that choose T1 small enough, we proved (5.5.3).

Convergence for small norm. We prove now that there exists Ar > 0, 0 < T2 ≤T1 such that

(5.5.4) ∥vn+1 − vn∥L∞([0,T2];Cr−1) ≤ Ar2−n, ∀n ∈ ℕ.

From equation (5.5.1), we have⎧⎨⎩ ∂t(vn+1 − vn) + vn ⋅ ∇(vn+1 − vn) = (vn−1 − vn) ⋅ ∇vn+B((vn − vn−1), vn) +B(vn−1, vn − vn−1),

(vn+1 − vn)∣t=0 = Δn+1(v0).

Since for T ≤ T1, we have

∥vn∥L∞([0,T ];Cr) ≤ 8C0∥v0∥Cr , ∀n ∈ ℕ.Proposition 5.4.2 give

∥B((vn−vn−1), vn)(t)+B(vn−1, vn−vn−1)(t)∥Cr−1 ≤ 8C0∥v0∥Cr∥vn−vn−1∥L∞([0,T ];Cr−1),

and some estimate for the term

∥(vn−1 − vn) ⋅ ∇vn(t)∥Cr−1 ≤ 8C0∥v0∥Cr∥vn − vn−1∥L∞([0,T ];Cr−1).

We use again Lemma 5.3.1 with V (t) same as above, we get

∥vn+1 − vn∥L∞([0,T ]; Cr) ≤ ∥Δn+1(v0)∥Cr−1 eC(r)T8C0∥v0∥Cr

+C(r)T eC(r)T8C0∥v0∥Cr∥vn − vn−1∥L∞([0,T1];Cr−1) .

Using induction hypothesis and the fact

∥Δn+1(v0)∥Cr−1 ≤ C(r)2−n∥v0∥Cr .Choose 0 < T2 ≤ T1 small such that

C(r)T2 eC(r)T28C0∥v0∥Cr ≤ 1

2.

Then, we can get (5.5.4) if we take

Ar ≥1

2C(r)∥v0∥Cr .

We have proved the Proposition 5.5.1.

5.5. EXISTENCE AND UNIQUENESS OF SOLUTION OF A MODEL EQUATION 75

Uniqueness of solution. We prove now the last part of Theorem 5.5.1, theuniqueness of Cauchy problem (5.4.3) : Let v, w ∈ L∞([0, T2]; Cr) be two solutionsof Cauchy problem (5.4.3), then⎧⎨⎩ ∂t(v − w) + v ⋅ ∇(v − w) = (w − v) ⋅ ∇w

+B(v, v − w) +B(v − w,w),(v − w)∣t=0 = v0 − w0.

Setting V (t) = ∥v(t)∥Cr + ∥w(t)∥Cr , then

∥(w − v) ⋅ ∇w +B(v, v − w) +B(v − w,w)∥Cr−1 ≤ V (t)∥(v − w)(t)∥Cr−1 .

Using the estimate for transport equation, we get

∥(v − w)(t)∥Cr−1 ≤ ∥v0 − w0∥Cr−1 eC∫ t0 V (s)ds.

So thatv0 − w0 = 0 =⇒ v(t)− w(t) = 0, ∀ t ∈ [0, T2].

CHAPTER 6

Existence of solution for shallow water equations

6.1. Shallow water equations

We consider in this chapter the following Cauchy problems for viscous shallow waterequations as follows:

(6.1.1)

⎧⎨⎩ ℎ(ut + (u ⋅ ∇)u)− �∇ ⋅ (ℎ∇u) + ℎ∇ℎ = 0,ℎt + div(ℎu) = 0,u∣t=0 = u0, ℎ∣t=0 = ℎ0;

where ℎ(x, t) is the height of fluid surface, u(x, t) = (u1(x, t), u2(x, t))t is the horizontalvelocity field, x = (x1, x2) ∈ ℝ2, and 0 < � < 1 is the viscous coefficients.

For more detail about Shallow water equation, see [73, 74].The equations form a quasi-linear hyperbolic-parabolic system. For the initial data

ℎ0(x), we suppose that it is a small perturbation of some positive constant ℎ0. Westudy the Cauchy problem (6.1.1) in the Sobolev function space.

The main theorems is :

Theorem 6.1.1. Let s > 0, u0, ℎ0 − ℎ0 ∈ H2+s(ℝ2), ∥ℎ0 − ℎ0∥H2+s << ℎ0. Thenthere exists a positive time T , an unique solution (u, ℎ) of Cauchy problem (6.1.1) suchthat

u, ℎ− ℎ0 ∈ L∞([0, T ];H2+s), ∇u ∈ L2([0, T ];H2+s).

Furthermore, there exists a constant c such that if ∥ℎ0 − ℎ0∥H2+s + ∥u0∥H2+s ≤ c, thenwe can choose T = +∞.

We use the Littlewood-Paley decomposition theory for Sobolev space to obtain thelosing energy estimates in Hs+2 for any s > 0, and we get the local existence of solutionfor all size initial data. Moreover, we also improve the global existence of solution andregularity for small initial date.

Dyadic estimates. In the low vertical frequencies estimates, we need the homo-geneous Sobolev spaces,

Hs(ℝ2) = {u ∈ S ′(ℝ2); ∥u∥2Hs =

∑j∈ℤ

22js∥Δju∥2L2 < +∞},

where Δj, j ∈ ℤ is homogeneous Littlewood-Paley decomposition, for the detail of thisnotation, see chapter ??.

For d = 2, we have that H2+s(ℝ2) ⊂ L∞(ℝ2) for any s > −1, and

∥f∥L∞(ℝ2) ≤ Cs∥f∥H2+s(ℝ2),

77

78 6. SHALLOW WATER EQUATIONS

Cs is the so-called Sobolev constant in ℝ2. We have also that, for any q ≥ 0,

∥Δqf∥L∞ ≤ C∥∇(Δqf)∥L2 ,

and∥Δqf∥L2 ≤ C2−q∥∇(Δqf)∥L2 .

In our equation, we have the products of 3 functions, so that we need the followingprecise estimates :

(6.1.2)∣(ab, c)L2∣ ≤ C∥a∥L∞∥b∥L2∥c∥L2 ,∣(ab, c)L2∣ ≤ C∥a∥H1/2∥b∥L2∥c∥H1/2 ,∥a∥2

H1/2 ≤ ∥a∥L2∥∇a∥L2 .

The proof of this estimates is similar to two functions case.In the prove of main theorem, we need to estimate the nonlinear term in the equa-

tions, this is so-called “Losing energy estimates”

Lemma 6.1.1. Let � > 1 and −1 ≤ k < +∞, then there exists C0 > 0 such thatfor all v,∇v, g,∇g ∈ H� , we have

∣∫ℝ2

Δk((v ⋅ ∇)g)Δkgdx∣ ≤ C0d2k2−2k�∥v∥H�+1∥g∥2

H� ,

with {dk} ∈ ℓ2 and ∥{dk}∥ℓ2 ≤ 1.

Lemma 6.1.2. .(a) Let � > 2 and −1 ≤ k < +∞, then there exists C0 > 0 such that for all

f, v, g, u,∇u ∈ H� , with ∥g∥L∞ ≤ 1/4, we have

∣∫ℝ2

Δk(∇f

1 + g∇v)Δkudx∣ ≤ C0d

2k2−2k�∥f∥H�∥v∥H� (1 + ∥g∥H� )∥u∥H�+1 ,

where ∥{dk}∥ℓ2 ≤ 1.(b) Let 1 < � < 2 and −1 ≤ k < +∞, then there exists C0 > 0 such that for all

f, g, u,∇u, v,∇v ∈ H� , with ∥g∥L∞ ≤ 1/4, we have

∣∫ℝ2

Δk(∇f

1 + g∇v)Δkudx∣ ≤ C0d

2k2−2k�∥f∥H� (1 + ∥g∥H� )U� (u, v),

with ∥{dk}∥ℓ2 ≤ 1, and

U� (u, v) =: ∥∇v∥L∞∥u∥H�+1 + ∥∇v∥H� (∥∇u∥H1 + ∥u∥H� ).

Lemma 6.1.3. .(a) Let � > 2 and −1 ≤ k < +∞, then there exists C0 > 0 such that for all

f, v, u,∇u, g1, g2 ∈ H� , with ∥g1∥L∞ , ∥g2∥L∞ ≤ 1/4, we have

∣∫ℝ2 Δk(

(g1−g2)(1+g1)(1+g2)

∇f∇v)Δkudx∣≤ C0d

2k2−2k�∥f∥H�∥v∥H�∥g1 − g2∥H�∥u∥H�+1 ,

with ∥{dk}∥ℓ2 ≤ 1.(b) Let 1 < � < 2 and −1 ≤ k < +∞, then there exists C0 > 0 such that for all

f, v, g1, g2, u,∇u, v,∇v ∈ H� , with ∥g1∥L∞ , ∥g2∥L∞ ≤ 1/4, we have

∣∫ℝ2 Δk(

(g1−g2)(1+g1)(1+g2)

∇f∇v)Δkudx∣≤ C0d

2k2−2k�∥f∥H�∥g1 − g2∥H�U� (u, v),

6.2. THE LOCAL EXISTENCE OF SOLUTION 79

with ∥{dk}∥ℓ2 ≤ 1, and U� (u, v) is as in Lemma 6.1.2, (b).

In the proofs of existence of global solutions, we need the following high verticalfrequencies estimates.

Lemma 6.1.4. Let � > 0, then there exists M > 0, C0 > 0 such that for allℎ, u, v,∇ℎ,∇u ∈ H� ,M ≤ k < +∞, with ∥ℎ∥L∞ ≤ 1/4, we have

∣∫ℝ2 Δk(

11+ℎ∇ℎ∇u)Δkvdx∣

≤ C0d2k2−2k� (1 + ∥ℎ∥H�+1)∥△u∥H�∥∇ℎ∥H�∥v∥H� ,

with ∥{dk}∥ℓ2 ≤ 1.

Lemma 6.1.5. Let � > 0, then there exists M > 0, C0 > 0 such that for allℎ ∈ H�+1 with ∥ℎ∥L∞ ≤ 1/4, and u ∈ H�+2,M ≤ k < +∞, we have

∣∫ℝ2 Δk(div(ℎu))Δk(Δℎ)dx∣

≤ C0d2k2−2k�∥∇ℎ∥H� (∥∇ℎ∥2

H� + ∥∇u∥2H�+1),

with ∥{dk}∥ℓ2 ≤ 1.

We will prove this five lemmas in the section 6.4.

6.2. The local existence of solution

In order to study the local existence of solution, we define the function set, (f, g) ∈X ([t1, t2], �, E1, E2) if (f, g) ∈ L∞([t1, t2], H�(ℝ2)),∇f ∈ L2([t1, t2], H�(ℝ2)) and

∥f∥2L∞([t1,t2],H�(ℝ2)) + �∥∇f∥2

L2([t1,t2],H�(ℝ2)) ≤ E21

∥g∥L∞([t1,t2],H�(ℝ2)) ≤ E2.

The main result of this section is the following local existence theorem for any initialdata :

Theorem 6.2.1. Let s > 0, (u0, ℎ0 − ℎ0) ∈ Hs+2(ℝ2) with ∥ℎ0 − ℎ0∥H2+s ≤ ℎ0

4Cs,

then there exist a positive time T and a solution

(u, ℎ− ℎ0) ∈ X ([0, T ], s+ 2, E1, E2)

for the Cauchy problem (6.1.1). Here E1 = 2∥u0∥Hs+2 , E2 = 2∥ℎ0− ℎ0∥Hs+2, and Cs isthe Sobolev constant.

For convenience sake, we take ℎ0 = 1. Substitute ℎ by 1 + ℎ in (6.1.1), we have⎧⎨⎩ ut + (u ⋅ ∇)u− �∇⋅((1+ℎ)∇u)1+ℎ

+∇ℎ = 0,ℎt + divu+ div(ℎu) = 0,u(x, 0) = u0(x), ℎ(x, 0) = ℎ0(x).

We suppose now ℎ0 ∈ Hs+2(ℝ2), ∥ℎ0∥H2+s ≤ 14Cs

, and E1 = 2∥u0∥Hs+2 , E2 = 2∥ℎ0∥Hs+2 .


The proof of Theorem 6.2.1 involves the method of successive approximations. Wedefine the sequence {un, ℎn} by following linear systems:

(6.2.1)

⎧⎨⎩(u1, ℎ1) = S2(u0, ℎ0),∂tun+1 − �Δun+1 = G1(un, ℎn),∂tℎn+1 + un∇ℎn+1 = G2(un, ℎn),(un+1, ℎn+1)∣t=0 = Sn+2(u0, ℎ0),

whereG1(un, ℎn) = �

1+ℎn∇ℎn∇un − un∇un +∇ℎn

G2(un, ℎn) = −(1 + ℎn)divun.

Since Sq are smooth operators, the initial data Sn+2(u0, ℎ0) are smooth functions.If (un, ℎn) ∈ X ([0, T ], s+ 2, E1, E2) and smooth, we have

∥ℎn∥L∞ ≤ Cs∥ℎn∥H2+s ≤ CsE2 = 2Cs∥ℎ0∥H2+s ≤ 2Cs4Cs≤ 1

2,

then G1(un, ℎn) and G2(un, ℎn) are also smooths functions. Note that the first equationin (6.2.1) is heat equations for un+1, and the second equation in (6.2.1) is transportequation for ℎn+1, then the existence of the smooth solutions for the Cauchy problems(6.2.1) is evident. We denote by Pn the application from (un, ℎn) to (un+1, ℎn+1) thesolution of problem (6.2.1).

Now the proof of theorem 6.2.1 is also in two steps : “Estimation for big norms”and “convergence for small norms”.

Estimation for big norms.

Proposition 6.2.1. Suppose that (u0, ℎ0) ∈ Hs+2(ℝ2) for s > 0 and ∥ℎ0∥Hs+2 ≤1

4Cs, then there exists a positive time T1 such that for any n ∈ ℕ, Pn is an applica-

tion from X ([0, T1], s + 2, E1, E2) to X ([0, T1], s + 2, E1, E2) for E1 = 2∥u0∥Hs+2 , E2 =2∥ℎ0∥Hs+2.

Proof : For convenience sake, we suppose that 1 ≤ E1 (the proof for E1 < 1 iseasy), and remark that 0 < E2 < 1, 0 < � < 1. We take now

T1 = min{(12

5K)−2,

�E22

16C20E

41

},

where K = ∥ℱ−1(')∥L1 . We prove the proposition by induction. Firstly, (u1, ℎ1) =S2(u0, ℎ0), then

∥u1∥Hs+2 ≤ ∥u0∥Hs+2 , ∥ℎ1∥Hs+2 ≤ ∥ℎ0∥Hs+2 ,

�∫ T1

0∥∇u1∥2

Hs+2d� ≤ �T1(125K)2∥u0∥2

Hs+2 ≤ ∥u0∥2Hs+2 .

Thus (u1, ℎ1) ∈ X ([0, T1], s+ 2, E1, E2).Now, we assume that (un, ℎn) ∈ X ([0, T1], s + 2, E1, E2) is valid, and prove that

Pn(un, ℎn) = (un+1, ℎn+1) ∈ X ([0, T1], s+ 2, E1, E2) is also valid.Applying the operator Δk to the equations (6.2.1), multiplying the first by Δkun+1,

and the second by Δkℎn+1, integration over ℝ2 yields

∂t∥Δkun+1∥2L2 + 2�∥∇Δkun+1∥2

L2 = 2∫ℝ2 ΔkG1(un, ℎn)Δkun+1dx,

∂t∥Δkℎn+1∥2L2 − 2

∫ℝ2 Δk(un∇ℎn+1)Δkℎn+1dx = 2

∫ℝ2 ΔkG2(un, ℎn)Δkℎn+1dx.

6.2. THE LOCAL EXISTENCE OF SOLUTION 81

By using Lemma 7.3.4, Lemma 6.1.2 (a) and hypotheses on (un, ℎn), we obtain

∂t∥Δkun+1∥2L2 + 2�∥Δk(∇un+1)∥2

L2 ≤ C0d2k2−2k(s+2)

×(∥ℎn∥Hs+2∥∇un+1∥Hs+2 + V1(t)(∥un+1∥Hs+2 + ∥∇un+1∥Hs+2)),(6.2.2)

∂t∥Δkℎn+1∥2L2 ≤ C0d

2k2−2k(s+2)

×(∥un∥Hs+2∥ℎn+1∥2Hs+2 + (1 + ∥ℎn∥Hs+2)∥∇un∥Hs+2∥ℎn+1∥Hs+2),(6.2.3)

where

V1(t) = ∥ℎn(t)∥Hs+2∥un(t)∥Hs+2(1 + ∥ℎn(t)∥Hs+2) + ∥un(t)∥2Hs+2 ≤

3

4E2

1 .

Multiplying both sides of (6.2.2) and (6.2.3) by 22k(s+2), and taking the sum over kgives respectively

∂t∥un+1∥2Hs+2 + �∥∇un+1∥2

Hs+2 ≤ ∥un+1∥2Hs+2 + 2C2

0E41�−1,

∂t∥ℎn+1∥2Hs+2 ≤ �E2

2

4E21∥∇un∥2

Hs+2 +5C2

0E21

�E22∥ℎn+1∥2

Hs+2 .

Taking integration from 0 to t yields

∥un+1(t)∥2Hs+2 + �

∫ t0et−�∥∇un+1(�)∥2

Hs+2d� ≤ ∥un+1(0)∥2Hs+2et + tet2C2

0E41�−1,

∥ℎn+1(t)∥2Hs+2 ≤ et5C

20E

21�−1E−2

2 (∥ℎn+1(0)∥2Hs+2 +

�E22

4E21

∫ t0∥∇un(t′)∥2

Hs+2dt′).

By the definition of (un+1, ℎn+1)∣t=0 we know that

∥un+1(0)∥Hs+2 ≤ ∥u0∥Hs+2 , ∥ℎn+1(0)∥Hs+2 ≤ ∥ℎ0∥Hs+2 .

Thus, the choose of T1 give that

∥un+1(t)∥2L∞([0,T1],Hs+2) + �∥∇un+1(�)∥2

L2([0,T1],Hs+2) ≤ E21

∥ℎn+1(t)∥2L∞([0,T1],Hs+2) ≤ E2

2 .

We have proved the proposition 6.2.1.

Convergence for small norm.

Proposition 6.2.2. Let (u0(x), ℎ0(x)) ∈ Hs+2(ℝ2) for s > 0 and ∥ℎ0∥Hs+2 ≤1

4Cs, then there exists a positive time T2(≤ T1) which independent of n, such that

{(un(x, t), ℎn(x, t))} is a Cauchy sequence in X ([0, T2], s + 1, E1, E2) if s ∕= 1, and inX ([0, T2], 2− ", E1, E2) for all 1 > " > 0 if s = 1.

Proof : From the equations in (6.2.1), we have

∂t(un+1 − un)− �Δ(un+1 − un) =∑6

j=1 Fj,(6.2.4)

∂t(ℎn+1 − ℎn) + un∇(ℎn+1 − ℎn) =∑3

j=1 Jj,(6.2.5)

where∑6j=1 Fj = 1

1+ℎn∇ℎn∇(un − un−1)

+ 11+ℎn∇(ℎn − ℎn−1)∇un−1 + ( 1

1+ℎn− 1

1+ℎn−1)∇ℎn−1∇un−1

−un∇(un − un−1)− (un − un−1)∇un−1 +∇(ℎn − ℎn−1),∑3j=1 Jj = (un − un−1)∇ℎn + (1 + ℎn)div(un − un−1) + (ℎn − ℎn−1)divun−1.


As in the proofs of proposition 6.2.1, applying the operator Δk to the equations (6.2.4)and (6.2.5), multiplying the first by Δk(un+1 − un), and the second by Δk(ℎn+1 − ℎn),then integrating over ℝ2, we obtain

∂t∥Δk(un+1 − un)∥2L2 + 2�∥Δk(un+1 − un)∥2

L2 =∑6

j=1

∫ℝ2 ΔkFjΔk(un+1 − un)dx,

∂t∥Δk(ℎn+1 − ℎn)∥2L2 =

∑3j=1

∫ℝ2 ΔkJjΔk(ℎn+1 − ℎn)dx.

Below we only consider the case of 0 < s < 1. By using Lemma 7.3.4, Lemma 6.1.2,Lemma 6.1.3 and the fact of ∥un(t)∥Hs+1 ≤ E1 and ∥ℎn(t)∥Hs+1 ≤ E2 when t ≤ T1, wehave that

∂t∥un+1 − un∥2Hs+1 + 2�∥∇(un+1 − un)∥2

Hs+1

≤ A0(∥un − un−1∥Hs+1 + ∥ℎn − ℎn−1∥Hs+1)×(∥un+1 − un∥Hs+1 + ∥∇(un+1 − un)∥Hs+1),

and

∂t∥ℎn+1 − ℎn∥2Hs+1 ≤ A0(∥ℎn+1 − ℎn∥2

Hs+1

+(∥un − un−1∥Hs+1 + ∥∇(un − un−1)∥Hs+1 + ∥ℎn − ℎn−1∥Hs+1)∥ℎn+1 − ℎn∥Hs+1),

where A0 is a constant, and A0 = O(E41E−22 ). By using Cauchy-Schwarz inequality, we

have that

(6.2.6)∂t∥un+1 − un∥2

Hs+1 + �∥∇(un+1 − un)∥2Hs+1 ≤ ∥un+1 − un∥2

Hs+1

+A2

0

�(∥un − un−1∥2

Hs+1 + ∥ℎn − ℎn−1∥2Hs+1),

(6.2.7)∂t∥ℎn+1 − ℎn∥2

Hs+1 ≤ A20∥ℎn+1 − ℎn∥2

Hs+1

+E2

2

4E21∥un − un−1∥2

Hs+1 +�E2

2

4E21∥∇(un − un−1)∥2

Hs+1 + ∥ℎn − ℎn−1∥2Hs+1 ,

where A20 = 4A2

0(1 +E2

1

�E22).

We prove now that there exist a positive time T2(≤ T1), such that for any n

∥un − un−1∥L∞([0,T2],Hs+1) + �∥∇(un − un−1)∥L2([0,T2],Hs+1) ≤ E12−n,∥ℎn − ℎn−1∥L∞([0,T2],Hs+1) ≤ E22−n.

(Cn)

We will prove (Cn) by induction on n. In fact, it is easy to see that (C1) is valid ifT2 ≤ T1.

We suppose now that (Cn) is holds and prove that (Cn+1) is valid by using theestimations (6.2.6) and (6.2.7). Taking integration from 0 to t on both side of (6.2.6),we deduce

∥(un+1 − un)(t)∥2Hs+1 + �

∫ t0et−t

′∥∇(un+1 − un)(t′)∥2Hs+1dt′

≤ et∥(un+1 − un)(0)∥2Hs+1 + tet

A20

�(E2

1 + E22)2−2n.

If T2 = min{T1, �(6A20)−1} and t ≤ T2, we have et ≤ 3/2, tet2

A20

�≤ 3/2, we have also

∥(un+1 − un)(0)∥Hs+1 ≤ 2−(n+1)∥Δn+1u0∥Hs+2 ≤ 1

2E12−(n+1).

Using (Cn), we obtain

(6.2.8) ∥(un+1 − un)∥2L∞([0,T2],Hs+1) + �∥∇(un+1 − un)∥2

L2([0,t2],Hs+1) ≤ E212−2(n+1).

6.3. THE GLOBAL EXISTENCE FOR SMALL INITIAL DATA 83

Taking integration from 0 to t on both side of (6.2.7), and using ((Cn), we deduce

∥(ℎn+1 − ℎn)(t)∥2Hs+1

≤ eA20t∥(ℎn+1 − ℎn)(0)∥2

Hs+1 + 2teA20tE2

22−2n.

Finally if T2 = min{T1, �(6A20)−1, A−2

0 } and t ≤ T2, we obtain

(6.2.9) ∥(ℎn+1 − ℎn)∥2L∞([0,T2],Hs+1) ≤ E2

22−2(n+1).

Proposition 6.2.2 is proved now with T2 = O(E−101 �2E6

2).

Regularity and uniqueness of solutions. From the Proposition 6.2.2, the ap-proximative sequence (un, ℎn) of problems (6.2.1) is a Cauchy sequence in X ([0, T2], s+1, E1, E2) with s > 0. So that the limiter (u, ℎ) is a solution of Cauchy problem (6.1.1).From the Proposition 6.2.1, this sequence is bounded in X ([0, T1], s+2, E1, E2), so thatit is also the Cauchy sequence in X ([0, T2], s′+2, E1, E2) for all s′ < s by interpolation,and the limiter is in X ([0, T2], s+2, E1, E2). So we have proved the existence of solutionfor Theorem 6.2.1.

The proofs of uniqueness of solution is similar with the proofs for the convergenceof approximative sequence. In fact, we consider

∂t(u− v)− �Δ(u− v) = G1(u, ℎ)−G1(v, g),(6.2.10)

∂t(ℎ− g)− u∇(ℎ− g) = (u− v)∇g +G2(u, ℎ)−G2(v, g),(6.2.11)

with initial data u(x, 0) = v(x, 0) = u0(x) ∈ Hs+2 and ℎ(x, 0) = g(x, 0) = ℎ0(x) ∈Hs+2.

As in the proofs of Proposition 6.2.2, we obtain that

(6.2.12)

∥(u− v)∥2L∞([0,T2],Hs+1) + �∥∇(u− v)∥2

L2([0,T2],Hs+1)

≤ 2∥(u− v)(0)∥2Hs+1 + 1

16(∥u− v∥2

L∞([0,T2],Hs+1)

+∥ℎ− g∥2L∞([0,T2],Hs+1) + �∥u− v∥2

L2([0,T2],Hs+1)),

and(6.2.13)

∥(g − ℎ)(t)∥2Hs+1 ≤ 2∥(ℎ− g)(0)∥2

Hs+1

+ 116

(∥u− v∥2L∞([0,T2],Hs+1) + ∥ℎ− g∥2

L∞([0,T2],Hs+1) + �∥u− v∥2L2([0,T2],Hs+1)).

Which give the uniqueness of solution.

6.3. The global existence for small initial data

We prove firstly a priori estimates for local solutions.

Theorem 6.3.1. (a priori estimate) Suppose that the problem (6.1.1) has a solution(u, ℎ) ∈ L∞([0, T ], Hs+1),∇u ∈ L2([0, T ], Hs+1(ℝ2)) for some T > 0 with initial datau0, ℎ0 ∈ Hs+1(ℝ2), s > 0, and

N(T ) = (∥u∥2L∞([0,T ];Hs+1) + ∥ℎ∥2

L∞([0,T ];Hs+1) + �∥∇u∥2L2([0,T ];Hs+1))

1/2 ≤ E0.


Then there exist positive constants " and C1 with "C1 ≤ E0, which are independent ofT such that, if N(T ) ≤ ", then

(6.3.1) N(T ) ≤ C1N(0).

A combination of local existence theorem 6.2.1 and above a priori estimate give thefollowing theorem.

Theorem 6.3.2. Suppose that u0, ℎ0 ∈ Hs+2(ℝ2), s > 0. Then there exists " > 0such that if

∥u0∥Hs+2 + ∥ℎ0∥Hs+2 ≤ ",

then the Cauchy problems (6.1.1) has a unique global solution

(u, ℎ) ∈ L∞([0,+∞[, Hs+2(ℝ2)),∇u ∈ L2([0,∞[, Hs+2(ℝ2)).

The proof of this theorem is standard.Remark: We get the global solution with index s+ 2, since we have only the local

solution with index s + 2 in theorem 6.2.1. But we have proved the a priori estimatefor small index s+ 1, so if we can get the local solution for s+ 1, we get also the globalsolution for small index s+ 1.

We prove now the theorem 6.3.1. We linearize the equations (6.1.1) on (ℎ, u) = (1, 0)as following

(6.3.2)

{ut − �△u+∇ℎ = H1,ℎt + divu = H2,

where {H1 = 1

1+ℎ∇ℎ∇u− (u ⋅ ∇)u,

H2 = −div(ℎu),

In the following, we will estimate (u, ℎ) under the a priori assumption

(6.3.3) N(T ) = ∥ℎ∥2L∞([0,T ],Hs+1) + ∥u∥2

L∞([0,T ],Hs+1) ≤ �0

where s > 0 and 0 < �0 << 1.Applying the operator Δk on (6.3.2), and multiplying first equation of (6.3.2) by

Δk(u−△u+�∇ℎ), second equation by Δk(ℎ−△ℎ), then summing them and integratingover ℝ2 yields

(6.3.4)

12∂t(∥uk∥2

H1 + ∥ℎk∥2H1) + �∥∇uk∥2

L2 + �∥△uk∥2L2 + �∥∇ℎk∥2

L2

=∫ℝ2(ΔkF1Δk(u−△u+ �∇ℎ) + ΔkF2Δk(ℎ−△ℎ))dx

−�∫ℝ2(∂tuk∇ℎk − �△uk∇ℎk)dx,

where 0 < � << 1, uk = △ku, ℎk = △kℎ.

High vertical frequencies estimates. Now we will give some estimates to theright hand of (6.3.4) for the case of high vertical frequencies. That means for certainM big enough, we study (6.3.4) for k > M . By lemma 6.1.4 we have

∣∫ℝ2 Δk(

11+ℎ∇ℎ∇u)ukdx∣

≤ C0d2k2−2ks∥u∥Hs(1 + ∥ℎ∥Hs+1)(∥△u∥2

Hs + ∥∇ℎ∥2Hs).(6.3.5)

6.3. THE GLOBAL EXISTENCE FOR SMALL INITIAL DATA 85

Similar we have

∣∫ℝ2 Δk(

11+ℎ∇ℎ∇u)△ukdx∣

≤ C0d2k2−2ks∥∇ℎ∥Hs(1 + ∥ℎ∥Hs+1)∥△u∥2

Hs ,(6.3.6)

and

(6.3.7)�∣∫ℝ2 Δk(

11+ℎ∇ℎ∇u)∇ℎkdx∣

≤ C0�d2k2−2ks∥∇ℎ∥Hs(1 + ∥ℎ∥Hs+1)(∥△u∥2

Hs + ∥∇ℎ∥2Hs).

We have also

∣∫ℝ2 Δk((u ⋅ ∇)u)△ukdx∣,

= (∣∫ℝ2 Δk(Tu∇u+ T∇uu)△uk +R(u,∇u)△ukdx∣

≤∑∣q−k∣≤N1

(∣∫ℝ2 Δk(Squ∇uq)△ukdx∣+ ∣

∫ℝ2 Δk(Sq(∇u)uq)△ukdx∣

+∑

q≥k−N2,j∈{−1,0,1} ∣∫ℝ2 Δk(uq∇uq−j)△ukdx∣

≤ C0d2k2−2ks(∥u∥L2∥△u∥2

Hs + ∥∇u∥L2∥∇u∥Hs∥△u∥Hs)≤ C0d

2k2−2ks∥u∥Hs+1∥∇u∥2

Hs+1 .

Similarly

�∣∫ℝ2 Δk((u ⋅ ∇)u)∇ℎkdx∣

≤ C0�d2k2−2ks∥u∥Hs+1(∥∇u∥2

Hs+1 + ∥∇ℎ∥2Hs).

Since ∥fq∥Hs ≤ ∥∇fq∥Hs for q ≥ 0, we can obtain that

∣∫ℝ2

Δk((u ⋅ ∇)u)ukdx∣ ≤ Cd2k2−2ks∥u∥Hs∥∇u∥2

Hs ,

and∣∫ℝ2 divΔk(uℎ)ℎkdx∣ = ∣

∫ℝ2 Δk(uℎ)∇ℎkdx∣

≤ Cd2k2−2ks(∥u∥L2 + ∥ℎ∥L2)(∥∇u∥2

Hs + ∥∇ℎ∥2Hs).

By lemma 6.1.5, we have

∣∫ℝ2

Δk(div(ℎu)△ℎkdx∣ ≤ Cd2k2−2ks∥∇ℎ∥Hs(∥∇ℎ∥2

Hs + ∥∇u∥2Hs+1).

It is easy to see

��∣∫ℝ2

△uk∇ℎkdx∣ ≤ C��d2k2−2ks("−1∥△u∥2

Hs + "∥∇ℎ∥2Hs).

Noting ∫ℝ2

(∂tuk)(∇ℎk)dx = ∂t

∫ℝ2

uk∇ℎkdx−∫ℝ2

uk∂t(∇ℎk)dx,

we have

�∣∫ℝ2 uk∂t(∇ℎk)dx∣

≤ C�d2k2−2ks(∥∇u∥2

Hs + (∥ℎ∥Hs+1 + ∥u∥Hs+1)(∥∇u∥2Hs + ∥∇ℎ∥2

Hs)).

and

�∣∫ t

0∂� (∫ℝ2 uk∇ℎkdx)d� ∣

≤ C�d2k2−2ks(∥u(t)∥Hs∥∇ℎ(t)∥Hs + ∥u(0)∥Hs∥∇ℎ(0)∥Hs).


Multiplying inequality (6.3.4) by 22ks and integrating over (0, t), we obtain(6.3.8)

∥uk(t)∥2Hs+1 + ∥ℎk(t)∥2

Hs+1 +∫ t

0(�∥∇uk(�)∥2

Hs+1 + �∥∇ℎk(�)∥2Hs)d�

≤ Cd2k(∥u(0)∥2

Hs+1 + ∥ℎ(0)∥2Hs+1) + C��d2

k

∫("−1∥∇u∥2

Hs+1 + "∥∇ℎ∥2Hs)d�

+Cd2k(∥ℎ∥L∞([0,T ],Hs+1) + ∥ℎ∥2

L∞([0,T ],Hs+1) + ∥u∥L∞([0,T ],Hs+1) + ∥u∥2L∞([0,T ],Hs+1))

×∫ t

0(∥∇u∥2

Hs+1 + ∥∇ℎ∥2Hs)d� + C�d2

k(∥u(t)∥2Hs + ∥ℎ(t)∥2

Hs+1).

Low vertical frequencies estimates. Now we will consider the low vertical fre-quencies. Denoting SM =

∑k<M Δk, and applying the operator SM on (6.3.2), and

multiplying first equation of (6.3.2) by Sk(u + �∇ℎ), second equation by Skℎ, thensumming them and integrating over ℝ2 yields

(6.3.9)

12∂t(∥SMu∥2

L2 + ∥SMℎ∥2L2) + �∥∇SMu∥2

L2 + �∥∇SMℎ∥2L2

=∫ℝ2(SM(F1)SM(u+ �∇ℎ) + SM(F2)SMℎ)dx−�∫ℝ2(∂tSMu∇SMℎ− �△SMu∇SMℎ)dx,

where 0 < � << 1. As in the proofs of (6.3.8), we will give some estimates to the righthand of (6.3.9). It is easy to see that

∣∫ℝ2

SM(∇ℎ

1 + ℎ∇u)SMudx∣ ≤ C∥ 1

1 + ℎ∥L∞∥∇ℎ∥L2∥u∥L∞∥∇u∥L2 ,

and

�∣∫ℝ2 SM( ∇ℎ

1+ℎ∇u− (u ⋅ ∇)u)SM(∇ℎ)dx∣

≤ C�(∥ 11+ℎ∥L∞∥∇ℎ∥2

L2∥∇u∥L∞ + ∥u∥L∞∥∇u∥L2∥∇ℎ∥L2).

We have also

∣∫ℝ2

SM((u ⋅ ∇)u)SMudx∣ ≤ C∥u∥2H1/2∥∇u∥L2 ≤ C∥u∥L2∥∇u∥2

L2 ,

and

∣∫ℝ2 SM(div(uℎ))SMℎdx∣ ≤ C∥u∥H1/2∥ℎ∥H1/2∥∇ℎ∥L2

≤ C(∥u∥L2 + ∥ℎ∥L2)(∥∇u∥2L2 + ∥∇ℎ∥2

L2).

Note that ∥SMΔu∥L2 ≤ 2M∥SM∇u∥L2 , we have

∣��∫ℝ2

SM(△u)SM(∇ℎ)dx∣ ≤ C��("−1∥∇u∥2L2 + "∥∇ℎ∥2

L2).

As in the above proofs, we have

�∣∫ℝ2 SMuSM(∂t(∇ℎ))dx∣ ≤ C�(∥ℎ∥L2 + ∥u∥L2)(∥∇u∥2

L2 + ∥∇ℎ∥2L2)

+C�("−1∥∇u∥2L2 + "∥∇ℎ∥2

L2),

and

�∣∫ t

0

∂t

∫ℝ2

SMuSM(∇ℎ)dxd� ∣ ≤ C�(∥u∥L2∥∇ℎ∥L2 + ∥u(0)∥L2∥∇ℎ(0)∥L2).

6.4. LOSING ENERGY ESTIMATES 87

Integrating both side of (6.3.9) over (0, t), and summing above estimates to the righthand of (6.3.9), we have

(∥SMu∥2L2 + ∥SMℎ∥2

L2) +∫ t

0(�∥∇SMu∥2

L2 + �∥∇SMℎ∥2L2)d�

≤ C(∥u∥L∞([0,T ],Hs+1) + ∥ℎ∥L∞([0,T ],Hs+1))∫ t

0(∥∇u∥2

Hs + ∥∇ℎ∥2Hs)d�

+C�(∥u(t)∥2L2 + ∥∇ℎ(t)∥2

L2) + C(∥u(0)∥2L2 + ∥∇ℎ(0)∥2

L2)

+C��∫ t

0("−1∥∇u∥2

L2 + "∥∇ℎ∥2L2)d�.

Since ∥SMf∥Hs ≤ 2Ms∥SMf∥L2 , we can write

(∥SMu∥2Hs+1 + ∥SMℎ∥2

Hs+1) +∫ t

0(∥∇SMu∥2

Hs+1 + �∥∇SMℎ∥2Hs)d�

≤ C(∥u∥L∞([0,T ],Hs+1) + ∥ℎ∥L∞([0,T ],Hs+1))∫ t

0(∥∇u∥2

Hs+1 + ∥∇ℎ∥2Hs)d�

+C�(∥u(t)∥2Hs + ∥∇ℎ(t)∥2

Hs) + C(∥u(0)∥2Hs + ∥∇ℎ(0)∥2

Hs)

+�∫ t

0("−1∥∇u∥2

Hs + "∥∇ℎ∥2Hs)d�.

Thus we obtain from (6.3.8) and above inequality

∥u(t)∥2Hs+1 + ∥ℎ(t)∥2

Hs+1 +∫ t

0(∥∇u(�)∥2

Hs+1 + �∥∇ℎ(�)∥2Hs)d�

≤ C(∥ℎ∥L∞([0,T ],Hs+1) + ∥u∥L∞([0,T ],Hs+1) + ∥ℎ∥2L∞([0,T ],Hs+1) + ∥u∥2

L∞([0,T ],Hs+1))

×∫ t

0(∥∇u∥2

Hs+1 + ∥∇ℎ∥2Hs)d� + C(∥u(0)∥2

Hs+1 + ∥ℎ(0)∥2Hs+1)

+C�(∥u(t)∥2Hs+1 + ∥ℎ(t)∥2

Hs+1) + C��∫ t

0("−1∥∇u∥2

Hs+1 + "∥∇ℎ∥2Hs)d�.

Taking " and � small enough, such that C" = 12

and C�"−1 = 14, we get

∥u(t)∥2Hs+1 + ∥ℎ(t)∥2

Hs+1 +∫ t

0(�∥∇u(�)∥2

Hs+1 + �∥∇ℎ(�)∥2Hs)d�

≤ C(∥ℎ∥L∞([0,T ],Hs+1) + ∥ℎ∥2L∞([0,T ],Hs+1) + ∥u∥L∞([0,T ],Hs+1) + ∥u∥2

L∞([0,T ],Hs+1))

×∫ t

0(�∥∇u∥2

Hs+1 + �∥∇ℎ∥2Hs)d� + C(∥u(0)∥2

Hs+1 + ∥ℎ(0)∥2Hs+1).

For fixed �, taking �0 small enough, such that C�0 <15

min{1, �}, we obtain

∥u(t)∥2Hs+1 + ∥ℎ(t)∥2

Hs+1 +∫ t

0(�∥∇u(�)∥2

Hs+1 + �∥∇ℎ(�)∥2Hs)d�

≤ C(∥u(0)∥2Hs+1 + ∥ℎ(0)∥2

Hs+1).

Which prove the theorem 6.3.1

6.4. Losing energy estimates

We prove now the losing energy estimates of the section 6.1, the proofs in thissection are technique.

Lemma 6.4.1. Let � > 1 and −1 ≤ k < +∞, then there exists C > 0 such that forall v,∇v, g,∇g ∈ H� , we have

∣∫ℝ2

Δk(v∇g)Δkgdx∣ ≤ Cd2k2−2k�∥v∥H�+1∥g∥2

H� ,

with {dk} ∈ ℓ2 and ∥{dk}∥ℓ2 ≤ 1.

Proof. By the paraproduct calculation, we have∫ℝ2 Δk(v∇g)Δkgdx =

∫ℝ2 Δk(T∇gv)Δkgdx+

∫ℝ2 Δk(Tv∇g)Δkgdx

+∫ℝ2 ΔkR(v,∇g)Δkgdx = I1 + I2 + I3.


Then there exists N1 > 0 such that for any fixed M > N1 and k > M ,

∣I1∣ ≤∑∣q−k∣≤N1

∥Sq(∇g)∥L∞∥Δqv∥L2∥Δkg∥L2

≤∑∣q−k∣≤N1

∥Sqg∥L∞∥Δq(∇v)∥L2∥Δkg∥L2 ≤ Cd2k2−2k�∥g∥2

H�∥v∥H�+1 .

Here we have used Sobolev inequality for ∥g∥L∞ since � > 1. For k ≤ M , by using∑∣q−k∣≤N1

∥Sq(∇g)∥L∞ ≤ C2M∥g∥L∞ , we can get the same results.For the term I2, since we want pass the operator ∇ from g to v, we rewrite

I2 =∑∣q−k∣≤N1

∫ℝ2 Δk(SqvΔq(∇g))Δkgdx

=∑∣q−k∣≤N1

(∫ℝ2 [Δk, Sqv]Δq(∇g)Δkgdx

+∫ℝ2(Sq − Sk)vΔkΔq(∇g)Δkgdx) +

∫ℝ2 SkvΔk(∇g)Δkgdx).

Note that the operator Δk are convolution operators in ℝ2, so

[Δk, Sqv]Δq(∇g) = 22k

∫ℝ2

(Sqv(x)− Sqv(y))f(2k(x− y))Δq(∇g)(y)dy,

where f(x) = (ℱ−1')(x). Using the fact ∣q − k∣ ≤ N1 and Hausdorff-Yang inequality,we have ∑

∣q−k∣≤N1∥[Δk, Sqv]Δq(∇g)∥L2

≤ C∑∣q−k∣≤N1

22k∥∇(Sqv)∥L∞2−k∥(2k⋅)f(2k⋅)∥L1∥Δq(∇g)∥L2

≤ Cdk2−k�∥∇v∥L∞∥g∥H� ,

Similar calculus for the other terms, we get

∣I2∣ ≤ Cd2k2−2k�∥∇v∥H�∥g∥2

H� .

Finally, for I3, there exists N1 > 0 such that

∣I3∣ ≤∑

q≥k−N2,j∈{−1,0,1} ∣∫ℝ2 Δk(ΔqvΔq−j(∇g))Δkgdx∣

≤ C∑

q≥k−N2∥Δqv∥L2∥Δq−j(∇g)∥L∞∥Δkg∥L2

≤ Cdk2−2k� (

∑q≥k−N2

dq2−(q−k)� )∥v∥H�+1∥g∥2

H� .

Denote d′k = (∑

q≥k−N2dq2−(q−k)� ), then {d′k} ∈ l2 since q > k and � > 1. For

convenience sake, we also denote d′k by dk below. Thus

∣I3∣ ≤ Cd2k2−2k�∥v∥H�+1∥g∥2

H� .

The lemma 6.4.1 is proved.

Lemma 6.4.2. (a) Let � > 2 and −1 ≤ k < +∞, then there exists C > 0 such thatfor all f, v, g, u,∇u ∈ H� , we have

∣∫ℝ2

Δk(∇f

1 + g∇v)Δkudx∣ ≤ Cd2

k2−2k�H0(g)∥f∥H�∥v∥H� (1 + ∥g∥H� )∥u∥H�+1 ,

with {dk} ∈ ℓ2, where

H0(g) = 1 + ∥(1 + g)−1∥L∞ + C0(∥g∥L∞),

(b) Let 1 < � < 2 and −1 ≤ k < +∞, then there exists C > 0 such that for allf, g, u,∇u, v,∇v ∈ H� , we have

∣∫ℝ2 Δk(

∇f1+g∇v)Δkudx∣

≤ Cd2k2−2k�H0(g)∥f∥H� (1 + ∥g∥H� )U� (u, v),


with {dk} ∈ ℓ2, where

U� (u, v) =: ∥∇v∥L∞∥u∥H�+1 + ∥∇v∥H� (∥∇u∥H1 + ∥u∥H� ).

Proof. (a) As the proof of Lemma 6.4.1, we have∫ℝ2

Δk(∇f

1 + g∇v)Δkudx =

∫ℝ2

Δk(T ∇f1+g∇v + T∇v

∇f1 + g

+R(∇f

1 + g,∇v))Δkudx.

For k > M , we have

∣∫ℝ2 Δk(T ∇f

1+g∇v)Δkudx∣ ≤

∑∣q−k∣≤N1

∥ ∇f1+g∥L∞∥Δq(∇v)∥L2∥Δku∥L2

≤∑∣q−k∣≤N1

∥ ∇f1+g∥L∞2q∥Δqv∥L22−k∥Δk∇u∥L2

≤ Cd2k2−2k�∥ 1

1+g∥L∞∥f∥H�∥v∥H�∥∇u∥H� .

For k ≤M , we have

∣∫ℝ2

Δk(T ∇f1+g∇u)Δkudx∣ ≤ Cd2

k2−2k�∥ 1

1 + g∥L∞∥f∥H�∥v∥H�∥u∥H� .

For the second term, we have

Δq(∇f

1 + g) = Δq(∇f)−Δq(T g

1+g∇f + T∇f

g

1 + g+R(∇f, g

1 + g)),

Thus we have

∣∫ℝ2 Δk(T∇v

∇f1+g

)Δkudx∣≤ Cd2

k2−2k�H0(g)∥∇v∥H�∥f∥H� (1 + ∥g∥H� )(∥∇u∥H� + ∥u∥H� ).

We have also

∣∫ℝ2 Δk(R(∇v, ∇f

1+g)Δkudx∣

≤∑

q≥k−N2,j∈{−1,0,1} ∣∫ℝ2 Δk(Δq(

∇f1+g

)Δq−j(∇v))Δkudx∣≤ Cd2

k2−2k�H0(g)∥∇v∥H�∥f∥H�∥u∥H�+1 .

The part (a) is proved.(b) Firstly, we write

∣∫ℝ2 Δk(

∇f1+g∇v)Δkudx∣ ≤ ∣

∫ℝ2 Δk(T ∇v

1+g∇f)Δkudx∣

+ ∣∫ℝ2 Δk(T∇f

∇v1+g

)Δkudx∣+ ∣∫ℝ2 ΔkR( ∇v

1+g,∇f))Δkudx∣,

The estimation for the first term and the third term is easy, so we discuss only thesecond term, we consider also two case, if k > M , we have

∣∫ℝ2

Δk(T∇f∇v

1 + g)Δkudx∣ ≤

∑∣q−k∣≤N1

∥Sq(∇f)∥L∞∥Δq(∇v

1 + g)∥L2∥Δku∥L2 .

Since 1 < � < 2, we have

∥Sq(∇f)∥L∞ ≤∑p≤q+2

22p∥Δpf∥L2 ≤ C2−q(�−2)∥f∥H� ,

and

∥ ∇v1 + g

∥H� ≤ ∥∇v∥H� (1 + ∥ g

1 + g∥H� ) ≤ ∥∇v∥H� (1 + ∥( 1

1 + g)2∥L∞∥g∥H� ),


one has

∣∫ℝ2

Δk(T∇f∇v

1 + g)Δkudx∣ ≤ Cd2

k2−2k�H0(g)(1 + ∥g∥H� )∥f∥H�∥∇v∥H�∥∇u∥H1 .

If k ≤M , it is easy to see that

∣∫ℝ2

Δk(T∇f∇v

1 + g)Δkudx∣ ≤ Cd2

k2−2k�H0(g)(1 + ∥g∥H� )∥f∥H�∥∇v∥H�∥u∥H� .

The lemma is proved.

Lemma 6.4.3. (a) Let � > 2 and −1 ≤ k < +∞, then there exists C > 0 such thatfor all f, v, u,∇u, g1, g2 ∈ H� , we have

∣∫ℝ2 Δk(

(g1−g2)(1+g1)(1+g2)

∇f∇v)Δkudx∣≤ Cd2

k2−2k�H1(g1, g2)∥f∥H�∥v∥H�∥g1 − g2∥H� (∥∇u∥H� + ∥u∥H� ),

with {dk} ∈ ℓ2, and

H1(g1, g2) = (1 + ∥(1 + g1)−1∥2L∞∥g1∥H� )(1 + ∥(1 + g2)−1∥2

L∞∥g2∥H� )+∥(1 + g1)−1∥2

L∞∥(1 + g2)−1∥2L∞ .

(b) Let 1 < � < 2 and −1 ≤ k < +∞, then there exists C > 0 such that for allf, v, g1, g2, u,∇u, v,∇v ∈ H� ,we have

∣∫ℝ2 Δk(

(g1−g2)(1+g1)(1+g2)

∇f∇v)Δkudx∣≤ Cd2

k2−2k�H1(g1, g2)∥f∥H�∥g1 − g2∥H�U� (u, v),

with {dk} ∈ ℓ2, and U� (u, v) as in Lemma 6.4.2, (b).

The proof of this lemma is similar to Lemma 6.4.2, just remark that if Fj =1

1+gj, Fj =

gj1+gj

(j = 1, 2), we have

F =g1 − g2

(1 + g1)(1 + g2)= (g1 − g2)F1F2 = (g1 − g2)(1− F1 − F2 + F1F2).

And we have the following estimates

∥F∥L∞ ≤ C∥g1 − g2∥L∞∥F1∥L∞∥F2∥L∞ ,∥ΔqF∥L2 ≤ Cd2

q2−2q�∥g1 − g2∥H� (1 + ∥F1∥2

L∞∥g1∥H� )(1 + ∥F2∥2L∞∥g2∥H� ).

Below we will consider the losing energy estimate for the case of high verticalfrequencies, i. e., k > M . Here, we assume that M > N1 +N2.

Lemma 6.4.4. Let � > 0 and M ≤ k < ∞, then there exists C > 0 such that forall g, u, v,∇g,∇u ∈ H� , we have

∣∫ℝ2 Δk(

11+ℎ∇ℎ∇u)Δkvdx∣

≤ Cd2k2−2k�H0(ℎ)(1 + ∥ℎ∥H�+1)∥△u∥H�∥∇ℎ∥H�∥v∥H� ,

with {dk} ∈ l2.

The proof of this lemma is similar with lemma 6.4.2 and the following lemma.


Lemma 6.4.5. Let � > 0 and M ≤ k < ∞, then there exists C > 0 such that forall g ∈ H�+1 and u ∈ H�+2, we have

∣∫ℝ2 Δk(div(ℎu))Δ(Δkℎ)dx∣

≤ Cd2k2−2k�∥∇ℎ∥H� (∥∇ℎ∥2

H� + ∥∇u∥2H�+1),

with {dk} ∈ l2.

Proof. Firstly we write

∣∫ℝ2 Δk(div(ℎu))Δ(Δkℎ)dx∣

≤ ∣∫ℝ2 Δk(∇ℎdivu)Δk(∇ℎ)dx∣+ ∣

∫ℝ2 Δk(ℎ∇(divu)Δk(∇ℎ)dx∣

+∣∫ℝ2 Δk(u∇ℎ)Δk(△ℎ)dx∣.

It is easy to estimate the first and the second terms by

Cd2k2−2k� (∥∇u∥L∞∥∇ℎ∥2

H� + ∥∇ℎ∥L2∥△u∥H�∥∇ℎ∥H� ),

for the third term, since we can’t control △ℎ, we first need to write∫ℝ2 Δk(Tu∇ℎ)Δk(△ℎ)dx =

∑∣q−k∣≤N1

∫ℝ2 Δk(Squ∇ℎq)Δk(△ℎ)dx

=∑∣q−k∣≤N1

∫ℝ2((SquΔkΔq(∇ℎ)) + [Δk, Squ]Δq(∇ℎ))Δk(△ℎ)dx

=∑∣q−k∣≤N1

∫ℝ2((Sq − Sk)uΔkΔq(∇ℎ) + [Δk, Squ]Δq(∇ℎ))Δk(△ℎ)dx

+∫ℝ2 Sku∇(Δkℎ)△(Δkℎ)dx = K1 +K2 +K3.

Since (Sq − Sk)u = −∑

q≤p≤k−1 Δpu,

K1 ≤ C∑

p ∥Δpu∥L∞∥∇Δkℎ∥L2∥△Δkℎ∥L2

≤ Cd2k2−2k�∥△u∥H�∥∇ℎ∥L2∥∇ℎ∥H� .

As the proof of lemma 6.4.1, we have

K2 ≤ Cd2k2−2k�∥∇u∥L∞∥∇ℎ∥2

H� .

Using the following calculus∫ℝ2

(Sku∇(Δkℎ))△(Δkℎ)dx =

∫ℝ2

(1

2div(Sku)∣∇(Δkℎ)∣2−

∑i,j

∂j(Skui)∂i(Δkℎ)∂j(Δkℎ))dx,

we get immediatelyK3 ≤ Cd2

k2−2k�∥∇u∥L∞∥∇ℎ∥2

H� .

The Lemma has been proved.

CHAPTER 7

Regularity of solutions

In this chapter, we study the regularity of solutions for linear and non linear partialdifferential equations. There are 2 notions about the regularity of solutions. First one isthe so-called hypo-ellipticity, the classical definition is due to L. Schwartz, for a linear(partial differential or pseudo-differential) operators P , we say that P is hypo-elliptic iffor any smooth function f , all weak solution (distribution sense) of equation Pu = f issmooth, where the smooth means C∞, analytic or Gevrey and in the interior of domain.This notion is come from the property of elliptic operators. We can naturally extendthis definition to the non linear equations. The second notion about the regularityof solutions is smoothing effect of Cauchy problem for evolution equation. Thatmeans, for the Cauchy problem, the initial date maybe not smooth, but the solution (ifexists) is smooth for any positive time. We consider those two notions in this chapter.

7.1. Hypo-ellipticity of Hormander’s operators

7.1.1. The Hormander condition. Let us consider a family of smooth vectorfields X = {Xj}0≤j≤m on an open domain Ω ⊂ ℝn, i. e.

Xj =n∑k=1

aj,k(x)∂xk , j = 0, ⋅ ⋅ ⋅ ,m

where the coefficients aj,k ∈ C∞(Ω). If X =∑n

k=1 ak(x)∂xk and Y =∑n

k=1 bk(x)∂xkare two vector fields (we use also the notation X(x) = (a1(x), ⋅ ⋅ ⋅ , an(x)) for a vectorfields), thanks to Leibnitz formula, the commutator is still a vector fields

[X, Y ] = X Y − Y X =n∑k=1

[X, Y ]k(x)∂xk

where for k = 1, ⋅ ⋅ ⋅ , n

[X, Y ]k(x) = Xbk(x)− Y ak(x) =n∑l=1

(al(x)(∂xlbk)(x)− bl(x)(∂xlak)(x)

).

For � = (�1, ⋅ ⋅ ⋅ , �p), 0 ≤ �j ≤ m, by iteration, we can define the commutators oforder ∣∣�∣∣ = p

[X�1 , ⋅ ⋅ ⋅ , [X�p−1 , X�p ] ⋅ ⋅ ⋅ ].

Definition 7.1.1. We say that the system X satisfy the Hormander condition onΩ, if for any x ∈ Ω, there exists rx such that

(7.1.1) Dim{

Span {[X�1 , ⋅ ⋅ ⋅ , [X�p−1 , X�p ] ⋅ ⋅ ⋅ ](x), ∣∣�∣∣ ≤ rx}}

= n .

93

94 7. REGULARITY OF SOLUTIONS

We say that the system X satisfy the Hormander condition of order p, if it satisfy theHormander condition on Ω and Supp {rx; x ∈ Ω} = p <∞. The following operator

H = X0 +m∑k=1

X2j

is called Hormander’s operator.

Remark that the condition (7.1.1) is equivalent to

(7.1.2) C−1∣�∣2 ≤∑∣∣�∣∣≤p

∣∣∣[X�1 , ⋅ ⋅ ⋅ , [X�p−1 , X�p ] ⋅ ⋅ ⋅ ](x) ⋅ �∣∣∣2 ≤ C∣�∣2

for all (x, �) ∈ Ω× ℝn.Here we give some examples :

∙ Laplace operator : △x =∑n

k=1 ∂2xk

, where X0 = 0, Xj = ∂xj , j = 1, ⋅ ⋅ ⋅ , nsatisfy the Hormander’s condition of order 1 on ℝn.∙ Heat operator : ∂t−△x, where we chose X0 = ∂t, Xj = ∂xj , j = 1, ⋅ ⋅ ⋅ , n, then

it satisfy the Hormander’s condition of order 1 on ℝt × ℝnx.

∙ Kolmogorov operator :

∂t + x ⋅ ∂y −△x

where we chose

X0 = ∂t + x ⋅ ∂y = ∂t +n∑j=1

xj∂yj , Xk = ∂xk , k = 1, ⋅ ⋅ ⋅ , n,

then this system satisfy the Hormander’s condition of order 2 on ℝt×ℝnx×ℝn

y .Since

[Xk, X0] = ∂yk , k = 1, ⋅ ⋅ ⋅ , n.

∙ Kohn-Laplace operator on the Heisenberg group ℍn ≈ ℝnx × ℝn

y × ℝs :

△ℍn =n∑j=1

{X2j + Y 2

j }

where

Xj = ∂xj + 2yj∂s, Yj = ∂yj − 2yj∂s,

then this system satisfy the Hormander’s condition of order 2 on ℝnx×ℝn

y×ℝs,where we have

[Xj, Yj] = 4∂s.

∙ A general model on ℝ3 : for k ≥ 1, X1 = ∂x1 + xk2∂x3 , X2 = ∂x2 , then thesystem {X1, X2} satisfy the Hormander’s condition of order k on ℝ3.

7.1. HYPO-ELLIPTICITY OF HORMANDER’S OPERATORS 95

7.1.2. Sub-elliptic estimates for the operator of sum of square of vectorfields. .

We study in this sub-section the following operators of sum of square vector fieldsm∑j=0

X2j .

The Hormander’s condition imply the so-called sub-elliptic estimates. Since weconsider only the local property, we suppose that the system of vector fields X is

defined on Ω such that Ω is a compact subset of Ω. By using a very power harmonicanalysis method (approximation by nilpotent groups), Rotshild-Stein [88] have provedthe following estimates.

Theorem 7.1.1. Assume that the system of vector fields X satisfy the Hormander’scondition of order p on Ω, then there exists a constant C such that

(7.1.3) ∥u∥2

H1p≤ C

{ m∑j=0

∥Xju∥2L2 + ∥u∥2

L2

},

for any u ∈ C∞0 (Ω).

Remark that (7.1.3) is equivalent to

(7.1.4) ∥u∥2

H2p≤ C

{∥

m∑j=0

X2j u∥2

L2 + ∥u∥2L2

},

for any u ∈ C∞0 (Ω).We give here a simple proof of (7.1.3) in the case of p = 2. We have firstly two

lemmas. The first one is just the pseudo-differential calculus.

Lemma 7.1.1. For any s, � ∈ ℝ and any smooth vector fields Z defined on Ω, wehave

∥[Λ�, Z]u∥Hs−� ≤ C∥u∥Hs


We have also

Lemma 7.1.2. Let X, Y two smooth vector fields defined on Ω, we have

∥[X, Y ]u∥2

H−12≤ C

{∥Xu∥2

L2 + ∥Y u∥2L2 + ∥u∥2

L2

}for any u ∈ C∞0 (Ω).

Proof. BY definition of commutators, we have

∥[X, Y ]u∥2

H−12 (ℝn)

=(

Λ−1[X, Y ]u, (XY − Y X)u)L2(ℝn)

=(X∗Λ−1[X, Y ]u, Y u

)L2(ℝn)

−(Y ∗Λ−1[X, Y ]u, Xu

)L2(ℝn)

=(

[X∗, Λ−1[X, Y ] ]u, Y u)L2(ℝn)

−(

[Y ∗, Λ−1[X, Y ] ]u, Xu)L2(ℝn)

+(

Λ−1[X, Y ]X∗u, Y u)L2(ℝn)

−(

Λ−1[X, Y ]Y ∗u, Xu)L2(ℝn)

.


By using X∗ = −X − divX and

[X∗, Λ−1[X, Y ], Λ−1[X, Y ] ∈ OpS0,

we obtain

∥[X, Y ]u∥2

H−12 (ℝn)

≤ C{∥u∥L2∥Y u∥L2 + ∥u∥L2∥Xu∥L2

+ ∥X∗u∥L2∥Y u∥L2 + ∥Y ∗u∥L2∥Xu∥L2

}which end the proof of Lemma. □

We can now prove the Theorem 7.1.1 for p = 2, by using (7.1.2), we have

∥u∥2

H12 (ℝn)

≤ C{∥u∥2

L2 +m∑j=0

∥Xju∥2

H−12

+∑∥[Xj, Xk]u∥2

H−12

}≤ C

{∥u∥2

L2 +m∑j=0

∥Xju∥2L2

},

which is (7.1.3).

We chose now u = �Λ1/2v, where v ∈ C∞0 (Ω) and � ∈ C∞0 (Ω), � = 1 on Ω. Thenby interpolations

∥v∥2H1(ℝn) ≤ C

{∥�Λ1/2v∥2

H12 (ℝn)

+ ∥(1− �)Λ1/2v∥2

H12 (ℝn)

}≤ C

{∥�Λ1/2v∥2

L2 +m∑j=0

∥Xj�Λ1/2v∥2L2 + ∥v∥2

L2

}≤ C

{∥�Λ1/2v∥2

L2 +m∑j=0

∥[Xj, �Λ1/2]v∥2L2 + ∥v∥2

L2

+(−

m∑j=0

X2j v, Λ1/2�2Λ1/2v

)L2

+m∑j=0

(Xjv, [Xj, Λ1/2�2Λ1/2]v

)L2

}≤ C"

{∥v∥2

L2 + ∥m∑j=0

X2j v∥2

L2 +m∑j=0

∥Xjv∥2L2

}+ "∥v∥2

H1 ,

where we used the fact: there exists � ∈ C∞0 (Ω), � = 1 on Supp v, then

(1− �)Λ1/2v = (1− �)Λ1/2� v,

and Supp (1− �) ∩ Supp � = ∅ imply

(1− �)Λ1/2� ∈ Op (S−∞).

We proved also (7.1.4) for p = 2.Similarly, we can get for any s ∈ ℝ,

(7.1.5) ∥[X, Y ]u∥2

Hs− 12≤ C

{∥Xu∥2

Hs + ∥Y u∥2Hs + ∥u∥2

Hs

},

7.1. HYPO-ELLIPTICITY OF HORMANDER’S OPERATORS 97

and

(7.1.6) ∥u∥2

Hs+ 1

p≤ C

{ m∑j=0

∥Xju∥2Hs + ∥u∥2

Hs

},


7.1.3. Sub-elliptic estimates for the Hormander’s operators. We considernow the Hormander operator H = X0 +

∑mj=1X

2j . We suppose in this subsection the

Hormander’s condition of order 2. But the sub-elliptic estimate keep true in generalcase.

Theorem 7.1.2. Assume that the system X satisfy the Hormander’s condition of

order 2 on Ω, then there exists a constant C such that

(7.1.7) ∥v∥2

H16≤ C

{∥H v∥2

L2 + ∥v∥2L2

}for any v ∈ C∞0 (Ω).

Proof. The condition (7.1.2) with p = 2 and (7.1.5) with s = −1/3 imply

∥v∥2

H16≤ C

{∥v∥2

L2 +m∑j=0

∥Xjv∥2

H−56

+∑

0≤j,k≤m

∥[Xj, Xk]v∥2

H−56

}≤ C

{∥v∥2

L2 +m∑j=1

∥Xjv∥2

H−13

+ ∥X0v∥2

H−56

+ +m∑k=1

∥[X0, Xk]v∥2

H−56

}.

Firstly (X0 v, v

)L2 =

(v, (−X0 − divX0) v

)L2 ,

we have

ℜe(X0 v, v

)L2 ≤ C∥v∥2

L2 .

We get immediately

m∑j=1

∥Xjv∥2

H−13≤

m∑j=1

∥Xjv∥2L2 ≤ C{ℜe

(−H v, v

)L2 + ∥v∥2

L2}.

On the other hand,

∥X0v∥2

H−56≤ ∥X0v∥2

H−23

=(X0 v, Λ−

43X0 v

)L2

= ℜe(H v, Λ−

43X0 v

)L2 −ℜe

( m∑j=1

X2j v, Λ−

43X0 v

)L2 .

Since

Λ−43X0 � ∈ Op(S−1/3),


we have ∣∣∣ m∑j=1

(X2j v, Λ−

43X0 v

)L2

∣∣∣ ≤ ∣∣∣ m∑j=1

(Xj v, Λ−

43X0X

∗j v)L2

∣∣∣+∣∣∣ m∑j=1

(Xj v, [X∗j , Λ−

43X0] v

)L2

∣∣∣≤ C

{ m∑j=1

∥Xj v∥2L2 + ∥v∥2

L2

}.

We study now the term

∥[X0, Xk]v∥2

H−56

=(

(X0Xk −XkX0) v, Λ−53 [X0, Xk]v

)L2

=(Xk v, Λ−

53 [X0, Xk]X

∗0 v)L2

+(Xk v, [X0 Λ−

53 [X0, Xk] ] v

)L2

−(X0 v, Λ−

53 [X0, Xk]X

∗k v)L2−(X0 v, [Xk Λ−

53 [X0, Xk] ] v

)L2

≤ C{∥Xkv∥2

L2 + ∥X0v∥2

H−23

+ ∥v∥2L2

},

here we use again

Λ−53 [X0, Xk]� ∈ Op(S−2/3).

Then we finish the proof by using the estimate

∥X0v∥2

H−23≤{∥Hv∥2

L2 + ∥v∥2L2

}.

□

By using the (7.1.7) we can prove the following hypoelliptic results (see [69, 102]).

Theorem 7.1.3. Let f ∈ C∞(Ω) and u ∈ D′(Ω) is a weak solution of equationHu = f , then u ∈ C∞(Ω).

7.2. Analytic smoothing effect of kinetic equations

7.2.1. Gevrey function spaces. We give now the definition of function spacesAs(Ω) where Ω is an open subset of ℝd.

Definition 7.2.1. For 0 < s < +∞, we say that f ∈ As(Ω), if f ∈ C∞(Ω), andthere exists C > 0, N0 > 0 such that

∥∂�f∥L2(Ω) ≤ C ∣�∣+1(�!)s, ∀ � ∈ ℕd, ∣�∣ ≥ N0.

If the boundary of Ω is smooth, by using Sobolev embedding theorem, we can replaceL2 norm by any Lp norm with 1 ≤ p ≤ +∞. On the whole space Ω = ℝd, it is alsoequivalent to

ec0(−Δ)12s (∂�0f) ∈ L2(ℝd)

for some c0 > 0 and �0 ∈ ℕd, where ec0(−Δ)12s is the Fourier multiplier defined by

ec0(−Δ)12s u(x) = ℱ−1

(ec0∣�∣

1s u(�)

).

7.2. ANALYTIC SMOOTHING EFFECT OF KINETIC EQUATIONS 99

∙ If s = 1, it is usual analytic function.∙ If s > 1, it is called Gevrey class function.∙ For 0 < s < 1, it is called ultra-analytic function.

Notice that all polynomial functions are ultra-analytic for any s > 0.

7.2.2. Ultra-analytic effect of Cauchy problem for heat equations. Weconsider the Cauchy problem for generalized heat equation (or say dissipative equation): {

∂tu− (−Δx) u = 0, x ∈ ℝd; t > 0

u∣t=0 = u0 ∈ L2(ℝd),

with 0 < < +∞, and

(−Δx) u = ℱ−1

(∣�∣2 u(�)

).

It is obvious that if u0 ∈ L2(ℝd) then, for any t > 0 , we have u(t, ⋅ ) = e−t(−Δx) u0 ∈A

12 (ℝd) is a solution of above Cauchy problem. So that if = k ∈ ℕ, there exists

C > 0 such that for any m ∈ ℕ,

∥(tm∂2kmx )u(t, ⋅ )∥L2(ℝd) ≤ Ckm∥(t(−Δx)

k)mu(t, ⋅ )∥L2(ℝd)

≤ ∥u0∥L2(ℝd)Ckmm !

≤ C2km+1((2km) !

) 12k ,

where ∂2kmx =

∑∣�∣=2km,�∈ℕd ∂

�x . We say that the diffusion operators ∂t − (−Δx)

possess

∙ the ultra-analytic effect property if > 1/2;∙ the analytic effect property if = 1/2;∙ the Gevrey effect property if 0 < < 1/2.

7.2.3. Kolmogorov operators. We study now the Cauchy problem of the fol-lowing generalized Kolmogorov operators

(7.2.1)

{∂tf + v ⋅ ∇xf + (−Δv)

� f = 0, (x, v) ∈ ℝ2d; t > 0f ∣t=0 = f0 ∈ L2(ℝ2d),

where 0 < � < ∞, the classical Kolmogorov operators is corresponding to � = 1.Denote by

f(t, �, �) = ℱx,v(f(t, x, v)),

the partial Fourier transformation of f with respect to (x, v) variable. Then, we have{∂∂tf(t, �, �)− � ⋅ ∇�f(t, �, �) = −∣�∣2�f(t, �, �), (�, �) ∈ ℝ2d, t > 0 ;

f ∣t=0 = ℱ(f0)(�, �).

Setting

w(t, �, �) = f(t, �, � − t�),

Then the above Cauchy problem is equivalent to

(7.2.2)

{∂∂tw(t, �, �) = −∣� − t�∣2�w(t, �, �), (�, �) ∈ ℝ2d, t > 0 ;

w∣t=0 = ℱ(f0)(�, �).


The explicit solution of above Cauchy problem is given by

w(t, �, �) = e−∫ t0 ∣�+s�∣

2�dsf0(�, � + t�).

Now we need to study the function∫ t

0∣� − s�∣2�ds, we prove the following estimate.

Lemma 7.2.1. For any � > 0, there exists a constant c� > 0 such that

(7.2.3)

∫ t

0

∣� − s�∣�ds ≥ c�(t∣�∣� + t�+1∣�∣�).

Remark : If � = 2, we can get above estimate by direct calculation. The followingsimple proof is due to Seiji Ukai.Proof : Setting s = t� and � = t�, we see that the estimate is equivalent to∫ 1

0

∣� − � �∣�d� ≥ c�(∣�∣� + ∣�∣�).

Since this is trivial when � = 0, we may assume � ∕= 0. If ∣�∣ < ∣�∣ then∫ 1

0

∣� − � �∣�d� ≥ ∣�∣�∫ 1

0

∣∣∣∣� − ∣�∣∣�∣∣∣∣∣� d�

= ∣�∣�{∫ ∣�∣/∣�∣

0

( ∣�∣∣�∣− �)�d� +

∫ 1

∣�∣/∣�∣

(� − ∣�∣∣�∣

)�d�

}

≥ ∣�∣�

� + 1min

0≤�≤1(��+1 + (1− �)�+1) =

∣�∣�

2�(� + 1)

≥ 1

2�+1(� + 1)(∣�∣� + ∣�∣�).

If ∣�∣ ≥ ∣�∣ then∫ 1

0

∣� − � �∣�d� ≥ ∣�∣�∫ 1

0

(1− � ∣�∣

∣�∣

)�d� ≥ ∣�∣�

∫ 1

0

(1− �

)�d�

=∣�∣�

� + 1≥ 1

2(� + 1)(∣�∣� + ∣�∣�).

Hence we obtain (7.3.43).So that we have

Theorem 7.2.1. Let f0 ∈ L2(ℝ2d) and f solution of Cauchy problem (7.2.1), then

ec�(t(−Δv)�+t2�+1(−Δx)�)f(t, ⋅, ⋅ ) ∈ L2(ℝ2d),

i. e. f(t, ⋅, ⋅ ) ∈ A1

(2�) (ℝ2d) for any t > 0.

Notice that this ultra-analytic (if � > 1/2) effect phenomenon is similar to heatequations of (x, v) variables. That is, this means v ⋅ ∇x + (−Δv)

� is equivalent to(−Δx)

� + (−Δv)� by time evolution in “some sense”, though the equation is only

transport for x variable.


We consider now a more complicate equation, the Cauchy problem for linear Fokker-Planck equation :

(7.2.4)

{ft + v ⋅ ∇xf = ∇v ⋅

(∇vf + vf

), (x, v) ∈ ℝ2d, t > 0 ;

f ∣t=0 = f0 .

This equation is a natural generalization of classical Kolmogorov equation, and a sim-plified model of inhomogeneous Landau equation (see [96]). The local property of thisequation is the same as classical Kolmogorov equation since the add terms ∇v ⋅ (vf)is a first order term, but for the studies of kinetic equation, v is velocity variable, andhence it is in full space ℝd

v. Then there occurs additional difficulty for analysis of thisequation.

The definition of weak solution in the function space L∞(]0, T [;L2(ℝ2dx, v))∩L1

1(ℝ2dx, v))

for the Cauchy problem is standard in the distribution sense, where for 1 ≤ p < +∞, l ∈ℝ

Lpl (ℝ2dx, v) =

{f ∈ S ′(ℝ2d); (1 + ∣v∣2)l/2f ∈ Lp(ℝ2d

x, v)}.

The existence of weak solution is similar to full Landau equation. We get also thefollowing ultra-analytic effect result.

Theorem 7.2.2. Let f0 ∈ L2(ℝ2dx, v) ∩ L1

1(ℝ2dx, v), 0 < T ≤ +∞. Assume that f ∈

L∞(]0, T [;L2(ℝ2dx, v) ∩L1

1(ℝ2dx, v)) is a weak solution of the Cauchy problem (7.2.4). Then,

for any 0 < t < T , we have

f(t, ⋅, ⋅ ) ∈ A1/2(ℝ2d).

Furthermore, for any 0 < T0 < T there exists c0 > 0 such that for any 0 < t ≤ T0, wehave

(7.2.5) ∥e−c0(t△v+t3△x)f(t, ⋅, ⋅ )∥L2(ℝ2d) ≤ ed2t∥f0∥L2(ℝ2d).

The ultra-analyticity results of above theorems are optimal for the smoothnessproperties of solutions. From these results, we obtain a good understanding for thehypoellipticity of kinetic equations (see [67]).

7.2.4. Ultra-analytic effect for spatial homogeneous Landau equations.We study the Cauchy problem for non linear Landau equation

(7.2.6)

{ft +

∑dj=1 vj ∂xjf = QL(f, f), (x, v) ∈ ℝd × ℝd, t > 0,

f ∣t=0 = f0,

where f(t, x, v) is the distribution of molecule depends on the time t ≥ 0, the spatialposition x ∈ ℝd and the velocity v ∈ ℝd, and

QL(f, g) = ∇v

(a(f) ⋅ ∇vg − b(f)g

),

is the Landau bilinear collision operators which is defined as follows : a(f) = (aij(f))and b(f) = (b1(f), ⋅ ⋅ ⋅ , bd(f) )

aij(f) = aij ★ f, bj(f) =d∑i=1

(∂viaij

)★ f , i, j = 1, ⋅ ⋅ ⋅ , d,


where the convolution is w. r. t. the variable v ∈ ℝd, and

aij(v) =

(�ij −

vivj∣v∣2

)∣v∣ +2, ∈ [−3, 1].

When = 0, we say it is the Maxwellian molecule case which is little easy for mathe-matical analysis.

For simplifier the problem, we suppose that the distribution of molecule is inde-pendent of the spatial position, say spatially homogeneous problem, then the problemis simplified to the following Cauchy problem for spatially homogeneous (non linear)Landau equation

(7.2.7)

{ft = Q(f, f) ≡ ∇v

(a(f) ⋅ ∇vf − b(f)f

), v ∈ ℝd, t > 0,

f ∣t=0 = f0,

We introduce also the notation, for l ∈ ℝ, Lpl (ℝd) = {f ; (1 + ∣v∣2)l/2f ∈ Lp(ℝd)} isthe weighted function space.

We prove the following ultra-analytic effect results for the non linear Cauchy prob-lem (7.2.7).

Theorem 7.2.3. Let f0 ∈ L2(ℝd)∩L1

2(ℝd) and 0 < T ≤ +∞. Suppose thatf ∈ L∞(]0, T [; L2(ℝd)

∩L1

2(ℝd)) and f(t, x) > 0 is a weak solution of the Cauchyproblem (7.2.7), then for any 0 < t < T , we have

f(t, ⋅ ) ∈ A1/2(ℝd),

and moreover, for any 0 < T0 < T , there exists c0 > 0 such that for any 0 < t ≤ T0

(7.2.8) ∥e−c0t△vf(t, ⋅ )∥L2(ℝd) ≤ ed2t∥f0∥L2(ℝd).

We refer to the works of C. Villani [96, 97] for the essential properties of homo-geneous Landau equations. We suppose the existence of weak solution f(t, v) > 0 inL∞(]0, T [;L1

2(ℝd)∩L2(ℝd)). The conservation of mass, momentum and energy reads,

d

dt

∫ℝdf(t, v)

⎛⎝ 1v∣v∣2

⎞⎠ dv ≡ 0.

Without loss of generality, we can suppose that∫ℝd f(t, v)dv = 1, unit mass∫

ℝd f(t, v)vjdv = 0, j = 1, ⋅ ⋅ ⋅ , d; zero mean velocity∫ℝd f(t, v)∣v∣2dv = T0, unit temperature∫

ℝd f(t, v)vjvkdv = Tj �jk,

d∑j

Tj = T0

Tj =∫ℝd f(t, v)v2

jdv > 0, j = 1, ⋅ ⋅ ⋅ , d; directional temperatures .


Then we have,

ajk(f) = �jk(∣v∣2 + T0 − Tj)− vjvk ;(7.2.9)

bj(f) = −vj ;(7.2.10)d∑j,k

ajk(f)�j�k ≥ C1∣�∣2 , ∀ (v, �) ∈ ℝ2d .(7.2.11)

where C1 = min1≤j≤d{T0 − Tj} > 0.Now for N > d

4+ 1 and 0 < � < 1/N, c0 > 0, t > 0 , set

G�(t, ∣�∣) =ec0t∣�∣

2

(1 + �ec0t∣�∣2) (1 + �c0t∣�∣2)N.

Since G�(t, ⋅ ) ∈ L∞(ℝd), we can use it as Fourier multiplier, denoted by

G�(t,Dv)f(t, v) = ℱ−1(G�(t, ∣�∣)f(t, �)

).

Then, for any t > 0,

G�(t) = G�(t,Dv) : L2(ℝd) → H2N(ℝd) ⊂ C2b (ℝd).

The object of this section is to prove the uniform bound (with respect to � > 0) of

∥G�(t,Dv)f(t, ⋅ )∥L2(ℝd).

Since f(t, ⋅ ) ∈ L2(ℝd)∩L1

2(ℝd) is a weak solution, we can take

G�(t)2f(t, ⋅ ) = G�(t,Dv)

2f(t, ⋅ ) ∈ H2N(ℝd),

as test function in the equation of (7.2.7), whence we have

1

2

d

dt∥G�(t)f(t, ⋅ )∥2

L2(ℝd) +d∑

j,k=1

∫ℝdaj k(f)

(∂vjG�(t)f(t, v)

)(∂vkG�(t)f(t, v)

)

=1

2

((∂tG�(t)

)f,G�(t)f

)L2(ℝd)

+d∑j=1

∫ℝd

(∂vj(vjf(t, v)

))G�(t)2f(t, v)

+d∑

j,k=1

∫ℝd

{ajk(f)

(G�(t)∂vjf(t, v)

)−G�(t)

(ajk(f)∂vjf(t, v)

)}×(∂vkG�(t)f(t, v)

).

To estimate the terms in the above equality, we prove the following two propositions.

Proposition 7.2.1. We have

∥∇vG�(t)f(t)∥2L2(ℝd)

≤ Cd∑

j,k=1

∫ℝdajk(f)

(∂vjG�(t,Dv)f(t, v)

)(∂vkG�(t,Dv)f(t, v)

).(7.2.12)

(7.2.13)∣∣∣((∂tG�(t)

)f, G�(t)f

)L2

∣∣∣ ≤ c0∥∇vG�(t)f(t)∥2L2 .


ℜed∑j=1

∫ℝd

(∂vj(vjf(t, v)

))G�(t)2f(t, v)(7.2.14)

≤ d

2∥G�(t)f(t)∥2

L2 + 2c0t∥∇vG�(t)f(t)∥2L2 .

Proof : The estimate (7.2.12) is exactly the elliptic condition (7.2.11). By using theFourier transformation, (7.2.13) is deduced from the following calculus

∂tG�(t, ∣�∣) = c0∣�∣2G�(t, ∣�∣)(

1

1 + �ec0t∣�∣2− N�

1 + �c0t∣�∣2

)= c0∣�∣2G�(t, ∣�∣) JN,�

where

∣JN,�∣ =∣∣∣∣ 1

1 + �ec0t∣�∣2− N�

1 + �c0t∣�∣2

∣∣∣∣ ≤ 1.

To treat (7.2.14), we use

(7.2.15) ∂�jG�(t, ∣�∣) = 2c0t�jG�(t, ∣�∣) JN,�.

Then, we have

ℜed∑j=1

∫ℝd

(∂vj(vjf(t, v)

))G�(t,Dv)2f(t, v)

= −ℜed∑j=1

∫ℝdvjG�(t,Dv)f(t, v)

(∂vjG�(t,Dv)f(t, v)

)

−ℜed∑j=1

∫ℝd

([G�(t,Dv), vj]f(t, v)

)(∂vjG�(t,Dv)f(t, v)

)

=d

2∥G�(t)f(t, ⋅)∥2

L2(ℝd) −ℜed∑j=1

∫ℝd



),

recall here [ ⋅, ⋅ ] is commutators.


Using Fourier transformation and (7.2.15), we have that for t > 0,

−d∑j=1

∫ℝ3



)dv

= −d∑j=1

∫ℝd

(G�(t,Dv)vjf(t, v)− vjG�(t,Dv)f(t, v)


)dv

=d∑j=1

∫ℝd

{i∂�j(G�(t, ∣�∣)f(t, �)

)−G�(t, ∣�∣)

(i∂�j f(t, �)

)}G�(t, ∣�∣) i�j f(t, �)d�

=d∑j=1

∫ℝ3

(∂�jG�(t, ∣�∣)

)f(t, �)�jG�(t, ∣�∣)f(t, �)d�

= 2c0t

∫ℝd∣�∣2∣G�(t, ∣�∣)f(t, �)∣2JN,� d� ≤ 2c0t

∫ℝd∣�∣2∣G�(t, ∣�∣)f(t, �)∣2d� ,

which give (7.2.14). The proof of Proposition 7.3.2 is now complete.

For the commutator term, the special structure of the operator implies

Proposition 7.2.2.d∑

j,k=1

∫ℝd

{ajk(f)

(G�(t,Dv)∂vjf(t, v)

)−G�(t,Dv)

(ajk(f)∂vjf(t, v)

)}×(∂vkG�(t,Dv)f(t, v)

)dv = 0.

Proof. We introduce now polar coordinates on ℝd� by setting r = ∣�∣ and ! =

�/∣�∣ ∈ Sd−1. Note that ∂/∂�j = !j∂/∂r + r−1Ωj where Ωj is a vector field on Sd−1,and (see [67], Proposition 14.7.1)

(7.2.16)d∑j=1

!jΩj = 0,d∑j=1

Ωj!j = d− 1 .

By using Fourier transformation, we have

−d∑

j,k=1

∫ℝd

{ajk(f)

(G�(t,Dv)∂vjf(t, v)

)−G�(t,Dv)

(ajk(f)∂vjf(t, v)

)}×(∂vkG�(t,Dv)f(t, v)

)=

∫ℝd

{ d∑j,k=1

�k

[ (�jkΔ� − ∂�k∂�j

), G�(t, ∣�∣)

]�j f(t, �))

}×G�(t, ∣�∣)f(t, �)d�.

Noting, in polar coordinates on ℝd� ,

Δ� =∂2

∂r2+d− 1

r

∂

∂r+

1

r2

d∑j=1

Ω2j ,


we have, denoting by G(r2) = G�(t, r),

d∑j,k=1

!k

[ (�jk

{ ∂2

∂r2+d− 1

r

∂

∂r

}−{

(!k∂/∂r + r−1Ωk)(!j∂/∂r + r−1Ωj)}), G(r2)

]!j

=[ ∂2

∂r2+d− 1

r

∂

∂r, G(r2)

]−[( d∑

k=1

(!2k∂/∂r+r

−1!kΩk)d∑j=1

(!2j∂/∂r+r

−1Ωj!j)), G(r2)

]

=[ ∂2

∂r2+d− 1

r

∂

∂r, G(r2)

]−[ ∂2

∂r2+

∂

∂r

d− 1

r, G(r2)

]= 0,

where we have used (7.2.16). Then we finish the proof of Proposition 7.2.2. □

In the above proof of Proposition 7.2.2, we have used the polar coordinates in thedual variable of v, which is essentially related to a form of the Landau operator withMaxwellian molecules. We notice that the same relation (in v variable) was describedby Villani [96] and Desvillettes-Villani [51, 52].

End of proof of Theorem 7.2.3 :

From Propositions 7.3.2 and 7.2.2, we get

1

2

d

dt∥G�(t)f(t, ⋅ )∥2

L2(ℝd) + (C1 −1

2c0 − 2c0t)∥∇vG�(t)f(t, ⋅ )∥2

L2(ℝd)

≤ d

2∥G�(t)f(t, ⋅ )∥2

L2(ℝd).

For any 0 < T0 < T , choose c0 small enough such that C1 − 12c0 − 2c0T0 ≥ 0. Then we

get

(7.2.17)d

dt∥G�(t)f(t, ⋅ )∥L2(ℝd) ≤

d

2∥G�(t)f(t, ⋅ )∥L2(ℝd).

Integrating the inequality (7.2.17) on ]0, t[, we obtain

(7.2.18) ∥G�(t)f(t, ⋅ )∥L2(ℝd) ≤ ed2t∥f0∥L2(ℝd).

Take limit � → 0 in (7.2.18). Then we get

(7.2.19) ∥e−c0t△vf(t, ⋅ )∥L2(ℝd) ≤ ed2t∥f0∥L2(ℝd)

for any 0 < t ≤ T0. We have now proved f(t, ⋅ ) ∈ A 12 (ℝd) and Theorem 7.2.3.

7.3. Gevrey effect for Kac’s equations

In this section, we consider the following Cauchy problem for spatially homogeneousnon linear Kac’s equation,

(7.3.1)

{∂f∂t

= K(f, f), v ∈ ℝ, t > 0,f ∣t=0 = f0 .

7.3. GEVREY EFFECT FOR KAC’S EQUATIONS 107

where f = f(t, v) is the nonnegative density distribution function of particles withvelocity v ∈ ℝ at time t. The right hand side of equation (7.3.1) is given by Kac’sbilinear collisional operator

K(f, g) =

∫ℝ

∫ �/2

−�/2�(�) {f(v′∗)g(v′)− f(v∗)g(v)} d�dv∗ ,

where

v′ = v cos � − v∗ sin �, v′∗ = v sin � + v∗ cos �.

We suppose that the cross-section kernel is non cut-off. To simplify the notations, wesuppose (see [49] for the precise description of cross-section kernel) that

(7.3.2) �(�) = C0∣ cos �∣∣ sin �∣1+2s

, −�2≤ � ≤ �

2,

where 0 < s < 1 and C0 > 0, then∫ �/2

−�/2�(�)d� = +∞,

and

(7.3.3)

⎧⎨⎩∫ �/2

−�/2�(�) ∣�∣ d� = Cs < +∞, 0 < s < 1/2,∫ �/2

−�/2�(�) �2 d� = Cs < +∞, 0 < s < 1.

Hereafter, use the following function spaces: For 1 ≤ p ≤ +∞, ℓ ∈ ℝ,

Lpℓ(ℝ) ={f ; ∥f∥Lpℓ =

(∫ℝ∣⟨v⟩ℓf(v)∣pdv

)1/p

< +∞}

where ⟨v⟩ = (1 + ∣v∣2)1/2.

L logL(ℝ) ={f ; ∥f∥L logL =

∫ℝ∣f(v)∣ log(1 + ∣f(v)∣)dv < +∞

}.

For k, ℓ ∈ ℝ,

Hkℓ (ℝ) =

{f ∈ S ′(ℝ); ⟨v⟩ℓf ∈ Hk(ℝ)

}.

We assume that the initial datum f0 ≡/ 0 satisfies the natural boundedness on themass, energy and entropy, that is,

(7.3.4) f0 ≥ 0,

∫ℝf0(v)

(1 + ∣v∣2 + log(1 + f0(v))

)dv < +∞.

In [44], L. Desvillettes has proved the existence of a nonnegative weak solution tothe Cauchy problem (7.3.1),

(7.3.5) f ∈ L∞([0,+∞[;L1k(ℝ)) ,

if f0 ∈ L1k(ℝ) for some k ≥ 2. The weak solution satisfies the conservation of mass

(7.3.6)

∫ℝf(t, v)dv =

∫ℝf0(v)dv, ∀t > 0,


the conservation of energy

(7.3.7)

∫ℝf(t, v)∣v∣2dv =

∫ℝf0(v)∣v∣2dv, ∀t > 0,

and also the entropy inequality

(7.3.8)

∫ℝf(t, v) log f(t, v)dv ≤

∫ℝf0(v) log f0(v)dv, ∀t > 0,

but does not conserve the momentum.L. Desvillettes proved also in [44] (see also [61]), the C∞-regularity of weak solutions

if f0 ∈ L1ℓ(ℝ) for any ℓ ∈ ℕ. This regularizing effect properties is now well-known for

non cut-off homogeneous Boltzmann equations (see also [4, 5, 50, 86]).Our result on the Gevrey regularity can be stated as follows.

Theorem 7.3.1. Assume that the initial datum f0 ∈ L12+2s ∩ L logL(ℝ), and the

cross-section � satisfy (7.3.2) with 0 < s < 12. For T0 > 0, if f ∈ L∞([0, T0];L1

2+2s ∩L logL(ℝ)) is a nonnegative weak solution of the Cauchy problem (7.3.1), then for any0 < s′ < s, there exists 0 < T∗ ≤ T0 such that

f(t, ⋅) ∈ G1

2s′ (ℝ)

for any 0 < t ≤ T∗.

Recall that Kac’s equation is obtained when one considers radially symmetric so-lutions of the spatially homogeneous Boltzmann equation for Maxwellian molecules(see [44]). The Cauchy problem for the spatially homogeneous Boltzmann equation isdefined by :

(7.3.9)∂g

∂t= Q(g, g), v ∈ ℝ3, t > 0 ; g∣t=0 = g0 ,

where the Boltzmann collision operator Q(g, f) is a bi-linear functional given by

(7.3.10) Q(g, f) =

∫ℝ3

∫S2

B (v − v∗, �) {g(v′∗)f(v′)− g(v∗)f(v)} d�dv∗ ,

for � ∈ S2 and where

v′ =v + v∗

2+∣v − v∗∣

2�, v′∗ =

v + v∗2− ∣v − v∗∣

2� .

The non-negative function B(z, �) called the Boltzmann collision kernel depends onlyon ∣z∣ and the scalar product < z

∣z∣ , � >. In most of the cases, the collision kernel B can

not be expressed explicitly. However, to capture its main property, it can be assumedto be in the form

B(∣v − v∗∣, cos �) = Φ(∣v − v∗∣)b(cos �), cos � =⟨ v − v∗∣v − v∗∣

, �⟩, −�

2≤ � ≤ �

2.

The Maxwellian case corresponds to Φ ≡ 1. Except for hard sphere model, the functionb(cos �) has a singularity at � = 0. We assume that

(7.3.11) sin � b(cos �) ≈ K�−1−2s when � → 0,

where K > 0, 0 < s < 1. Remark that the solution of Boltzmann equation satisfiesalso the conservation of mass, energy and the entropy inequality.


A function g is radially symmetric with respect to v ∈ ℝ3, if it satisfy the property

g(t, v) = g(t, Av), v ∈ ℝ3

for any rotation A in ℝ3. We proved the following results.

Theorem 7.3.2. Assume that the initial datum g0 ∈ L12+2s ∩ L logL(ℝ3), g0 ≥ 0

is radially symmetric. Let Φ ≡ 1 and let b satisfy (7.3.11) with 0 < s < 12. If g is a

nonnegative radially symmetric weak solution of the Cauchy problem (7.3.9) such thatg ∈ L∞(]0,+∞[;L1

2+2s ∩ L logL(ℝ3)) , then

g(t, ⋅ ) ∈ G1

2s′ (ℝ3v)

for any t > 0 and any 0 < s′ < s.

Remark that for the non cut-off spatially homogeneous Boltzmann equation, wehave the H∞-regularizing effect of weak solutions (see also [50, 86, 5]). Namely if f isa weak solution of the Cauchy problem (7.3.9) and the cross section b satisfy (7.3.11),then we have f(t, ⋅) ∈ H+∞(ℝ) for any 0 < t.

7.3.1. Fourier analysis of Kac’s operators. We will now be interested in study-ing the Fourier analysis of the Kac’s collision operator. This is a key step in the regu-larity analysis of weak solutions. For simplification of notations, we use (⋅ , ⋅) insteadof (⋅ , ⋅)L2(ℝv). We have firstly the following coercivity estimate deduced from the noncut-off of collision kernel.

Proposition 7.3.1. Assume that the cross-section is non cut-off, satisfies the as-sumption (7.3.2). Let f ≥ 0, f ∕= 0, f ∈ L1

1(ℝ)∩L logL(ℝ), then there exists a constantcf > 0, depending only on �, ∥f∥L1

1, and ∥f∥LLogL, such that

(7.3.12) −(K(f, g), g

)≥ cf∥g∥2

Hs(ℝv) − C∥f∥L1∥g∥2L2

for any smooth function g ∈ H1(ℝ).

Recall the following weak formulation for collision operators(K(f, g), ℎ

)=

∫∫ℝ2

∫ �2

−�2

�(�)f(v∗)g(v)(ℎ(v′)− ℎ(v)

)d�dv∗dv,

for suitable functions f, g, ℎ with reals values. Then(−K(f, g), g

)=

1

2

∫∫ℝ2

∫ �2

−�2

�(�)f(v∗)(g(v′)− g(v)

)2

d�dv∗dv

−1

2

∫∫ℝ2

∫ �2

−�2

�(�)f(v∗)(g(v′)2 − g(v)2

)d�dv∗dv .


The second term of right hand side can be estimated by using the Cancellation lemmaof [2]. But in the Maxwellien case, by an appropriate change of variable, we then have,∣∣∣∣∣12

∫ℝ2

∫∫ �2

−�2

�(�)f(v∗)(g(v′)2 − g(v)2

)d�dv∗dv

∣∣∣∣∣=

∣∣∣∣∣12∫∫

ℝ2

∫ �2

−�2

�(�)f(v∗)g(v)2( 1

cos �− 1)d�dv∗dv

∣∣∣∣∣≤ C

∫∫ℝ2

∫ �2

−�2

∣∣ sin(�)∣∣−1−2s

∣∣∣ sin(�2

)∣∣∣2∣f(v∗)∣g(v)2d�dv∗dv

≤ C∥f∥L1∥g∥2L2 .

The coercivity term in Hs is deduced from the following positive term,

1

2

∫ℝ2

∫ �2

−�2

�(�)f(v∗)(g(v′)− g(v)

)2

d�dv∗dv.

Here we need the Bobylev formula, i. e. the Fourier transform of collision operators :

(7.3.13) ℱ(K(f, g)

)(�) =

1

2�

∫ �2

−�2

�(�){f(� sin �)g(� cos �)− f(0)g(�)

}d� ,

for suitable functions f and g and by using both properties (1) and (2) and the unifomintegrability of ft. (see [2, 5, 86]). From the above formula, we can get also thefollowing upper bound estimates (see [86]). For m, ℓ ∈ ℝ, and for suitable functionsf, g, we have

(7.3.14) ∥K(f, g)∥Hmℓ (ℝv) ≤ C∥f∥L1

ℓ++2s(ℝv)∥g∥Hm+2s

(ℓ+2s)+(ℝv) ,

where �+ = max{�, 0}.

To study the Gevrey regularity of the weak solution, as in [86, 83], we consider theexponential type mollifier. For 0 < � < 1, c0 > 0 and 0 < s′ < s, we set

G�(t, �) =ec0 t ⟨�⟩

2s′

1 + �ec0 t ⟨�⟩2s′

where⟨�⟩ = (1 + ∣�∣2)

12 , � ∈ ℝ.

Then, for any 0 < � < 1,

(7.3.15) G�(t, �) ∈ L∞(]0, T [×ℝ),

and

(7.3.16) lim�→0

G�(t, �) = ec0 t ⟨�⟩2s′

.

Denote by G�(t, Dv), the Fourier multiplier of symbol G�(t, �),

G� g(t, v) = G�(t, Dv)g(t, v) = ℱ−1�→ v

(G�(t, �)g(t, �)

).

Then our aim is to prove the uniform boundedness (with respect to 0 < � < 1) of theterm ∥G�(t, Dv)f(t, ⋅)∥L2(ℝ) for the weak solution of the Cauchy problem (7.3.1). In


what follows, we will use the same notation G� for the pseudo-differential operatorsG�(t,Dv) and also its symbol G�(t, �).

Lemma 7.3.1. Let T > 0, c0 > 0, We have that for any 0 < � < 1 and 0 ≤ t ≤T, � ∈ ℝ,

∣∂tG�(t, �)∣ ≤ c0⟨�⟩2s′G�(t, �),

∣∂�G�(t, �)∣ ≤ 2s′ c0 t ⟨�⟩2s′−1G�(t, �)

and ∣∣∂2�G�(t, �)

∣∣ ≤ C⟨�⟩2(2s′−1)G�(t, �)

with C > 0 independent of �.

In fact, we have the following formulas

(7.3.17) ∂tG�(t, �) = c0⟨�⟩2s′G�(t, �)

1

1 + �ec0t⟨�⟩2s′ ,

(7.3.18) ∂�G�(t, �) = 2s′ c0 t (1 + ∣�∣2)s′−1� G�(t, �)

1

1 + �ec0t⟨�⟩2s′ ,

and

∂2�G�(t, �) =

(2s′ c0 t (1 + ∣�∣2)s

′−1�)2

G�(t, �)1− �ec0t⟨�⟩2s

′(1 + �ec0t⟨�⟩2s

′)2

+ 2s′ c0 t(

(1 + ∣�∣2)s′−1 + 2(s′ − 1)�2(1 + ∣�∣2)s

′−2)G�(t, �)

1

1 + �ec0t⟨�⟩2s′ .

(7.3.19)

Lemma 7.3.2. There exists C > 0 such that for all 0 < � < 1 and � ∈ ℝ, we have,

(7.3.20) ∣G�(�)−G�(� cos �)∣ ≤ C sin2(�/2)⟨�⟩2s′G�(� cos �)G�(� sin �),

and

(7.3.21)∣∣(∂�G�

)(�)−

(∂�G�

)(� cos �)

∣∣ ≤ C sin2(�/2)⟨�⟩(4s′−1)+

G�(� cos �)G�(� sin �),

where (4s′ − 1)+ = max{4s′ − 1, 0}.

Proof. For the estimate (7.3.20), we have, by using the Taylor formula

G�(�)−G�(� cos �) =(� − � cos �

) ∫ 1

0

(∂�G�

)(� cos � + �(� − � cos �))d�

where �� = � cos � + �(� − � cos �). Then (7.2.15) implies

∣Gt, �(�)−G�(t, � cos �)∣ ≤ 4s′ c0 t ∣�∣ sin2(�/2)

∫ 1

0

G�(t, �� )⟨�� ⟩2s′−1d� .

For 0 ≤ � ≤ 1 and −�/4 ≤ � ≤ �/4,√

2

2∣�∣ ≤ ∣�� ∣ = ∣� cos � + �(� − � cos �)∣ ≤ ∣�∣,

which implies, for 0 < 2s′ < 1, that there exists Cs′ > 0 such that

⟨�� ⟩2s′ ≤ ⟨�⟩2s′ , ⟨�� ⟩2s

′−1 ≤ Cs⟨�⟩2s′−1.


On the other hand, G�(t, �) = G�(t, ∣�∣) is increasing with respect to ∣�∣, since for � > 0,∂�G�(t, �) > 0, then

G�(t, �� ) ≤ G�(t, �).

By using

∣�∣2 = ∣� cos �∣2 + ∣� sin �∣2,and

(1 + a+ b)2s′ ≤ (1 + a)2s′ + (1 + b)2s′ , (1 + �e�)(1 + �e�) ≤ 3(1 + �e�+�),

we get

(7.3.22) G�(�) ≤ 3G�(� cos �)G�(� sin �).

Thus

∣G�(�)−G�(� cos �)∣ ≤ C sin2(�/2)⟨�⟩2s′G�(� cos �)G�(� sin �).

We have proved the estimate (7.3.20) when ∣�∣ ≤ �/4. If �/4 ≤ ∣�∣ ≤ �/2, we have

∣G�(�)−G�(� cos �)∣ ≤ ∣G�(�)∣+ ∣G�(� cos �)∣ ≤ 2∣G�(�)∣≤ 6 G�(� cos �)G�(� sin �) ≤ C sin2(�/2)G�(� cos �)G�(� sin �).

For the estimate (7.3.21), by using (7.3.19), we have that if ∣�∣ ≤ �/4,∣∣(∂�G�

)(�)−

(∂�G�

)(� cos �)

∣∣ =

∣∣∣∣(� − � cos �)

∫ 1

0

(∂2�G�

)(�� )d�

∣∣∣∣≤ C∣�∣ sin2(�/2) ⟨�⟩2(2s′−1)

∫ 1

0

G�(�� )d�

≤ C sin2(�/2) ⟨�⟩4s′−1G�(� sin �)G�(t, � cos �).

The case �/4 ≤ ∣�∣ ≤ �/2 is similar to (7.3.20). Thus, we have proved Lemma 7.3.2. □

We now study the commutators of Kac’s collision operators with the above mollifieroperators.

Proposition 7.3.2. Assume that 0 < s′ < 1/2, Let f, g ∈ L21(ℝv) and ℎ ∈ Hs′(ℝv),

then we have that ∣∣∣(G�K(f, g), ℎ)−(K(f, G� g), ℎ

)∣∣∣≤ C ∥G� f∥L2

1(ℝ) ∥G� g∥Hs′ (ℝ)∥ℎ∥Hs′ (ℝ),(7.3.23)

and ∣∣∣((v G�)K(f, g), ℎ)−(K(f, (v G�) g), ℎ

)∣∣∣≤ C

(∥f∥L1

1(ℝ) + ∥G� f∥L21(ℝ)

)∥G� g∥Hs′

1 (ℝ)∥ℎ∥Hs′ (ℝ).(7.3.24)

Proof. By definition, we have, for a suitable function F ,

(7.3.25) ℱ(G� F )(�) = G�(t, �)F (�),

and

(7.3.26) ℱ((v G�)F )(�) = i∂�(G� F

)(�) = i(∂�G�)(�) F (�) + iG�(�) (∂�F )(�).


By using the Bobylev formula (7.3.13) and the Plancherel formula,

(2�)1/2{(G�K(f, g), ℎ

)−(K(f, G� g), ℎ

)}=

∫ℝ�

∫ �2

−�2

�(�)G�(�){f(� sin �)g(� cos �)− f(0)g(�)

}d� ℎ(�)d�

−∫ℝ�

∫ �2

−�2

�(�){f(� sin �)

(ℱ(G� g)

)(� cos �)− f(0)

(ℱ(G� g)

)(�)}d� ℎ(�)d�

=

∫ℝ�

∫ �2

−�2

�(�)f(� sin �){G�(�)−G�(� cos �)

}g(� cos �) ℎ(�) d� d� .

The above formula can be justified by the cutoff approximation of collision kernel �(�),then (7.3.20) and (7.3.3) imply

∣∣∣(G�K(f, g), ℎ)−(K(f, G� g), ℎ

)∣∣∣≤C

∫ℝ�

∫ �2

−�2

�(�) sin2(�/2)∣G�(� sin �) f(� sin �)∣

× ∣G�(� cos �) g(� cos �)∣ ⟨�⟩2s′ ∣ℎ(�)∣ d� d�

≤C∥∣G� f∥L∞(ℝ�)

∫ �2

−�2

�(�) sin2(�/2)

×

(∫ℝ�⟨�⟩2s′ ∣G�(� cos �)g(� cos �)∣2 d�

)1/2

∥ℎ∥Hs′ (ℝ) d�

≤C∥∣G� f∥L1(ℝv)

∫ �2

−�2

�(�)sin2(�/2)

∣ cos �∣1/2+s′d � ∥ ⟨ ⋅ ⟩s′ G�g∥L2(ℝ�) ∥ℎ∥Hs′ (ℝ)

≤C∥∣G� f∥L21(ℝv)∥G� g∥Hs′ (ℝ) ∥ℎ∥Hs′ (ℝ) ,

where we have used the following continuous embedding

L2�(ℝ) ⊂ L1(ℝ), � > 1/2.

We have proved (7.3.23).


To treat (7.3.24), by using (7.3.26), we similarly have,

(2�)1/2{(

(v G�)K(f, g), ℎ)−(K(f, (v G�) g), ℎ

)}=

∫ℝ�

∫ �2

−�2

�(�){i∂�

(G�(�)f(� sin �)g(� cos �)

)− f(� sin �)ℱ

((v G�) g

)(� cos �)

}ℎ(�) d� d�

=i

∫ℝ�

∫ �2

−�2

�(�) sin � (∂�f)(� sin �)G�(�)g(� cos �) ℎ(�) d� d�

+i

∫ℝ�

∫ �2

−�2

�(�)f(� sin �){∂�(G�(�)g(� cos �)

)−(∂� (G� g)

)(� cos �)

}ℎ(�) d� d�

= (I) + (II).

For the term (I), we have

∣(I)∣ ≤∫ℝ�

∫ �2

−�2

�(�) ∣ sin �∣ ∣(∂�f)(� sin �)∣∣∣G�(� cos �)g(� cos �)

∣∣ ∣∣ℎ(�)∣∣ d� d�

+

∫ℝ�

∫ �2

−�2

�(�) ∣ sin �∣ ∣(∂�f)(� sin �)∣∣∣∣G�(�)−G�(� cos �)

∣∣∣∣g(� cos �)∣∣∣ℎ(�)

∣∣ d� d�≤ I1 + I2.

Firstly, (7.3.3) with the hypothesis 0 < s < 1/2 implies that

I1 ≤ ∥∂�f∥L∞(ℝ�)

∫ℝ�

∫ �2

−�2

�(�)∣ sin �∣∣∣G�(� cos �) g(� cos �)

∣∣ ∣∣ℎ(�)∣∣ d� d�

≤ C∥f∥L11(ℝv)

∫ �2

−�2

�(�)∣ sin �∣∣ cos �∣1/2

d� ∥ℎ∥L2(ℝ�)

×(∫

ℝ�

∣∣G�(� cos �) g(� cos �)∣∣2d(� cos �)

)1/2

≤ C∥f∥L11(ℝv)∥G� g∥L2(ℝv) ∥ℎ∥L2(ℝv) .


For the term I2, by using (7.3.20) we have the following estimates which are also truefor 0 < s < 1),

I2 ≤∫ℝ�

∫ �2

−�2

�(�)∣ sin �∣ sin2(�/2) ∣G�(� sin �)(∂�f)(� sin �)∣

×∣∣G�(� cos �) g(� cos �)

∣∣ ⟨�⟩2s′ ∣∣ℎ(�)∣∣ d� d�

≤ C∥⟨ ⋅ ⟩s′G� g∥L∞(ℝ�)

∫ �2

−�2

�(�)∣ sin �∣ sin2(�/2)

∣ sin �∣1/2d� ∥ℎ∥Hs′ (ℝv)

×(∫

ℝ�

∣∣G�(� sin �) (∂�f)(� sin �)∣∣2d(� sin �)

)1/2

≤ C∥⟨Dv⟩s′G� g∥L1(ℝv)

∫ �2

−�2

�(�)∣ sin �∣ sin2(�/2)

∣ sin �∣1/2d� ∥ℎ∥Hs′ (ℝv)

×(∫

ℝ�

∣∣G�(� sin �) (∂�f)(� sin �)∣∣2d(� sin �)

)1/2

≤ C∥G� g∥Hs′1 (ℝv)∥G� (v f)∥L2(ℝv) ∥ℎ∥Hs′ (ℝv) .

Moreover, for a suitable function F , we have

G� (v F ) = v G� F + [G�, v]F,

and

ℱ([G�, v]F

)(�) = i(∂�G�)(�)F (�).

Then the symbolic calculus (7.2.15) implies that, for 0 < 2s′ < 1, we have

(7.3.27) ∥G� (v F )∥H�(ℝv) ≤ C∥G� F∥H�1 (ℝv)

for any � ≥ 0, then

(7.3.28) ∣(I)∣ ≤ C{∥f∥L1

1(ℝv) + ∥G� f∥L21(ℝv)

}∥G� g∥Hs′

1 (ℝv) ∥ℎ∥Hs′ (ℝv)

On the other hand, for the term (II), we have

∂�(G�(�)g(� cos �)

)−(∂� (G� g)

)(� cos �) =

{G�(�)−G�(� cos �)

}(∂�g)(� cos �)

+G�(�)(

cos � − 1)(∂�g)(� cos �) +

{(∂�G�)(�)− (∂�G�)(� cos �)

}g(� cos �)

= A1 + A2 + A3.

Thus

∣(II)∣ ≤ C

∫ℝ�

∫ �2

−�2

�(�)∣f(� sin �)∣∣∣A1 + A2 + A3

∣∣ ∣∣ℎ(�)∣∣ d� d� .


We study now the above 3 terms on the right-hand side. By using (7.3.20),∫ℝ�

∫ �2

−�2

�(�)∣f(� sin �)∣ ∣A1∣∣∣ℎ(�)

∣∣ d� d�≤∫ℝ�

∫ �2

−�2

�(�) sin2(�/2) ∣G�(� sin �)f(� sin �)∣

×∣∣G�(� cos �) (∂�g)(� cos �)

∣∣ ⟨�⟩2s′ ∣∣ℎ(�)∣∣ d� d�

≤ C∥G� f∥L1(ℝv)

∫ �2

−�2

�(�)sin2(�/2)

∣ cos �∣1/2d� ∥G� (v g)∥Hs′ (ℝv) ∥ℎ∥Hs′ (ℝv)

≤ C∥G� f∥L21(ℝv)∥G� (v g)∥Hs′ (ℝv) ∥ℎ∥Hs′ (ℝv) .

The estimate (7.3.22) and cos � − 1 = −2 sin2(�/2) imply∫ℝ�

∫ �2

−�2

�(�)∣f(� sin �)∣ ∣A2∣∣∣ℎ(�)

∣∣ d� d�≤ C

∫ℝ�

∫ �2

−�2

�(�) sin2(�/2) ∣G�(� sin �)f(� sin �)∣

×∣∣G�(� cos �) (∂�g)(� cos �)

∣∣ ∣∣ℎ(�)∣∣ d� d�

≤ C∥G� f∥L21(ℝv)∥G� (v g)∥L2(ℝv) ∥ℎ∥L2(ℝv) .

Finally, the hypothesis 0 < s < 1/2 implies (4s′ − 1)+ < 2s′, then (7.3.21) yields,∫ℝ�

∫ �2

−�2

�(�)∣f(� sin �)∣ ∣A3∣∣∣ℎ(�)

∣∣ d� d�≤ C

∫ℝ�

∫ �2

−�2

�(�) sin2(�/2) ∣G�(� sin �)f(� sin �)∣

× ⟨�⟩2s′∣∣G�(� cos �) g(� cos �)

∣∣ ∣∣ℎ∣∣ d� d�≤ C∥G� f∥L2

1(ℝv)∥G� g∥Hs′ (ℝv) ∥ℎ∥Hs′ (ℝv) .

By summing the above 3 estimates,

(7.3.29) ∣(II)∣ ≤ C∥G� f∥L21(ℝv)∥G� g∥Hs′

1 (ℝv) ∥ℎ∥Hs′ (ℝv) .

Proof of Proposition 7.3.2 is established. □

Remark 7.3.1. In the proof of estimate for the term I1 and the last term of (II),we have used crucially the restrict assumption 0 < s < 1/2.

7.3.2. Sobolev regularizing effect of weak solutions. We will first give anH+∞-regularizing effect results for Kac’s equation. The following Theorem is moreprecise than Theorem 1.1 of [86] where the homogeneous Boltzmann equation withMaxwellian molecules has been studied.

Theorem 7.3.3. Assume that the initial datum f0 ∈ L12+2s ∩ L logL(ℝ), and the

cross-section � satisfy (7.3.2 ) with 0 < s < 12. If f ∈ L∞(]0,+∞[;L1

2+2s ∩ L logL(ℝ))


is a nonnegative weak solution of the Cauchy problem (7.3.1), then f(t, ⋅) ∈ H+∞2 (ℝ)

for any t > 0.

Remark 7.3.2. 1) This is a H+∞-smoothing effect results for the Cauchy problem,it is different from that of [44] where their assumption is that all moments of the initialdatum are bounded.

2) The results of theorem 7.3.3 is also true if we assume the following Debye-Yukawatype collision kernel :

�(�) = C0∣ cos �∣∣ sin �∣

(log ∣�∣−1

)m, 0 < m.

To prove the Theorem 7.3.3, we use, as in [86], the mollifier of polynomial type

M�(t, �) = ⟨�⟩tN−1(1 + �∣�∣2)−N0 ,

for 0 < � < 1, t ∈ [0, T0] and 2N0 = T0N + 4.The idea is the same as the section 3 of [86], but now we need to estimate the

commutators with weighted ⟨v⟩2. It is analogous to the computation of precedingsection. We give here only the main points of the proof,

Lemma 7.3.3. We have that for any 0 < � < 1 and 0 ≤ t ≤ T0, � ∈ ℝ,

∣∂tM�(t, �)∣ ≤ N log(⟨�⟩)M�(t, �).

For −�/4 ≤ � ≤ �/4,

∣M�(�)−M�(� cos �)∣ ≤ C sin2(�/2)M�(� cos �) ,∣∣(∂�M�

)(�)−

(∂�M�

)(� cos �)

∣∣ ≤ C sin2(�/2)⟨�⟩−1M�(� cos �),

and ∣∣(∂2�M�

)(�)−

(∂2�M�

)(� cos �)

∣∣ ≤ C sin2(�/2)⟨�⟩−2M�(� cos �),

where the constant C depends on T0, N , but is independents of 0 < � < 1.

We prove also this Lemma by using the Taylor formula, and for any k ∈ ℕ,∣∣∂k�M�(�)∣∣ ≤ Ck⟨�⟩−kM�(�), � ∈ ℝ

with Ck depends on T0, N , but is independents of 0 < � < 1. Moreover, for the poly-nomial mollifier, we can substitute the inequality (7.3.22) by the following inequality,

(7.3.30) M�(�) ≤ CM�(� cos �) , −�4≤ � ≤ �

4,

here again C depending on N0, T , and independents of � > 0. We have therefore

Proposition 7.3.3. Assume that 0 < s < 1/2, we have that∣∣∣((vM�

)K(f, g), ℎ

)−(K(f, (vM�) g), ℎ

)∣∣∣≤ C ∥f∥L1

1(ℝ) ∥M� g∥L21(ℝ)∥ℎ∥L2(ℝ),

(7.3.31)

and ∣∣∣((⟨v⟩2M�

)K(f, g), ℎ

)−(K(f, (⟨v⟩2M�) g), ℎ

)∣∣∣≤ C ∥f∥L1

2(ℝ) ∥M� g∥L22(ℝ)∥ℎ∥L2(ℝ),

(7.3.32)


The proof of (7.3.31) is similar to (7.3.24) where we substitute Lemma 7.3.2 byLemma 7.3.3, and replace (7.3.22) by (7.3.30). Consider now the estimate (7.3.32), wehave, as in the proof of the proposition 7.3.2,

(2�)1/2{(

(v2M�)K(f, g), ℎ)−(K(f, (v2M�) g), ℎ

)}=−

∫ℝ�

∫ �2

−�2

�(�) sin2 � (∂2� f)(� sin �)M�(�)g(� cos �) ℎ(�) d� d�

−2

∫ℝ�

∫ �2

−�2

�(�) sin � (∂�f)(� sin �)(∂�(M�(�)g(� cos �)

))ℎ(�) d� d�

−∫ℝ�

∫ �2

−�2

�(�)f(� sin �){∂2�

(M�(�)g(� cos �)

)−(∂2� (M� g)

)(� cos �)

}ℎ(�) d� d�

= B1 +B2 +B3.

Then

∣B1∣ ≤ C ∥∂2� f∥L∞(ℝ) ∥M� g∥L2(ℝ)∥ℎ∥L2(ℝ) ≤ C ∥f∥L1

2(ℝ) ∥M� g∥L2(ℝ)∥ℎ∥L2(ℝ),

and for 0 < 2s < 1,

∣B2∣ ≤ C ∥f∥L11(ℝ)

(∥M� g∥L2(ℝ) + ∥M� (v g)∥L2(ℝ)

)∥ℎ∥L2(ℝ).

The term B3 is evidently more complicate, but the idea is the same, we omit here theircomputations.

Using the continuous embedding

L1ℓ(ℝ) ⊂ H−1

ℓ (ℝ),

the upper bounded (7.3.14) with m = −2, ℓ = 2 and 0 < 2s < 1 imply ,

∥K(g, ℎ)∥H−22 (ℝv) ≤ C∥g∥L1

2+2s(ℝv)∥ℎ∥H−2+2s2+2s (ℝv) ≤ C∥g∥L1

2+2s(ℝv) ∥ℎ∥L12+2s(ℝv) .

Let f ∈ L∞(]0,+∞[;L12+2s(ℝ)) be a weak solution of the Cauchy problem (7.3.1), then

we can take

f1 = M�(t,Dv)⟨v⟩4M�(t,Dv)f ∈ L∞([0, T0];H5−2+2s(ℝ)) ,

as test functions of the Cauchy problem (7.3.1). By using similar manipulations as in[86], we can obtain the regularity with respect to t variable, to simplify the notationswe suppose that f1 ∈ C1([0, T0];H5

−2+2s(ℝ)). We have(∂tf(t, ⋅ ), f1(t, ⋅ )

)L2(ℝv)

=(K(f , f), f1

)L2(ℝv)

.

Then Lemma 7.3.3, Proposition 7.3.3, the coercivity estimate (7.3.12) and the conser-vations (7.3.6), (7.3.7), (7.3.8) imply that

d

dt∥M�f(t)∥2

L22(ℝv) + cf0∥M�f(t)∥2

Hs2(ℝv)

≤Cf0∥ log1/2(∣Dv∣)M�f(t)∥2L2

2(ℝv) + C ∥f0∥L12(ℝ) ∥M� f(t)∥2

L22(ℝ).

We now use the following interpolation inequality, for any small " > 0

(7.3.33) ∥ log1/2(∣Dv∣)M�f(t)∥2L2

2(ℝv) ≤ "∥M�f(t)∥2Hs

2(ℝv) + C"∥M�f(t)∥2L2

2(ℝv).


Then for t ∈ [0, T0],d

dt∥M�f(t)∥2

L22(ℝv) ≤ C1 ∥M� f(t)∥2

L22(ℝ)

where C1 depends on T0, N , but independents of 0 < � < 1. So that for t ∈ [0, T0],

∥M�f(t)∥L22(ℝv) ≤ eC1 t ∥M� f(0)∥L2

2(ℝ) ≤ eC1 t ∥f0∥H−12 (ℝ) ≤ eC1 t ∥f0∥L1

2(ℝ).

We have therefore proved for t ∈ [0, T0],

(1 + ∣Dv∣2)tN−1f(t, ⋅ ) ∈ L22(ℝ).

Since we can choose arbitrary N > 0 and T0 > 0, we have proved Theorem 7.3.3.

7.3.3. Gevrey regularizing effect of solutions. Theorem 7.3.3 implies that theweak solution of the Cauchy problem (7.3.1) satisfies f ∈ L∞([t0, T0[;H1

2 (ℝ)) for anyt0 > 0. Then f is a solution of the following Cauchy problem :{

∂f∂t

= K(f, f), v ∈ ℝ, t > t0,f ∣t=t0 = f(t0, ⋅ ) ∈ H1

2 (ℝ).

We now study the local Gevrey regularizing effect of the Cauchy problem, andsuppose that the initial datum is f0 ∈ H1

2 ∩ L12(ℝ). We state this result as the:

Theorem 7.3.4. Assume that the initial datum f0 ∈ H12 ∩ L1

2(ℝ), and the cross-section � satisfy (7.3.2) with 0 < s < 1

2. For T0 > 0, if f ∈ L∞([0, T0];H1

2 ∩ L12(ℝ)) is

a nonnegative weak solution of the Cauchy problem (7.3.1), then for any 0 < s′ < s,

there exists 0 < T∗ ≤ T0 such that f(t, ⋅) ∈ G1

2s′ (ℝ) for any 0 < t ≤ T∗. More precisely,there exists c0 > 0,

ec0t⟨Dv⟩2s′

f ∈ L∞([0, T∗]; L21(ℝ)).

Remark 7.3.3. The above Gevrey smoothing effect property of Cauchy problem isfor any weak solution f ∈ L∞([0, T0];H1

2 ∩ L12(ℝ)), so that we don’t need to use the

uniqueness of solution for Kac’s equation.

We prove the above theorem by construction of a priori estimates for the mollifiedweak solution. Take f ∈ L∞(]0, T0[;H1

2 ∩ L12(ℝ)) to be a weak solution of the Cauchy

problem (7.3.1), then (7.3.14) with m = ℓ = 0 implies that, (recall the assumption0 < s < 1/2)

K(f, f) ∈ L∞(]0, T0[;L2(ℝv)).

So that we need to choose a test function ' ∈ C1([0, T0]; L2(ℝv)) to make sense(K(f, f), '

)L2(ℝv)

.

The right way is to choose a mollified weak solution f , we first have

f(t, ⋅ ) =(G�(t,Dv)⟨v⟩2G�(t,Dv)f

)(t, ⋅ ) ∈ L∞(]0, T0[; H1(ℝ)).

Here again we suppose that f ∈ C1([0, T0]; H1(ℝv)), and study the equation of (7.3.1)in the following weak formulation

(7.3.34)(∂tf(t, ⋅ ), f(t, ⋅ )

)L2(ℝv)

=(K(f , f), f

)L2(ℝv)

.


First, the left hand side term is(∂tf(t, ⋅ ), f(t, ⋅ )

)L2(ℝv)

=1

2

d

dt∥G�f(t)∥2

L21(ℝv)

−((∂tG�

)(t,Dv)f(t, ⋅ ), G�(t,Dv)f(t, ⋅ )

)L2(ℝv)

−(v(∂tG�

)(t,Dv)f(t, ⋅ ), v G�(t,Dv)f(t, ⋅ )

)L2(ℝv)

.

Then we estimate the two terms on right hand side by using the following lemma.

Lemma 7.3.4. There exists C > 0 such that

(7.3.35)

∣∣∣∣((∂tG�

)(t,Dv)f(t, ⋅ ), G�(t,Dv)f(t, ⋅ )

)L2(ℝv)

∣∣∣∣ ≤ C∥G� f∥2Hs′ (ℝv)

,

and

(7.3.36)

∣∣∣∣(v (∂tG�

)(t,Dv)f(t, ⋅ ), v G�(t,Dv)f(t, ⋅ )

)L2(ℝv)

∣∣∣∣ ≤ C∥G� f∥2Hs′

1 (ℝv).

Proof. (7.3.35) can be deduced directly from (7.3.17) by using the Plancherelformula.

For (7.3.36), we have∣∣∣∣(v (∂tG�

)(t,Dv)f(t, ⋅ ), v G�(t,Dv)f(t, ⋅ )

)L2(ℝv)

∣∣∣∣=C

∣∣∣∣∫ℝ

(∂�

(c0⟨�⟩2s

′G�(t, �)

1

1 + �ec0t⟨�⟩2s′ f(t, �)

))ℱ(v G�f

)(t, � )d�

∣∣∣∣≤C

∫ℝ⟨�⟩2s′

∣∣∣∂�(G�(t, �)f(t, �))∣∣∣ ∣∣ℱ(v G�f

)(t, � )

∣∣ d�+ C

∫ℝ

∣∣∣∣∂�(⟨�⟩2s′ 1

1 + �ec0t⟨�⟩2s′

)∣∣∣∣ ∣∣G�(t, �)f(t, �)∣∣ ∣∣ℱ(v G�f

)(t, � )

∣∣ d�≤C∥G� f∥2

Hs′1 (ℝv)

,

where we use the fact that∣∣∣∣∂�(⟨�⟩2s′ 1

1 + �ec0t⟨�⟩2s′

)∣∣∣∣ ≤ C⟨�⟩2s′ .

Hence Lemma 7.3.4 is proved □

Then (7.3.34) and Lemma 7.3.4 give

(7.3.37)1

2

d

dt∥G�f(t)∥2

L21(ℝv) −

(K(f , f), f

)L2(ℝv)

≤ C∥G� f∥2Hs′

1 (ℝv).


On the other hand, we have(K(f, f ), f

)L2(ℝv)

=(G�K(f, f ), (1 + v2)G�f

)L2(ℝv)

=(K(f, G�f ), G�f

)L2(ℝv)

+(K(f, v G�f ), v G�f

)L2(ℝv)

+(G�K(f, f )−K(f, G�f ), G�f

)L2(ℝv)

+(v G�K(f, f )−K(f, v G�f ), v G�f

)L2(ℝv)

.

Then Proposition 7.3.2 implies∣∣∣∣(G�K(f, f )−K(f, G�f ), G�f)L2(ℝv)

∣∣∣∣ ≤ C∥G�f∥L21(ℝv)∥G�f∥2

Hs′ (ℝv)

and ∣∣∣∣(v G�K(f, f )−K(f, v G�f ), v G�f)L2(ℝv)

∣∣∣∣≤ C

(∥f∥L1

1(ℝ) + ∥G� f∥L21(ℝ)

)∥G� f∥2

Hs′1 (ℝ)

.

The Proposition ?? implies

−(K(f, G�f), G�f

)≥ cf∥G�f∥2

Hs(ℝv) − C∥f∥L1(ℝv)∥G�f∥2L2(ℝv),

−(K(f, v G�f), v G�f

)≥ cf∥v G�f∥2

Hs(ℝv) − C∥f∥L1(ℝv)∥v G�f∥2L2(ℝv).

Since

∥G�f∥2Hs

1(ℝv) ≤ ∥G�f∥2Hs(ℝv) + ∥v G�f∥2

Hs(ℝv) + C∥G�f∥2L2

1(ℝv)

By summing all the above estimates and (7.3.37), we obtain

d

dt∥G�f(t)∥2

L21(ℝv) + cf(t)∥G�f(t)∥2

Hs1(ℝv)

≤ C∥G�f(t)∥2Hs′

1 (ℝv)+ C∥f(t)∥L1(ℝv)∥G�f(t)∥2

L21(ℝv)

+ C(∥f(t)∥L1

1(ℝ) + ∥G� f(t)∥L21(ℝ)

)∥G� f(t)∥2

Hs′1 (ℝ)

.

(7.3.38)

End of proof of Theorem 7.3.4By using (7.3.6) and (7.3.7), we have

∥f(t)∥L1(ℝv) + ∥f(t)∥L12(ℝv) ≤ C∥f0∥L1

2(ℝv), cf(t) ≥ cf0 > 0.

Then (7.3.38) yields

d

dt∥G�f(t)∥2

L21(ℝv) + cf0∥G�f(t)∥2

Hs1(ℝv)

≤ Cf0∥G�f(t)∥2Hs′

1 (ℝv)+ C ∥G� f(t)∥L2

1(ℝ) ∥G� f(t)∥2Hs′

1 (ℝ).

(7.3.39)

We now need the following interpolation inequality, for 0 < s′ < s and any � > 0,

(7.3.40) ∥u∥2Hs′ ≤ �∥u∥2

Hs + �−s′s−s′ ∥u∥2

L2 .


Then for any small " > 0,

Cf0∥G�f(t)∥2Hs′

1 (ℝv)≤ "∥G�f(t)∥2

Hs1(ℝv) + C",f0∥G�f(t)∥2

L21(ℝv)

and

C ∥G� f(t)∥L21(ℝ)∥G�f(t)∥2

Hs′1 (ℝv)

≤ "∥G�f(t)∥2Hs

1(ℝv) + C"∥G� f(t)∥s′s−s′+2

L21(ℝ)

.

We finally get from (7.3.39),that for any 0 < " and 0 < s′ < s, there exists C" > 0such that

d

dt∥G�f(t)∥2

L21(ℝv) + (cf0 − 2")∥G�f∥2

Hs1(ℝv)

≤ C",f0∥G�f(t)∥2L2

1(ℝv) + C"∥G� f(t)∥s′s−s′+2

L21(ℝ)

.

We choose 0 < 2" ≤ cf0 , we get

(7.3.41)d

dt∥G�f(t)∥L2

1(ℝv) ≤ C1∥G�f(t)∥L21(ℝv) + C2∥G� f(t)∥

s′s−s′+1

L21(ℝ)

, t ∈ [0, T0],

with C1, C2 > 0 and independent of � > 0. Then

d

dt

(e−C1t∥G�f(t)∥L2

1(ℝv)

)≤ C2e

C1t(e−C1t∥G�f(t)∥L2

1(ℝv)

) s′s−s′+1

where C1 = s′ C1

s−s′ , thus for t ∈]0, T0]∫ t

0

d

d�

(e−C1�∥G�f(�)∥L2

1(ℝv)

)− s′s−s′

d� ≥ C2

C1

(1− eC1t

).

So that, for 0 < � < 1,

∥G�f(t)∥L21(ℝv) ≤

˜C1 e

C1t∥f0∥L21(ℝv)(

C1 + C2

(1− eC1t

)∥f0∥

s′s−s′

L21(ℝv)

) s−s′s′

.

We now choose 0 < T∗ ≤ T0 small enough so that(C1 + C2

(1− eC1t

)∥f0∥

s′s−s′

L21(ℝv)

) s−s′s′ ≥ C3 > 0, t ∈ [0, T∗],

then by compactness and by taking limit � → 0, we have for t ∈ [0, T∗],

(7.3.42) ∥ec0t⟨Dv⟩2s′

f∥2L∞(]0,T∗[; L2

1(ℝv)) ≤ eC1T∗

˜C1

C3

∥f0∥2L2

1(ℝv).

We therefore have proved Theorem 7.3.4.


7.3.4. Radially symmetric Boltzmann equations. We consider now the Boltz-mann collision operators (7.3.10). In the Maxwellien case, the Bobylev’s formula takesthe form

(7.3.43) ℱ(Q(g, f)

)(�) =

∫S2

b

(�

∣�∣⋅ �){

g(�−)f(�+)− g(0)f(�)}d�

where � ∈ ℝ3,

�+ =� + ∣�∣�

2, �− =

� − ∣�∣�2

.

On the other hand

∣�+∣2 = ∣�∣21 + �

∣�∣ ⋅ �2

, ∣�−∣2 = ∣�∣21− �

∣�∣ ⋅ �2

,

so that if we define � by

cos � =�

∣�∣⋅ �,

we obtain

∣�+∣2 = ∣�∣2 cos2

(�

2

), ∣�−∣2 = ∣�∣2 sin2

(�

2

).

We now consider the radially symmetric function with respect to v ∈ ℝ3, namely thefunction satisfy the property

ℎ(v) = ℎ(Av), v ∈ ℝ3

for any proper orthogonal 3 × 3 matrix A, then ℎ(v) = ℎ(0, 0, ∣v∣). Denote by ℱℝ3

the Fourier transformation in ℝ3 and ℱℝ1 the Fourier transformation in ℝ1. Thenℱℝ3(ℎ)(�) is also radially symmetric with respect to � ∈ ℝ3, and it is in the form

ℱℝ3(ℎ)(�) = ℱℝ3(ℎ)(0, 0, ∣�∣) =

∫ℝe−i∣�∣v3

(∫ℝ2

ℎ(v1, v2, v3)dv1dv2

)dv3.

So that

(7.3.44) ℱ−1ℝ1

(ℱℝ3(ℎ)(0, 0, ⋅ )

)(u) =

∫ℝ2

ℎ(v1, v2, u)dv1dv2

is an even function in ℝ, and we have

Lemma 7.3.5. Assume that ℎ ∈ L1k(ℝ3), ℎ ≥ 0 is a radially symmetric function for

certain k ≥ 0, and uniformly integrable in ℝ3, then

ℱ−1ℝ1

(ℱℝ3(ℎ)(0, 0, ⋅ )

)∈ L1

k(ℝ)

is a nonnegative even function, and uniformly integrable in ℝ.

Proof. By using (7.3.44), it is evident that ℎ ∈ L1k(ℝ3) implies ℱ−1

ℝ1

(ℱℝ3(ℎ)(0, 0, ⋅ )

)∈

L1k(ℝ), and ℎ ≥ 0 implies ℱ−1

ℝ1

(ℱℝ3(ℎ)(0, 0, ⋅ )

)≥ 0. Hence we need only to check the

uniform integrability of ℱ−1ℝ1

(ℱℝ3(ℎ)(0, 0, ⋅ )

)in ℝ. Since ℎ ∈ L1(ℝ3), for any " > 0,

there exits R0 > 0 such that∫{v∈ℝ3; ∣v∣≥R0}

∣ℎ(v1, v2, v3)∣dv1dv2dv3 <"

2.


The uniform integrability of ℎ in ℝ3 imply that, there exists �1 > such that∫B

∣ℎ(v1, v2, v3)∣dv1dv2dv3 <"

2,

for any B ⊂ ℝ3 with ∣B∣ ≤ �1. Choose new �0 = �1(R20)−1, then for any A ⊂ ℝ, if

∣A∣ ≤ �0, we have∫A

∣ℱ−1ℝ1

(ℱℝ3(ℎ)(0, 0, ⋅ )

)(u)∣du ≤

∫ℝ2×A

∣ℎ(v1, v2, v3)∣dv1dv2dv3

≤∫{(v1,v2,v3)∈ℝ3; ∣v1∣≤R0,∣v2∣≤R0,v3∈A}

∣ℎ(v1, v2, v3)∣dv1dv2dv3 +"

2< ",

because of

∣{(v1, v2, v3) ∈ ℝ3; ∣v1∣ ≤ R0, ∣v2∣ ≤ R0, v3 ∈ A}∣ ≤ R20∣A∣ ≤ �1.

□

Remark 7.3.4. In the proof of above Lemma, if ℎ ∈ L logL(ℝ3) then ℎ is uni-formly integrable in ℝ3 with �1 depends only on ", ∥ℎ∥L logL(ℝ3) and ∥ℎ∥L1(ℝ3). There-

fore, ℱ−1ℝ1

(ℱℝ3(ℎ)(0, 0, ⋅ ) is uniformly integrable in ℝ1 with �0 also depends only on

", ∥ℎ∥L logL(ℝ3) and ∥ℎ∥L1(ℝ3).

End of proof of Theorem 7.3.2Suppose now g ∈ L∞(]0,+∞[;L1

2+2s ∩ L logL(ℝ3)) is a non negative radially sym-metric weak solution of the Cauchy problem (7.3.9). Setting, for t ≥ 0, u ∈ ℝ,

(7.3.45) f(t, u) = ℱ−1ℝ1

(ℱℝ3(g)(t, 0, 0, ⋅ )

)(u) =

∫ℝ2

g(t, v1, v2, u)dv1dv2 ,

hereafter, the time variable t is always considered as parameters for the Fourier trans-formation, then f(t, u) is an even function with respect to u ∈ ℝ, and

f(t, �) = ℱℝ1

(f(t, ⋅ )

)(�) = ℱℝ3(g)(t, 0, 0, �).

So that the Bobylev’s formula (7.3.43) give, for � ∈ ℝ3,(7.3.46)

ℱℝ3

(Q(g, g)

)(�) =

∫ �2

−�2

�(∣�∣){f(t, ∣�∣ sin(�/2))f(t, ∣�∣ cos(�/2))− f(t, 0)f(t, ∣�∣)

}d�

where

�(∣�∣) =1

2∣ sin �∣b(cos �).

Then the right hand side of (7.3.46) is Fourier transformation of Kac’s operator K(f, f).We have proved that if g(t, v) is a non negative radially symmetric weak solution ofthe Cauchy problem (7.3.9), then f(t, u) is a weak solution of the Cauchy problem ofKac’s equation :

(7.3.47)

{∂f∂t

(t, u) = K(f, f)(t, u),f(0, u) = f0(u) =

∫ℝ2 g0(v1, v2, u)dv1dv2 ,


or equivalently in the Fourier variable:

{∂f∂t

(t, �) =∫ �

2

−�2�(∣�∣)

{f(t, � sin(�/2))f(t, � cos(�/2))− f(t, 0)f(t, �)

}d�,

f(0, �) = f0(�) = g0(0, 0, �).

Under the assumption of Theorem 7.3.2 for g(t, v), Lemma 7.3.5 and Remark 7.3.4implies that f(t, u) satisfy the hypothesis of Theorem 7.3.1 except f belong to L logLwhich substituted by the uniform integrability of f = ft( ⋅ ) in ℝ. Then we applyTheorem 7.3.1 to the Cauchy problem (7.3.47), thus there exists T∗ > 0 such that for0 < t ≤ T∗,

ec0t⟨ ∣� ∣ ⟩2s′

f(t, �) = ec0t⟨ ∣� ∣ ⟩2s′ℱℝ3(g)(t, 0, 0, �) ∈ H1(ℝ� ).

It remain to prove the Gevrey smoothing effect in the global time interval. Kac’sequation shares with the homogeneous Boltzmann equation for Maxwellian moleculesthe existence and uniqueness theory for the Cauchy problem. We take 0 < t0 < t1 ≤ T∗,and consider the Cauchy problem (7.3.47) with even initial datum f(t1, �). The Sobolevembedding H1(ℝ) ⊂ L∞(ℝ) imply that

∥ec0t1⟨ ⋅ ⟩2s′

f(t1, ⋅ )∥L∞(ℝ) ≤ C∥ec0t1⟨ ⋅ ⟩2s′

f(t1, ⋅ )∥H1(ℝ)

≤ C∥ec0t1⟨ ∣Du∣ ⟩2s′

f(t1, ⋅ )∥L21(ℝ) < +∞

Now the following propagation of Gevrey regularity results deduces the Gevrey smooth-ing effect in the global time interval.

Theorem 7.3.5. (Theorem 2.3 of [49])Let f0 be a non negative, even function, satisfying

sup�∈ℝ

(∣f0(�)∣ec1⟨�⟩2s

)< +∞,

for some c1 > 0 and the cross-section � satisfying (7.3.2) with 0 < s < 1. Then the

solution of the Cauchy problem (7.3.1) satisfies f(t, ⋅ ) ∈ G12s (ℝ)) for any t ≥ 0.

In conclusion, if g ∈ L∞(]0,+∞[;L12+2s ∩ L logL(ℝ3)) is a non negative radially

symmetric weak solution of the Cauchy problem (7.3.9), then under the assumptionof Theorem 7.3.2, we have proved that for any fixed 0 < t < +∞, there exists c0 > 0such that

ec0⟨∣Du∣⟩2s′

f(t, ⋅ ) ∈ L2(ℝ)


where f is the function defined by (7.3.45). We can finish now the proof by the followingestimations, for fixed t > 0,

∥ec02⟨∣Dv ∣⟩2s

′

g(t, ⋅ )∥2L2(ℝ3) =

∫ℝ3

∣∣∣e c02 ⟨∣�∣⟩2s′ℱℝ3(g)(t, �1, �2, �3)∣∣∣2 d�

=

∫ℝ3

∣∣∣e c02 ⟨∣�∣⟩2s′ℱℝ3(g)(t, 0, 0, ∣�∣)∣∣∣2 d�

= C

∫ ∞0

∣∣∣e c02 ⟨�⟩2s′ f(t, �)∣∣∣2 � 2d�

≤ C

∫ ∞0

∣∣∣ec0⟨�⟩2s′ f(t, �)∣∣∣2 d�

≤ C∥ec0⟨∣Du∣⟩2s′

f(t, ⋅ )∥2L2(ℝ) < +∞.

We finished the proof of Theorem 7.3.2.

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