fourier analysis of non cutoﬀ boltzmann...

.

Fourier Analysis

of Non Cutoff Boltzmann Equations

-

Chao-Jiang XU

Universite de Rouen, France

Contents

Introduction 1

Chapter I. Sobolev spaces 31.1. Fourier transformation 31.2. Fractional derivative formulas 41.3. Definition of Sobolev space 61.4. Logarithmic Sobolev space 81.5. Sobolev embedding Theorem 101.6. Compactness and interpolation inequalities 141.7. Sobolev space on an open domain and trace Theorem 19

Chapter II. Pseudo-differential operators 212.1. Symbols class 212.2. Asymptotic expansion 222.3. Definition of pseudo-differential operators 242.4. Algebra of pseudo-differential operators 252.5. Continuity in Sobolev spaces 27

Chapter III. Boltzmann equations without angular cutoff 313.1. Boltzmann equations 313.2. Fourier transformation of collision operators 373.3. Coercivity estimates 393.4. Functional estimate of collision operators 44

Chapter IV. Spatially homogeneous Boltzmann equations 514.1. Weak solution of Cauchy problems 514.2. Smoothness effect of Cauchy problems 544.3. Gevrey regularity effect for linearized equations 57

Chapter V. A model of kinetic equations 655.1. Subelliptic estimates 655.2. C∞-regularity of weak solutions 705.3. Gevrey hypoellipticity 755.4. Semi-linear equations 77

Chapter VI. Uncertainty principle and kinetic equation 796.1. Uncertainty principle 796.2. Hypoellipticity of kinetic equations 836.3. Weighted hypoelliptic estimates 86

I

II CONTENTS

Chapter VII. Linearized inhomogeneous Boltzmann equations 897.1. Cauchy problem for linearized Boltzmann equation 897.2. Hypoellipticity of linearized Boltzmann equation 907.3. Regularity of weak solutions 957.4. Linearized inhomogeneous Landau equations 98

Bibliography 105

Index 111

Introduction

There have been extensive mathematical studies on kinetic equations, especially onthe Boltzmann equation, one of the most fundamental equations in Statistical Physics.However, most of the works are based on the Grad’s angular cut-off assumption whichremoves the singularity in cross section. On the other hand, the non-cutoff Boltzmannoperator behaves like a singular integral operator with leading order as a fractionalLaplace operator. This was formulated explicitly by P.-L. Lions in the beginning of1990s by using the compactness of solutions. Around 2000s, some elegant formulationswere obtained by Alexandre-Desvillettes-Villani-Wennberg. In some sense, the noncutoff collision operator behaves like a diffusion terms of velocity variables. So that itpermit to prove the smoothing effect of Cauchy problem for the spatially homogeneousBoltzmann equations.

Recently, in a series collaborative works by R. Alexandre, Y. Morimoto, C.-J. Xu,S. Ukai and T. Yang, we use Fourier analysis to study the non cutoff Boltzmannequations. The object of this lecture is to present the main results of this collaborationworks.

The plan of this lecture notes is as following :In the chapter I and II, we recall some basic properties of Sobolev space and pseudo-

differential operators, then in the chapter III, we introduce the Boltzmann equationsand the Fourier transformation of collision operators. We prove also the coercivityestimate due to the singularity of collision kernel.

In the Chapter IV, we study the spatially homogeneous Boltzmann equations, weobtained an optimal estimate for smoothing effect of Cauchy problem. As in thestudy of the hypoellipticity of (linear or non linear) partial differential equations, thekey steps for the regularity of weak solutions is the analysis of commutators betweenthe regularizer operators (for mollifier the weak solutions) and the (nonlinear) collisionoperators. It is a hard part of the microlocal analysis theory. We use a very weakregularizer operators for the Debye-Yukawa type potential where the collision operatorlike a logarithmic of Laplace operator, and we choose a very power regularizer operatorsfor the Gevrey regularity of linearized homogeneous equations.

In the spatially inhomogeneous case, the interaction between the kinetic part andnonlinear collision part is very complicate. For simplifier the problem, we consider, inthe chapter V, a model of kinetic equations, it is like the hypoelliptic Kolomogroveoperators, but a pseudo-differential version. We proved that it is also hypoelliptic in aglobal version by the classical commutators estimate of pseudo-differential calculus.

Finally we consider the linearized inhomogeneous non cutoff Boltzmann equationsin chapter VII, we obtained also the smoothing effect results of Cauchy problem for allvariables. Namely, there exists the regularizing effect of the kinetic part of Boltzmann

1

2 INTRODUCTION

operators: the non cutoff collision operator deduce the regularizer effect for velocityvariables, then the kinetic operators deduce the regularity of the time and the positionvariables from the smoothness of velocity variables. This phenomenon was alreadyobserved in the H-Theorem of P.-L. Lions, but we prove now a generalized version ofthe uncertainty principle which is the main results of chapter VI. Our results is moreeasy to adapt into kinetic equations by iteration to deduce the high order regularity.

In this lecture, we focus only to the regularity of solutions, and limited in themathematical Maxwellian case. We does not consider here the existence of weaksolutions. This lecture notes is in some sense the reproduction of several recent papersin collaboration with R. Alexandre, Y. Morimoto, C.-J. Xu, S. Ukai and T. Yang.

This lectures notes is the collections of a series of lectures which I given during thesummers of 2007 at “Morning side Center of Mathematics” of Chinese Academy ofSciences in Beijing, and also at Wuhan University. I would like to thanks to ProfessorZHANG Ping and Professor CHEN Hua for their invitations at Beijing and Wuhan.

Rouen, France. March 2008

CHAPTER I

Sobolev spaces

In this chapter, we recall without detail some basic properties of Sobolev spacesand the important inequalities which we will use to the analysis of kinetic equations.

1.1. Fourier transformation

We use the following notations in this lectures notes : For α = (α1, · · · , αd) ∈Nd, x = (x1, · · · , xd) ∈ Rd, setting

|α| = α1 + · · ·+ αd, |x| = (|x1|2 + · · ·+ |xd|2)1/2,

and

xα = xα11 · · ·xα1

1 , ∂αx = ∂α1

x1· · · ∂αd

xd, Dα

x =(1i∂x1

)α1 · · ·(1i∂xd

)αd .

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

F(f)(ξ) = f(ξ) =

∫Rd

e−ix · ξf(x)dx,

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

∂αξ f(ξ) = (−ix)αf(ξ), F

(Dα

xf)(ξ) = ξαf(ξ),

and

(1 + |ξ|2)k∂αf(ξ) =

∫Rd

e−ix · ξ(I −4x)k(−ix)αf(x)dx,

where we use the identity

(I −4x)ke−ix · ξ = (1 + |ξ|2)ke−ix · ξ.

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 · ξf(x)g(ξ)dxdξ = 〈Ff, g〉.

We define also the inverse Fourier transformation by

F(f)(ξ) =

∫Rd

eix · ξf(x)dx.

Theorem 1.1.1. .

(1) If we have f, f ∈ L1(Rd), then

f = (2π)−dF(f).

3

4 I. SOBOLEV SPACES

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

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

Blanchrel Formula :

(2π)−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|α|≤k

(1 + |x|)k|∂αf(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(δa)(ξ) = e−ia · ξ.(2) F(1)(ξ) = (2π)dδ0(ξ).(3) Let z ∈ C such that Re z > 0 then

F(e−z| · |2

)(ξ) =

(πz

)d/2

e−|ξ|2

4z ,

where z1/2 = |z|1/2e−iθ/2 if z = |z|e−iθ with θ ∈]− π/2, π/2[.(4) If 0 < σ < d, then

F(

1

| · |σ

)(ξ) = cd|ξ|σ−d,

with

cd =

∫Sd−1

∫ ∞

0

e−erω1rd−1−σdrdω.

(5)

F(

1

1 + x2

)(ξ) = πe−|ξ|.

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

F(u A) =1

| detA|

((Fu)(t

A−1))

.

And

F(u ? v) = F(u)F(v)

when the convolution u ? v is well-defined.

1.2. Fractional derivative formulas

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

|Dx|α g(x) = F−1(| · |αg( · )

)(x).

Then, we have the following formula.

1.2. FRACTIONAL DERIVATIVE FORMULAS 5

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

(1.2.1) |Dx|α g(x) = Cα

∫Rd

g(x)− g(x+ x′)

|x′|n+αdx′

with Cα 6= 0 being a complex constant depending only on α and the dimension d.

In fact, using Fourier inverse formula∫Rd

g(x)− g(x+ x′)

|x′|d+αdx′ = (2π)−d

∫Rd

F(g)(η) ei x·η(∫

Rn

1− e−i x′·η

|x′|d+αdx′)dη

On the other hand, it is clear that∫Rd

1− e−i x′·η

|x′|d+αdx′ = |η|α

∫Rd

1− e−i u· η|η|

|u|d+αdu.

Observing∫

Rd1−e

i u· η|η|

|u|d+α du 6= 0 is a complex constant depending only on α and the

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

In the study of Debye-Yukawa type potential Boltzmann equations and infinitelydegenerate elliptic equations, we need also the logarithmic type derivative. For s > 0and g ∈ S(Rd), we define

(log 〈Dx〉)s g(x) = F−1((log 〈| · |〉)sg( · )

)(x),

where for ξ ∈ Rd,

(log 〈| ξ |〉)s =(12

log(e+ | ξ |2))s.

We will use the notation

(1.2.2) log ρ =

log ρ, if 0 < ρ < 1 ;0, if 1 ≤ ρ .

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)(ξ)| ≤ C0(log 〈|ξ|〉)s−1, ∀ ξ ∈ 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 ξ ∈ Rd,

F(c )(ξ) =(log 〈 |ξ| 〉

)−s∫|z|≤1

(1− e−iz·ξ

)| log |z| |s−1

|z|ndz.

The estimate of this function is exactly the same as the proof of Proposition 1.4.1.

6 I. 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 + |ξ|2)s/2f(ξ) ∈ 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); |ξ|sf(ξ) ∈ L2(Rd).

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

(u, v)Hs =

∫Rd

(1 + |ξ|2)su(ξ)¯v(ξ)dξ.

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 + |ξ|2)sdξ).

We study now the equivalent definition of Sobolev spaces.

Proposition 1.3.2. Let m ∈ N, then

Hm(Rd) = f ∈ L2(Rd); ∂αf ∈ L2(Rd),∀|α| ≤ m,

and the norm is equivalent to ( ∑|α|≤m

‖∂αf‖2L2(Rd)

)1/2

.

Here, we use the simple fact,

F(∂αu) = (−iξ)αu(ξ)

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

C−1m

(1 +

∑0<|α|≤m

|ξα|2)≤ (1 + |ξ|2)m ≤ Cm

(1 +

∑0<|α|≤m

|ξα|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 7

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+ σ,m ∈ N, 0 < σ < 1, then

Hs(Rd) =∂αf ∈ L2(Rd),∀|α| ≤ m;∫∫

Rd×Rd

|∂βf(x+ y)− ∂βf(x)|2

|y|d+2σdxdy < +∞,∀|β| = m

,

and the norm is equivalent to( ∑|α|≤m

‖∂αf‖2L2(Rd) +

∑|β|=m

∫∫Rr×Rd

|∂βf(x+ y)− ∂βf(x)|2

|y|d+2σdxdy

)1/2

.

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

|f(x+ y)− f(x)|2

|y|d+2sdxdy =

∫Rd

|f(ξ)|2∫

Rd

|eiy · ξ − 1|2

|y|d+2sdydξ,

and ∫Rd

|eiy · ξ − 1|2

|y|d+2sdy = |ξ|2s

∫Rd

|eiy · ξ|ξ| − 1|2

|y|d+2sdy = cs|ξ|2s.

It is clary that

cs =

∫Rd

|eiy · ξ|ξ| − 1|2

|y|d+2sdy

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

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

Proposition 1.3.4. .

(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, uα ∈ L2(Rd), |α| = m such that

u = u0 +∑|α|=m

∂αuα.

8 I. SOBOLEV SPACES

In fact, we can use the following algebraic identity

|ξ|2m =∑

1≤j1,··· ,jm≤d

ξ2j1· · · ξ2

jm=∑|α|=m

(iξ)α(−iξ)α.

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

u(ξ) = θ(ξ)u(ξ) +∑|α|=m

(−iξ)αvα(ξ)

with

vα(ξ) = (1− θ(ξ))(iξ)α

|ξ|2mu(ξ).

Thenu0 = F−1(θu), uα = F−1vα ∈ L2(Rd),

andu = u0 +

∑|α|=m

∂αuα.

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

• δ0 ∈ 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)(ξ) = (2π)dδ0(ξ) /∈ L2loc(Rd).

1.4. 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 1.4.1. Let s > 0, we define the following logarithmic Sobolev’s space.

Hslog(Rd) = u ∈ L2(Rd);

(log 〈ξ〉

)s |u(ξ)| ∈ L2(Rd),

where 〈ξ〉 = (e+ |ξ|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 [84, 86]),

(1.4.1) ‖(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.

For the Debye-Yukawa type potential collision operators, we have same type oflower bounded estimate for the collision operators, see the Proposition 3.3.2 of Section3.3.

1.4. LOGARITHMIC SOBOLEV SPACE 9

In the chapter VI, we need the logarithmic derivative norm in L2.

Proposition 1.4.1. Let s > 1/2, then for any u ∈ S(Rd)

(1.4.2) ||(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·ξ − 1|2|u(ξ)|2dξ

)dz =

∫I(ξ)|u(ξ)|2dξ,

where

I(ξ) =

∫|z|≤1

|eiz·ξ − 1|2| log |z| |2s−1

|z|ddz.

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

C−1I(ξ) ≤ | log < ξ > |2s ≤ C(I(ξ) +O(1)).

When |ξ| ≤ 1, we have by the Taylor’s expansion that |eiz·ξ − 1|2 ≤ |z|2. Thus,

|I(ξ)| ≤∫|z|≤1

| log |z| |2s−1

|z|d−2dz = cd−1

∫ 1

0

ρ(|log ρ|)2s−1 dρ = O(1),

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

I(ξ) =

∫|y|≤|ξ|

∣∣∣eiy·ω − 1|2| log(|y||ξ|

) ∣∣∣2s−1

|y|ddy

=

∫ |ξ|

0

∣∣∣ log(

ρ|ξ|

) ∣∣∣2s−1

ρ

(∫Sd−1

|eiρ(ϑ·ω) − 1|2dϑ)dρ

=

∫ 1

0

· · · dρ+

∫ |ξ|

1

· · · dρ := I1(ξ) + I2(ξ),

where we have used the polar coordinate (ρ, ϑ). Since I(ξ) is rotationally invariant, wecan simply take ω = (1, 0, · · · , 0) so that∫

Sd−1

|eiρ(ϑ·ω) − 1|2dϑ = cd−2

∫ 2π

0

|eiρϑ1 − 1|2dϑ1 =cd−2

ρ

∫ 2πρ

0

|eit − 1|2dt,

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

c1 =

∫ 2π

0

|eit − 1|2dt > 0.

10 I. SOBOLEV SPACES

If ρ ≥ 1, then we have

c1cd−2/2 ≤ c1cd−2[ρ]

ρ≤∫

Sd−1

|eiρ(ϑ·ω) − 1|2dϑ ≤ c1cd−2[ρ+ 1]

ρ≤ 2c1cd−2.

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

∫ |ξ|

1

∣∣∣ log(

ρ|ξ|

) ∣∣∣2s−1

ρdρ =

∫ log |ξ|

0

τ 2s−1dτ = (log |ξ|)2s/(2s),

where we have used the change of variables τ = log( |ξ|

ρ

). If ρ ≤ 1, then the Taylor

expansion implies ∣∣∣∣∫Sd−1

|eiρ(ϑ·ω) − 1|2dϑ∣∣∣∣ ≤ cd−1ρ

2.

Since | log( |ξ|

ρ

)| = | log ρ|+ | log |ξ|| when ρ ≤ 1 and |ξ| ≥ 1, we have

|I1(ξ)| ≤ cd−1

∫ 1

0

ρ(| log ρ|+ | log |ξ||)2s−1dρ = O((log < ξ >)2s−1

).

Combining all the above estimates completes the proof of the proposition 1.4.1.

1.5. Sobolev embedding Theorem

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

Ck,σ(Rd) =∂αu ∈ L∞(Rd), |α| ≤ k; sup

|x−y|6=0

|∂βu(x)− ∂βu(y)||x− y|σ

< +∞, |β| = k,

with the norm

‖u‖Ck,σ =∑|α|≤k

‖∂αu‖L∞(Rd) +∑|β|=k

sup|x−y|6=0

|∂βu(x)− ∂βu(y)||x− y|σ

.

It is Banach space.

Theorem 1.5.1. (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 + σ, kN, σ ∈]0, 1[, we have continuous embedding

Hs(Rd) ⊂ Ck,σ(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(ξ) =φ(ξ)|ξ|−d

1 + log(2 + |ξ|),

where φ ∈ 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) .

1.5. SOBOLEV EMBEDDING THEOREM 11

Proof of Theorem 1.5.1 : (a) By density of S(Rd) in Hs(Rd), we prove the followingSobolev inequality

(1.5.1) ‖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

λp−1dλdx = p

∫ ∞

0

λp−1M(|u| > λ

)dλ,

where M( · ) 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∞ ≤ (2π)−d‖F(u1,A)‖L1 ≤ (2π)−d

∫B(0,A)

(1 + |ξ|2)−s/2(1 + |ξ|2)s/2|u(ξ)|dξ

≤ (2π)−d‖u‖Hs

(∫B(0,A)

(1 + |ξ|2)−sdξ

)1/2

≤ Cs‖u‖HsAd2−s.

The triangle inequality implies that for any positive A,

x ∈ Rd; |u(x)| > λ ⊂ x ∈ Rd; |u1,A(x)| > λ⋃x ∈ Rd; |u2,A(x)| > λ.

We choose

A = Aλ =

(λ

4Cs‖u‖Hs

) pd

,

thenMx ∈ Rd; |u1,Aλ

(x)| > λ = 0.

And we have

‖u‖pLp(Rd)

= p

∫ ∞

0

λp−1M(|u2,Aλ

| > λ)dλ.

It is well known that

M(|u2,Aλ

| > λ)

=

∫(|u2,Aλ

|>λ) dx

≤∫(

|u2,Aλ|>λ) 4|u2,Aλ

(x)|2

λ2dx ≤

4‖u2,Aλ‖2

L2

λ2.

So that

‖u‖pLp(Rd)

≤ 4p

∫ ∞

0

λp−3‖u2,Aλ‖2

L2dλ

≤ 4p(2π)−d

∫ ∞

0

λp−3

∫|ξ|≥Aλ

|u(ξ)|2dξdλ.


By the definition of Aλ, we have

|ξ| ≥ Aλ ⇐⇒ λ ≤ Cξ = 4Cs‖u‖Hs |ξ|dp .

Fubini Theorem implies that

‖u‖pLp(Rd)

≤ 4p(2π)−d

∫Rd

(∫ Cξ

0

λp−3dλ

)|u(ξ)|2dξ

≤ 4p(2π)−d

p− 2

(4Cs‖u‖Hs

)p−2∫

Rd

|ξ|d(p−2)

p |u(ξ)|2dξ.

As s = d(12− 1

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

(1.5.2) ‖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

(1.5.3) ‖u‖L∞ ≤ (2π)−d‖u‖L1 ≤ Cs‖u‖Hs

where ∫(1 + |ξ|2)−sdξ = 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

|ξ||u1,A(ξ)|dξ ≤ ‖u‖Hs

(∫|ξ|≤A

|ξ|2−2sdξ

)1/2

≤ Cs‖u‖HsA1−s+ d2 .

On the other hand

‖u2,A‖L∞ ≤∫

Rd

|u2,A(ξ)|dξ ≤ ‖u‖Hs

(∫|ξ|≥A

|ξ|−2sdξ

)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,(1.5.3) implies

|u(x)− u(y)| ≤ 2‖u‖L∞ ≤ Cs‖u‖Hs |x− y|s−d2 .

We get finally,

(1.5.4) ‖u‖Cs− d

2≤ Cs‖u‖Hs .

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

1.5. SOBOLEV EMBEDDING THEOREM 13

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)

(∫|ξ|≤A

|ξ|2−ddξ)1/2

≤ CRA‖u‖Hd/2(Rd).

1

|B|1/2‖u2,A‖L2(Rd) ≤ C(AR)−d/2

(∫|ξ|≥A

|ξ|d|u(ξ)|2dξ)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,

(1.5.5) ‖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 1.5.2. 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 + |ξ|2)su(ξ)

)∈ H−s(Rd),

and for any ϕ ∈ D(Rd \ 0 ),

0 = (u, ϕ)Hs =

∫Rd

us(ξ) ¯ϕ(ξ)dξ = 〈us, ϕ〉.

This implies that Supp us ⊂ 0, i. e.

us =∑|α|≤N

aα∂αδ0 ∈ H−s(Rd).

Now −s ≥ −d/2 implies that us = 0 which proved Theorem 1.5.2.

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

such that

(1.5.6)

∫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

(1.5.7)

∫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 1.4.1, we have the followinglogarithmic Sobolev inequality (see [84]). If s > 1/2, there exists Cs > 0 such that

(1.5.8)

∫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 [106]).

Lemma 1.5.1. 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.

1.6. Compactness and interpolation inequalities

We have the following interpolation results. If s1 < s2, s = (1− θ)s1 + θs2, θ ∈]0, 1[and u ∈ Hs2(Rd), then we have the convexity inequality

(1.6.1) ‖u‖Hs ≤ ‖u‖1−θHs1‖u‖θ

Hs2

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

(1.6.2) ‖u‖Hs ≤ ε‖u‖Hs2 + ε−θ

1−θ ‖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 1.6.1. 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),

1.6. COMPACTNESS AND INTERPOLATION INEQUALITIES 15

for any s < s′.We can prove this Theorem by the interpolation inequality (1.6.2). But we present

here the following more general compactness theorem.

Theorem 1.6.2. Let s < s′, then the multiplication by a function of S(Rd) is acompact operator from Hs′(Rd) into Hs(Rd).

Proof : Let φ ∈ S(Rd) and un ⊂ Hs′(Rd) such that ‖un‖Hs′ ≤ 1, we have toprove that we can extract a subsequence and unk

such that φunk converge in

Hs(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 element

u ∈ Hs′(Rd) with ‖u‖Hs′ ≤ 1. Denote by vk = unk− u, we have to prove

supk‖vk‖Hs′ ≤ C =⇒ φvk → 0 in Hs(Rd).

Now for any R > 0,∫(1 + |ξ|2)s|F(φvk)(ξ)|2dξ ≤

∫|ξ|≤R

(1 + |ξ|2)s|F(φvk)(ξ)|2dξ∫|ξ|≥R

(1 + |ξ|2)s−s′(1 + |ξ|2)s′|F(φvk)(ξ)|2dξ

≤∫|ξ|≤R

(1 + |ξ|2)s|F(φvk)(ξ)|2dξ +1

(1 +R2)s′−s‖φvk‖2

Hs′ .

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

1

(1 +R2)s′−s‖φvk‖2

Hs′ ≤ε

2.

On the other hand,

F(φvk)(ξ) = (2π)−d

∫φ(ξ − η)vk(η)dη =

∫(1 + |η|2)s′ψξ(η)vk(η)dη

where for any ξ ∈ Rd,

ψξ(η) = (2π)−d(1 + |η|2)−s′φ(ξ − η) ∈ S(Rd).

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

limk→∞

F(φvk)(ξ) = 0.

If we have the estimate

(1.6.3) sup|ξ|≤R, k∈N

|F(φvk)(ξ)| ≤M < +∞,

then Lebesgue dominant theorem implies that

limk→∞

∫|ξ|≤R

(1 + |ξ|2)s|F(φvk)(ξ)|2dξ = 0

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


We prove now (1.6.3), it is clear that∣∣F(φvk)(ξ)∣∣ ≤ (2π)−d

∫ ∣∣φ(ξ−η)∣∣ ∣∣vk(η)

∣∣dη ≤ ‖vk‖Hs′

(∫(1+ |η|2)−s′|φ(ξ−η)|2dη

)1/2

.

Since φ ∈ S(Rd), we have

|φ(ξ − η)|2 ≤ CN0(1 + |ξ − η|2)−N0

with N0 = d/2 + |s′|+ 1. Thus, for any |ξ| ≤ R,∫(1 + |η|2)−s′|φ(ξ − η)|2dη ≤ CR + 2N0CN0

∫|η|≥2R

(1 + |η|2)−d/2−1dη ≤MR <∞

where we have used

ξ, η ∈ Rd, |ξ| ≤ R, |η| ≥ 2R =⇒ |η − ξ| ≥ |η|/2.We have finish the proof of Theorem 1.6.2.

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

Theorem 1.6.3. Let s > 0 and K a compact subset of Rd, then there exists CK,s > 0such that

(1.6.4) C−1‖u‖2Hs ≤

∫|ξ|2s|u(ξ)|2dξ ≤ 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 thatHs

K(Rd) = HsK(Rd)

with equivalent norms.

The inequality (1.6.4) is known by Poincare inequality. The classical form is fors = 1 and present as

(1.6.5) ‖u‖L2 ≤ CK‖∇u‖L2 ,

for all u ∈ H1K(Rd).

Proof : The right inequality is obvious. We prove the left inequality of (1.6.4) byabsolve , we suppose that there exists a sequence un of Hs

K(Rd) such that

‖un‖Hs = 1, limn→∞

∫|ξ|2s|un(ξ)|2dξ = 0.

By Rellich theorem, there exists a subsequence unk which converges to an element

u ∈ L2K(Rd) with ‖u‖L2 = 1. By Cauchy-Schwarz inequality,

|unk(ξ)− u(ξ)| ≤ (2π)−d

∫K

|unk(x)− u(x)|dx ≤ CK‖unk

− u‖L2 .

Then unkconverges uniformly to u. So Lebesgue dominant theorem implies that for

any R > 0, ∫|ξ|≤R

|ξ|2s|u(ξ)|2dξ = 0,

this implies that u = 0 and thus u = 0, we have proved (1.6.4) by contradiction.

1.6. COMPACTNESS AND INTERPOLATION INEQUALITIES 17

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

Proposition 1.6.1. Let 0 ≤ t ≤ s, there exists a constant C such that for any δ > 0

(1.6.6) ‖u‖Ht ≤ C δs−t ‖u‖Hs

for any u ∈ HsBδ

(Rd), where Bδ = x ∈ Rd; |x− x0| ≤ δ.

Remark : The classical Poincare inequality (1.6.5) is holds for the function u ∈ H1(Rd)such that Supp u ⊂ M = x = (x1, · · · , xd) ∈ Rd; |x1| ≤ δ (so that no necessarycompact). And the constant CK in (1.6.5) can be as Cd δ.

From (1.6.4), we can also take the homogeneous norms in (1.6.6).

Proof of Proposition 1.6.1: By translation, we can take x0 = 0. Setting v(x) =u(δx), 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(ξ) = δ−du

(ξ

δ

).

We have finish the proof of Proposition 1.6.1 by using (1.6.4) and By dilatation.

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

Definition 1.6.1. (1) For p ∈ [1,+∞[ and r > 0, we define the function space

Lpr(Rd) = f ∈ Lp(Rd); ‖f‖p

Lpr(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‖2Hs

r (Rd) =

∫Rd

|Λsf(x)|2 < x >2r dx,

where

Λsf = F−1(1 + |ξ|2)s/2f(ξ)

).

• Lpr(Rd) is Banach space, but Lp

(logL

)r(Rd) is not, since

‖f‖Lp(log L)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‖2Hk

r (Rd) =∑|α|≤k

∫Rd

|∂αf(x)|2 < x >2r dx.

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

‖f‖2Hs

k(Rd) =∑|α|≤k

∫Rd

(1 + |ξ|2)s|F(xαf)(ξ)|2dξ.

We have the following interpolation inequality for the weighted in Sobolev space(see [45]).

Lemma 1.6.1. For any s ∈ R+, r ∈ N, ε > 0. Then, there exits a constant Ks,r,ε,d >0 such that for any f ∈ S(Rd),

(1.6.7) ‖f‖2Hs

r (Rd) ≤ Ks,r,ε,d‖f‖Hs−ε2r (Rd)‖f‖Hs+ε

0 (Rd).

Remark : By density, (1.6.7) is holds for any f ∈ Hs−ε2r (Rd)

⋂Hs+ε

0 (Rd).

Proof : Write

‖f‖2Hs

r (Rd) =∑|α|≤r

∫Rd

(1 + |ξ|2)s|F(xαf)(ξ)|2dξ

≤∑|α|≤r

∣∣∣∣∫Rd

(1 + |ξ|2)s(∂αf

)(ξ)(∂α ¯f)(ξ)dξ

∣∣∣∣≤

∑|α|≤r

∣∣∣∣∫Rd

∂α((1 + |ξ|2)s

(∂αf

)(ξ))

¯f(ξ)dξ

∣∣∣∣ .Notice that

∂α((1 + |ξ|2)sg(ξ)

)=∑β≤α

Pα−β(ξ)((1 + |ξ|2)s−(|α−β|))∂βg(ξ)

where Pα−β(ξ) are polynomials of degree |α−β|. Then there exits constants Kα,β suchthat ∣∣∣Pα−β(ξ)

(1 + |ξ|2

)−|α−β| ≤ Kα,β.

1.7. SOBOLEV SPACE ON AN OPEN DOMAIN AND TRACE THEOREM 19

We obtain the estimate

‖f‖2Hs

r (Rd) ≤∑|α|≤r

∑β≤α

∣∣∣∣∫Rd

Pα−β(ξ)(1 + |ξ|2)s−(|α−β|)(∂α+β f)(ξ)

¯f(ξ)dξ

∣∣∣∣≤

∑|α|≤r

∑β≤α

∫Rd

∣∣∣Pα−β(ξ)(1 + |ξ|2)s−(|α−β|)−ε∣∣∣1/2∣∣∂α+β f(ξ)

∣∣×∣∣∣Pα−β(ξ)(1 + |ξ|2)s−(|α−β|)+ε

∣∣∣1/2∣∣f(ξ)∣∣dξ

≤∑|α|≤r

∑β≤α

Kα,β

(∫Rd

(1 + |ξ|2)s+ε|f(ξ)|2dξ)1/2

×(∫

Rd

(1 + |ξ|2)s−ε|F(xα+βf(x)

)(ξ)|2dξ

)1/2

.

The lemma is proven.

We have also

Lemma 1.6.2. Let r > 0, ε > 0, then there exists a constant Cε > 0 such that forany f ∈ S(Rd),

(1.6.8) ‖f‖2L2

r(Rd) ≤ Cε‖f‖L12r(Rd)‖f‖H

d/2+ε0 (Rd)

.

This is a direct application of Sobolev embedding (1.5.3),∫Rd

(1 + |x|2)r|f(x)|2dx ≤ ‖f‖L12r‖f‖L∞ ≤ Cε‖f‖L1

2r‖f‖

Hd/2+ε0

.

1.7. Sobolev space on an open domain and trace Theorem

Definition 1.7.1. 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

(1.7.1) ‖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); ∂αu ∈ L2(Ω), |α| ≤ k

.

This is also a Hilbert space. We have

Proposition 1.7.1. If Ω is a regular open domain of Rd, k ∈ N, then

(1.7.2) 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 [30]. 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 1.7.1. Let s ∈ R, s > 12. The trace operator γ : C∞

0 (Rd) → C∞0 (Rd−1

defined byγ(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 δ0 /∈ H−1/2(R). If d ≥ 2, take g ∈ S(Rd−1),we consider the sequence ukk∈N ⊂ S ′(Rd) defined by

uk(ξ) = 12≤|ξ1|≤k

|ξ1|−1(

log |ξ1|)− 3

4g(ξ′).

Then‖uk‖2

H1/2(Rd) ≤ 4 log 2 ‖g‖2H1/2(Rd−1).

But

uk(0, x′) = (2π)−1 g(x′)

∫2≤|ξ1|≤k

|ξ1|−1(

log |ξ1|)− 3

4dξ1

= (π)−1 g(x′)

∫ k

2

r−1(

log r)− 3

4dr = Ck g(x

′)

where Ck = 4(π)−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 γ can’t be extended into a continuous maps fromH1/2(Rd) into L2(Rd−1).

CHAPTER II

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 [14, 66, 70, 71, 95, 96].

2.1. Symbols class

Recall that ifP (x,D) =

∑|α|≤m

aα(x)Dαx

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

P (x,D)u(x) = (2π)−d

∫Rd

eix · ξP (x, ξ)u(ξ)dξ,

whereP (x, ξ) =

∑|α|≤m

aα(x)ξαx

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

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 α, β ∈ N, we have, for all x, ξ ∈ Rd,

(2.1.1) |∂αξ ∂

βxa(x, ξ)| ≤ Cα,β(1 + |ξ|)m−|α|.

Sm is called the space of symbols of order m. We write S−∞ =⋂Sm, S∞ =

⋃Sm.

It is clair that

a ∈ Sm ⇒ ∂αξ ∂

βxa ∈ Sm−|α|; a ∈ Sm1 , b ∈ Sm2 ⇒ ab ∈ Sm1+m2 .

Some examples :

(1) P (x, ξ) =∑

|α|≤m aα(x)ξα ∈ Sm if aα ∈ C∞b (Rd), where

C∞b (Rd) = f ∈ C∞(Rd); ∂αf ∈ L∞(Rd), ∀α ∈ Nd.

We say that P (x, ξ) is a differential symbol of order m.(2) Let a(ξ) ∈ C∞(Rd \ 0) a (positive) homogeneous function of order m ∈ R in

the sense : ∀λ > 0, a(λξ) = λma(ξ), then a(ξ) = (1− χ(ξ))a(ξ) ∈ Sm where

χ ∈ C∞0 (Rd); χ(ξ) = 1, |ξ| ≤ 1

2and χ(ξ) = 0, |ξ| ≥ 1.

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

21

22 II. PSEUDO-DIFFERENTIAL OPERATORS

(3) If a ∈ S(Rd), then a(ξ) ∈ S−∞.(4) The function eix · ξ 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, ξ) ∼∑

aj(x, ξ).

If for any k ≥ 0,

a(x, ξ)−k∑

j=0

aj(x, ξ) ∈ Smk+1 .

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

(positive) homogeneous of order m − j for |ξ| ≥ 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, ξ) ∼∑

aj(x, ξ).

We need the classical Borel Lemma.

Lemma 2.2.1. Let bjj∈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

bjχ(λjx)xj

j!

with λj +∞ to choose, where χ is cutoff function near to0. For j > k, we have∣∣∣∣ dk

dxk

(bjχ(λjx)

xj

j!

)∣∣∣∣ =

∣∣∣∣∣k∑

l=0

C lkbjχ

k−l(λjx)λk−lj

xj−l

(j − l)!

∣∣∣∣∣ ≤ C|bj|λk−jj

1

(j − k)!.

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

2.2. ASYMPTOTIC EXPANSION 23

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

a(x, ξ) =∑

aj(x, ξ) =∑

(1− χ(εjξ))aj(x, ξ),

with εj 0 to choose. Since

(1− χ(εjξ)) = 0, if |ξ| ≤ 1

2εj

.

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

R ≤ 1

2εj

,

then for any |ξ| ≤ R and for all j ≥ j0

aj(x, ξ) = (1− χ(εjξ))aj(x, ξ) = 0.

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

(2.2.1) |∂αξ ∂

βx aj(x, ξ)| ≤ 2−j(1 + |ξ|)1+mj−|α|,

for any x, ξ ∈ Rd and any |α|+ |β| ≤ j. In fact, by Leibnitz formula,

∂αξ ∂

βx aj(x, ξ) = (1− χ(εjξ))∂

αξ ∂

βxaj(x, ξ)−

∑0<γ≤α

Cγαε

|γ|j (∂γχ)(εjξ))∂

α−γξ ∂β

xaj(x, ξ),

then|∂α

ξ ∂βx aj(x, ξ)| ≤ Cα,β,jεj(1 + |ξ|)1+mj−|α|

with

Cα,β,j = supx,ξ∈Rd

|1− χ(εjξ)|εj|ξ|

|∂αξ ∂

βxaj(x, ξ)|(1 + |ξ|)−(mj−|α|)

+∑

0<γ≤α

Cγα|εjξ||γ|(∂γχ)(εjξ))|∂α−γ

ξ ∂βxaj(x, ξ)|(1 + |ξ|)−(mj−|α−γ|)

.

Since aj ∈ Smj , we choose 0 < εj such that

εj sup|α|+|β|≤j

Cα,β,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 α, β ∈ Nd, k ∈ N, we have to prove

(2.2.2)∣∣∣∂α

ξ ∂βx

(a(x, ξ)−

k∑j=0

aj(x, ξ))∣∣∣ ≤ Cα,β,j(1 + |ξ|)mk+1−|α|.

We choose N ≥ |α|+ |β| big such that mN + 1 ≤ mk+1, then (2.2.1) implies that∣∣∣∂αξ ∂

βx

(a(x, ξ)−

N−1∑j=0

aj(x, ξ))∣∣∣ ≤ (1 + |ξ|)mk+1−|α|.

On the other hand, we have

a(x, ξ)−k∑

j=0

aj(x, ξ) =(a(x, ξ)−

N−1∑j=0

aj(x, ξ))

+N−1∑

j=k+1

aj(x, ξ)

+k∑

j=0

(aj(x, ξ)− aj(x, ξ)

).

Then∑N−1

j=k+1 aj(x, ξ) ∈ 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) = (2π)−d

∫Rd

eix · ξa(x, ξ)u(ξ)dξ.

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, ξ) =k∑

|α|≤m

aα(x)ξα,

where aα ∈ C∞b (Rd), |α| ≤ k. And a(x,D)u ∈ S(Rd) for any u ∈ S(Rd). More

generally, 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 xj

are

[a(x,D), Dj] = i(∂xj

a)(x,D), [a(x,D), xj] = −i

(∂ξja)(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 25

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

|a(x,D)u(x)| ≤ (2π)−d

∫Rd

(1 + |ξ|)m|u(ξ)|dξ supx,ξ∈Rd

(|a(x, ξ)|(1 + |ξ|)−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(∂xj

a)(x,D)u(x).

Since F(xju) = −Dju(ξ), we have that

a(x,D)(xju) = xja(x,D)u− i(∂ξja)(x,D)u(x).

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

ξ ∂β′

x a)(x,D)xα′′Dβ′′u; α′ + α′′ = α, β′ + β′′ = β.

Hence xαDβa(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) = (2π)−d

∫Rd

ei(x−y) · ξa(x, ξ)dξ,

a(x,D)u(x) =

∫K(x, x− y)u(y)dy

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

a(x, ξ) =

∫K(x, x− y)e−iy · ξdy.

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) =

∫Rd

u(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, ξ) ∈ Sm and

a∗(x, ξ) ∼∑α|

1

α!∂α

ξ Dαx a(x, ξ).


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, ξ) = (2π)−d

∫e−iy · ηa(x− y, ξ − η)dydη,

and the asymptotic expansion deduced from Taylor formula. We omit the detail ofproof, and send to [14, 66, 96].

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, ξ) ∼

∑α

1

α!

(∂α

ξ a1

)(x, ξ)

(Dα

x a2

)(x, ξ).

We have again the oscillatory integral(a1]a2

)(x, ξ) = (2π)−d

∫e−i(x−y) · ξ−η)a1(x, η)a2(y, ξ)dydη.

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, ξ) =1

i

a1, a2

(x, ξ) + rm1+m2−2(x, ξ),

where rm1+m2−2 ∈ Sm1+m2−2, andf, g

(x, ξ) =

d∑j=1

((∂ξj

f)(x, ξ)(∂xjg)(x, ξ)− (∂xj

f)(x, ξ)(∂ξjg)(x, ξ)

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

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, ξ)| ≥ c|ξ|m, ∀ |ξ| ≥ 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, ξ)− 1 ∈ S−∞ and (b ] a)(x, ξ)− 1 ∈ S−∞.

2.5. CONTINUITY IN SOBOLEV SPACES 27

Proof : In fact, if we take

b0(x, ξ) =(1− χ(ξ))

am(x, ξ)∈ S−m

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

(a ] b0)(x, ξ) = 1− r(x, ξ),

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.

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) =

∫Rd

K(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)| ≤ (2π)−d

∫|a(x, ξ)|dξ ≤ C1

∫(1 + |ξ|)−d−1dξ ≤ C2.

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

[xj1 , [xj2 · · · , [xjd, a(x,D)]]] = i|α|

(∂α

ξ a)(x,D),

it is a pseudo differential operators of order −d− 1− |α|, 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, ξ)|2.Then, by Lemma 2.1.1,

c(x, ξ) = (M − |a(x, ξ)|2)1/2 ∈ S0,

since M − |a(x, ξ)|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 ξ but instead some decay as x→∞.

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

supξ∈Rd

∑|α|≤d+1

∫|∂α

xa(x, ξ)|dx ≤M < +∞.

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

In fact, we have

F(a(x,D)u(x)

)(η) =

∫A(η − ξ, ξ)u(ξ)dξ,

where

A(η, ξ) = (2π)−d

∫e−ix · ηa(x, ξ)dx.

By hypothesis(1 + |η|)d+1|A(η, ξ)| ≤ CM.

2.5. CONTINUITY IN SOBOLEV SPACES 29

which implies that∫|A(η − ξ, ξ)|dη ≤ CM,

∫|A(η − ξ, ξ)|dξ ≤ 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−s

x (Λsxu)‖L2 ≤ C‖Λs

xu‖L2 ∀ Λsxu ∈ S(Rd).

Since Λs−mx a(x,D)Λ−s

x = a(x,D) is a pseudo-differential operator of order 0, it iscontinuous 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(ξ) ≥ 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, ξ) ≥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, ξ) ≥ c|ξ|2m, ∀ |ξ| ≥ 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, ξ) =1

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

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

b(x, ξ)− 3

4c(1 + |ξ|2)m ≥ ε > 0, ∀ |ξ| ≥ C.

Then

d(x, ξ) =

(b(x, ξ)− 3

4c(1 + |ξ|2)m

)1/2 (1− χ(C−1ξ)

)∈ Sm.

Using the composition formula of Theorem 2.4.2,

(d∗ ] d)(x, ξ) = b(x, ξ)− 3

4c(1 + |ξ|2)m + r(x, ξ)

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 (1.6.2) deduced

‖u‖Hm−1 ≤ ε‖u‖Hm + Cε,N‖u‖H−N .

ChooseC0ε =

c

4.

We have proved the Proposition 2.5.1.

CHAPTER III

Boltzmann equations without angular cutoff

The mathematical theory of Boltzmann equations have a very long history andimmense works by asymptotic analysis, numerical analysis and also functional analysis.In this lecture, we focus to the cross section with singulary kernel, we say withoutcutoff. We consider only the so-called mathematical Maxwellian case. For the generalpresentation of Boltzmann equations, see the standard references [28, 29]. We stateour problem in 3-dimension case, but it is true in any dimension.

3.1. Boltzmann equations

We consider the following Boltzmann equation,

(3.1.1) ft(t, x, v) + v · ∇xf(t, x, v) = Q(f, f)(t, x, v)

where f(t, x, v) is the unknown function and it represent the (probability) density offinding particles at time t ≥ 0, located around position x ∈ R3, with velocity closeto v ∈ R3. So that f(t, x, v) ≥ 0. The right hand side, called Boltzmann collisionoperator is an operator acting only with respect to velocity variable v, which is bilinearand given by

(3.1.2) Q(g, f) =

∫R3

∫S2

B (v − v∗, σ) g(v′∗)f(v′)− g(v∗)f(v) dσdv∗ ,

where B(v − v∗, σ) is a given positive function, depending only on the interaction lawbetween particles. For convenience, we choose the σ−representation to describe thepost- and pre-collisional velocities, that is, for σ ∈ S2,

(3.1.3) v′ =v + v∗

2+|v − v∗|

2σ, v′∗ =

v + v∗2

− |v − v∗|2

σ .

The non-negative function B(z, σ) called the cross-section depends only on the relativevelocity |z| = |v − v∗| and the scalar product < z

|z| , σ >= cos θ.

3.1.1. Cross sections. In the almost cases, the collision kernel B cannot be ex-pressed explicitly, but to capture its main feature, it can be assumed to be of the form

(3.1.4) B(|v − v∗|, cos θ) = Φ(|v − v∗|)b(cos θ), cos θ =⟨ v − v∗|v − v∗|

, σ⟩, 0 ≤ θ ≤ π

2.

The cross section is never integrable for σ ∈ S2. The singularity for θ close zerocorresponding to so called grazing collision.

This singularity implies huge difficulties in the mathematical treatment of Boltz-mann equation, and this explains why H. Grad has introduced a cutoff assumption for

31

32 III. BOLTZMANN EQUATIONS WITHOUT ANGULAR CUTOFF

such collisional cross sections, namely, remove the singularity of kernel b by multipli-cations of a cutoff function. Then σ 7→ B(|z|, σ) was integrable on the unit sphere S2.

If the inter-molecule potential satisfies the classical inverse-power law potentialU(ρ) = ρ−(γ−1), γ > 2, then,

(3.1.5) Φ(|v − v∗|) = |v − v∗|γ−5γ−1 ; sin θ b(cos θ) ≈ Kθ−1−2α when θ → 0,

where K > 0, 0 < α = 1γ−1

< 1. So that the kernel function b present the strong

singularity, and it is never integrable on the unit sphere S2. We say non cutoff crosssection.

The Maxwellian molecule case corresponds to γ = 5 and Φ = 1. Finally, note thatwhen γ = 2, it is the Coulomb potential, and it leads to a differential equation, namelythe Fokker-Planck-Landau equation.

In the end of this section, we do a asymptotic calculus for a Debye-Yukawa typepotentials where the potential function is given by

(3.1.6) U(ρ) = ρ−1e−ρs

, with s > 0.

In some sense, it is a model between the Coulomb potential corresponding to s = 0and the inverse power law potential. In fact, the classical Debye-Yukawa potential iswhen s = 1. We will show that the collision cross-section of this kind of potentials hasthe singularity in the following form,

(3.1.7) b(cos θ) ≈ Kθ−2(log θ−1

) 2s−1, when θ → 0.

In this lecture, we focus to the non cutoff case. To concentrate on the singularityarising from the grazing collisions and to avoid the difficulty coming from the rela-tive velocity in the cross-section, we will only consider the mathematical Maxwellianmolecule type case, that is, we take Φ = 1 .

3.1.2. Basic properties of Boltzmann’s kernel. We recall some fundamentalfacts,

• The conservation of momentum is

(3.1.8) v + v∗ = v′ + v′∗.

• The conservation of kinetic energy is

(3.1.9)|v|2

2+|v∗|2

2=|v′|2

2+|v′∗|2

2.

In the cutoff case, namely when B is locally integrable, we can decompose Q =Q+ −Q−, with

(3.1.10)

Q+(g, f) =

∫R3

∫S2

B (v − v∗, σ) g(v′∗)f(v′)dσdv∗ ;

Q−(g, f) =

∫R3

∫S2

B (v − v∗, σ) g(v∗)f(v)dσdv∗.

We shall use the so-called pre/post collisional change of variables (v, v∗, σ) 7→ (v′, v′∗, σ)and also the change of variables (v, v∗, σ) 7→ (v∗, v, σ), which ensures that for all func-tion f(v, v∗, v

′, v′∗, σ), one has (at the formal level, i. e. if two sides of following formulas

3.1. BOLTZMANN EQUATIONS 33

have make sense.) :∫∫R3×R3

∫S3

f(v, v∗, v′, v′∗, σ)dσdv∗dv =

∫∫R3×R3

∫S3

f(v′, v′∗, v, v∗, σ)dσdv∗dv,

and ∫∫R3×R3

∫S3

f(v, v∗, v′, v′∗, σ)dσdv∗dv =

∫∫R3×R3

∫S3

f(v∗, v, v′∗, v

′, σ)dσdv∗dv.

As an immediate consequence of those formulas, we get the following various weakformulations for Boltzmann’s kernel(3.1.11)∫

R3

Q(g, f)(v)ϕ(v)dv =

∫∫R3×R3

∫S2

B (v − v∗, σ) g(v∗)f(v) ϕ(v′)− ϕ(v) dσdv∗dv.

Formally, we can justifier this formula by a change of variable(formally) (v, v∗, σ) 7→(v′, v′∗, σ) for Q+. By using the change of variables (v, v∗, σ) 7→ (v∗, v, σ), we get also

(3.1.12)

∫R3

Q(f, f)(v)ϕ(v)dv =1

2

∫∫R3×R3

∫S2

B (v − v∗, σ) f(v)f(v∗)×

×ϕ(v′∗) + ϕ(v′)− ϕ(v′)− ϕ(v) dσdv∗dv,and(3.1.13)∫

R3

Q(f, f)(v)ϕ(v)dv = −1

4

∫∫R3×R3

∫S2

B (v − v∗, σ) f(v′)f(v′∗)− f(v)f(v∗)×

×ϕ(v′∗) + ϕ(v′)− ϕ(v′)− ϕ(v) dσdv∗dv.

3.1.3. Conservations laws. If we put in formula (3.1.12) the test function asϕ(v) = 1, vj, j = 1, 2, 3 and |v|2/2, we have

(3.1.14)

∫R3

Q(f, f)(v)

1vj|v|22

dv = 0.

So that we get the the conservation of mass, momentum and energy, i.e. if f(t, x, v) isa solution of equation (3.1.1), we have that for any times t1, t2

(3.1.15)

∫∫R3×R3

f(t1, x, v)dxdv =

∫∫R3×R3

f(t2, x, v)dxdv ;∫∫R3×R3

f(t1, x, v)vjdxdv =

∫∫R3×R3

f(t2, x, v)vjdxdv, j = 1, 2, 3;∫∫R3×R3

f(t1, x, v)|v|2

2dxdv =

∫∫R3×R3

f(t2, x, v)|v|2

2dxdv.

Defining the entropy dissipation by

D(f) = −∫

R3

Q(f, f)(v) log f(v)dv,

we get

(3.1.16) D(f) =1

4

∫∫R3×R3

∫S2

B (v − v∗, σ) f(v′)f(v′∗)− f(v)f(v∗)×


× log

(f(v′∗)f(v′)

f(v)f(v∗)

)dσdv∗dv.

We observe that D(f) ≥ 0. It is also possible to prove that

D(f) = 0 ⇔ Q(f, f)(v) ≡ 0

⇔ ∃C ≥ 0, T > 0, v0 ∈ R3, f(v) = C exp

(−|v − v0|2

2T

).

So that the stationary solution of Boltzmann equations is Maxwellian functions. Thisis also a part of Boltzmann’s H-theorem.

In particular, we denote the absolute Maxwellian distribution by

(3.1.17) µ(v) = (2π)−32 e−

|v|22 ,

we have Q(µ, µ) = 0.

3.1.4. Landau equations. We consider also the Landau equation which is also awell-known kinetic equation naturally associated with the Boltzmann equation.

The (full) spatially inhomogeneous Landau equation reads

(3.1.18) ft + v · ∇xf = ∇v

(a(f) · ∇vf − b(f)f

)≡ Q(f, f),

where a = (aij) and b = (b1, · · · , b3) are defined as follows (convolution is w.r.t. thevariable v)

aij(f) = aij ? f, bj(f) =3∑

i=1

(∂viaij

)? f , i, j = 1, 2, 3,

with

aij(v) =

(δij −

vivj

|v|2

)|v|γ+2, γ ∈ [−3, 1].

Now the collision operator Q is elliptic with respect to v variables. There is alsothe conservation of mass, momentum and energy for Landau equation. Again theMaxwellian molecule case which corresponds to γ = 0. See [61, 33] for more references.

3.1.5. Kac’s equation. We present here the Kac’s equation (see [43, 56]), it isa simplification of Boltzmann equation to one dimension case. The Kac’s operator isdefined with

(3.1.19) Q(g, f)(v) =

∫R

∫ 2π

0

(g(v sin θ + v∗ sin θ)f(v cos θ − v∗ sin θ)

−g(v∗)f(v))β(|θ|)dθdv∗ .

The non negative function β is also called the cross section non cutoff with the singu-larity.

(3.1.20) β(|θ|) ≈ Kθ−1−2α, when θ → 0

for K > 0, α ∈]0, 1[.This operator is very close to the Boltzmann operators for Maxwellian molecules

when it restricted to the radially symmetric functions. Mass and energy are conservedfor this operator, but not momentum.

3.1. BOLTZMANN EQUATIONS 35

3.1.6. Debye-Yukawa potential. Following the computation given in [28, 102],we will give an asymptotic description of the Boltzmann collision kernel B(z, σ) forthe Debye-Yukawa potential U(ρ) = ρ−1e−ρs

. Here, ρ is the distance between twointeracting particles, z = v−v∗ is relative velocity, σ ∈ S2 and < z

|z| , σ >= cos(π−2ϑ),

θ = π − 2ϑ is the deviation angle. Let p ≥ 0 be the impact parameter which is afunction of ϑ and z. Then Boltzmann collision cross-section is defined by

(3.1.21) B(|z|, ϑ) = |z|s(|z|, ϑ) =|z|s(|z|, ϑ)

4 cosϑ= |z| p

2 sin 2ϑ

∂p

∂ϑ,

where s(|z|, ϑ) is called the differential scattering cross-section.If ρ and ϕ are the radial and angular coordinates in the plane of motion, then

the impact parameter p(V, ϑ) is determined by the conservation of energy and angularmomentum respectively:

1

2

(ρ2 + ρ2ϕ2

)+ U(ρ) =

1

2V 2 + U(σ), (ρ ≤ σ),

ρ2ϕ = pV 2.

Here the relative speed is now denoted by V = |v − v∗|. As usual, it is impossible togive an explicit expression of solutions to this nonlinear system of ordinary differentialequations. Hence, in the following, we will study the singular behavior of the solutionsaround the grazing collisions, that it, the situation when θ ∼ 0.

By using ϕ as the independent variable to eliminate the time derivative, afterintegration, we have

ϑ =1√2V p

∫ σ

ρ0

ρ−2[V 2

2

(1− p2

ρ2

)− U(ρ) + U(σ)

]−1/2dρ+ sin−1

( pσ

),

where ρ0 is the smallest distance between two particles which satisfies

1

2V 2(1− p2

ρ20

)= U(ρ0)− U(σ) > 0.

Note that p < ρ0 < ρ ≤ σ. By the transformation u = pρ, we have

ϑ =

∫ u0

p/σ

[1− u2 − 2

V 2

(U(pu

)− U(σ)

)]−1/2du+ sin−1

( pσ

),

where u0 = p/ρ0 satisfies

1− u20 −

2

V 2

(U( pu0

)− U(σ)

)= 0.

Therefore,

θ

2=

π

2− ϑ =

π

2−∫ u0

p/σ

[1− u2 − 2

V 2

(U(pu

)− U(σ)

)]−1/2du− sin−1

( pσ

)=

∫ 1

0

dt√1− t2

−∫ p/σ

0

dt√1− t2

−∫ u0

p/σ

[1− u2 − 2

V 2

(U(pu

)− U(σ)

)]−1/2du.


By setting u = u0t, we get

θ

2=

∫ 1

p/σ

dt√1− t2

−∫ 1

pu0σ

[1− u2

0t2 − 2

V 2

(U( pu0t

)− U(σ)

)]−1/2u0dt

=

∫ 1

p/σ

dt√1− t2

−∫ 1

pu0σ

[1− t2 +

2

V 2u20

(U( pu0

)− U

( pu0t

))]−1/2dt

= −∫ p/σ

pu0σ

dt√1− t2

+

∫ 1

pu0σ

1√1− t2

[1−

(1 +

2U( pu0

)− 2U( pu0t

)

(1− t2)V 2u20

)−1/2]dt,

where we have used the fact that

1− u20

u20

=2

V 2u20

(U(

p

u0

)− U(σ)).

It is clear that there is no explicit formula for θ = θ(p, V ). To study its asymptoticbehavior when θ ∼ 0, we let σ → ∞ which is equivalent to let p → ∞. In this limit,we have u0 ≈ 1 and(

1 +2U( p

u0)− 2U( p

u0t)

(1− t2)V 2u20

)−1/2 ≈ 1−U( p

u0)− U( p

u0t)

(1− t2)V 2u20

.

Thus,

θ

2≈∫ 1

0

1√1− t2

U(p)− U(pt)

(1− t2)V 2dt.

By plugging U(ρ) = ρ−1e−ρsinto the above integral, we have

θ

2≈ 1

V 2pe−ps

∫ 1

0

(1− t2)−3/2(1− te−ps(t−s−1)

)dt.

Since

0 ≤ ∂

∂p

(∫ 1

0

(1− t2)−3/2(1− te−ps(t−s−1)

)dt

)=

∫ 1

0

(1− t2)−3/2t(t−s − 1)sps−1e−ps(t−s−1)dt ≤ Cssps−1,

it holds that

0 < c0 ≤∫ 1

0

(1− t2)−3/2(1− te−ps(t−s−1)

)dt ≤ Csp

s + c0.

where c0 =∫ 1

0(1− t2)−3/2(1− t)dt. Finally, for p→∞ (equivalently θ → 0), we have

log θ ≈ −K ′ps.

In summary, we have the Boltzmann collision cross-section for the Debye-Yukawa typepotentials as

(3.1.22) B(V, θ) = − V

sin θ

∂p2

∂θ≈ KV θ−2

(log θ−1

) 2s−1,

3.2. FOURIER TRANSFORMATION OF COLLISION OPERATORS 37

for some constant K > 0 when θ ∼ 0. Note that the cross-section B(V, θ) satisfies forany s > 0,∫ π/2

0

B(V, θ) sin θdθ = +∞, and

∫ π/2

0

B(V, θ) sin2 θdθ < +∞.

So that it is a non cutoff cross section and not satisfy the Grad’s cutoff assumption.

3.2. Fourier transformation of collision operators

In this section, we consider the Fourier transformation of Boltzmann collision op-erator Q(g, f) with respect to v variables. The variables (t, x) are considered as pa-rameters, so no appear in the notations. We prove the Bobylev formula which enableto express directly F(Q(g, f)) in terms of F(g) and F(f). Recall that this formulais true only for Maxwellian case, namely the collision kernel B depends only on thesecond variable.

3.2.1. Bobylev’s identity.

Theorem 3.2.1. Suppose that the Boltzmann collision kernel B = b(cos θ). Thenthe following formulas hold

(3.2.1) F(Q+(g, f)

)(ξ) =

∫S2

b

(ξ

|ξ|· σ)g(ξ−)f(ξ+)dσ ,

(3.2.2) F(Q−(g, f)

)(ξ) =

∫S2

b

(ξ

|ξ|· σ)g(0)f(ξ)dσ ,

and

(3.2.3) F(Q(g, f)

)(ξ) =

∫S2

b

(ξ

|ξ|· σ)

g(ξ−)f(ξ+)− g(0)f(ξ)dσ

where

(3.2.4) ξ+ =ξ + |ξ|σ

2, ξ− =

ξ − |ξ|σ2

.

Remark : The formulas (3.2.1) and (3.2.2) make sense only for cutoff cross sectionb(λ), but The formula (3.2.3) make sense for non cutoff cross section b(λ) if the functionsg and f are enough regular. In fact, we get firstly (3.2.3) for the cutoff cross sectionb(λ)χε(λ), where χε(λ) = 0 if |λ− 1| < ε, then take ε→ 0.

Proof : The change of variable (v, v∗, σ) 7→ (v′, v′∗, σ) for Q+ deduce that for all testfunction ϕ(v), holds∫

R3

Q+(g, f)(v)ϕ(v)dv =

∫R3

∫R3

∫S2

b

(v − v∗|v − v∗|

· σ)g(v∗)f(v)ϕ(v′)dσdv∗dv.

Putting

ϕ(v′) = e−iv′ · ξ = e−i v+v∗2

· ξe−i|v−v∗|σ

2· ξ

in the above identity, we get

F(Q+(g, f)

)(ξ) =

∫R3

∫R3

∫S2

b

(v − v∗|v − v∗|

· σ)g(v∗)f(v)e−i v+v∗

2· ξe−i

|v−v∗|σ2

· ξdσdv∗dv.


A key remark by Bolylev is that

(3.2.5)

∫S2

b

(v − v∗|v − v∗|

· σ)e−i

|v−v∗|σ2

· ξdσ =

∫S2

b

(ξ

|ξ|· σ)e−i

|ξ|σ2

· (v−v∗)dσ .

This is a consequence of the general equality∫S2

F (k · σ, ` · σ)dσ =

∫S2

F (` · σ, k · σ)dσ

for any k, ` ∈ S2, since there exists an isometry on S2 maps k to `. Thus

F(Q+(g, f)

)(ξ) =

∫R3

∫R3

∫S2

b

(ξ

|ξ|· σ)g(v∗)f(v)e−i v+v∗

2· ξe−i

|ξ|σ2

· (v−v∗)dσdv∗dv

=

∫R3

∫R3

∫S2

b

(ξ

|ξ|· σ)g(v∗)f(v)e−iv · ξ+

e−iv∗ · ξ−dσdv∗dv

=

∫S2

b

(ξ

|ξ|· σ)g(ξ−)f(ξ+)dσ,

this is (3.2.1). For (3.2.2), we just need to use∫S2

b (k · σ) dσ =

∫S2

b (` · σ) dσ

for any k, ` ∈ S2. We get

F(Q−(g, f)

)(ξ) =

∫R3

∫R3

∫S2

b

(v − v∗|v − v∗|

· σ)g(v∗)f(v)e−iv · ξdσdv∗dv

=

∫R3

∫R3

∫S2

b

(ξ

|ξ|· σ)g(v∗)f(v)e−iv · ξdσdv∗dv = g(0)f(ξ)

∫S2

b

(ξ

|ξ|· σ)dσ.

Recall that the cross section is cutoff and integrable.

Remark : The Bobylev identity of Theorem 3.2.1 is hold only for Maxwellien case.In the general case, that means the collision kernel is in the form

B

(|v − v∗|,

v − v∗|v − v∗|

· σ).

There exists a general form of (3.2.5) as following :

(3.2.6)

∫S2

B

(|v − v∗|,

v − v∗|v − v∗|

· σ)e−i

|v−v∗|σ2

· ξdσ

=

∫S2

B

(|v − v∗|,

ξ

|ξ|· σ)e−i

|ξ|σ2

· (v−v∗)dσ .

By using this formula, we get

(3.2.7) F(Q+(g, f)

)(ξ) =

1

(2π)3/2

∫R3

∫S2

B

(|ξ∗|,

ξ

|ξ|· σ)g(ξ−+ξ∗)f(ξ+−ξ∗)dξ∗dσ

where

B (|ξ|, cos θ) =

∫R3

B (|q|, cos θ) e−iq · ξdq

denote the Fourier transformation of B in the relative velocity variable.

3.3. COERCIVITY ESTIMATES 39

The formula (3.2.7) can be use to study the non Maxwellien case.

3.2.2. Radially symmetric case and Kac’s operator. Remark that we have

|ξ+|2 = |ξ|21 + ξ

|ξ| · σ2

, |ξ−|2 = |ξ|21− ξ

|ξ| · σ2

,

so that if we define θ by

cos θ =ξ

|ξ|· σ,

we obtain

|ξ+|2 = |ξ|2 cos2

(θ

2

), |ξ−|2 = |ξ|2 sin2

(θ

2

).

Then if the functions g and f are radially symmetric, we get, for ξ ∈ R,

(3.2.8) F(Q+(g, f)

)(ξ) =

∫ π2

−π2

β(|θ|)g(ξ sin(θ/2))f(ξ cos(θ/2))dθ ,

(3.2.9) F(Q−(g, f)

)(ξ) =

∫ π2

−π2

β(|θ|)g(0)f(ξ)dθ ,

and

(3.2.10) F(Q(g, f)

)(ξ) =

∫ π2

−π2

β(|θ|)g(ξ sin(θ/2))f(ξ cos(θ/2))− g(0)f(ξ)

dθ

where

β(|θ|) =1

2sin |θ|b(cos θ).

Direct calculation show that (3.2.10) is the Fourier transformation of Kac’s operatordefined in (3.1.19) (A very good exercise).

3.3. Coercivity estimates

3.3.1. Subelliptic estimate. We consider now the non cutoff Maxwellian crosssection with the following hypothesis

(3.3.1) B(|v − v∗|, cos θ)) = b(cos θ); sin θ b(cos θ) ≈ Kθ−1−2α when θ → 0,

where K > 0, 0 < α < 1, and b(λ) = 0 if |λ| ≤ 1/2. We prove now the followingsubeliptic estimate.

Proposition 3.3.1. Assume that the collision kernel B satisfies the assumption(3.3.1) and g ≥ 0, g ≡/ 0, g ∈ L1

2

⋂L logL. Then there exists a constant Cg > 0

depending only on B, ‖g‖L12

and ‖g‖L log L such that

(3.3.2) ‖Λαf‖2L2(R3) ≤ Cg

(−Q(g, f), f)L2(R3) + ‖f‖2

L2(R3)

for any smooth function f ∈ H2(R3), where Λ = (1 + |Dv|2)1/2.


For f ∈ H2(R3), we have

(−Q(g, f), f)L2(R3) = −∫

R6

∫S2

b(k · σ)g(v∗)f(v)(f(v′)− f(v)

)dσdv∗dv

=1

2

∫R6

∫S2

b(k · σ)g(v∗)(f(v′)− f(v)

)2

dσdv∗dv

− 1

2

∫R6

∫S2

b(k · σ)g(v∗)(f(v′)2 − f(v)2

)dσdv∗dv.

Now the proof of Proposition 3.3.1 is reduced to the proof of the following 2 lemmas.For the last term, we can estimate by the cancelation lemma (Corollary 2 of [6]), butin the Maxwellian case, we have a direct proof.

Lemma 3.3.1. (Cancelation Lemma)Then there exists a constant C > 0 depending only on b such that for any g ∈ L1

2

and f ∈ H2(R3) ∩ L22,

(3.3.3)

∣∣∣∣12∫

R6

∫S2

b(k · σ)g(v∗)(f(v′)2 − f(v)2

)dσdv∗dv

∣∣∣∣ ≤ C‖g‖L1‖f‖2L2 .

Remark : In the right hand side, we need only to suppose g ∈ L1 and f ∈ L2. Butthe well-defined of left hand side need to suppose more conditions on the function.

Proof : For ε > 0 very small, setting

bε(cos θ) = b(cos θ)χε(cos θ)

where χε ∈ C∞(R), χε(λ) = 0 if |λ− 1| < ε. Then we have

limε→0

∫R6

∫S2

bε(k · σ)g(v∗)(f(v′)2 − f(v)2

)dσdv∗dv

=

∫R6

∫S2

b(k · σ)g(v∗)(f(v′)2 − f(v)2

)dσdv∗dv,

for any g ∈ L12 and f ∈ H2(R3)∩L2

2, where k = (v−v∗)/|v−v∗|. In fact, if 0 < α < 1/2,we use Taylor formula and |v′ − v| = |v − v∗| sin(θ/2), two side are well-defined andLebesgue dominant Theorem deduced the results. If 1/2 ≤ α < 1, we use the symmetricpropriety of v and v′.

On the other hand, by the change of variables

v 7→ v′ =v + v∗

2+|v − v∗|

2σ

for fixed v∗ and σ whose Jacobian is∣∣∣ ∂v∂v′

∣∣∣ =8∣∣∣I + k ⊗ σ

∣∣∣ =8

|1 + k · σ|=

4

cos2(θ/2), θ ∈ [0,

π

2],

we get∫R6

∫S2

bε(k · σ)g(v∗)f(v′)2dσdv∗dv =

∫R6

∫S2

bε(k · σ)g(v∗)f(v)2 4

cos2(θ/2)dσdv∗dv.


Take k′ ∈ S2 such that k′ · σ = cos(θ/2), then∫S2

bε(k · σ)4

cos2(θ/2)dσ =

∫S2

bε

(2(k′ · σ)2 − 1

) 4

(k′ · σ)2dσ

= |S2|∫ π/4

−π/4

bε

(cos(2θ′)

)sin θ′

4

(cos θ′)2dθ′

= |S2|∫ π/4

−π/4

bε

(cos(2θ′)

)sin(2θ′)

1

(cos θ′)3d(2θ′)

= |S2|∫ π/2

−π/2

bε

(cos(θ)

)sin(θ)

1

(cos(θ/2))3dθ.

Then the hypothesis (3.3.1) implies∫ π/2

−π/2

sin θb(cos θ)

(1

cos3(θ/2)− 1

)dθ <∞.

We have then proved Lemma 3.3.1.

We study now the coercivity term.

Lemma 3.3.2. There exists a constant Cg > 0, depending only on b, ‖g‖L11

and

‖g‖L log L such that

(3.3.4) ‖Λαf‖2L2 ≤ Cg

∫R6

∫S2

b(k · σ)g(v∗)(f(v′)− f(v)

)2

dσdv∗dv + ‖f‖2L2

.

The proof of this lemma is similar to that of Theorem 1 in [6]. In fact, by takingthe Fourier transform on the collision operator and applying the Bobylev identity, wehave ∫

R6

∫S2

b(k · σ)g(v∗)(f(v)− f(v′)

)2

dσdv∗dv

= (2π)−3

∫R3

∫S2

b( ξ|ξ|

· σ)g(0)|f(ξ)|2 + g(0)|f(ξ+)|2

−2Re g(ξ−)f(ξ+)¯f(ξ)

dσdξ

≥ 1

2(2π)3

∫R3

|f(ξ)|2∫

S2

b( ξ|ξ|

· σ)(g(0)− |g(ξ−)|)dσ

dξ .

Then we can complete the proof of lemma 3.3.2 by following Lemma.

Lemma 3.3.3. Suppose that b satisfies assumption (3.3.1).Then there exists a pos-itive constant Cg depending only on b, ‖g‖L1

1and ‖g‖L log L such that

(3.3.5)

∫S2

b( ξ|ξ|

· σ)(g(0)− |g(ξ−)|)dσ ≥ C−1

g |ξ|2α

for all |ξ| ≥ 1.

This lemma is itself a consequence of the following two lemmas.


Lemma 3.3.4. Then there exists a positive constant C ′g depending only on ‖g‖L1

1

and ‖g‖L log L such that

(3.3.6) g(0)− |g(ξ) ≥ C ′g(|ξ|2

∧1)

for all ξ ∈ R3, where (|ξ|2∧

1) = min|ξ|2, 1.

Lemma 3.3.5. Suppose that b satisfies assumption (3.3.1).Then there exists a pos-itive constant C0 depending only on b such that

(3.3.7)

∫S2

b( ξ|ξ|

· σ)(|ξ−|2

∧1)dσ ≥ C0

|ξ|2α, if |ξ| ≥ 1

|ξ|2, if |ξ| ≤ 1.

Proof of Lemma 3.3.4: Since g ∈ L1 and g ≥ 0, we have firstly

g(0)− |g(ξ)| ≥ 0.

For fix ξ ∈ R3, there exists some θ ∈ R such that

g(0)− |g(ξ)| =

∫R3

g(v)(1− cos(v · ξ + θ))dv = 2

∫R3

g(v) sin2

(v · ξ + θ

2

)dv

≥ 2 sin2 ε

∫|v|≤r;∀p∈Z,|v · ξ+θ−2pπ|≥2ε

g(v)dv

≥ 2 sin2 ε

‖g‖L1(R3) −

‖g‖L11

r−∫

|v|≤r;∀p∈Z,|v · ξ|ξ|+

θ−2pπ|ξ| |≤2 ε

|ξ|

g(v)dv

≥ 2 sin2 ε

‖g‖L1(R3) −‖g‖L1

1

r− sup

|A|≤ 4ε|ξ| (2r)2

(1+

r|ξ|π

) ∫A

g(v)dv

.

If |ξ| ≥ 1, we obtain (3.3.7) with

C1 = 2 sin2 ε

‖g‖L1(R3) −

‖g‖L11

r− sup

|A|≤4ε(2r)2+ 2επ

(2r)3

∫A

g(v)dv

,

where ε > 0, r > 0 being chosen in such a way that this quantity is positive, this isensure by ‖g‖L1 > 0 , ‖g‖L1

1< +∞ and ‖g‖L log L < +∞. In fact, we choose firstly

r0 > 0 big enough such that

‖g‖L1(R3) −‖g‖L1

1

r0≥ 1

2‖g‖L1(R3) > 0.

Since g ∈ L logL implies that g is uniformly integrable, namely there exists δ0 > 0such that

sup|A|≤δ0

∫A

g(v)dv ≤ 1

4‖g‖L1(R3),

finally we choose ε0 > 0 small enough such that

4ε0(2r0)2 +

2ε0

π(2r0)

3 ≤ δ0.


Then we get

C1 =1

2sin2(ε0)‖g‖L1(R3) > 0.

If |ξ| ≤ 1, we put ε = ν|ξ| and setting now

C2 = 2ν2 inf|ξ|≤1

∣∣∣∣sin2(ν|ξ|)ν2|ξ|2

∣∣∣∣‖g‖L1(R3) −

‖g‖L11

r− sup

|A|≤4ν(2r)2(1+ r

π

) ∫A

g(v)dv

,

where ν > 0, r > 0 being chosen in such a way that this quantity is positive. And wechoose finally C ′

g = minC1, C2.

Proof of Lemma 3.3.5: We first note that

|ξ−|2 =|ξ|2

2

(1− ξ

|ξ|· σ)

= |ξ|2 sin2

(θ

2

).

Passing to spherical coordinates, there exists θ0 > small enough such that∫S2

b( ξ|ξ|

· σ)(|ξ−|2

∧1)dσ = 2π

∫ π/2

−π/2

sin θb(cos θ)(|ξ|2 sin2

(θ

2

) ∧1)dθ

≥ Kπ

∫ θ0

0

(|ξ|2θ2

4

∧1

)dθ

θ1+2α

≥ Kπ|ξ|2α

∫ θ0|ξ|

0

(θ2

4

∧1

)dθ

θ1+2α

≥ Kπ|ξ|2α

∫ θ0

0

(θ2

4

∧1

)dθ

θ1+2α,

here we consider the case |ξ| ≥ 1. We get (3.3.7) with

C0 =1

4Kπ

∫ θ0

0

θ1−2αdθ.

If |ξ| ≤ 1, we have∫ θ0

0

(|ξ|2θ2

4

∧1

)dθ

θ1+2α

≥∫ θ0

0

(|ξ|2θ2

4

)dθ

θ1+2α

≥ |ξ|2∫ θ0

0

θ1−2αdθ.

So that we have proved Lemma 3.3.5.


3.3.2. Logarithmic estimate for Debye-Yukawa potential. We consider nowthe non cutoff Maxwellian cross section for Debye-Yukawa potential with the followinghypothesis

(3.3.8) B(|v − v∗|, cos θ) = b(cos θ) ≈ Kθ−2(log θ−1

)mwhen θ → 0,

with K > 0,m > 0. We prove now the following logarithmic regularity estimate.

Proposition 3.3.2. Assume that the collision kernel B satisfies the assumption(3.3.8) and g ≥ 0, g ≡/ 0, g ∈ L1

2

⋂L logL. Then there exists a constant Cg > 0 de-

pending only on B, ‖g‖L12and ‖g‖L log L such that for any smooth function f ∈ H2(R3),

(3.3.9) ‖ (log Λ)m+1

2 f‖2L2(R3) ≤ Cg

(−Q(g, f), f)L2(R3) + ‖f‖2

L2(R3)

,

where Λ = (e+ |Dv|2)1/2.

Analogue to the proof of Proposition 3.3.1, we just to prove Lemma 3.3.5 in loga-rithmic version. We have

Lemma 3.3.6. Suppose that b satisfies assumption (3.3.8).Then there exists thepositive constants R0, C0 depending only on b such that

(3.3.10)

∫S2

b( ξ|ξ|

· σ)(|ξ−|2

∧1)dσ ≥ C0

(log < ξ >

)m+1, if |ξ| ≥ R0

|ξ|2, if |ξ| ≤ R0

.

Cheek again the proof of Lemma 3.3.5, we need only to calculus the term∫ θ0

0

(|ξ|2θ2

4

∧1

)dθ

θ (log θ−1)−m .

If |ξ| ≤ R0 and R0 > 1, we have∫ θ0

0

(|ξ|2θ2

4

∧1

)dθ

θ (log θ−1)−m ≥ 1

4|ξ|2

∫ R−10 θ0

0

θ(log θ−1

)mdθ = C0|ξ|2.

If |ξ| ≥ R0 for R0 big enough, we have∫ θ0

0

(|ξ|2θ2

4

∧1

)dθ

θ (log θ−1)−m ≥∫ θ0

√2

|ξ|

θ−1(log θ−1

)mdθ

≥ ((m+ 1)(

log|ξ|2

)m+1

− ((m+ 1)(

log θ−10

)m+1

≥ C0

(log < ξ >

)m+1.

We have proved Lemma 3.3.6.

3.4. Functional estimate of collision operators

We study now the functional estimate of collision operators. By using only theFourier transformation, we obtain firstly a weaker estimate on the collision operatorwhich is not optimal w.r.t. either the weight or the Sobolev index for the Maxwelliantype cross-section in a straightforward way.

3.4. FUNCTIONAL ESTIMATE OF COLLISION OPERATORS 45

3.4.1. Functional estimate for Maxwellian case.

Lemma 3.4.1. Let 0 < s < 1/2. Then for any ρ ∈ R,

(3.4.1) ‖Q(g, f)‖Hρ−1(R3v) ≤ C‖g‖L1

1(R3v)‖f‖Hρ

1 (R3v).

Proof: By using the Bobylev identity, we have

|Fv(Q(g, f)(ξ)| =∣∣∣∣∫

S2

b(g(ξ−)f(ξ+)− g(0)f(ξ)

)dσ

∣∣∣∣≤∣∣∣∣∫

S2

b(g(ξ−)− g(0)

)f(ξ+)dσ

∣∣∣∣+ ∣∣∣∣∫S2

bg(0)(f(ξ+)− f(ξ)

)dσ

∣∣∣∣≤∫

S2

∫ 1

0

b|ξ−| |(∇g)(τξ−)| |f(ξ+)|dτdσ

+

∫S2

∫ 1

0

b|ξ−| |g(0)| |(∇f)(ξ + τ(ξ+ − ξ))|dτdσ

Note that

|ξ−| = |ξ| sin θ2≤ C|ξ|θ, |ξ|

2≤ |ξ+| ≤ |ξ|, 0 ≤ θ ≤ π

2.

and that for 0 < s < 1/2, ∫S2

bθdσ ≤ C

∫ π/2

0

θ−2sdθ < +∞.

By Schwarz, we then get∫R3

ξ

< ξ >2ρ |Fv(Q(g, f)(ξ)|2 dξ

≤C‖Og‖2L∞

∫R3

ξ

∫S2

< ξ >2ρ+2 θ−1−2s|f(ξ+)|2dσdξ

+C|g(0)|2∫

R3ξ

∫S2

∫ 1

0

< ξ >2ρ+2 θ−1−2s|(∇f)(ξ + τ(ξ+ − ξ))|2dτdσdξ

=C(‖Og‖2

L∞J1 + |g(0)|2J2

).

The estimate of the integral J1 is done by the help of the change of variables ξ → ξ+.The Jacobian is computed, with k = ξ/|ξ|, as∣∣∣∂(ξ+)

∂(ξ)

∣∣∣ =∣∣∣12I +

1

2σ ⊗ k

∣∣∣ =1

8(1 + k · σ) =

1

4cos2 θ

2.

Here, we shall notice that after this change of variable, θ plays no longer the role of thepolar angle because the “pole” k now moves with σ and hence the measure dσ is nolonger given by sin θ dθ dφ. However, the situation is rather good because if we takek+ = ξ+/|ξ+| as a new pole which is independent of σ, then the new polar angle ψdefined by cosψ = k+ · σ satisfies

ψ =θ

2, dσ = sinψdψdφ, ψ ∈ [0,

π

4],


and thus θ works almost as the polar angle, giving

J1 ≤ C

∫R3

ξ+

D0(ξ+)|f(ξ+)|2dξ+

with

D0(ξ+) =

∫S2

< ξ >2ρ+2 θ−1−2sdσ

≤ C < ξ+ >2ρ+2

∫ π/4

0

ψ−1−2s sinψdψ ≤ C < ξ+ >2ρ+2

which impliesJ1 ≤ C‖f‖2

Hρ+1 .

In order to estimate J2, first, put

ξτ = ξ + τ(ξ+ − ξ)

and notice that for any τ ∈ [0, 1] and θ ∈ [0, π/2],√

2

2|ξ| ≤ |ξτ | ≤

3√2|ξ|

and ∣∣∣∣∂(ξτ )

∂(ξ)

∣∣∣∣ = |((2− τ)/2)I + (τ/2)σ ⊗ k|

= ((2− τ)/2)3 + ((2− τ)/2)2(τ/2)k · σ,so that the Jacobian of the change of variables

ξτ → ξ

is uniformly bounded away from zero for any τ ∈ [0, 1] and θ ∈ [0, π2]. Here, the same

remark follows for the pole and polar angle. In this case, the new pole is kτ = ξτ/|ξτ |for which the polar angle defined by cosψ = kτ · σ is easily seen to satisfy

θ

2≤ ψ ≤ θ, dσ = sinψdψdφ.

Hence, we get

J2 ≤ C

∫ 1

0

∫R3

ξτ

D1(ξτ , τ)|∇f(ξτ )|2dξτdτ

with

D1(ξτ , τ) =

∫S2

< ξ >2ρ+2 θ−1−2sdσ

≤ C < ξτ >2ρ+2

∫ π/4

0

ψ−1−2s sinψdψ ≤ C < ξτ >2ρ+2

which implies

J2 ≤ C

∫ 1

0

‖f‖2Hρ+1

1dτ = C‖f‖2

Hρ+11,

and this completes the proof of the lemma.


3.4.2. Functional estimate for non Maxwellian case. For non-Maxwelliancase, we can also directly prove the no optimal estimate on Rn, n ≥ 2,

Lemma 3.4.2. Let 0 < s < 1/2. We consider the non-Maxwellian cross kernelB(Z, cos θ) = |Z|κb(cos θ). Suppose that

sinn−2 θb(cos θ) ∼ θ−1−2s.

Then, for any sufficiently small ε > 0, there is a constant C > 0 such that for any f, g,it holds that

(3.4.2) ‖Q(f, g)‖H−2s−ε(Rnv ) ≤ C‖f‖L2

2s+ n2 +ε+κ

(Rnv )‖g‖L2

2s+ n2 +ε+κ

(Rnv ).

Proof : Write

< Q(f, g), h > =

∫∫∫|v − v∗|κb(cos θ)(f ′∗g

′ − f∗g)hdvdv∗dσ

=

∫∫∫|v − v∗|κbf∗g(h′ − h)dvdv∗dσ

=

∫∫∫|v′−v|<1

+

∫∫∫|v′−v|>1

≡ J1 + J2

Let η > 0. Since |v′ − v| = |v − v∗| sin(θ/2), we have, for γ ≥ 0,

|J1| ≤∫∫∫

|v′−v|<1

|v − v∗|κb|f∗||g||v′ − v|η |h′ − h|

|v′ − v|ηdvdv∗dσ

≤ C

∫∫∫|v′−v|<1

bθη(|v|+ |v∗|)η+κ|f∗||g||h′ − h||v′ − v|η

dvdv∗dσ

≤ C∫∫∫

|v′−v|<1

b θη−γ(|v|+ |v∗|)2(η+κ)|f∗|2|g|2dvdv∗dσ1/2

×∫∫∫

|v′−v|<1

b θη+γ( |h′ − h||v′ − v|η

)2

dvdv∗dσ1/2

≤ C‖f‖L2η+κ‖g‖L2

η+κH0

∫ π/2

0

θ−1−2s+η−γdθ,

where the last integral converges if

(3.4.3) −2s+ η − γ > 0,

and by the ”singular” change of variables v∗ → v′ for which dv∗ ∼ θ−2dv′ ,

H20 =

∫∫∫|v′−v|<1

b θη+γ |h′ − h|2

|v′ − v|2ηdvdv∗dσ

≤ C

∫∫∫|v′−v|<1

θ−1−2s+η+γ−(n−2)−2 |h′ − h|2

|v′ − v|2ηdvdv′dσ

= C

∫∫|v′−v|<1

D0(v, v′)|h′ − h|2

|v′ − v|2ηdvdv′.


Here

D0(v, v′) =

∫Sn−1

θ−1−2s+η+γ−(n−2)−2dσ.

Notice that an additional singularity appears due to this change of variables. Actually,the situation is much worse:

After this change of variables, k = (v − v∗)/|v − v∗| is a function ofv, v′, σ so that θ plays no longer the role of polar angle because thepole “k” moves with σ, and hence the measure dσ is no longer givenby sinn−2 θdθdφ. Thus, we need a new pole which is independent ofσ in order to carry out the integration in σ.

A possible (and actually the best) choice is k′ = (v′ − v)/|v′ − v| for which the polarangle ψ defined by cosψ = k′ · σ satisfies (cf. Figure 1 of p. 337, [6])

θ = π − 2ψ, ψ ∈ [π/4, π/2],

dσ = sinn−2 ψdψdω′, ω′ ∈ Sn−2.

This shows that the measure dσ does not kill the extra singularity of b(cos θ) comingfrom sinn−2 θ. Nevertheless, if we assume

(3.4.4) −2s− n+ η + γ > 0,

then, we can have

D0(v, v′) ≤ C

∫ π/2

π/4

(π

2− ψ)−1−2s−n+η+γ sinn−2 ψdψ ≤ C0 (∃C0 > 0,∀v, v′),

and hence, by the change of variable v′ → z = v′ − v

H20 ≤ C

∫∫R2n∩|v′−v|<1

|h′ − h|2

|v′ − v|2ηdvdv′

=

∫∫|z|<1

|h(v + z)− h(v)|2

|z|2ηdvdz =

∫Rn

|h(ξ)|2S(ξ)dξ,

where, for |ξ| ≥ 1,

S(ξ) =

∫|z|<1

|eiξ·z − 1|2

|z|2ηdz = |ξ|2η−n

∫|z|<|ξ|

|ei ξ|ξ| ·z − 1|2

|z|2ηdz

≤ Cη ×|ξ|2η−n, n

2< η < n

2+ 1,

1, η < n2.

.

Adding (3.4.3) and (3.4.3) yields

−4s− n+ 2η > 0 or η >n

2+ 2s.

This require n2

+ 2s < η < n2

+ 1 or s < 12. Now we choose η = n

2+ 2s + ε and γ = n

2.

Then, (3.4.3) and (3.4.4) hold and we get

|J1| ≤ C‖f‖L2η+κ‖g‖L2

η+κ‖h‖Hη−n/2 = C‖f‖L2

2s+ n2 +ε+κ

‖g‖L22s+ n

2 +ε+κ‖h‖H2s+ε .


On the other hand, for β > 0

|J2| ≤∫∫∫

|v′−v|>1

b|v′ − v|β|f∗||g|(|h′|+ |h|)dvdv∗dσ

≤ C

∫∫∫θ−1−2s−(n−2)+β(< v∗ >

β |f∗|)(< v >β |g|)(|h′|+ |h|)dvdv∗dσ

The integral containing |h| is evaluated, by Schwarz, from above by

C

∫ π/2

0

θ−1−2s+βdθ‖f‖L1β‖g‖L2

β‖h‖L2 .

The integral containing |h′| is computed by the help of the change of variablesv → v′ introduced in [6] for which dv ∼ dv′. Unlike the previous one, however, noadditional singularity arises although θ again does not play the role of polar angle. Inthis case, the best choice of pole is k′′ = (v′ − v∗)/|v − v∗|, being independent of σ, forwhich the polar angle ψ defined by cosψ = k′′ · σ satisfies

ψ = θ/2, dσ = sinn−2 ψdψdφ ψ ∈ [0, π/4].

This implies ψ ∼ θ and∫θ−1−2s−(n−2)+βdσ ≤ C

∫ π/4

0

ψ−1−2s−(n−2)+β sinn−2 ψdψ ≤ C

∫ π/4

0

ψ−1−2s+βdψ

Hence the computation with |h′| is the same with |h|, to conclude

J2 ≤ C

∫ π/2

0

θ−1−2s+βdθ‖f‖L1β‖g‖L2

β‖h‖L2 .

The last integral is finite if β > 2s. Take β = 2s+ ε/2. Then

‖f‖L1β≤ C‖f‖L2

β+ n2 +ε/2

≤ C‖f‖L22s+ n

2 +ε.


3.4.3. Optimal estimate. We also need the boundedness of the collision operatorin the Sobolev space,

Lemma 3.4.3. Let 0 < s < 1/2. Then for any ρ ∈ R, we have

(3.4.5) ‖Q(g, f)‖Hρ−2s(R3v) ≤ C‖g‖W ρ,1

2s (R3v)‖f‖Hρ

2s(R3v);

Where W ρ,12s (R3

v) is usual Sobolev space in L1 with a weight W2s

Remark : The estimate in (3.4.5) is optimal with respect to the order of derivative( exact order of 2s) and also with respect to the order of weight.

The proof of estimate (3.4.5) use a very power harmonic analysis tools: the Littlewood-Paley decomposition, see [2] for the general non-Maxwellian case and estimate in Besovspace.

If we take ρ = s in (3.4.5), we get∣∣(Q(g, f), f)L2(R3v)

∣∣ ≤ C‖g‖W ρ,12s (R3

v)‖f‖Hs2s(R3

v)‖f‖Hs(R3v).


Thanks to the coercivity estimate (3.3.2), if g ≥ 0, g ≡/ 0, g ∈ L12

⋂L logL, the singu-

larity of collision kernel

sin θ b(cos θ) ≈ Kθ−1−2s when θ → 0

with 0 < s < 1 implies that for any smooth functions f ,

(3.4.6) −(Q(g, f), f

)L2(R3

v)≈ Cg

((−4v)

sf, f)

L2(R3v)

where the operator (−4v)s is a Fourier multiplier of symbol |ξ|2sχ(ξ) + |ξ|2(1− χ(ξ)),

with χ ∈ C∞(Rn), 0 ≤ χ ≤ 1, χ(ξ) = 1 if |ξ| ≥ 2 and χ(ξ) = 0 if |ξ| ≤ 1.This is the motivation that we consider in the Chapiter V, the following linear

equations

(3.4.7) ∂tg + v · ∇xg + σ(−4v)sg = F

as a model of inhomogeneous Boltzmann equations.

CHAPTER IV

Spatially homogeneous Boltzmann equations

We consider now a simple case of the Boltzmann equation, the distribution ofparticles is spatially homogeneous, namely the density f(t, x, v) is independents on thespatial position variable x ∈ R3, then the equation (3.1.1) simplifier to

∂f

∂t= Q(f, f),

and f = f(t, v). For more detail about this equation, see [28, 45, 101]. This chapteris concerned with the smoothing effect of the singular integral kernel. There are twomain results in this chapter. One is about the Debye-Yukawa type potential, we getthe following smoothness effect results: For any N ∈ N and t ≥ 0

‖ 〈|D|〉Nt−2 f(t)‖L2(R3) ≤ C1 eC0t‖f0‖L1

where C0, C1 depending only on N, ‖f0‖L11

.Another problem is concerned with the Gevrey regularity of linearized equations

for the inverse power laws. We get also a very strong smoothness effect estimate: Forany t ∈ [0, T ],

‖et〈|D|〉α 〈|D|〉−2 f(t)‖L2(R3) ≤ C1eC0t‖f0‖L1

2(R3).

The above smoothness effect estimates of Cauchy problem are very strong, theyare uniformly estimates on the closed interval [0, T ], it is difficult to get this type ofuniformly estimate for inhomogeneous equations even in a model case of chapter V.

4.1. Weak solution of Cauchy problems

We consider the Cauchy problem of spatially homogeneous nonlinear Boltzmannequation

(4.1.1)∂f

∂t= Q(f, f), x ∈ R3, v ∈ R3, t > 0 ; f |t=0 = f0,

where f = f(t, v) ≥ 0 on [0,∞)× R3v represents the particle distribution function. In

the following, we assume that the initial datum f0 ≡/ 0 satisfies the natural bounded onthe mass, energy and entropy, that is,

(4.1.2) f0 ≥ 0,

∫R3

f0(v)(1 + |v|2 + log(1 + f0(v)))dv < +∞.

Let us recall the definition of weak solution for the Cauchy problem (4.1.1), cf.[101].

51

52 IV. SPATIALLY HOMOGENEOUS BOLTZMANN EQUATIONS

Definition 4.1.1. Let f0(v) ≥ 0 be a function defined on R3 with finite mass,energy and entropy. f(t, v) is called a weak solution of the Cauchy problem (4.1.1), ifit satisfies the following conditions:

f(t, v) ≥ 0, f(t, v) ∈ C(R+;D′(R3)) ∩ L1([0, T ];L12(R3)), f(0, v) = f0(v);∫

R3

f(t, v)ψ(v)dv =

∫R3

f0(v)ψ(v)dv for ψ = 1, vj, |v|2;

f(t, v) ∈ L1(R3) logL1(R3),

∫R3

f(t, v) log f(t, v)dv ≤∫

R3

f0 log f0dv, ∀t ≥ 0;∫R3

f(t, v)ϕ(t, v)dv −∫

R3

f0ϕ(0, v)dv −∫ t

0

dτ

∫R3

f(τ, v)∂τϕ(τ, v)dv

=

∫ t

0

dτ

∫R3

Q(f, f)(τ, v)ϕ(τ, v)dv,

where ϕ(t, v) ∈ C1(R+;C∞0 (R3)). Here, the right hand side of the last integral given

above is defined by∫R3

Q(f, f)(v)ϕ(v)dv =1

2

∫R6

∫S2

Bf(v∗)f(v)(ϕ(v′) + ϕ(v′∗)− ϕ(v)− ϕ(v∗))dvdv∗dσ.

Since the existence of weak solution was already proved in [101]. And the conser-vation law implies that the weak solution satisfies

(4.1.3) f ≥ 0,

∫R3

f(t, v)(1 + |v|2 + log(1 + f(t, v)))dv < +∞,

for any t ∈ [0, T ]. We are going to prove the following theorem on the regularity ofweak solutions.

Theorem 4.1.1. Assume that the initial datum f0 satisfies (4.1.2) and the collisioncross-section satisfies

(4.1.4) B(|v − v∗|, cos θ) = b(cos θ) ≈ Kθ−2(log θ−1

)mwhen θ → 0,

with K > 0,m > 0. Let f be a weak solution of Cauchy problem (4.1.1). Then for any0 < t ≤ T , we have f(t, ·) ∈ H+∞(R3).

Remark 4.1.1. Note that m > 0 corresponds to 0 < s < 2 in (3.1.22). In [8, 45],the H+∞(R3) regularity of weak solutions was proved under the condition :

(4.1.5) B(|v − v∗|, cos θ) = b(cos θ) ≈ Kθ−2−2α when θ → 0,

where K > 0, 0 < α < 1. We say the Mathematical Maxwellian case. Notice that thecondition (4.1.4) is much weaker than (4.1.5) and the theorem 4.1.1 shows that it stillleads to H+∞(R3) regularity on the weak solutions. Moreover, the following proof onthe regularity of weak solutions is more straightforward and illustrative than the previ-ous methods. Even though the assumption (4.1.4) on the cross-section is mathematicalbecause the exact cross-section depends also on the relative velocity, the following anal-ysis reveals the smoothing effect of the singularity in the collision operator on the weaksolution to the Boltzmann equation.

4.1. WEAK SOLUTION OF CAUCHY PROBLEMS 53

Let f be a weak solution of the Cauchy problem (4.1.1). For any fixed T0 > 0, weknow that f(t, · ) ∈ L1(R3) ⊂ H−2(R3) for all t ∈ [0, T0]. For t ∈ [0, T0], N > 0 and0 < δ < 1, set

Mδ(t, ξ) =(1 + |ξ|2

)Nt−42 ×

(1 + δ|ξ|2

)−N0

,

with N0 = NT0

2+ 2. Then, for any δ ∈]0, 1[

Mδ(t,Dv)f ∈ L∞([0, T0];W2,∞(R3)),

whose norm is bounded above from Cδ‖f0‖L1 .We want use M2

δ f ∈ L∞([0, T0];W2,∞(R3)) as test function in the definition of weak

solution,

Lemma 4.1.1. We have

(4.1.6) Mδf ∈ C([0, T0];L2(R3)),

and for any t ∈]0, T0], we have

1

2

∫R3

f(t)M2δ (t)f(t)dv − 1

2

∫ t

0

∫R3

f(τ)(∂tM

2δ (τ)

)f(τ)dvdτ(4.1.7)

=1

2

∫R3

f0M2δ (0)f0dv +

∫ t

0

(Q(f, f)(τ),M2

δ (τ)f(τ))

L2dτ.

Proof : In Definition 4.1.1, taking ϕ(t, v) = ψ(v) ∈ C∞0 (R3), we get∫

R3

f(t)ψdv −∫

R3

f(s)ψdv =

∫ t

s

dτ

∫R3

Q(f(τ), f(τ))ψdv , 0 ≤ s ≤ t ≤ T0 .

We can set ψ = M2δ f(t),M2

δ f(s) because they belong to L∞([0, T ];W 2,∞(R3)). Bytaking the sum, we obtain∫

R3

f(t)M2δ f(t)dv −

∫R3

f(s)M2δ f(s)dv =

∫R3

f(t)(M2

δ (t)−M2δ (s)

)f(s)dv

+

∫ t

s

dτ

∫R3

Q(f(τ), f(τ))(M2

δ f(t) +M2δ f(s)

)dv .

Since the integrand of the first term on the right is estimated by |t − s|C ′δ||f0||L1f(t)

and in addition, the collision integral term is bounded by Cδ′′||f0||L1||f ||2L1

2, we obtain

(4.1.6), namely Mδf ∈ C([0, T0];L2(R3)). In Definition 4.1.1 , we can rewrite the term∫ t

0

dτ

∫R3

f(τ, v)∂τϕ(τ, v)dv

= limh→0

∫ t

0

dτ

∫R3

(f(τ, v) + f(τ + h, v))ϕ(τ + h, v)− ϕ(τ, v)

2hdv

for ϕ(t, v) ∈ C1(R+;C∞0 (R3)), by noting f ∈ C(R+;D′). Let

ϕ(t) ≡M2δ (t)f(t),

in the above equation, then its right hand side equals to

limh→0

∫ t

0

dτ

∫R3

(Mδf)2(τ + h)− (Mδf)2(τ)

2hdv


+

∫ t

0

dτ

∫R3

f(τ)f(τ + h)(Mδ)

2(τ + h)− (Mδ)2(τ)

2hdv

.

It follows from (4.1.6) that

limh→0

∫ t

0

dτ

∫R3

(Mδf)2(τ + h)− (Mδf)2(τ)

2hdv

= limh→0

1

2h

∫ t+h

t

dτ −∫ h

0

dτ

∫R3

(Mδf)2(τ)dv

=1

2

∫R3

(Mδf)2(t)dv − 1

2

∫R3

(Mδf)2(0)dv.

Hence, we obtain (4.1.7) because the Lebesgue convergence theorem shows that

limh→0

∫ t

0

dτ

∫R3

f(τ)(Mδ)

2(τ + h)− (Mδ)2(τ)

2hf(τ + h)dv

=1

2

∫ t

0

∫R3

f(τ)(∂tM

2δ (τ)

)f(τ)dvdτ.

4.2. Smoothness effect of Cauchy problems

By using Proposition 3.3.2, we have

(4.2.1) ‖ (log Λ)m+1

2 Mδ f‖2L2 ≤ Cf

(−Q(f,Mδf),Mδf)L2 + ‖Mδf‖2

L2

,

where the constant Cf is independent of δ ∈]0, 1[.To apply this logarithmic regularity estimate to the nonlinear Boltzmann equation,

we need to estimate the commutators of the pseudo-differential operator Mδ(t,Dv) andthe nonlinear operator Q(f, ·) which is given in the following lemma.

Lemma 4.2.1. Under the hypothesis of Theorem 4.1.1, we have that

(4.2.2) |(Q(f,Mδf),Mδf)L2 − (Q(f, f),M2δ f)L2| ≤ Cf‖Mδf‖2

L2

with a constant Cf > 0 independent of 0 < δ < 1.

Proof. By applying Proposition 3.3.2 to the function Mδf ∈ H2, we have

(−Q(f,Mδf),Mδf)L2(R3) +O(‖Mδf‖2L2)

=1

2(2π)3

∫R3

∫S2

b( ξ|ξ|

· σ)f(0)M2

δ (t, ξ)|f(ξ)|2 + f(0)M2δ (t, ξ+)|f(ξ+)|2

− 2Re f(ξ−)Mδ(t, ξ+)f(ξ+)Mδ(t, ξ)

¯f(ξ)

dσdξ,

By the Bobylev identity, we also have(Q(f, f),M2

δ f)

L2=

∫R6

∫S2

b(k · σ)f(v∗)f(v)(M2

δ f(v′)−M2δ f(v)

)dv∗dσdv

=1

(2π)3

∫R3

∫S2

b( ξ|ξ|· σ)f(ξ−)M2

δ (t, ξ)f(ξ+)¯f(ξ)− f(0)M2

δ (t, ξ)|f(ξ)|2dσdξ.

4.2. SMOOTHNESS EFFECT OF CAUCHY PROBLEMS 55

Thus, (Q(f, f),M2

δ f)

L2= − 1

2(2π)3

∫R3

∫S2

b( ξ|ξ|

· σ)f(0)M2

δ (t, ξ)|f(ξ)|2

+ f(0)M2δ (t, ξ+)|f(ξ+)|2 − 2Re f(ξ−)Mδ(t, ξ

+)f(ξ+)Mδ(t, ξ)¯f(ξ)

dσdξ

+1

2(2π)3

∫R3

∫S2

b( ξ|ξ|

· σ)f(0)M2

δ (t, ξ+)|f(ξ+)|2 − f(0)M2δ (t, ξ)|f(ξ)|2

+ 2Re f(ξ−)Mδ(t, ξ)f(ξ+)¯f(ξ)

[Mδ(t, ξ)−Mδ(t, ξ

+)]dσdξ.

=(Q(f,Mδf),Mδf

)L2

+O(‖Mδf‖2L2)

+1

2(2π)3

∫R3

∫S2

b( ξ|ξ|

· σ)f(0)M2

δ (t, ξ+)|f(ξ+)|2 − f(0)M2δ (t, ξ)|f(ξ)|2

+ 2Re f(ξ−)Mδ(t, ξ)f(ξ+)¯f(ξ)


+)]dσdξ.

Hence, it remains to show that∣∣∣∣∫R3

∫S2

bf(0)M2

δ (t, ξ+)|f(ξ+)|2 − f(0)M2δ (t, ξ)|f(ξ)|2dσdξ

∣∣∣∣ ≤ Cf‖Mδf‖2L2 ,

and(4.2.3)∣∣∣∣∫

R3

∫S2

bRe f(ξ−)Mδ(t, ξ)f(ξ+)

¯f(ξ)


+)]dσdξ

∣∣∣∣ ≤ Cf‖Mδf‖2L2 .

The first estimate can be obtained by using the argument for the cancellation lemmagiven in [6] because

|∫

R3

∫S2

bf(0)M2

δ (t, ξ+)|f(ξ+)|2 − f(0)M2δ (t, ξ)|f(ξ)|2|

= (2π)|∫

R3

∫ π/2

−π/2

sin θ b(cos θ)f(0)M2δ (t, ξ)|f(ξ)|2

[ 1

cos3(θ/2)− 1]dθdξ|

≤ C0‖f‖L1‖Mδf‖2L2 .

To prove the second estimate, we need to show that

(4.2.4) |Mδ(t, ξ+)−Mδ(t, ξ)| ≤ N02

(NT0+4)/2 sin2 θ

2Mδ(t, ξ

+).

For this, recall

ξ+ =ξ + |ξ|σ

2, |ξ+|2 = |ξ|2 cos2 θ

2,

ξ

|ξ|· σ = cos θ,

and the collision kernel is supported in |θ| ≤ π/2. Then

|ξ|2

2≤ |ξ+|2 ≤ |ξ|2, |ξ|2 − |ξ+|2 = |ξ−|2 = sin2 θ

2|ξ|2.

Denote

Mδ(t, s) = (1 + s)Nt−4

2 × (1 + δs)−N0 , s = |ξ|2, s+ = |ξ+|2,


so thatMδ(t, ξ) = Mδ(t, |ξ|2).

Then, there exists s+ < s < s such that

Mδ(t, s)− Mδ(t, s+) =∂Mδ

∂s(t, s)(s− s+).

Note that s− s+ = s sin2 θ2

and

∂Mδ

∂s(t, s) =

(Nt− 4)

1

2(1 + s)−N0

δ

1 + δs

Mδ(t, s).

By usings

1 + s,

δs

1 + δs≤ 1,

and ∣∣∣∣∣ Mδ(t, s)

Mδ(t, s+)

∣∣∣∣∣ ≤ 2(NT0+4)/2,

we have

|Mδ(t, s)− Mδ(t, s+)| ≤ N02(NT0+4)/2 sin2 θ

2Mδ(t, s+),

which gives (4.2.1). Now the second estimate in (4.2.4) can be proved as follows,∣∣∣∣∫ bRe f(ξ−)Mδ(t, ξ)f(ξ+)

¯f(ξ)


+)]dσdξ

∣∣∣∣≤ C

∫R3

∫S2

b(cos θ) sin2 θ

2|f(ξ−)|Mδ(t, ξ

+)|f(ξ+)|Mδ(t, ξ)|f(ξ)|dσdξ

≤ C

∫R3

∫ π/2

−π/2

b(cos θ) sin2 θ

2sin θ|f(ξ−)|Mδ(t, ξ

+)|f(ξ+)|Mδ(t, ξ)|f(ξ)|dθdξ

≤ C‖f‖L1‖Mδf‖2L2 .

If Mδ(t,Dv) is replaced by the differential operator Dkv , then the commutator is

given by Leibniz formula. Therefore, in some sense, this lemma is a microlocal versionof the computation given in [45]. We are now ready to prove Theorem 4.1.1.

Proof of Theorem 4.1.1:From Lemma 4.1.1, we use (4.1.7) as definition of weak solution. It follows from

(4.2.1) and (4.2.2) that

(4.2.5) ‖ (log Λ)m+1

2 Mδf‖2L2 ≤ Cf

(−Q(f, f),M2

δ f)L2 + ‖Mδf‖2L2

.

Since(∂tMδ)(t, ξ) = N log < ξ > Mδ(t, ξ),

we obtain∣∣∣∣∫ t

0

∫R3

f(τ)(∂tM

2δ (τ)

)f(τ)dvdτ

∣∣∣∣ ≤ 2N

∫ t

0

‖ (log Λ)12 (Mδf)(τ)‖2

L2dτ.

4.3. GEVREY REGULARITY EFFECT FOR LINEARIZED EQUATIONS 57

This together with (4.1.7) and (4.2.5), imply

‖(Mδf)(t)‖2L2 + 1

2Cf

∫ t

0‖ (log Λ)

m+12 (Mδf)(τ)‖2

L2dτ ≤

‖Mδ(0)f0‖2L2 + 2N

∫ t

0‖ (log Λ)

12 (Mδf)(τ)‖2

L2dτ +∫ t

0‖(Mδf)(τ)‖2

L2dτ.

For m > 0, by interpolation, we have for any ε > 0,

‖(Mδf)(t)‖2L2 +

( 1

2Cf

− ε)∫ t

0

‖ (log Λ)m+1

2 (Mδf)(τ)‖2L2dτ

≤ ‖Mδ(0)f0‖2L2 + Cε,N

∫ t

0

‖(Mδf)(τ)‖2L2dτ.

By choosing ε = 14Cf

> 0, there exists Cf,N > 0 depending only on Cf , N, T0 and being

independent of δ ∈]0, 1[, such that for any t ∈]0, T0],

‖Mδ(t)f(t)‖2L2 ≤ ‖Mδ(0)f0‖2

L2 + Cf,N

∫ t

0

‖Mδ(τ)f(τ)‖2L2dτ.

Then Gronwall inequality yields

‖(Mδf)(t)‖2L2 ≤ eCf,N t‖Mδ(0)f0‖2

L2 .

Since ‖Mδ(t)f(t)‖2L2 = ‖(1− δ4)−N0f(t)‖2

HNt−4(R3), and

‖Mδ(0)f0‖2L2 = ‖(1− δ4)−N0f0‖2

H−4(R3) ≤ ‖f0‖2H−4(R3) ≤ C0‖f0‖2

L1 ,

we obtain

‖(1− δ4)−N0f(t)‖2HNt−4(R3) ≤ CeCf,N t‖f0‖2

L1 ,

where the constant C > 0 is independent of δ. By letting δ → 0, we obtain, for anyN ∈ N,

(4.2.6) ‖f(t)‖2HNt−4(R3) ≤ C1 e

C0t‖f0‖2L1

where C0, C1 depending only on N and ‖f0‖L11. The negative index −4 is due to at the

time of t = 0 the embedding f0 ∈ L1(R3) ⊂ H−4(R3).Finally, for any given t > 0, since N can be arbitrarily large, we have

f(t) ∈ H+∞(R3).

And this completes the proof of Theorem 4.1.1.

4.3. Gevrey regularity effect for linearized equations

To have a more precise description on the regularity, we consider now the Gevreyregularity of solutions with cross-section satisfying (4.1.5). Notice that while localsolutions having the Gevrey regularity have been constructed in [98] for initial datahaving Gevrey regularity, the result given here is concerned with the production of theGevrey regularity for weak solutions whose initial data have no regularity (for moredetail, see [87]).


Before stating the result, we now recall the definition of Gevrey regularity. Fors ≥ 1, u ∈ Gs(R3) which is the Gevrey class function space with index s, if there existsC > 0 such that for any k ∈ N,

(4.3.1) ‖Dku‖L2(R3) =( ∑|β|=k

‖Dβu‖2L2(R3)

) 12 ≤ Ck+1(k!)s,

or equivalently, there exist ε0 > 0 such that

(4.3.2) eε0<|D|>1/s

u ∈ L2(R3).

Note that G1(R3) is usual analytic function space.

We first linearize the Boltzmann equation near the absolute Maxwellian distribution

(4.3.3) µ(v) = (2π)−32 e−

|v|22 .

Since Q(µ, µ) = 0, we have

Q(µ+ g, µ+ g) = Q(µ, g) +Q(g, µ) +Q(g, g).

SetLg = Q(µ, g) +Q(g, µ),

where L is the usual linearized collision operator. Then, consider the linear Cauchyproblem

(4.3.4)∂g

∂t= Lg, v ∈ R3, t > 0 ; g|t=0 = g0.

The result on Gevrey regularity can be stated as follows.

Theorem 4.3.1. Assume that the initial datum in (4.3.4) satisfies g0 ∈ L12(R3),

and Q is defined by the Maxwellian collision cross-section B satisfying (4.1.5) with0 < α < 1. For T0 > 0, if g ∈ L1([0, T0];L

12(R3)) ∩ L∞([0, T0];L

1(R3)) is a weaksolution of the Cauchy problem (4.3.4), then g(t, ·) ∈ G1/α(R3) for any 0 < t ≤ T0.

We don’t study the existence of weak solution in this lecture notes, see [87] for moredetail about the existence of weak solutions. We focus now to the Gevrey regularity ofweak solution.

Under the assumption (4.1.5) on the cross-section, the coercivity estimate (3.3.4)in Lemma 3.3.2 write as following:

(4.3.5) ‖Λαf‖2L2 ≤ Ch

(−Q(h, f), f)L2 + ‖f‖2

L2

,

for any f ∈ H2 and h ≥ 0, h ≡/ 0, h ∈ L11 ∩L logL. Here, the constant Ch > 0 depends

only on ‖h‖L11

and ‖h‖L log L. For 0 < δ < 1, set

Gδ(t, ξ) =et〈|ξ|〉α

1 + δet〈|ξ|〉α .

Then if g ∈ L1([0, T0];L12(R3)), we have

Gδ(t,Dv) 〈|Dv|〉−4 g ∈ L1([0, T0];H2(R3));

G2δ(t,Dv) 〈|Dv|〉−8 g ∈ L1([0, T0];W

2,∞(R3)),


and

‖ΛαGδ(t,Dv) 〈|D|〉−4 g‖2L2 ≤ Cµ

(−Q(µ,Gδ 〈|D|〉−4 g), Gδ 〈|D|〉−4 g)L2

+‖Gδ(t,Dv) 〈|D|〉−4 g‖2L2

,(4.3.6)

where the constant Cµ > 0 is independent on δ.As in the previous section, the following lemma gives the estimate on the commu-

tator of the pseudo-differential operator Gδ(t,Dv) 〈|Dv|〉−4 and the collision operatorQ(µ, ·).

Lemma 4.3.1. For the function g and with the notations given above, we have

|(Q(µ,Gδ 〈|D|〉−4 g), Gδ 〈|D|〉−4 g)L2 − (Q(µ, g), G2δ 〈|D|〉

−8 g)L2|(4.3.7)

≤ Cµ‖Gδ 〈|D|〉−4 g‖L2‖ΛαGδ 〈|D|〉−4 g‖L2 ,

and

(4.3.8) |(Q(g, µ), G2δ 〈|D|〉

−8 g)L2| ≤ Cµ‖g‖L12‖Gδ 〈|D|〉−4 g‖L2 ,

where Cµ > 0 is independent of 0 < δ < 1.

Proof. For (4.3.7), we choose Gδ(t,Dv) 〈|D|〉−4 g ∈ H2(R3) as the test function.Without loss of generality and for simplicity of notations, we drop the regularizedoperator 〈|D|〉−4 in the following calculation because it does not create extra difficulty.In fact, the main problem is to estimate following term,∣∣∣∣∫ b

( ξ|ξ|

· σ)Re µ(ξ−)g(ξ+)Gδ(t, ξ)¯g(ξ)

[Gδ(t, ξ)−Gδ(t, ξ

+)]dσdξ

∣∣∣∣ .Notice that the weight Gδ(t, ξ) is an exponential function, so that we will show thefollowing estimate

(4.3.9) |Gδ(t, ξ+)−Gδ(t, ξ)| ≤ C sin2 θ

2〈ξ〉αGδ(t, ξ

−)Gδ(t, ξ+),

where the constant C > 0 depends only on α and T0. For this, set

Gδ(s) =s

1 + δs.

Note that ddsGδ(s) > 0 and

Gδ(t, ξ) = Gδ

(et(1+|ξ|2)α/2

).

By recalling |ξ|2 = |ξ+|2 + |ξ−|2 and |ξ−|2 = |ξ|2 sin2 θ2, we have

|Gδ(t, ξ+)−Gδ(t, ξ)| =

∣∣∣ ∫ 1

0

exp t(1 + |ξ|2 + τ(|ξ+|2 − |ξ|2))α/2

(1 + δ exp t(1 + |ξ|2 + τ(|ξ+|2 − |ξ|2))α/2)2

× tα

2(1 + |ξ|2 + τ(|ξ+|2 − |ξ|2))α/2−1dτ

∣∣∣ |ξ−|2≤ CGδ(t, ξ)(1 + |ξ|2)α/2 sin2 θ

2,


where we have used 12|ξ|2 ≤ |ξ|2 + τ(|ξ+|2− |ξ|2) ≤ |ξ|2. Notice that for 0 < α < 1, 0 <

δ < 1, and for any a, b ≥ 0, we have

(1 + a+ b)α ≤ (1 + a)α + (1 + b)α , (1 + δea)(1 + δeb) ≤ 3(1 + δea+b).

Then,

Gδ

(et(1+|ξ+|2+|ξ−|2)α/2

)≤ Gδ

(et(1+|ξ+|2)α/2+t(1+|ξ−|2)α/2

)≤ 3Gδ(t, ξ

+)Gδ(t, ξ−),

which gives (4.3.9). Therefore, we have∣∣∣∣∫R3

∫S2

bRe µ(ξ−)g(ξ+)Gδ(t, ξ)¯g(ξ)

[Gδ(t, ξ)−Gδ(t, ξ

+)]dσdξ

∣∣∣∣≤ C

∫R3

∫S2

b sin2 θ

2|Gδ(t, ξ

−)µ(ξ−)|Gδ(t, ξ+)|g(ξ+)| 〈ξ〉αGδ(t, ξ)|g(ξ)|dσdξ

≤ C‖Gδµ‖L1‖Gδg‖L2‖ΛαGδg‖L2 ,

so that (4.3.7) follows.We now turn to prove (4.3.8). By using Bobylev identity, and µ(ξ) = µ(ξ+)µ(ξ−), µ(0) =

1, we have

|(Q(g, µ), G2δg)L2| =

∣∣∣∣∫ b(g(ξ−)µ(ξ+)− g(0)µ(ξ)

)G2

δ(t, ξ)¯g(ξ)dσdξ

∣∣∣∣=

∣∣∣∣∫ b(g(ξ−)− g(0)µ(ξ−)

)Gδ(t, ξ)µ(ξ+)Gδ(t, ξ)¯g(ξ)dσdξ

∣∣∣∣≤

∣∣∣∣∫ b g(0)(µ(ξ−)− µ(0)


∣∣∣∣+

∣∣∣∣∫ b(g(ξ−)− g(0)


∣∣∣∣ .For the first term in the last inequality, since

|µ(ξ−)− µ(0)| ≤ |ξ−|2 ≤ |ξ|2 sin2 θ

2,

we have ∣∣∣∣∫ b g(0)(µ(ξ−)− µ(0)


∣∣∣∣≤ ‖g‖L1

∣∣∣∣∫ b(cos θ) sin2 θ

2Gδ(t, ξ)|ξ|2µ(ξ+)Gδ(t, ξ)¯g(ξ)dθdξ

∣∣∣∣≤ CT0‖g‖L1‖Gδg‖L2 ,

where CT0 = 4‖Gδ(2T0, D)|D|2µ‖L2 . While for the second term, when 0 < α < 1/2,the estimate

|g(ξ−)− g(0)| ≤ ‖ 5 g‖L∞|ξ−| ≤ ‖g‖L11|ξ| sin θ

2,


gives ∣∣∣∣∫R3

∫S2

b(g(ξ−)− g(0)


∣∣∣∣≤ ‖g‖L1

1

∣∣∣∣∫R3

∫S2

b(cos θ)∣∣ sin θ

2

∣∣Gδ(t, ξ)|ξ|µ(ξ+)Gδ(t, ξ)¯g(ξ)dθdξ

∣∣∣∣≤ CT0‖g‖L1

1‖Gδg‖L2 .

On the other hand, when 1/2 ≤ α < 1, the above simple calculation does notwork. Instead, we need to use the symmetry in the integral according to the geometricstructure of

ξ+ =ξ + |ξ|σ

2, ξ− =

ξ − |ξ|σ2

, cos θ = |ξ|−1 〈ξ, σ〉 .

For a fixed ξ 6= 0, denote the unit vector σ = Rθ

(ξ|ξ|

)as a rotation of the unit vector

ξ|ξ| by an angle θ. Moreover, denote σ = R−θ

(ξ|ξ|

)and

ξ+ =ξ + |ξ|σ

2, ξ− =

ξ − |ξ|σ2

.

Then we have,|ξ+| = |ξ+|, |ξ−| = |ξ−|, |ξ|−1 〈ξ, σ〉 = cos θ.

With these notations, the integral can be estimated as follows,∫R3

∫S2

b( ξ|ξ|

· σ)(g(ξ−)− g(0)


=

∫R3

∫S2

b( ξ|ξ|

· σ)(g(ξ−)− g(0)


=1

2

∫R3

∫S2

b(cos θ)(g(ξ−) + g(ξ−)− 2g(0)

)Gδ(t, ξ)µ(ξ+)Gδ(t, ξ)¯g(ξ)dσdξ.

Here we have used the fact that dσ = dσ and µ(ξ+) = µ(ξ+). Notice that ξ− and ξ−

are symmetric with respect to ξ so that we can denote them by

ξ− = ~a+~b, ξ− = ~a−~b,with

|~a| = sinθ

2|ξ−| = sin2 θ

2|ξ|, |~b| = sin

θ

2|ξ+| = sin

θ

2cos

θ

2|ξ|.

Thus,

|g(ξ−) + g(ξ−)− 2g(0)| = |g(~a+~b)− 2g(~a) + g(~a−~b) + 2(g(~a)− g(0))|≤ ‖g‖L1

2|~b|2 + 2‖g‖L1

1|~a|.

Finally, for 1/2 ≤ α < 1, we have∣∣∣∣∫R3

∫S2

b(g(ξ−)− g(0)


∣∣∣∣≤ (‖g‖L1

1+ ‖g‖L1

2)

∣∣∣∣∫R3

∫S2

b(cos θ) sin2 θ

2Gδ(t, ξ)(|ξ|+ |ξ|2)µ(ξ+)Gδ(t, ξ)¯g(ξ)dθdξ

∣∣∣∣≤ CT0(‖g‖L1

1+ ‖g‖L1

2)‖Gδg‖L2 .


Therefore, we have obtained (4.3.8) and then completes the proof of the lemma.

We are now ready to prove the second main result in this section.

Proof of Theorem 4.3.1. By the same argument as given in Section 4.2, we see that

G2δ(t,Dv) 〈|Dv|〉−8 g ∈ L∞([0, T0];W

2,+∞(R3))

whose norm is bounded byCδ sup

[0,T0]

‖g(t)‖L1 ,

and moreover,Gδ(t,Dv) 〈|Dv|〉−4 g ∈ C([0, T0];L

2(R3)),

by using g ∈ L1([0, T0], L12). Hence, set

ϕ(t) = G2δ(t,Dv) 〈|Dv|〉−8 g(t, v)

in the last equation of Definition 4.1.1. By a similar argument as the one for (4.1.7),we have

1

2

∫R3

|Gδ(t) 〈|D|〉−4 g(t)|2dv − 1

2

∫R3

|Gδ(0) 〈|D|〉−4 g0|2dv

−1

2

∫ t

0

∫R3

g(τ)(∂tG

2δ(τ)

)〈|D|〉−8 g(τ)dvdτ(4.3.10)

=

∫ t

0

(Lg(τ), G2

δ(τ) 〈|D|〉−8 g(τ)

)L2dτ,

for any t ∈ [0, T0]. On the other hand, it follows from (4.3.6), (4.3.7) and (4.3.8) that

(4.3.11) ‖ΛαGδ 〈|D|〉−4 g‖2L2 ≤ Cµ

(−Lg,G2

δ 〈|D|〉−8 g)L2

+‖Gδ 〈|D|〉−4 g‖2L2 + ‖g‖2

L12

.

Combining (4.3.10) and (4.3.11) implies

‖Gδ(t) 〈|D|〉−4 g(t)‖2L2 +

1

2Cµ

∫ t

0

‖ΛαGδ(τ) 〈|D|〉−4 g(τ)‖2L2dτ ≤

‖Gδ(0) 〈|D|〉−4 g0‖2L2 +

∣∣∣∣∫ t

0

∫R3

g(τ)(∂tG

2δ(τ)

)〈|D|〉−8 g(τ)dvdτ

∣∣∣∣+

∫ t

0

‖Gδ(τ) 〈|D|〉−4 g(τ)‖2L2dτ +

∫ t

0

‖g(τ)‖2L1

2dτ.

Since|∂tGδ(t, ξ)| ≤ Gδ(t, ξ) < ξ >α,

we have ∣∣∣∣∫ t

0

∫R3

g(τ)(∂tG

2δ(τ)

)〈|D|〉−8 g(τ)dvdτ

∣∣∣∣≤ 2

∫ t

0

‖ΛαGδ(τ) 〈|D|〉−4 g(τ)‖L2‖Gδ(τ) 〈|D|〉−4 g(τ)‖L2dτ,

and‖Gδ(0) 〈|D|〉−4 g0‖2

L2 ≤ ‖ 〈|D|〉−4 g0‖2L2 ≤ C‖g0‖2

L1 .


Thus, for any ε > 0, we have

‖Gδ 〈|D|〉−4 g(t)‖2L2 +

( 1

2Cµ

− ε)∫ t

0

‖ΛαGδ(τ) 〈|D|〉−4 g(τ)‖2L2dτ

≤ C0‖g0‖2L1 + Cε

∫ t

0

‖Gδ(τ) 〈|D|〉−4 g(τ)‖2L2dτ + C1

∫ t

0

‖g(τ)‖2L1

2dτ.

By choosing ε = 14Cµ

> 0, the above inequality shows that there exists a constant

C2 > 0 independent of δ ∈]0, 1[, such that for any t ∈]0, T0]

‖Gδ(t) 〈|D|〉−4 g(t)‖2L2

≤ C0‖g0‖2L1 + C2

∫ t

0

‖Gδ(τ) 〈|D|〉−4 g(τ)‖2L2dτ +

∫ t

0

‖g(τ)‖2L1

2dτ

.

Then the Gronwell inequality yields

‖Gδ(t) 〈|D|〉−4 g(t)‖2L2 ≤ C0e

C2t‖g0‖2L1 + C2e

C2t

∫ t

0

e−C1τ‖g(τ)‖2L1

2dτ,

where the positive constants C0 and C2 are independent of δ. By letting δ → 0, weget, for any t ∈ [0, T ],

(4.3.12) ‖et〈|D|〉α 〈|D|〉−4 g(t)‖2L2 ≤ C0e

C2t‖g0‖2L1 + ‖g‖2

L2([0,T ];L12(R3)).

Hence, by noticing

e−t〈|ξ|〉α 〈|ξ|〉8 ≤ Cαt− 8

α ,

for any fixed 0 < t ≤ T0, we have

e12t〈|Dv |〉αg(t, v) ∈ L2.

And this completes the proof of Theorem 4.3.1.

CHAPTER V

A model of kinetic equations

The inhomogeneous Boltzmann equations is much more complicate then homoge-neous case, the main difficulty is to take care of the interplay between the transportoperator, i.e. ∂t + v · ∇x and the collision operator acting only on the microscopicvelocity variable v.

Before the study for the full non linear and non homogeneous Boltzmann equations.In this chapter, we study the model of kinetic type equations given in (3.4.7), themotivation is to understand the smoothing effect of singularity of collision kernel.

5.1. Subelliptic estimates

we consider the following Cauchy problem for a model of kinetic equations

(5.1.1)

Pu = ∂tu+ v · ∇xu+ σ(−4v)

αu = f,u|t=0 = u0

where (x, v) ∈ R2d, 0 ≤ t ≤ T and 0 < σ0 ≤ σ, σ ∈ C∞b . The operator (−4v)

α = |Dv|2α

for α ∈ R, where |Dv|α is a Fourier multiplier of symbol |ξ|αχ(ξ) + |ξ|(1− χ(ξ)), withχ ∈ C∞(Rd), 0 ≤ χ ≤ 1, χ(ξ) = 1 if |ξ| ≥ 2 and χ(ξ) = 0 if |ξ| ≤ 1. If α = 1, this is alinear Vlasov-Fokker-Planck equation(see [64, 65]). When 0 < α < 1 the equation isa linearlized model of Boltzmann equation given in (3.4.7).

The existence of weak solution for Cauchy problem was proved in [85].

Theorem 5.1.1. If f ∈ L1(]0, T [;Hs(R2d)) for some 0 < T < ∞, s ≥ 0 andu0 ∈ Hs(R2d). Then the Cauchy problem of equation (5.1.1) admits an unique weaksolution

u ∈ L∞(]0, T [;Hs(R2d)), (−4v)α/2u ∈ L2(]0, T [;Hs(R2d)).

In this lecture, we consider only the regularity of weak solution. We first studythe commutators of this operator with functions in C∞

b and unbounded function (thecoefficients of our operators P ) vk, k = 1, · · · , d. We give the following technical lemma.

Lemma 5.1.1. Let Ω be an open (unbounded) domain of ]T1, T2[×R2d, a ∈ C∞b (Ω), β ∈

R+. Then there exists C > 0 depending only on the bounded-ness of function a andtheir derivation such that

(5.1.2) ‖[a, |Dv|β]f‖L2(Ω) ≤ C‖ |Dv|(β−1)f‖L2(Ω) + ‖f‖L2(Ω),for any f ∈ C∞

0 (Ω).

Moreover, we have that [vk, |Dv|β] is a Fourier multiplier and

(5.1.3) ‖[vk, |Dv|β]f‖L2(Ω) ≤ |β| ‖ |Dv|(β−1)f‖L2(Ω) + C‖f‖L2 ,

for any f ∈ C∞0 (Ω), k = 1, · · · , d.

65

66 V. A MODEL OF KINETIC EQUATIONS

Proof : Now |Dv|βχ(Dv) ∈ Op(Sβ(Rdv)), then [a, |Dv|βχ(Dv)] is a pseudo-differential

operators of order (β − 1), its symbol isn∑

k=1

(∂vka) (β|ξ|β−2 (iξk)χ(ξ) + |ξ|βi∂ξk

χ(ξ)) + rβ(x, ξ)

where rβ ∈ Sβ−2, which deduces that

‖[a, |Dv|βχ(Dv)]f‖L2 ≤ C‖|Dv|(β−1)f‖L2 + ‖v‖2L2+ ‖|Dv|(1− χ(Dv))f‖L2 .

For the terms |Dv|β(1− χ(Dv)), we just use the bounded-ness of a and |ξ|β(1− χ(ξ))to get the L2 bounded-ness. This is the reason why we give (5.1.3) for unboundedfunction vk. Direct calculation gives

vk(|Dv|βf)(v) = F−1(Dξk

((|ξ|βχ(ξ) + |ξ|(1− χ(ξ))

)f(ξ)

))= F−1

((Dξk

(|ξ|αχ(ξ) + |ξ|(1− χ(ξ))

))f(ξ)

+(|ξ|αχ(ξ) + |ξ|(1− χ(ξ))

)Dξk

f(ξ))

= F−1((Dξk

(|ξ|αχ(ξ) + |ξ|(1− χ(ξ))

))f(ξ)

)+(|Dv|β(vk f)

)(v).

But, for |ξ| ≥ 1,

|Dξk|ξ|β| ≤ |β||ξ|β−1,

and for |ξ| ≤ 2, |Dξk|ξ|| ≤ C. We get that

‖[vk, |Dv|β]f‖L2 ≤ |β|‖|Dv|(β−1)f‖L2 + C‖f‖L2 .


We study now the sub-elliptic estimates. Without loss of generality, we suppose inthe following that σ = σ0 > 0 is constant, and consider the operators

P = ∂t + v · ∇x + σ0(−4v)α

on an open domain Ω ⊂ Rt×Rdx×Rd

v. In the application, we will take Ω =]a, b[×Rdx×Rd

v,so that P are not a pseudo-differential operators in Ω.

We put

Λ =(1 + |Dt|2 + |Dx|2 + |Dv|2

)1/2,

andX0 = Λ−1/3(∂t + v · ∇x), Xj = Λα−1∂vj

, j = 1, · · · , d.Then Xj ∈ Op(Sα(Rt × Rd

x × Rdv)), j = 1, · · · , d is a family of pseudo-differential

operators. But X0 is not a pseudo-differential operator (of order 2/3) on ]a, b[×R2d,since the coefficient x is not bounded on ]a, b[×R2d. We will pay more attention totreat this term ( see [85]).

Proposition 5.1.1. If Ω is an open domain of Rt × Rdx × Rd

v, 1/3 < α < 1, thenthere exists C > 0 such that

(5.1.4)n∑

j=1

‖Xju‖2L2 ≤ C

Re(Pu, u) + ‖u‖2

L2

,

5.1. SUBELLIPTIC ESTIMATES 67

for any u ∈ C∞0 (Ω). For the X0, we have

(5.1.5) ‖X0u‖2L2 ≤ C

3∑

k=1

Re(Pu,Aku) + ‖u‖2L2

,

for any u ∈ C∞0 (Ω), where

A1 ∈ Op(S01,0(R2d+1)), A2 = (Λ−1/3 + Λ−1)X0, A3 = −(∂t + v · ∇x)Λ

−1X0.

Proof : For any u ∈ C∞0 (Ω), the integration by parts deduces immediately

(5.1.6) Re(Pu, u) = Re(σ0|Dv|αu, |Dv|αu) = σ0‖|Dv|αu‖2L2 .

Then a direct calculation gives

‖Xju‖L2 = ‖Λα−1Dvju‖L2 ≤ ‖|Dv|αu‖L2 + C‖u‖L2 .

We have proved (5.1.4). In the future proof, we need also the estimate (5.1.6). Puttingnow w = A2u = Λ−1/3X0u, we have

‖X0u‖2L2 = Re(Pu,w)−Re(σ0|Dv|2αu,w).

Since |Dv|ασ0|Dv|α is a positive operator on L2, it follows that∣∣∣Re (σ0|Dv|2αu,w)∣∣∣ ≤ Re (σ0|Dv|αu, |Dv|αu) +Re(σ0|Dv|αw, |Dv|αw)

We get

(5.1.7) ‖X0u‖2L2 ≤ |Re(Pu,w)|+ |Re (σ0|Dv|αw, |Dv|αw)|+ C‖|Dv|αu‖2

L2 .

We study now the term

Re (σ0|Dv|αw, |Dv|αw = Re (σ0|Dv|2αw,w) = Re(Pw,w)= Re (Pu,A3u) + Re ([Λ−2/3, v] · ∂x(∂t + v · ∂x)u,w)

+ Re (σ0[|Dv|2α, Λ−1/3X0]u,w).

But we have

[Λ−2/3, v] = −2

3Λ−2/3−2∂v,

and

[|Dv|2α, Λ−1/3X0] = Λ−2/3[|Dv|2α, v] ·Dx.

We use now (5.1.3), to deduce that∣∣∣Re(σ0[|Dv|2α,Λ−1/3X0]u,w)∣∣∣ ≤ C(‖|Dv|αu‖L2 + ‖u‖L2)‖X0u‖L2 .

It is easier for

|Re ([Λ−2/3, v] · ∂x(∂t + v · ∂x)u,w)| ≤ C‖w‖2L2

and‖w‖2

L2 = ((∂t + v · ∇x)u,Λ−1X0u)

≤ |Re (Pu,Λ−1X0u)|+ |(σ0|Dv|2αu,Λ−1X0u)|≤ |Re (Pu,Λ−1X0u)|+ C||Dv|αu‖2

L2 + 116‖X0u‖2

L2 .

We get finally the desired estimate (5.1.5).


We study now the microlocal regularity,

[Xj, X0] = Λα−1/3−1∂xj+ (α− 1)Λα−1/3−3∂v · ∂x∂vj

= Λα−1/3(Λ−1∂xj

)+ (α− 1)Λ−1/3Λ0Xj,

where Λ0 = Λ−2∂v · ∂x is a pseudo-differential operator of order 0. So that

Λα/2−1/6(Λ−1∂xj

)(5.1.8)

= Λ−α/2+1/6[Xj, X0]− (α− 1)Λ−α/2−1/6Λ0Xj,

and(5.1.9)

Λα/2−1/6(Λ−1∂t

)= Λα/2−1/6Λ−1

(∂t + v · ∇x

)− Λα/2−1/6Λ−1

(v · ∇x

)= Λα/2−1/6−2/3X0 −

∑nj=1

(Λ−α/2+1/6[Xj, X0]vj − (α− 1)Λ−α/2−1/6Λ0Xjvj

).

We recall

(5.1.10) Λα(Λ−1∂vj

)= Xj.

We have proved the following subelliptic estimates.

Proposition 5.1.2. With same notations of Proposition 5.1.1, there exists C > 0such that

(5.1.11)n∑

j=1

∥∥∥Λα2− 1

6−1∂xj

u∥∥∥2

L2≤ C

3∑k=1

Re (Pu,Aku) + ‖u‖2L2

.

For the differentiation with respect to variable t, we have

(5.1.12)∥∥∥Λα

4− 1

12−1∂tu

∥∥∥2

L2≤ C

3∑k=1

Re(Pu,Aku) + ‖ < v > u‖2L2

,

and

(5.1.13)∥∥∥Λα

4− 1

12u∥∥∥2

L2≤ C

3∑k=1

Re(Pu,Aku) + ‖ < v > u‖2L2

.

for any u ∈ C∞0 (Ω), where < v >= (1 + |v|2)1/2.

Remark 5.1.1. By density, the estimates of above proposition are also true for anyu ∈ H2

0 (Ω).

Proof : Since ‖(A+ iB)u‖2L2 ≥ 0, it follows that

(u, i[A,B]u) ≤ 3

2

(‖Au‖2

L2 + ‖Bu‖2L2

)+

1

2

(‖(A− A∗)u‖2

L2 + ‖(B −B∗)u‖2L2

)for any operators A, B.

Putting A = −i[Xj, X0]∗Λ−α+1/3Xj and B = X0, we have [Xj, X0]

∗ = −[Xj, X0]and

i[A,B] = −[ [Xj, X0]Λ−α+1/3Xj, X0]

= [Xj, X0]∗Λ−α+1/3[Xj, X0]− [ [Xj, X0]Λ

−α+1/3, X0]Xj

= [Xj, X0]∗Λ−α+1/3[Xj, X0] + Λ−1/3Xj,

5.1. SUBELLIPTIC ESTIMATES 69

where Λ−1/3 is a pseudo-differential operator of order −1/3.We obtain, from (5.1.8),∥∥∥Λα

2− 1

6−1(∂xj

u)∥∥∥2

L2≤

∥∥∥Λ−α2+ 1

6 [Xj, X0]u∥∥∥2

L2+ C‖Xju‖2

L2

≤ C(‖Xju‖2

L2 + ‖X0u‖2L2 + ‖v‖2

L2

).

Those estimates for j = 1, · · · , d together with (5.1.4) and (5.1.5) show (5.1.11).

To prove (5.1.12), we take w = Λα2− 1

6−2(∂tu). Then we have that

‖Λα4− 1

12−1(∂tu)‖2

L2

= Re (Pu, Λ0u)− Re (σ0|Dv|2αu, Λ0u)− Re (v · ∇xu, Λα2− 1

6−2(∂tu)),

where Λ0 = Λα2− 1

6−2∂t ∈ Op(S0(R2d+1)). Then

|Re (σ0|Dv|2αu, Λ0u)| ≤ C‖Dv|αu‖2L2 + ‖u‖2

L2.

For the last term, we have

|Re (v · ∇xu, Λα2− 1

6−2(∂tu))| =

∣∣∣∣∣d∑

k=1

Re (Λ−1∂t(vku), Λα2− 1

6−1(∂xk

u))

∣∣∣∣∣≤ C

d∑k=1

‖vku‖L2‖Λα2− 1

6−1(∂xk

u)‖L2 ,

we get the desired estimate (5.1.12) by using (5.1.11) and (5.1.6).Finally, combination of (5.1.4), (5.1.11) and (5.1.12) give (5.1.13). We have proved

Proposition 5.1.2.

We can use directly (5.1.9) to get

‖Λα2− 1

6−1(∂tu)‖2

L2 ≤ ‖Λα2− 1

6−2/3X0u‖2

L2 +d∑

j=1

‖Λα2− 1

6−1(∂yj

(xju))‖2L2 ,

then (5.1.11) with test function (vku) and (5.1.5) deduce the following estimate

(5.1.14)

∥∥∥Λα2− 1

6−1∂tu

∥∥∥2

L2≤ C

3∑k=1

Re(Pu,Aku)

+d∑

j=1

3∑k=1

Re(P (vju), Ak(vju)) + ‖ < v > u‖2L2

,

which gives also the following hypoelliptic estimate with weight

(5.1.15)

∥∥∥Λα2− 1

6u∥∥∥2

L2≤ C

3∑k=1

Re (Pu,Aku)

+d∑

j=1

3∑k=1

Re(P (vju), Ak(vju)) + ‖ < v > u‖2L2

,


5.2. C∞-regularity of weak solutions

By combination the estimate(5.1.4), (5.1.5) and (5.1.13), we get the following sub-elliptic estimate :

(5.2.1)∥∥∥Λα

4− 1

12u∥∥∥2

L2(R2d+1)≤ C

‖ < v > Pu‖2

L2(R2d+1) + ‖ < v > u‖2L2(R2d+1)

,

for any u ∈ H20 (]0, T [×R2d). Here we used the fact

‖A1u‖L2 ≤ C‖u‖L2 , ‖(A2+ < v >−1 A3)u‖L2 ≤ C‖X0u‖L2 .

For δ > 0, we set

Λδ =(1 + δ(|Dt|2 + |Dx|2 + |Dv|2)

)1/2.

We will use the following notations : for ϕ, ψ ∈ C∞0 , we say ϕ ⊂⊂ ψ if ψ = 1 in a

neighborhood of supp ϕ.We prove the following results.

Proposition 5.2.1. Let 1/3 < α < 1 and s ≥ 0. Assume that f, < v > f ∈Hs(]a, b[×R2d). Let u ∈ Hs(]a, b[×R2d) be a weak solution of equation Pu = f on]a, b[×R2d such that < v > u ∈ Hs(]a, b[×R2d). Then for any ϕ, ψ ∈ C∞

0 (]a, b[), ϕ ⊂⊂ ψand 0 < δ < 1, there exists a constant C > 0 independent of δ such that

(5.2.2)

∥∥∥Λα4− 1

12ψ(Λδ)−2Λs(ϕu)

∥∥∥2

L2(R2d+1)≤ C

‖Λs(< v > ψf)‖2

L2(R2d+1)

+‖Λs(< v > ψu)‖2L2(R2d+1)

+ ‖Λsψf‖2L2(R2d+1)

+ ‖Λsψu‖2L2(R2d+1)

,

with some ψ ∈ C∞0 (]a, b[), ψ ⊂⊂ ψ. Here the cut-off function are only for t variable.

Take the limit δ → 0 in (5.2.2). Then it deduces

ψΛs(ϕu) ∈ Hα4− 1

12 (R2d+1),

because the commutator [ψ, Λs] is a pseudo-differential operator of order s−1. We haveobtained a gain of regularity of order 1

4(α − 1

3) for (weak) solution with a supplement

condition < v > f ∈ Hs(]a, b[×R2d) and < v > u ∈ Hs(]a, b[×R2d).

Proof of Proposition 5.2.1 We will choose u1 = ψΛ−2δ Λs(ϕu) as test function in

(5.1.5) and (5.1.13), (if we consider the partial Sobolev space, we take u1 = ψΛ−2δ Λs

x,v(ϕu)as test function)∥∥∥Λα

4− 1

12ψ(Λδ)−2Λs(ϕu)

∥∥∥2

L2+ ‖X0ψ(Λδ)

−2Λs(ϕu)‖2L2

≤ C∑3

k=1Re(Pψ(Λδ)−2Λs(ϕu), Akψ(Λδ)

−2Λs(ϕu))

+‖ < v > ψ(Λδ)−2Λs(ϕu)‖2

L2

.

We calculate the commutator terms

[P, ψΛ−2δ Λsϕ]u = ∂t(ψΛ−2

δ Λsϕ)ψu+ ψ[v, Λ−2δ Λs] · ∇x(ϕu)

which is a pseudo-differential operator of order s for (x, v) variables. And moreover

< v > [P, ψΛ−2δ Λsϕ] < v >−1

5.2. C∞-REGULARITY OF WEAK SOLUTIONS 71

= [P, ψΛ−2δ Λsϕ]+ < v > [[P, ψΛ−2

δ Λsϕ], < v >−1],

where the second term is also a pseudo-differential operator of order s. There existsC > 0 independent of δ such that

‖ < v > [P, ψΛ−2δ Λsϕ]u‖L2 ≤ C‖ψΛs < v > ψu‖L2 ,

and

‖ < v > ψΛ−2δ ΛsϕPu‖L2 ≤ ‖ψΛs < v > ψPu‖L2 .

We get the estimate (5.2.2) directly from (5.2.1).

We study now the regularity of < v >−1 u.

Theorem 5.2.1. Let 1/3 < α < 1 and s ≥ 0. We suppose that f ∈ Hs(]a, b[×R2d))and u ∈ Hs(]a, b[×R2d) is a weak solution of equation Pu = f on ]a, b[×R2d. Then wehave

< v >−1 u ∈ Hs+α4− 1

12 (]a′, b′[×R2d),

for any a < a′ < b′ < b.

The proof of this theorem uses the following hypoelliptic estimate with weight< v >−1.

Proposition 5.2.2. Suppose that 1/3 < α < 1. Then there exists C > 0 such that

(5.2.3)

∥∥∥Λα4− 1

12 (< v >−1 u1)∥∥∥2

L2+ ‖X0(< v >−1 u1)‖2

L2

≤ C 4∑

k=1

Re(< v >−1 Pu1, Ak(< v >−1 u1) + ‖u1‖2L2

for any u1 ∈ C∞

0 (]a, b[×R2d), where A1, A2, A3 are the same operators as in Proposition5.1.1 and A4 =< v >2.

Proof : Putting the test function < v >−1 u1 in (5.1.5) and (5.1.13), we have that∥∥∥Λα4− 1

12 (< v >−1 u1)∥∥∥2

L2+ ‖X0(< v >−1 u1)‖2

L2

≤ C 3∑

k=1

Re(P (< v >−1 u1), Ak(< v >−1 u1) + ‖u1‖2L2

.

We need to estimate the commutator term

3∑k=1

|([P, < v >−1]u1, Ak(< v >−1 u1))|

by th right hand side of (5.2.3). Since < v >−1∈ C∞b (Rn), Lemma 5.1.1 implies

‖[P, < v >−1]u1‖2L2 = ‖σ0[|Dv|2α, < v >−1]u1‖2

L2 ≤ C(‖|Dv|αu1‖2

L2 + ‖u1‖2L2

).


By using Cauchy-Schwarz inequality,

(5.2.4)

2∑k=1

|([P, < v >−1]u1, Ak(< v >−1 u1))|

≤ C(‖|Dv|αu1‖2

L2 + ‖u1‖2L2

)+ 1

1000‖X0(< v >−1 u1)‖2

L2

≤ C(|Re (Pu1, u1) + ‖u1‖2

L2

)+ 1

1000‖X0(< v >−1 u1)‖2

L2 .

For the last term, we have

([P, < v >−1]u1, A3(< v >−1 u1))

= −σ0([|Dv|2α, < v >−1]u1, ∂tΛ−1X0(< v >−1 u1))

−d∑

j=1

σ0(vj[|Dv|2α, < v >−1]u1, ∂xjΛ−1X0(< v >−1 u1)).

The estimation of σ0([|Dv|2α, < v >−1]u1, ∂tΛ−1X0(< v >−1 u1)) is the same as (5.2.4).

On the other hand, for j = 1, · · · , d,

vj[|Dv|2α, < v >−1] = vj|Dv|2α < v >−1 −vj < v >−1 |Dv|2α

= [vj, |Dv|2α] < v >−1 +[|Dv|2α, vj < v >−1]

=< v >−1 [vj, |Dv|2α] + [[vj, |Dv|2α], < v >−1] + [|Dv|2α, vj < v >−1].

Since [vj, |Dv|2α] is a Fourier multiplier, we use Lemma 5.1.1 with the functions <v >−1, vj < v >−1∈ C∞

b (Rn),

|(vj[|Dv|2α, < v >−1]u1, ∂xjΛ−1X0(< v >−1 u1))|

≤ C(‖|Dv|αu1‖2

L2 + ‖u1‖2L2

)+ 1

1000‖X0(< v >−1 u1)‖2

L2

≤ C(|Re(Pu1, u1) + ‖u1‖2

L2

)+ 1

1000‖X0(< v >−1 u1)‖2

L2 .

We proved finally (5.2.3).

Similarly to Proposition 5.2.1, we have the following :

Proposition 5.2.3. Let 1/3 < α < 1 and s ≥ 0. Assume that f ∈ Hs(]a, b[×R2d).If u ∈ Hs(]a, b[×R2d) is a (weak) solution of equation Pu = f in ]a, b[×R2d, then forany ϕ, ψ ∈ C∞

0 (]a, b[), ϕ ⊂⊂ ψ and 0 < δ < 1, there exists a C > 0 independent of δsuch that ∥∥∥Λα

4− 1

12 < v >−1 ψ(Λδ)−2Λs(ϕu)

∥∥∥2

L2(R2d+1)(5.2.5)

≤ C‖Λsψf‖2

L2(R2d+1) + ‖Λsψu‖2L2(R2d+1)

,

with some ψ ∈ C∞0 (]a, T [), ψ ⊂⊂ ψ.

We just choose u1 = ψΛ−2δ Λs(ϕu) as test function in (5.2.3). The estimation of

commutator term

[P, ψΛ−2δ Λsϕ]u = ∂t(ψΛ−2

δ Λsϕ)ψu+ ψ[v,Λ−2δ Λs] · ∇x(ϕu)

5.2. C∞-REGULARITY OF WEAK SOLUTIONS 73

is the same as in the proof of Proposition 5.2.1. We get also∥∥∥|Dv|α < v >−1 ψ(Λδ)−2Λs(ϕu)

∥∥∥2

L2(5.2.6)

≤ C‖ < v >−1 Λs(ψf)‖2

L2 + ‖Λsψu‖2L2

,

and

(5.2.7)∥∥X0 < v >−1 ψ(Λδ)

−2Λs(ϕu)∥∥2

L2 ≤ C‖Λsψf‖2

L2 + ‖Λsψu‖2L2

.

Proof of Theorem 5.2.1. Take the limit δ → 0 in (5.2.5). Then it deduces

< v >−1 ψΛs(ϕu) ∈ Hα4− 1

12 (R2d+1),

because the commutator [ψ, Λs] is a pseudo-differential operator of order s − 1. Weobtain a gain of regularity for < v >−1 u.

We have also proved that

u ∈ Hs+α4− 1

12loc (]a, b[×R2d)

since for any ϕ ∈ C∞0 (]a, b[), ψ ∈ C∞

0 (Rdv), we have

‖Λs+α4− 1

12 (ϕ(t)ψ(v)u)‖L2

≤ ‖ψ(v)Λs+α4− 1

12 (ϕ(t)u)‖L2 + C‖Λs(ϕ(t)u)‖L2

≤ C‖ < v >−1 Λs+α

4− 1

12 (ϕ(t)u)‖L2 + ‖Λs(ϕu)‖L2

≤ C

‖Λs+α

4− 1

12 (< v >−1 ϕ(t)u)‖L2 + ‖Λs(ϕu)‖L2

.

High order regularity

We prove now the high order regularity of weak solution of Cauchy problem (5.1.1).

Theorem 5.2.2. Let 1/3 < α < 1 and f ∈ Hs(]a, b[×R2d)) for s ≥ 0. If u ∈L2(]a, b[×R2d) is a weak solution of equation Pu = f on ]a, b[×R2d, then there existsk0 ∈ N such that

< v >−k0−1 u ∈ Hs+α4− 1

12 (]a′, b′[×R2d),

for any a < a′ < b′ < b. In particular, if f ∈ H∞(]a, b[×R2d)), then u ∈ C∞(]a, b[×R2d)).

Remark that k0 is in order of [4s(α− 13)−1] + 1. We can also use the local estimate

of Proposition 5.3.1 to deduce a local regularity results for 0 < α < 1, but we focushere to global estimate, so we suppose that 1/3 < α < 1.

We have to study the commutator of < v >−k with |Dv|2α and Λs. We give firstlythe following lemma.

Lemma 5.2.1. Let B ∈ Op(Sm(R2d+1)), then, for any k ∈ N,

(5.2.8) [< v >−k, B] =< v >−k−1 B1+ < v >−k−2 B2,

where B1 ∈ Op(Sm−1(R2d+1)) with symbolv

< v >· ∂ξB1, and B2 ∈ Op(Sm−2(R2d+1)).


For the commutator with Fourier multiplier |Dv|2α, we have that, for k = 2`, ` ∈ N,

(5.2.9) [< v >−k, |Dv|2α] =< v >−1 F1 < v >−k + < v >−2 F2 < v >−k,

where F1 is an operators of form

F1 =n∑

j=1

ajAj(Dv), with aj ∈ C∞b (Rn), Aj(ξ) = Dξj

(|ξ|2αχ2(ξ)

),

and F2 ∈ L(L2, L2) is a finite sum of form a(v)b(Dv) with a, b ∈ C∞b (Rn).

Remark 5.2.1. In the application, if we take B = Λ−mδ , 0 ≤ m, 0 < δ < 1

an uniformly bounded family in Op(S0(R2d+1)), then B1 ∈ Op(S−1(R2d+1)), B2 ∈Op(S−2(R2d+1)) is also uniformly bounded. We remark also

‖F1w‖L2 ≤ C‖|Dv|αw‖L2 ≤ C‖Λαxw‖L2 + ‖w‖L2.

Proposition 5.2.4. Suppose that 1/3 < α < 1 and k ∈ N, there exists C > 0 suchthat ∥∥∥Λα

4− 1

12 (< v >−k−1 u1)∥∥∥2

L2+∥∥X0(< v >−k−1 u1)

∥∥2

L2(5.2.10)

≤ C 4∑

k=1

Re (< v >−k−1 Pu1, Ak(< v >−k−1 u1) + ‖ < v >−k u1‖2L2

for any u1 ∈ C∞

0 (]a, b[×R2d), where A1, · · · , A4 are the same as in Proposition 5.2.2 .

By density, (5.2.10) is true for any u1 ∈ H20 (]a, b[×R2d). The proof of this Proposi-

tion is similar to that of 5.2.2.

Proof of Theorem 5.2.2Take the limit δ → 0 in (5.2.5). Then it deduces that the weak solution has the

following regularity

< v >−1 (ϕu) ∈ Hα4− 1

12 (R2d+1).

We have proved the Theorem 5.2.2 if s = 0.

We shall prove higher order regularity by induction.

Proposition 5.2.5. Let ε0 = α4− 1

12> 0 and u ∈ L2(]a, b[×R2d). Suppose that for

some k0 ∈ N we have

Λkε0(< v >−k ϕ(t)u) ∈ L2(R2d+1) and(5.2.11)

Λkε0(< v >−k ϕ(t)Pu) ∈ L2(R2d+1)

for any ϕ ∈ C∞0 (]a, b[) and 0 ≤ k ≤ k0. Then we have that

(5.2.12) Λ(k0+1)ε0(< v >−k0−1 ϕ(t)u) ∈ L2(R2d+1)

for any ϕ ∈ C∞0 (]a, b[).

5.3. GEVREY HYPOELLIPTICITY 75

Consider now s > 0 in Theorem 5.2.2 and take k0 ∈ N such that k0ε0 ≤ s. Thenwe have by hypothesis of Theorem 5.2.2 that f ∈ Hk0ε0(]a, b[×R2d) ⊂ Hs(]a, b[×R2d).Since < v >−k0∈ C∞

b (Rnx) we have < v >−k0 f ∈ Hk0ε0 (]a, b[×R2d). We prove finally,

by induction results of Proposition 5.2.5, that

< v >−k0−1 ϕ(t)u ∈ Hs+ε0(R2d+1)

with k0 = [sε−10 ] + 1. We have proved Theorem 5.2.2.

Proof of Proposition 5.2.5. The proof is similar to that of Proposition 5.2.4. Wechoose u1 = ψΛ−2−k0ε0

δ Λk0ε0(ϕu) ∈ H20 (]a, b[×R2d) as test function in (5.2.10). We have∥∥Λε0(< v >−k0−1 ψΛ−2−k0ε0

δ Λk0ε0(ϕu))∥∥2

L2 +∥∥X0(< v >−k0−1 u1)

∥∥2

L2

≤ C 4∑

j=1

Re(< v >−k0−1 PψΛ−2−k0ε0

δ Λk0ε0(ϕu), Aj(< v >−k0−1 u1))

+‖ < v >−k0 u1‖2L2

.

For the commutator terms,

[P, ψΛ−2−k0ε0δ Λk0ε0ϕ]u =

∂t(ψΛ−2δ Λk0ε0ϕ)ψu+ ψ[x, Λ−2−k0ε0

δ Λk0ε0 ] · ∇y(ϕu),

we have immediately

4∑j=1

∣∣∣( < v >−k0−1 [P, ψΛ−2−k0ε0δ Λk0ε0ϕ]u, Aj(< v >−k0−1 u1)

)∣∣∣≤ C

‖Λk0ε0 < v >−k0 (ψu)‖2

L2 + ‖ψu‖2L2

+ 1

1000‖X0(< v >−k0−1 u1))‖2

L2 .

Finally we prove, ∥∥Λε0(< v >−k0−1 ψΛ−2−k0ε0δ Λk0ε0(ϕu))

∥∥2

L2

≤ C‖Λk0ε0 < v >−k0 ϕPu‖2

L2(R2d+1)

+‖Λk0ε0(< v >−k0 ϕu)‖2L2 + ‖ϕu‖2

L2

.

Taking δ → 0, we have proved Proposition 5.2.5, since [< v >−k0−1 ψ, Λk0ε0 ] is apseudo-differential operator of order k0ε0 − 1.

5.3. Gevrey hypoellipticity

We present now some results about the local Gevrey regularity of solution of kineticequation

(5.3.1) Pu = ∂tu+ v · ∇xu+ σ(−4v)αu = f.

Since we study the interior regularity of solution, we consider the above equation onfull space Rt × Rd

x × Rdv. For the detail of this section, see [34].

For this kinetic equation, it is difficult (impossible!) to prove the global Gevreyregularity effect as in (4.3.12). We recall here the definition of local Gevrey class


function. Let U be an open subset of Rn and 1 ≤ s < +∞, we say that f ∈ Gsloc(U) if

for any compact subset K of U , there exists a constant CK , such that for all α ∈ Nn,

‖∂αf‖L2(K) ≤ C|α|+1K (α!)s.(5.3.2)

If W is a closed subset of Rn, Gsloc(W ) denote the restriction of Gs

loc(W ) on W where

W is an open neighborhood of W .We say an operator A is Gs-hypoelliptic in U if u ∈ D′(U), A u ∈ Gs

loc(U) impliesu ∈ Gs

loc(U).We have now the following result.

Theorem 5.3.1. Let 0 < α < 1 and δ = maxα4, α

2− 1

6. Then the operator P

given by (5.3.1) is Gs-hypoelliptic in R2n+1 for any s ≥ 2δ

, provided the coefficientσ(t, x, v) > 0 and belongs to Gs

loc(R2n+1).

Similar to the C∞-hypoellipticity of operators P . We need the following subellipticestimate.

Proposition 5.3.1. Let 0 < α < 1 and K be a compact subset of ]0, T [×R2d. Forany r ≥ 0, there exists a constant CK,r, such that for any f ∈ C∞

0 (K),

(5.3.3) ‖u‖2Hr+δ ≤ CK,r ‖Pu‖2

Hr + ‖u‖2L2 ,

where δ = maxα/4, α/2− 1/6.

The proof of this Proposition is by using the Kohn-Fourier multiplier, and a verycareful computation of commutators. This Proposition improve the results of Proposi-tion 5.1.2 for the index up to 0 < α ≤ 1/3. This proposition deduces also the followingresults as in the section 5.2.

Theorem 5.3.2. Let 0 < α < 1 and f ∈ Hsloc(R2d+1)) for s ≥ 0. If u ∈ L2(R2d+1)

is a weak solution of equation Pu = f on R2d+1, then

u ∈ Hs+δloc (R2d+1).

In particular, if f ∈ C∞(R2d+1), then u ∈ C∞(R2d+1).

But this regularity results is strictly a local results, the estimate (5.3.3) is strictlya local estimate, that means that it is depends the compact K, therefore the estimate(5.1.15) is holds on non compact domain Ω =]a, b[×R2d. We want just point outthat the commutators of a cutoff function of v variable with the Boltzmanncollision operator is a very delicate analysis.

For the local Gevrey regularity of Theorem 5.3.1, the proof is standard by thederivation of equations and multiplication of cutoff function. We omit here the detailof proof. We point out just the key steps, the estimation of commutators of operatorswith “Gevrey cutoff” functions, see for examples [35, 37].

For 0 ≤ ρ < 1, set

Ωρ = v ∈ Rd; |v|2 < 1− ρ.

5.4. SEMI-LINEAR EQUATIONS 77

Now for any N ∈ N, N ≥ 2 and any 0 < ρ < 1, there exists the following cutoff function

(5.3.4)

ϕρ,N ∈ C∞0 (ΩN−1

Nρ)

ϕρ,N(v) = 1, v ∈ Ωρ,

supv∈Rd

|∂αϕρ,N(v)| ≤ Cα

(N

ρ

)|α|.

We study now the commutators of above cutoff function with operator |Dv|α .

Lemma 5.3.1. There exists a constant Cα,d, such that for any 1 ≤ κ ≤ d+ 3, andany f ∈ S(Rd),

(5.3.5) ‖[|Dv|α, ϕρ,N ]f‖Hκ ≤ Cα,d

(Nρ

)α‖f‖Hκ +(Nρ

)κ+α‖f‖L2

and

(5.3.6) ‖[|Dv|α, [|Dv|α, ϕρ,N ] ]f‖Hκ ≤ Cα,d

(Nρ

)2α‖f‖Hκ +(Nρ

)κ+2α‖f‖L2

.

To prove this Lemma, we use the fractional derivative formula (1.2.1), for anyg ∈ S(Rd),

|Dv|α g(v) = Cα

∫Rd

g(v)− g(v − v′)

|vd+αdv′.

Then

|Dv|α(ϕρ,N(v)f(v)

)= Cα

∫Rd

f(v)ϕρ,N(v)− f(v − v)ϕρ,N(v − v)

|v|d+αdv

= ϕρ,N(v)|Dv|α(f(v)

)+ Cα

∫Rn

f(v − v)(ϕρ,N(v)− ϕρ,N(v − v)

)|v|d+α

dv,

this gives the formula of commutators[|Dv|α, ϕρ,N(v)

]f(v) = Cα

∫Rd

f(v − v)(ϕρ,N(v)− ϕρ,N(v − v)

)|v|d+α

dv.(5.3.7)

Then we can deduces (5.3.1) by Holder estimate of cutoff function.

5.4. Semi-linear equations

We consider now the following semi-linear Cauchy problems

(5.4.1)

Pu = F (u)u|t=0 = u0

with F ∈ C∞(R) and F (0) = 0. We get the following results .

Theorem 5.4.1. If s > d, 1/3 < α < 1 and u0 ∈ Hs(R2d), then there exists a T > 0such that the Cauchy problem (5.4.1) admits a solution.

u ∈ C0([0, T [;Hs(R2d)), (−4v)α/2u ∈ L2(]0, T [;Hs(R2d)),


andu ∈ H+∞

loc (]0, T [×R2d) ⊂ C∞(]0, T [×R2d).

More precisely, for the regularity we see that for any m ∈ N there exists an m0 ∈ Nsuch that

< v >−m0 u ∈ Hm(]a, b[×R2d),

for any 0 < a < b < T .

Remark : We have also same regularity results in Gevrey class as Theorem 5.3.1. Ifσ, F ∈ Gs, s ≥ 2

δ, then u ∈ Gs. see [34].

In [85], we have proved that the Cauchy problem (5.4.1) admits a weak solutionu ∈ L∞(]0, T [;Hs(R2d)) if u0 ∈ Hs(R2d)) and s > d. By using Sobolev embeddingtheorem, the condition s > d implies that u ∈ L∞(]0, T [;Hs(R2d)) ∩ L∞(]0, T [×R2d).Now Lemma 1.5.1 ensures the stability in Sobolev space by nonlinear composition.

We prove the following proposition for nonlinear hypoellipticity. It deduces imme-diately the regularity of weak solution of Theorem 5.4.1.

Proposition 5.4.1. Suppose that 1/3 < α < 1 and F ∈ C∞(R), F (0) = 0. Letu ∈ L2(]a, b[×R2d) ∩ L∞(]a, b[×R2d) be a weak solution of equation Pu = F (u) in]a, b[×R2d. Then for any m ∈ N, there exists m0 ∈ N such that

< v >−m0 u ∈ Hm(]a′, b′[×R2d),

for any a < a′ < b′ < b. In particular, we have that u ∈ C∞(]a, b[×R2d).

Proof : We prove also this proposition by induction. By hypothesis, we have that

u ∈ L2(]a, b[×R2d) ∩ L∞(]a, b[×R2d), then F (t, x, y) = F (u(t, x, y)) ∈ L2(]a, b[×R2d).Proposition 5.2.5 with k = 0 deduces that for any ϕ ∈ C∞

0 (]0, T [), there exists aconstant C > 0 and ψ ∈ C∞

0 (]0, T [) with ϕ ⊂⊂ ψ such that

(5.4.2)∥∥< v >−1 ϕu

∥∥2

Ha/4−1/12 ≤ C‖ψF‖2

L2 + ‖ψu‖2L2

.

We suppose now for some k ∈ N and any ϕ ∈ C∞0 (]0, T [),

< v >−k ϕu ∈ Hkε0(R2d+1),

here ε0 = 14

(α− 1

3

)> 0. We want to prove that

< v >−k−1 ϕu ∈ H(k+1)ε0(R2d+1).

But from Proposition 5.2.5, we need only to prove that

‖Λkε0 < v >−k ϕF (u)‖2L2(R2d+1) ≤ C

‖Λkε0 < v >−k ψu‖2

L2(R2d+1) + ‖ψu‖2L2

,

with the constant C as in Lemma 1.5.1. The proof of this estimate is also the same asthat of Lemma 1.5.1. We just remark that, for the nonlinear function

F (x, u1) =< v >−k F (< v >k u1),

if u1 ∈ Hk0ε0 and < v >k u1 ∈ L∞, then for any λ ∈ Nn,

|∂λv F (v, u1)| ≤ Cλ,

and F ((v, 0) = 0.

CHAPTER VI

Uncertainty principle and kinetic equation

Before the consideration for the inhomogeneous Boltzmann equation, we concen-trate to the kinetic part. We present, in this chapter, a new methods for the smoothnesseffect of solution of kinetic equation, i. e. if the solution of a kinetic equations hasalready some regularity with respect to velocity variables, we can get a gain of reg-ularity with respect to time and space variables. We look the kinetic equations as aHormander type equations, so that we prove first a generalized version of Fefferman’suncertainty principle, then we applies this principle to prove the the hypoellpiticity ofkinetic equations.

6.1. Uncertainty principle

We state now the generalized version of the uncertainty principle as follows :

From now on, the variable v ∈ Rn corresponds to the microscopic velocity in thecontext of kinetic equations. Let a+ and a− be two non-negative functions of variablev ∈ Rn, with a− ∈ L∞(Rn). Set

as(Dv) = |Dv|s , for 0 < s < 1 ;as,log(Dv) = (log < Dv >)s, for 1

2< s .

Our aim is to prove the following type coercivity estimate under some reasonableconditions on the functions a+ and a− defined on Rn. That is, there exists a constantc > 0 such that

(6.1.1) Re((a2(Dv) + a+)f, f)L2(Rn) ≥ c(a−f, f)L2(Rn) for all f ∈ S(Rn),

where a(Dv) can be either as(Dv) or as,log(Dv).We introduce now some notations. For the variable r > 0, we set

(6.1.2) a(r) = r−2s if a = as, and a(r) = |log r|2s−1 if a = as,log,

where log r is given in (1.2.2).For a given cube Q in Rn, we denote its side length by l(Q). Let Q∗ be any cube

such that Q ⊂ Q∗ with l(Q∗) = 2l(Q). For each pair of such cubes Q and Q∗ in Rn,we define the following subset

(6.1.3) E(Q,Q∗) =v ∈ Q∗; a+(v) ≥ ||a−||L∞(Q) − a(l(Q))

.

The main assumption on the functions a+ and a− is that there exists a uniformconstant κ > 0 such that

(H0) inf

|E(Q,Q∗)||Q∗|

; all cubes Q ⊂ Q∗ with 2l(Q) = l(Q∗)

≥ κ,

79

80 VI. UNCERTAINTY PRINCIPLE AND KINETIC EQUATION

where |E| stands for the Lebesgue measure of a set E ⊂ Rn, and the infimum istaken over all pairs of cubes Q and Q∗ in Rn satisfying the property : Q ⊂ Q∗ with2l(Q) = l(Q∗).

With these notations, the generalized uncertainty principle can be stated as follows.

Theorem 6.1.1. Uncertainty PrincipleLet a− ∈ L∞(Rn) and a+, a− ≥ 0. If Condition (H0) holds for some κ > 0, then

there exists a constant C > 0 depending only on κ and n such that

(6.1.4)

∫Rn

a−(v)|f(v)|2dv ≤ C

∫Rn

(|a(Dv)f(v)|2 + a+(v)|f(v)|2

)dv,

for any f ∈ S(Rn), where a(Dv) can be either as(Dv) or as,log(Dv)

Remark : As for the uncertainty principle in the above formulation, a more preciseresult was proved for the case s = 1 in [83].

Adapted dyadic covering. For k ∈ Z, we consider the lattice Λk = 2−kZn formedby the points in Rn whose coordinates are multiples of 2−k. Let Dk be the collection ofcubes with side length 2−k and vertices located at the points in Λk. The cubes belonging

to D =∞∪−∞

Dk are called dyadic cubes. If a cube Q ∈ D is obtained by dividing of a

bigger dyadic cube Q′ with double side length, we call Q′ the mother cube of Q. Ofcourse, we say also that the cube Q is the child of Q′. To be deterministic, the cubeQ is demi-open and given by Q = [2−k~j; 2−k(~j + ~1)[, where ~j ∈ Zn, ~1 = (1, ..., 1), then

the mother cube is given by Q′ = [2−k+1 Int (2−1~j); 2−k+1( Int (2−1~j) +~1)[.

Definition 6.1.1. For a constant A > 0 and a function 0 ≤ a−(v) ∈ L∞(Rn), weset

F(A, a−) =Q ∈ D; ||a−||L∞(Q) ≤ A a(l(Q))

where the function a is given in (6.1.2). Furthermore, we denote by M(A, a−) the setof dyadic cubes Q ∈ F(A, a−) but not their mother cubes.

The following lemma follows directly from the definition and the fact that a isdecreasing.

Lemma 6.1.1. If a cube Q′ ∈ F(A, a−), then any child cube Q of Q′ belong toF(A, a−).

Since a− ∈ L∞, we have that Dk ⊂ F(A, a−) if k large enough. Moreover, thecollection of cubes in M(A, a−) is a non-overlapping covering on Rn, that is

(6.1.5) Rn = ∪Q∈M(A,a−)

Q, |Qk ∩Qj| = 0, Qk, Qj ∈M(A, a−), k 6= j.

For the non-overlapping covering M(A, a−), we now construct another locally uni-form finite covering M(A, a−) on Rn by taking into account Hypothesis (H0).

Proposition 6.1.1. Under the hypothesis of Theorem 6.1.1, there exists a constantAs depending only on s, such that for all A ≥ As, and any cube Qj ∈M(A, a−), there

exists another cube Qj satisfying the following conditions:

(1) Qj ⊂ Qj ⊂ Q′j, where Q′

j denotes the mother cube of Qj;

6.1. UNCERTAINTY PRINCIPLE 81

(2) There exists a dual pair of cubes (Q,Q∗), Q having the side length 12l(Qj) such

that, by setting Ej = Qj ∩ E(Q,Q∗), one has

(6.1.6) |Ej| ≥κ

2|Qj|,

and

(6.1.7) a+(v) ≥ C1 a(l(Qj)) ≥ C2 ‖a−‖L∞(Qj), for any v ∈ Ej.

(3) The collection Qj = M(A, a−) is a locally finite covering of Rn, and thenumber of overlapping is order O(κ−n) which is uniform in Rn.

Proof: Firstly, note that if Qj ∈ M(A, a−) belongs to Dk, then there is a uniquemother cube Q′

j ∈ Dk−1. For the constant κ > 0 in Condition (H0) , we choose aninteger N > 0 satisfying

κ

4<

n

2N≤ κ

2.

Then we only need to consider the following two distinct cases:• Case 1. Q′

j does not contain any cube belonging to M(A, a−) with side length

smaller than l(Qj)/2N . In this case, we choose Qj = Q′

j.• Case 2. There is at least one dyadic cube belonging to M(A, a−) in Q′

j whose

side length is smaller than 2−N l(Qj). We then choose one of such cubes denoted byQk(j) such that Q′

k(j) is closest to Qj. Note that this choice is not necessarily unique.

Here, the distance between two points v = (v1, · · · , vn) and w = (w1, · · · , wn) is definedby ‖v − w‖∞ ≡ maxj|vj − wj|. Then, we define Qj as the unique largest cube which

satisfies Qj ⊂ Qj ⊂ Q′j and does not intersect the interior of Q′

k(j).

With the above choice of Qj, let us check the statement of Proposition 6.1.1.We start with Case 2. In this case, first note that for any cube Q containing Q′

k(j),we have, by Lemma 6.1.1,

(6.1.8) ||a−||L∞(Q) > A a(l(Q)).

Now, take a cube Q with side length l(Qj)/2, such that Q′k(j) ⊂ Q ⊂ Q′

j, with maximum

value of |Q ∩ Qj|. Then choose a cube Q∗ ⊃ Q with l(Q∗) = 2l(Q) = l(Qj) and with

maximum value of |Q∗∩ Qj|. Note that the choice of Qj implies |Q∗∩ Qcj| ≤ n2−N |Q∗|.

From the Condition (H0), for the subset Ej = Qj ∩ E(Q,Q∗), we have

|Ej| ≥ (κ− n2−N)|Q∗| ≥ κ

2|Qj|,

and for any v ∈ Ej, the estimate

a+(v) ≥ (A− 1)a(l(Q)) ≥ (A− 1)a(l(Qj)) ≥(A− 1)

A||a−||L∞(Qj)

holds because of (6.1.8) and the fact that Qj ∈M(A, a−). If we choose A bigger than1, it follows that there exists a constant C > 0 such that

(6.1.9) a+(v) ≥ Ca(l(Qj)) ≥C

A||a−||L∞(Qj), ∀ v ∈ Ej,


and Ej ⊂ Qj satisfying

(6.1.10) |Ej| ≥κ

2n+1|Qj|.

We now turn to consider Case 1. In this case, since Qj = Q′j /∈M(A, a−), we have

||a−||L∞(Q′j)> A a(l(Q′

j)).

Let us choose a dyadic cube Qm(j) ⊂ Q′j with the same side length as Qj, and satisfying

||a−||L∞(Qm(j)) = ||a−||L∞(Q′j). Since there exists a (explicitly depending on s) constant

c0 > 0 such that

a(2r) ≥ c0 a(r), if 0 < r < 1/2,

we have

||a−||L∞(Qm(j)) > Ac0 a(l(Qm(j))).

By choosing Q = Qm(j), for property (2) of the Proposition, we get

a+(v) ≥ (Ac0 − 1)a(l(Qj)) ≥Ac0 − 1

A||a−||L∞(Qj), ∀ v ∈ Ej,

and also Ej ⊂ Qj with |Ej| ≥ κ|Qj|. Then it is sufficient to take a larger constant

A if needed such that Ac0 > 1. Since Qj meets only those cubes Qk contained in Q′j

whose side lengths are bigger than 2−N−1l(Qj), the covering Qj is locally finite, andthe number of overlapping estimated above is bounded by O(2nN) = O(κ−n). Thiscompletes the proof of Proposition 6.1.1.

Estimation of integration over the cubes. Except for the case s = 1, it followsfrom Propositions 1.3.3 and 1.4.1, that∫

Rnv

|a(Dv)f(v)|2dv ≥ c1

∫Rn

v×Rnw

|f(v)− f(w)|2a(|v − w|)|v − w|n

dvdw

≥ c2

∫∑

j Qj×∑

k Qk

|f(v)− f(w)|2a(|v − w|)|v − w|n

dvdw

≥ c2∑

j

a(l(Qj)

|Qj|

∫Qj×Qj

|f(v)− f(w)|2dvdw,

where c1, c2 are positive constants depending only on κ, n and s. In the case s = 1,the above inequality also holds because for any cube Q, we have∫

Q×Q|f(v)− f(w)|2dvdw ≤

∫Q×Q

|v − w|2∫ 1

0

|∇u(v + θ(w − v)|2dθ

≤ l(Q)2

∫Q

∫ 1/2

0

dθ

(∫Q

|∇f(v + θ(w − v)|2dv)dw

+l(Q)2

∫Q

∫ 1

1/2

dθ

(∫Q

|∇f(v + θ(w − v)|2dw)dv.

By the change of variables z = v+ θ(w− v) ( v → z, w → z respectively), the last twoterms can be estimated by 2nl(Q)2|Q|

∫Q|∇f(z)|2dz.

6.2. HYPOELLIPTICITY OF KINETIC EQUATIONS 83

It follows from Proposition 6.1.1 that

a(l(Qj)

|Qj|

∫Qj×Qj

|f(v)− f(w)|2dvdw

≥ c0a(l(Qj))

|Qj|

∫Qj×Ej

|f(v)− f(w)|2dvdw

≥ c0a(l(Qj))

|Qj|

∫Qj×Ej

(1

2|f(v)|2 − |f(w)|2

)dvdw

≥ c0a(l(Qj))|Ej|2|Qj|

∫Qj

|f(v)|2dv − c0a(l(Qj))

∫Ej

|f(w)|2dw

≥ c0κ

2n+2

∫Qj

a−(v)|f(v)|2dv − c0C

∫Ej

a+(w)|f(w)|2dw,

which leads to the desired estimate by summing over the index j. This completes theproof of Theorem 6.1.1.

6.2. Hypoellipticity of kinetic equations

The first application of the generalized uncertainty principle is concerned with somehypoelliptic estimates on solutions of kinetic equations.

Consider a transport equation in the form of

(6.2.1) ft + v · ∇xf = g ∈ D′(R2n+1),

where (t, x, v) ∈ R1+n+n = R2n+1. The induced regularity w.r.t. the space and timevariables arising from the transport operator and the regularity w.r.t. the microscopicvelocity variable can be stated as follows.

Theorem 6.2.1. Assume that g ∈ H−s′(R2n+1), for some 0 ≤ s′ < 1. Letf ∈ L2(R2n+1) be a weak solution of the kinetic equation (6.2.1) such that as(Dv)f ∈L2(R2n+1) for some 0 < s ≤ 1. Then it follows that

Λs(1−s′)/(s+1)x f

(1 + |v|2)ss′/2(s+1),

Λs(1−s′)/(s+1)t f

(1 + |v|2)s/2(s+1)∈ L2(R2n+1).

Similarly, if g ∈ H−s′(R2n+1), for some 0 ≤ s′ < 1, and as,log(Dv)f ∈ L2(R2n+1) forsome s > 1/2, it follows that

(log Λx)s−1/2f

(1 + |v|2)s′/2,

(log Λt)s−1/2f

(1 + |v|2)1/2∈ L2(R2n+1),

where Λ• = (1 + |D•|2)1/2.

This theorem can be proven once the following estimate is obtained

‖A(v,Dt, Dx)f‖2L2(R2n+1) ≤ C

‖as(Dv)f‖2

L2(R2n+1)(6.2.2)

+‖X0f‖2H−s′ (R2n+1)

+ ‖f‖2L2(R2n+1)

,

for all f ∈ S(R2n+1), where X0 ≡ (∂t + v · ∇x) and

A(v,Dt, Dx)f =< Dx >

s(1−s′)/(s+1) f

(1 + |v|2)ss′/2(s+1)+< Dt >

s(1−s′)/(s+1) f

(1 + |v|2)s/2(s+1).


We denote by τ , η and ξ the dual (Fourier) variables corresponding to t, x andv respectively. Furthermore, we use standard notations from the pseudo-differentialoperator theory.

Choose χ(τ, η, ξ) ∈ S0(R1+2n), such that χ = 1 on Γ = (τ, η, ξ) ∈ R1+2nτ,η,ξ ; |τ |2 +

|η|2 ≥ 1 + |ξ|2/2 , and χ = 0 outside of Γ = |τ |2 + |η|2 ≤ 1 + |ξ|2. Denote by χ(D)the pseudo-differential operator with symbol χ. We have

‖A(v,Dt, Dx)f‖L2(R2n+1) ≤ ‖A(v,Dt, Dx)χ(D)f‖L2(R2n+1)

+ ‖A(v,Dt, Dx)(1− χ(D))f‖L2(R2n+1),

and it is straightforward to check that

‖A(v,Dt, Dx)(1− χ(D))f‖L2(R2n+1) ≤ ||as(Dt)(1− χ(D))f ||L2(R2n+1)

+||as(Dx)(1− χ(D))f ||L2(R2n+1) ≤ 2||as(Dv)(1− χ(D))f ||L2(R2n+1)

≤ C(||as(Dv)f ||L2(R2n+1) + ||f ||L2(R2n+1)) .

Since

‖[< D >−s′ X0, χ(D)]f‖L2(R2n+1) ≤ C||f ||L2(R2n+1),

it is therefore enough to prove that

‖A(v,Dt, Dx)χ(D)f‖2L2(R2n+1) ≤ C

‖as(Dv)χ(D)f‖2

L2(R2n+1)(6.2.3)

+‖X0χ(D)f‖2H−s′ (R2n+1)

+ ‖f‖2L2(R2n+1)

.

As supp Ft,x,v(χ(D)f) ⊂ Γ, we deduce that∫Rn

v

(∫Rn+1

τ,η

|τ + v · η|2

(1 + τ 2 + |η|2)s′|Ft,x(χ(D)f)(v, τ, η)|2dτdη

)dv ≤ ||X0χ(D)f ||2

H−s′ (R2n+1),

where F• denotes the Fourier transform with respect to the variable •.Since the operator as(Dv) is independent of t and x variables, (6.2.3) will follow if

we can prove that

(6.2.4) c(a−(v, τ, η)u, u)L2(Rn) ≤ Re((a2(Dv) + a+(v, τ, η))u, u

)L2(Rn)

, u ∈ S(Rn),

for any (τ, η) ∈ R1+n when |τ | , |η| ≥ R0 for some large constant R0 > 0. Here,

(6.2.5) a−(v, τ, η) = c0|τ |2s(1−s′)/(s+1)

(1 + |v|2)s/(s+1)+ c0

|η|2s(1−s′)/(s+1)

(1 + |v|2)ss′/(s+1)= a−1 (v, τ) + a−2 (v, η),

for some constant c0 > 0, and

(6.2.6) a+(v, τ, η) = 1 +|τ + v · η|2

(1 + τ 2 + |η|2)s′.

We check the Condition (H0) for the uncertainty principle given in Theorem 6.1.1,now (τ, η) ∈ R1+n is considered as some parameter such that |τ | , |η| ≥ R0, for somelarge constant R0 > 0.

Let τ/|η| = p ∈ R \ 0 and η/|η| = ω ∈ Sn−1. We consider Rn in the two regions:

Ω = |τ | ≥ 2(1 + |v|2)1/2|η| = |p| ≥ 2(1 + |v|2)1/2

6.2. HYPOELLIPTICITY OF KINETIC EQUATIONS 85

and Ωc = |p| ≤ 2(1 + |v|2)1/2. Obviously, there exists a constant c > 0 such that

(6.2.7)|τ + v · η|2

(1 + τ 2 + |η|2)s′≥

|τ |2(1−s′) ≥ c(1 + |v|2)(1−s′)|η|2(1−s′), in Ω,

c(τ/|η|+ v · η/|η|)2

(1 + |v|2)s′|η|2(1−s′), in Ωc.

We now check the Condition (H0) for the functions a+(v, τ, η), a−1 (v, τ) and a−2 (v, η).For given p ∈ R, ω ∈ Sn−1, introduce the hyperplane in Rn

v , Pp,ω = v ∈ Rn : p+v ·ω =0.

Consider a dual pair of cubes Q ⊂ Q∗ ⊂ Rnv with l(Q) = 2l(Q∗).

• Case A. We assume that Q∗ ⊂ Ω. We have by (6.2.7) that

(6.2.8) a+(v, τ, η) ≥ |τ |2(1−s′) ≥ c(1 + |v|2)(1−s′)|η|2(1−s′), if v ∈ Q∗ ⊂ Ω.

In this case, we have E(Q,Q∗) = Q∗ if we choose c0 = 1.

• Case B. If Q∗ ⊂ Ωc, then

a+(τ, η, v) ≥ c|η|2(1−s′)

(1 + |v|2)s′(p+ v · ω)2 + 1, if v ∈ Q∗.

Then for any Q∗ and Pp,ω, we have

(6.2.9) |E∗| = |v ∈ Q∗; |p+ v · ω| ≥ 1

2l(Q)| ≥

(1

4

)n

|Q∗|,

and thus, for v ∈ E∗,

|η|2(1−s′)

(1 + |v|2)s′(p+ v · ω)2 ≥ |η|2(1−s′)

(1 + |v|2)s′(1

2l(Q))2.

Now if l(Q) ≥ (1 + |v|2)s′/2(s+1)|η|(s′−1)/(s+1), we have

|η|2(1−s′)

(1 + |v|2)s′(l(Q))2 ≥ |η|2s(1−s′)/(s+1)

(1 + |v|2)ss′/(s+1),

and if l(Q) ≤ (1 + |v|2)s′/2(s+1)|η|(s′−1)/(s+1), we have

|η|2s(1−s′)/(s+1)

(1 + |v|2)ss′/(s+1)≤ (l(Q))−2s = as(l(Q)).

Thus, it follows that E∗ ⊂ E(Q,Q∗) for a+ and a−2 .The cases when Q∗ ∩ Ω 6= ∅ and Q∗ ∩ Ωc 6= ∅ is just a combination of the above

two cases, so that we have checked the Condition (H0) by using (6.2.9) with κ = 4−n

for some small constant c0 > 0 depending only on s, s′ .To check condition (H0) for a−1 , we use the same argument and the fact that

a+(v, τ, η) ≥ c|τ |2(1−s′)

1 + |v|2(p+ v · ω)2 + 1, if v ∈ Ωc.


Finally, we have obtained

‖as(Dv)(χ(D)f

)‖2

L2(R2n+1) + ‖X0

(χ(D)f

)‖2

H−s′ (R2n+1)

≥ c

∫Rn

v

(∫Rn

|τ |2s(1−s′)/(s+1)

(1 + |v|2)s/(s+1)|Ft,x(χ(D)f)(v, τ, η)|2dτdη

+ c

∫Rn

|η|2s(1−s′)/(s+1)

(1 + |v|2)ss′/(s+1)|Ft,x(χ(D)f)(v, τ, η)|2dτdη

)dv,

and this completes the first part of Theorem 6.2.1.For the second part, if s > 1/2, we set

a−(v, τ, η) = c1( ˜log(|τ |))2s−1

(1 + |v|2)+ c2

( ˜log(|η|))2s−1

(1 + |v|2)s′= a−1 (v, τ) + a−2 (v, η),

and take the same function a+(v) as defined above. Then, we introduce the set

E∗ = v ∈ Q∗ ; min(1, (p+ v · ω)2)|η|2(1−s′) ≥ cs′| ˜log(|η|)|2s−1 − | ˜log(l(Q)

)|2s−1.

By considering the cases l(Q) ≤ |η|(s′−1)| ˜log(|η|)|s−1/2 and l(Q) ≥ |η|(s′−1)| ˜log(|η|)|s−1/2

respectively, we get |E∗| ≥ ν|Q∗| for some positive constant ν, which verifies condition(H0) for a+(v, τ, η) and a−2 (v, η). The same method also works for a−1 (v, τ). Therefore,the proof of Theorem 6.2.1 is completed.

6.3. Weighted hypoelliptic estimates

For the applications we have in mind, we shall also need the following result involv-ing moment weights.

Theorem 6.3.1. Assume that g ∈ H−s′(R2n+1), for some 0 ≤ s′ < 1. Letf ∈ L2(R2n+1) be a weak solution of the kinetic equation (6.2.1) such that < v >l

f, as(Dv)(< v >l f

)∈ L2(R2n+1) for some 0 < s ≤ 1 and any l ∈ N. Then it follows

that for any δ > 0 such that s′ + δ < 1, we have

Λs(1−s′−δ)/(s+1)x < v >l f

(1 + |v|2)ss′/2(s+1),

Λs(1−s′−δ)/(s+1)t < v >l f

(1 + |v|2)s/2(s+1)∈ L2(R2n+1),

for any l ∈ N.

Proof : It is enough to prove : For any value M(δ, l) ≥ (l + δ)(1− s′)/δ, one has

‖A(v,Dt, Dx) < v >l f‖2L2(R2n+1) ≤ C

‖as(Dv) < v >l f‖2

L2(R2n+1)(6.3.1)

+‖X0f‖2H−s′ (R2n+1)

+ ‖ < v >M(δ,l) f‖2L2(R2n+1)

,

for any f ∈ S(R2n+1), where

A(v,Dt, Dx) =< Dx >

s(1−s′−δ)/(s+1)

(1 + |v|2)ss′/2(s+1)+< Dt >

s(1−s′−δ)/(s+1)

(1 + |v|2)s/2(s+1).

6.3. WEIGHTED HYPOELLIPTIC ESTIMATES 87

We replace the function a+(v, τ, η) defined in the proof of Theorem 6.2.1 by thefollowing function

a+(v, τ, η) = c|η|2(1−s′−δ)

(1 + |v|2)s′(1 + |v|2)1Ω + (p+ v · ω)21Ωc + 1.

It follows that, for f = χ(D)f and f(v, τ, η) = Ft,x(f)(v, τ, η),∫Rn+1

τ,η

(∫Rn

v

|as(Dv) < v >l f(v, τ, η)|2 + a+(v)| < v >l f(v, τ, η)|2dv)dτdη

≤ ‖as(Dv) < v >l f ||2L2(R2n+1) + ||X0f ||2H−s′ (R2n+1)+ || < v >M(δ,l) f ||2L2(R2n+1).

If we decompose Rnv into Ωη = |η|δ ≥< v >l and Ωc

η, then we have∫Rn+1

τ,η

(∫Rn

v

a+(v)| < v >l f |2dv)dτdη ≤

∫Rn+1

τ,η

(∫Ωη

a+(v)|f |2dv

)dτdη

+

∫Rn+1

τ,η

(∫Ωc

η

| < v >M(δ,l) f |2dv

)dτdη + || < v >l f ||2L2(R2n+1)

with M(δ, l) ≥ (l + δ)(1− s′)/δ. Then, by setting

a−(v) = c|η|2s(1−s′−δ)/(s+1)

(1 + |v|2)ss′/(s+1),

and by a similar argument used for Theorem 6.2.1, we can estimate∫Rn+1

τ,η

(∫Rn

v

a−(v)| < v >l f |2dv)dτdη,

by replacing |η| by |η|(1−s′−δ)/(1−s′) and replacing f by < v >l f .

Remark 6.3.1. We shall apply the above result by choosing δ = 12(1− s′). On the

other hand, we can also choose a−2 as

a−2 (v, η) = c(1 + |v|2)s(l−s′)/(s+1)|η|2s(1−s′)/(s+1).

In this way, we keep the same order of differentiation as the one in Theorem 6.2.1, butsome weight in v is lost.

CHAPTER VII

Linearized inhomogeneous Boltzmann equations

We attack now the challenge problem, inhomogeneous Boltzmann equations, butwe consider here only the linearized inhomogeneous Boltzmann equation, we limit alsoin Maxwellian case. We prove as in homogeneous case that the singularity of collisionkernel (non cutoff case) implies the smoothness of weak solutions. But as we pointout in the introduction of chapter IV, we get only the interior regularity on the openinterval ]0, T [, not the uniformly regularity estimate up to t = 0. Our power argumentfor the regularity of t and x variables is the uncertainty principle of chapter VI, it isvalid on full space Rt × R3

x, so that the cutoff function on t variable are necessary inthis chapter. This is the key different point of this chapter with respect that of chapterIV.

7.1. Cauchy problem for linearized Boltzmann equation

We now come to the Cauchy problem for the linearized Boltzmann equation :

(7.1.1)∂f

∂t+ v · ∂xf = Lf, x, v ∈ R3, t > 0 ; f |t=0 = f0,

where

Lf = Q(µ, f) +Q(f, µ),

is the linearized collision operator. We recall the non cutoff mathematical Maxwellianassumption :

(7.1.2) B(|v − v∗|, cos θ)) = b(cos θ) ≈ Kθ−2−2α when θ → 0,

implies the coercivity estimate.

(7.1.3) ‖Λαf‖2L2(R3) ≤ Cg

(−Q(g, f), f)L2(R3) + ‖f‖2

L2(R3)

.

The definition of weak solution of linear Cauchy problem (7.1.1) is similar to theone for the nonlinear Boltzmann equation. We recall here.

Definition 7.1.1. With an initial datum f0(x, v) ∈ L2(R3x;L

11(R3

v)), f(t, x, v) iscalled a weak solution of the Cauchy problem (7.1.1) if it satisfies:

f(t, x, v) ∈ C(R+;D′(R6)) ∩ L1([0, T0];L2(R3

x;L11(R3

v))) ∩ L∞([0, T0];L1(R6));

f(0, x, v) = f0(x, v);∫R6 f(t, x, v)ϕ(t, x, v)dxdv −

∫R6 f0(x, v)ϕ(0, x, v)dxdv

−∫ t

0dτ∫

R6 f(τ, x, v)(∂τ + v · ∂x)ϕ(τ, x, v)dxdv =∫ t

0dτ∫

R6 f(τ, x, v)L∗(ϕ)(τ, x, v)dxdv,

89

90 VII. LINEARIZED INHOMOGENEOUS BOLTZMANN EQUATIONS

for any test function ϕ(t, x, v) ∈ C1(R+;C∞0 (R6)). Here, L∗ is the (formal) adjoint

operator of L given by

L∗(ϕ) =

∫R3

∫S2

B (v − v∗, σ)µ(v∗) ϕ(v′) + ϕ(v′∗)− ϕ(v)− ϕ(v∗) dσdv∗.

Note that the above definition also makes sense for ϕ ∈ L∞([0, T0];W2,∞(R6)) as

test function.For the existence of weak solution. We have the following Theorem. For the proof

of theorem, we send to [11, 12].

Theorem 7.1.1. Suppose that the collision kernel B satisfies (7.1.2) with 0 < s < 12,

and that the initial datum satisfies

< v >l f0 ∈ L2(R3 × R3), for any l ∈ N.

Then there exists an unique weak solution f of the linearized Boltzmann equation (7.1.1)such that

< v >l f ∈ L∞ ∩ L2(]0, T [×R6), for any l ∈ N and for any T > 0.

Remark : The assumption that all the moments of the initial data are bounded inL2(R6) is in fact not necessary for the existence. We can only assume that the initialmoments of order L with any L ≥ 3 are bounded in L2(R6). Then, there exists aunique weak solution to the linearized Boltzmann equation with moments up to thesame order bounded for any positive . Here, we require that all the moments arebounded in L2(R6) just to ensure that the solution becomes H+∞ w.r.t. all variablesfor any positive time, and also to simplify the exposition of the proof.

Same theorem is holds for space homogeneous Boltzmann equations (4.3.4).

7.2. Hypoellipticity of linearized Boltzmann equation

We prove now an energy estimate for weak solution of Cauchy problem.

Proposition 7.2.1. Assume that f0 ∈ L2l , for some l ∈ N with l > 5

2. Let f ∈

L∞(]0, T [;L2l ) be a weak solution of the Cauchy problem (7.1.1) for some T > 0. Then

Λsv(Wlf) ∈ L2(]0, T [;L2(R6)),

and there exist C1, C2 > 0 which are independent of f such that for any t ∈]0, T [,

(7.2.1) ||f(t)||2L2l+ C1

∫ t

0

||ΛsvWlf(τ)||2L2(R6)dτ ≤ C2

∫ t

0

||f(τ)||2L2ldτ + ||f0||2L2

l.

Where Wl = (1 + |v|2)1/2.

Remark : From the energy estimate (7.2.1), we get immediately the uniqueness ofweak solution to the Cauchy problem (7.1.1).

The energy estimate (7.2.1) deduces the partial regularity w.r.t. the variable v.

7.2. HYPOELLIPTICITY OF LINEARIZED BOLTZMANN EQUATION 91

As in chapter IV, we need to mollifier the weak solution f and take it as testfunctions into the definition of weak solution. Let 0 < δ < 1 and N0 ∈ N such thatN0 >

92. Set

Mδ(Dx, Dv) = (1 + δ(|Dx|2 + |Dv|2))−N0 .

Now the commutator of the mollified Mδ and the collision operator Q(µ, · ) is estimatedby the following lemma.

Lemma 7.2.1. There exists a constant C > 0 independent of δ > 0 such that forany g ∈ L2(R6), we have

(7.2.2)∣∣∣(MδQ(µ, g)−Q(µ,Mδg),Mδg

)L2(R6)

∣∣∣ ≤ C||Mδg||2L2(R6).

Proof: Since Mδg ∈ H2(R6), we have

−(Q(µ,Mδg),Mδg

)L2(R6)

=

= −∫

R3x

∫R6

v∗,v

∫S2

bµ(v∗)Mδg(x, v)(Mδg(x, v

′)−Mδg(x, v))dσdv∗dvdx

=1

2

∫R3

x

∫R6

v∗,v

∫S2

bµ(v∗)(Mδg(x, v

′)−Mδg(x, v))2

dσdv∗dvdx

−1

2

∫R3

x

∫R6

v∗,v

∫S2

bµ(v∗)((Mδg(x, v

′))2 − (Mδg(x, v)

)2)dσdv∗dvdx.

According to the cancelation lemma 3.3.1 (see also Corollary 2 of [6]), we have∣∣∣∣∣∫

R9x,v∗,v

∫S2

bµ(v∗)((Mδg(x, v

′))2 − (Mδg(x, v)

)2)dσdv∗dvdx

∣∣∣∣∣≤ C‖µ‖L1(R3)‖Mδg‖2

L2(R6).

By taking the partial Fourier transform Fv w.r.t. the variable v with the dual variabledenoted by ξ, the Bobylev identity (3.2.3) gives

1

2

∫R3

x

∫R6

v∗,v

∫S2

bµ(v∗)(Mδg(x, v

′)−Mδg(x, v))2

dσdv∗dvdx

=1

2(2π)3

∫R3

x

∫R3

ξ

∫S2

b( ξ|ξ|

· σ)µ(0)|Mδ(Dx, ξ)Fvg(x, ξ)|2 +

+µ(0)|Mδ(Dx, ξ+)|Fvg(x, ξ

+)|2 +

−2Re µ(ξ−)(Mδ(Dx, ξ

+)Fvg)(x, ξ+)

(Mδ(Dx, ξ+)Fvg

)(x, ξ+)

dσdξdx.


On the other hand, Bobylev identity again also gives(Q(µ, g),M2

δ g)

L2(R6)

=

∫R3

∫R6

∫S2

bµ(v∗)g(x, v)(M2

δ g(x, v′)−M2

δ g(x, v))dv∗dσdvdx

=1

(2π)3

∫R3

x

∫R3

ξ

∫S2

bµ(ξ−)Fvg(x, ξ

+)− µ(0)Fvg(x, ξ)

× M2δ (Dx, ξ)Fvg(x, ξ)dσdξdx.

Thus,(Q(µ, g),M2

δ g)

L2(R6)=(Q(µ,Mδg),Mδg

)L2(R6)

+O(‖Mδg‖2L2)

+1

2(2π)3

∫R3

x

∫R3

ξ

∫S2

bµ(0)∣∣Mδ(Dx, ξ

+)Fvg(x, ξ+)∣∣2 − ∣∣Mδ(Dx, ξ)Fvg(x, ξ)

∣∣2dσdξdx+

1

(2π)3Re

∫R3

x

∫R3

ξ

∫S2

bµ(ξ−)Mδ(Dx, ξ)Fvg(x, ξ

+)−Mδ(Dx, ξ+)Fg(x, ξ+)

× Mδ(Dx, ξ)Fvg(x, ξ)dσdξdx = (A) + (B),

where we have used the fact that as operators we have M∗δ (Dx, ξ) = Mδ(Dx, ξ) in

L2(R3x). Now, since∫ π/2

−π/2

sin θ b(cos θ)[ 1

cos3(θ/2)− 1]dθ ≤ C < +∞,

one has

|(A)| =∣∣∣ ∫

R6

∫S2

b∣∣Mδ(Dx, ξ

+)Fvg(x, ξ+)∣∣2

−∣∣Mδ(Dx, ξ)Fvg(x, ξ)

∣∣2dσdξdx∣∣∣ ≤ C‖Mδg‖2L2(R6) .

For the term (B), by taking the Fourier transformation w.r.t. the variable x, we have

|(B)| = 1

2(2π)6

∣∣∣ ∫R6

∫S2

bµ(ξ−)Mδ(η, ξ)−Mδ(η, ξ

+)

×Fx,v(g)(η, ξ+) Mδ(η, ξ)Fx,vg(η, ξ)dσdξdη

∣∣∣.By using

|Mδ(η, ξ+)−Mδ(η, ξ)| ≤ CN0 sin2 θ

2Mδ(η, ξ

+)

in the above estimate, we complete the proof of Lemma 7.2.1.

To proceed further, we shall need some mollifier for the weighted function Wl. Let0 < δ1, δ2 < 1 and N0 ∈ N such that N0 >

92. Set

Wl,δ1 = Wle−δ1

|v|22 , Mδ2(Dx, Dv) = (1 + δ2(|Dx|2 + |Dv|2))−N0 .

7.2. HYPOELLIPTICITY OF LINEARIZED BOLTZMANN EQUATION 93

If f ∈ L∞([0, T ];L2(R3x;L

2l (R3

v))), then

Wl,δ1M2δ2

(Dx, Dv)Wl,δ1f ∈ L∞(]0, T [;W 2,∞(R6)).

We have now the following Lemma.

Lemma 7.2.2. For the collision kernel (7.1.2), assume 0 < s < 12. Then, for each

l ∈ N with l > 52, it follows that

(Lf,Wl,δ1M2δ2

(Dx, Dv)Wl,δ1f)L2(R6) + ||ΛsvMδ2Wl,δ1f(t)||2L2(R6)

≤ C||Mδ2(Dx, Dv)Wl,δ1f ||L2(R6)

||f ||L2(R3

x,L11(R3

v))

+||Wlf ||L2(R6) + ||f ||L2(R3x,L2

2(R3v))

,

for some constant C > 0 which is independent of f ∈ L2(R3x, L

2l (R3

v)) and δ1, δ2 > 0.

Proof: Write

(Lf,Wl,δ1M2δ2Wl,δ1f)L2(R6) = I1 + I2 + I3,

where

I1 ≡(Q(µ,Mδ2Wl,δ1f),Mδ2Wl,δ1f

)L2(R6)

,

I2 ≡(Q(f, µ),Wl.δ1M

2δ2Wl,δ1f

)L2(R6)

,

and

I3 ≡(Mδ2Wl,δ1Q(µ, f)−Q(µ,Mδ2Wl,δ1f),Mδ2Wl,δ1f

)L2(R6)

.

Now, by using the coercivity estimate (7.1.3), I1 is estimated by

(7.2.3) C−1µ ||Λs

vMδ2Wl,δ1f(t)||2L2(R6) ≤ −I1 + Cµ||Mδ2Wl,δ1f(t)||2L2(R6).

For I2, we split it into two parts as follows,

I2 =

∫b (f ′∗µ

′ − f∗µ)WlM2δ2Wl,δ1f

=

∫b f∗µW ′

l,δ1(M2

δ2Wl,δ1f)′ −Wl,δ1M

2δ2Wl,δ1f

=

∫b f∗(Wl,δ1µ)(M2

δ2Wl,δ1f)′ −M2

δ2Wl,δ1f)

+

∫b f∗µ((Wl,δ1)

′ −Wl,δ1)(M2δ2Wl,δ1f)′

= I2,a + I2,b.

By using the estimates (3.4.1) of Lemma 3.4.1 with ρ = 1, we have

|I2,a| = |(Q(f,Wl,δ1µ),M2δ2Wl,δ1f)|

≤∫

R3x

‖Q(f,Wl,δ1µ)‖L2(R3v)‖M2

δ2Wl,δ1f‖L2(R3

v)dx

≤ C

∫R3

x

‖f‖L11(R3

v)‖Wl+1,δ1µ‖H1(R3v)‖Mδ2Wl,δ1f‖L2(R3

v)dx

≤ C||f ||L2(R3x;L1

1(R3v))||Mδ2Wl,δ1f ||L2(R6

x,v),


where C is a generic positive constant, and we have used 0 < s < 1/2 and the fact that

‖Wl+1,δ1µ‖H1(R3v) ≤ Cl,

for some positive constant Cl independent of δ1 > 0.For I2,b, note that

µ|(Wl,δ1)′ −Wl,δ1| ≤ µ|W ′

l −Wl|+ µWl

∣∣∣∣e−δ1|v′|2

2 − e−δ1|v|22

∣∣∣∣≤ C

(θµ1/2(1 + |v∗|)W ′

l−1 + 1+ µ1/2δ1∣∣|v′|2 − |v|2∣∣)

≤ Cθµ14 (v)(Wl(v∗) +Wl(v

′) + δ1|v∗|2) ≤ Cθµ18 (v)Wl(v∗),

where we have used

|v′ − v|2 =1

2|v − v∗|2(1− cos θ).

Since 0 < s < 12, b(cos θ)θ is integrable in θ. The Cauchy-Schwarz inequality gives

|I2,b| ≤ C||Mδ2Wl,δ1f ||L2(R6)||Wlf ||L2(R6).

Hence,

(7.2.4) |I2| ≤ C(||f ||L2(R3x;L1

1(R3v)) + ||Wlf ||L2)||Mδ2Wl,δ1f ||L2 .

Next, we write I3 = I3,a + I3,b, where

I3,a = (Wl,δ1Q(µ, f)−Q(µ,Wl,δ1f),M2δ2Wl,δ1f) =

∫bµ∗f((Wl,δ1)

′−Wl,δ1)(M2δ2Wl,δ1f)′.

Since

µ∗|(Wl,δ1)′ −Wl,δ1 | ≤ µ∗|W ′

l −Wl|+ µ∗W′l

∣∣∣∣e−δ1|v′|2

2 − e−δ1|v|22

∣∣∣∣≤ C(θµ1/2

∗ (1 + |v|)(Wl−1 +W ′l−1) + µ∗W

′l

(e−δ1

|v′|22 + e−δ1

|v|22

)δ1∣∣|v′|2 − |v|2∣∣

≤ Cθµ1/4∗ Wl

1 +

W ′l

Wl

δ1(1 + |v|2)(e−δ1

|v′|22 + e−δ1

|v|22

)≤ Cθµ1/4

∗ Wl

1 +

W ′l−2

Wl−2

δ1(1 + |v′|2)e−δ1|v′|2

2 +W ′

l

Wl

δ1(1 + |v|2)e−δ1|v|22 )

≤ Cθµ1/8

∗ Wl,

we have

|I3,a| ≤ C||Wlf ||L2||M2δ2Wl,δ1f ||L2 ≤ C||Wlf ||L2||Mδ2Wl,δ1f ||L2 .

Finally, since

I3,b =(Mδ2Q(µ,Wl,δ1f)−Q(µ,Mδ2Wl,δ1f),Mδ2Wl,δ1f

)L2(R6)

,

by appling Lemma 7.2.1 with g = Wl,δ1f ∈3 (R6), we get

|I3,b| ≤ C||Mδ2Wl,δ1f ||2L2(R6).

Therefore, the proof for Lemma 7.2.2 is completed.

7.3. REGULARITY OF WEAK SOLUTIONS 95

Proof of Proposition 7.2.1:By using the notations in the previous Lemma, if f ∈ L∞([0, T ];L2(R3

x;L2l (R3

v))) isa weak solution to the Cauchy problem (7.1.1), then

Wl,δ1M2δ2

(Dx, Dv)Wl,δ1f ∈ L∞(]0, T [;W 2,∞(R6)).

We can then use this function as a test function (with some mollification in order totake care of the regularity w.r.t. the variable t) in Definition 7.1.1 of weak solution.Then it follows that

d

dt||Mδ2Wl,δ1f(t)||2L2(R6) + 2

([v, Mδ2 ]∂xWl,δ1f, Mδ2Wl,δ1f

)L2(R6)

= 2(Lf,WlM2δWl,δ1f)L2(R6).

A direct computation gives

[v, Mδ2 ] · ∂x = −N0(1 + δ2(|Dx|2 + |Dv|2))−N0−1δ2∂v · ∂x.

Thus

(7.2.5)∣∣∣([v, Mδ2 ] · ∂xWl,δ1f, Mδ2Wl,δ1f

)L2(R6)

∣∣∣≤ CN0||Mδ2Wl,δ1f ||2L2(R6) + C||Wl,δ1f ||2L2(R6).

By using Lemma 7.2.2 for l > 52, we then have

d

dt||Mδ2Wl,δ1f ||2L2 + C1||Λs

vMδ2Wl,δ1f ||2L2

≤ C||Mδ2Wl,δ1f ||2L2 + ||Wlf ||2L2

.

Thus

||Mδ2Wl,δ1f(t)||2L2(R6) + C1

∫ t

0

||ΛsvMδ2Wl,δ1f(τ)||2L2dτ

≤ C2

∫ t

0

||Wlf ||2L2 + ||Wlf0||2L2 .

By letting δ1, δ2 tend to 0, we get

||Wlf(t)||2L2 + C1

∫ t

0

||ΛsvWlf(τ)||2L2dτ ≤ C2

∫ t

0

||Wlf(τ)||2L2dτ + ||Wlf0||2L2 .

The proof of the Proposition 7.2.1 is then completed.

7.3. Regularity of weak solutions

We now turn to the interior regularity of the weak solution. We have the followingresults.

Theorem 7.3.1. Assume that f is a weak solution of the linearized Boltzmannequation such that

< v >l f ∈ L2(]0, T [×R6), for all l ∈ N.Then, for any l ∈ N and for any 0 < δ < T ′ < T , we have

< v >l f ∈ H+∞(]δ, T ′[×R6).


Let f be a weak solution of the Cauchy problem (7.1.1), and φ = φ(t) ∈ C∞0 (]0, T [)

be a smooth cutoff function. Then

(7.3.1)∂(φ(t)f)

∂t+ v · ∂x(φ(t)f) = L(φ(t)f) + φtf, (t, x, v) ∈ R7.

We follows the proof of Proposition 7.2.1, and using now the mollified

Mδ(Dt, Dx, Dv) = (1 + δ(|Dt|2 + |Dx|2 + |Dv|2)−N0 ,

with N0 >112, we have firstly

Proposition 7.3.1. Assume that f is a weak solution of the equation (7.1.1), suchthat

Wlf ∈ L2(]0, T [×R6), ∀l ∈ N.Then, for any l ∈ N and any cutoff function φ ∈ C∞

c (]0, T [), we have

||ΛsvWl(φ(t)f)||2L2(R7) ≤ C||Wlf ||2L2(]0,T [×R6).

We now consider the regularity of the right hand side of the equation (7.3.1), that is,the collision operator. We use the following results of Lemma 3.4.1. Let 0 < s < 1/2,for any ρ ∈ R,

(7.3.2) ‖Q(g, f)‖Hρ−1(R3v) ≤ C‖g‖L1

1(R3v)‖W1f‖Hρ(R3

v).

We now apply the uncertainty principle to the transport equation (7.3.1).

Proposition 7.3.2. Assume that f is a weak solution of equation (7.3.1) such thatWlf ∈ L2(]0, T [×R6) for any l ∈ N with l > 5/2. Then, we have

(7.3.3) Λks1t,x Wl(f) ∈ L2(R7), ∀k, l > 5

2,

where s1 = s2

2(s+1)> 0, f = φ(t)f .

For the proof of this Proposition, let us first recall that if f is a weak solution ofthe linearized Boltzmann equation, one has, by Proposition 7.3.1

ΛsvWl(f) ∈ L2(R7).

An application of (7.3.2), (see Lemma 3.4.1), with ρ = s, l > 5/2, gives

||Λs−1v

L(f) + φtf

||L2(R7) ≤ C

(||Λs

vW1(f)||L2(R7) + ||Wlf ||L2([0,T ]×R6)

).

We then apply Corollary 6.3.1 of the uncertainty principle with s′ = 1− s, δ = s/2,to get

||Wl−1Λs1t,xf ||2L2(R7) ≤ C||Λs

vWlf ||2L2(R7) + ||Λs−1v (Lf)||2L2(R7) + ||Wl(s)f ||2L2(]0,T [×R6).

Thus, it follows that

WlΛs1t,x(f) ∈ L2(R7),

for all l ∈ N and any cutoff function φ ∈ C∞(]0, T [) with f = φ(t)f .

Next, by setting f1 = Λs1t,xf , we see that

Wlf1, Λs1t,xφtf ∈ L2(R7), ∀l ∈ N,

7.3. REGULARITY OF WEAK SOLUTIONS 97

and f1 satisfies (in the weak sense) the following linearized Boltzmann equation,

(7.3.4)∂f1

∂t+ v · ∂xf1 = Lf1 + Λs1

t,xφtf.

Thus, by applying again Proposition 7.3.1, we have

ΛsvWl(f1) ∈ L2(R7),

and using again (7.3.2) with ρ = s, l > 5/2, we have

||Λs−1v

L(f1) + Λs1

t,xφtf||L2(R7) ≤ C(||Λs

vW1(f1)||L2(R7) + ||Wl(s)f ||L2([0,T ]×R6).

It follows from the same argument as for f that

WlΛ2s1t,x (φ(t)f) = WlΛ

s1t,x(f1) ∈ L2(R7),

for any l ∈ N and any cutoff function φ ∈ C∞0 (]0, T [). By induction, we get

(7.3.5) WlΛks1t,x (φ(t)f) ∈ L2(R7), ∀k, l > 5

2.

and thus Proposition 7.3.2 is proved.

End of the proof for Theorem 7.3.1. Now, fix p, l ∈ N, and set

f2 = Λpt, x(φ(t)f) and Λm

δ = (1 + |Dv|2)m2 (1 + δ|Dv|2)−N0 ,

for 0 < δ < 1,m ∈ R+ with a large constant N0 > 7/2+m. Then we have from (7.3.5),

Wl f2 and Wl Λpx(φtf) ∈ L2(R7) , ∀l ∈ N.

By takingΛm

δ W2l Λm

δ f2 ∈ H2(R7)

as a test function in the weak formulation of the following equation

(7.3.6)∂f2

∂t+ v · ∂x(f2) = L(f2) + g,

where g = Λpt,x(φtf), one has, with ( , ) being the inner product of L2(R7),(

Wl[v, Λmδ ]∂xf2, WlΛ

mδ f2

)=

(WlΛ

mδ Lf2, WlΛ

mδ f2

)+(g, Λm

δ W2l Λm

δ f2

)= (A) + (B) + (C),(7.3.7)

where(A) = (Wl(L(Λm

δ f2), WlΛmδ f2),

(B) = (Λmδ Lf2 − L(Λm

δ f2), W2l Λm

δ f2),

and|(C)| ≡

∣∣∣(g, Λmδ W

2l Λm

δ f2

)∣∣∣ ≤ C||WlΛmδ f2||L2(R7)||WlΛ

mδ g||L2(R7).

For the terms (A) and (B), one can proceed as in the proof of Proposition 7.3.1 to get

C−1||ΛsvWlΛ

mδ f2||2L2(R7) ≤ −(A)+C||WlΛ

mδ f2||L2(R7)(||Λm

δ f2||L2(R4t,x,L1

1(R3v))+||WlΛ

mδ f2||L2(R7))

and|B| ≤ C(||Λm

δ f2||L2(R7)||W 2l Λm

δ f2||L2(R4t,x,L1

1(R3v)) + ||W 2

l Λmδ f2||2L2(R7)).

For the commutator, a direct computation gives

[v, Λmδ ] · ∂x = −N0Λ

mδ (1 + δ|Dv|2)−1δ ∂v · ∂x +mΛm−2

δ ∂v · ∂x.


Thus,∣∣∣(Wl[v, Λmδ ]∂xf2, WlΛ

mδ f2

)∣∣∣ ≤ C√

δ ||WlΛmδ Λxf2||L2(R7)

+ |m| ||WlΛm−1δ Λxf2||L2(R7)

||WlΛ

mδ f2||L2(R7).

By combining the above estimates, (7.3.7) implies that for any l > 5/2,

||ΛsvWlΛ

mδ f2||2L2(R7) ≤ C

||W 2

l Λmδ f2||2L2(R7) + ||WlΛ

mδ g||2L2(R7) + ||WlΛ

m−1δ Λxf2||2L2(R7)

,

where the constant C > 0 is independent of δ. Note that the function g is like f2 butwith different cutoff function because φt ∈ C∞

0 (]0, T [). By setting m = ks, k ∈ N, byusing induction on k ≥ 0, we can prove

(7.3.8) WlΛksv Λp

t, x(φ(t)f) ∈ L2(R7),

for any p ∈ N and any cutoff function φ ∈ C∞0 (]0, T [). This completes the proof of

Theorem 7.3.1.

7.4. Linearized inhomogeneous Landau equations

In this section, we will study the regularity of solutions to the Landau equation asanother application of the generalized uncertainty principle. We recall the inhomoge-neous Landau equation

(7.4.1) ft + v · ∇xf = ∇v

(a(f) · ∇vf − b(f)f

)≡ Q(f, f),

where a = (aij) and b = (b1, · · · , b3) are defined as follows (convolution is w.r.t. thevariable v)

aij(f) = aij ? f, bj(f) =3∑

i=1

(∂viaij

)? f , i, j = 1, 2, 3,

with

aij(v) =

(δij −

vivj

|v|2

)|v|γ+2, γ ∈ [−3, 1].

Again the Maxwellian molecule case which corresponds to γ = 0. See [61, 33] for morereferences.

We consider the linearized Landau equation (7.4.1) around the normalized Maxwelliandistribution µ(v). Since µ is the equilibrium state which implies that Q(µ, µ) = 0, thelinearized Landau operator takes the form

(7.4.2) Lf = Q(µ, f) +Q(f, µ)

= ∇v

(a(µ) · ∇vf − b(µ)f

)+∇v

(a(f) · ∇vµ− b(f)µ

).

We then consider the following Cauchy problem

(7.4.3) ft + v · ∇xf = Lf, x, v ∈ R3, t > 0 ; f |t=0 = f0.

The definition of weak solution in the function space L2(]0, T [×R3;L12(R3

v)) for theCauchy problem is standard in the distribution sense.

The following is the existence of weak solution for Linearized Landau Equation.We send the proof of the following Theorem to [12].

7.4. LINEARIZED INHOMOGENEOUS LANDAU EQUATIONS 99

Theorem 7.4.1. Assume that

< v >l f0 ∈ L2(R3 × R3), for any l ∈ N.Then there exists a unique weak solution f of The Cauchy problem (7.4.3) such that

< v >l f, < v >l ∇vf ∈ L∞ ∩ L2(]0, T [;L2(R6)),

for any l ∈ N and for any T > 0 .

We study now the regularity of solution for Linearized Landau equation.

Theorem 7.4.2. Assume that f is a weak solution of the linearized Landau equationsuch that

< v >l f ∈ L2(]0, T [×R6), for any l ∈ N.Then, for any l ∈ N and for any 0 < δ < T ′ < T , we have

< v >l f ∈ H+∞(]δ, T ′[×R6).

The coefficients of the linearized Landau operator can be expressed as follows (seealso [46]) by recalling that convolution is w.r.t. the variable v

aij(µ) = aij ? µ = δij(|v|2 + 1)− vivj,

bj(µ) =3∑

i=1

(∂viaij

)? µ = −vj, j = 1, 2, 3,

aij(f) = aij ? f = δij|v|2∫

R3v

f(v∗)dv∗ − 2δij

3∑k

vk

∫R3

v

f(v∗)v∗kdv∗

+δij

∫R3

v

f(v∗)|v∗|2dv∗ − vivj

∫R3

v

f(v∗)dv∗ + vi

∫R3

v

f(v∗)v∗jdv∗

+vj

∫R3

v

f(v∗)v∗idv∗ −∫

R3v

f(v∗)v∗iv∗jdv∗,

bj(f) = −vj

∫R3

v

f(v∗)dv∗ +

∫R3

v

f(v∗)v∗jdv∗.

In particular, it follows that3∑

ij=1

aij(µ)ξiξj ≥ |ξ|2, for all ξ ∈ R3,(7.4.4)

|aij(f)(t, x, v)| ≤ C(1 + |v|2)‖f‖L12(R3

v)(t, x),(7.4.5)

|bj(f)(t, x, v)| ≤ C(1 + |v|)‖f‖L11(R3

v)(t, x).(7.4.6)

Let us introduce the operators

L0f = ∇v

(a(µ) · ∇vf − b(µ)f

), G(f) = ∇v

(a(f) · ∇vµ− b(f)µ

).

In particular, L0 is a second order partial differential operator, while G is a convolutiontype operator. It is straigtforward to show that for any l ∈ N, there exists Cl > 0 suchthat

(7.4.7) ‖Wl G(f)‖2L2(]0,T [×R6) ≤ Cl‖Wl f‖2

L2(]0,T×R6)),


where again Wl =< v >l = (1 + |v|2)l/2.By introducing the operator

P = ∂t + v · ∂x − L0,

we consider the Cauchy problem

(7.4.8) P (f) = G(f), x, v ∈ R3, t > 0 , f |t=0 = f0.

We will first give the following hypoelliptic estimate of the operator P . Denote byX0 = ∂t + v · ∂x the kinetic part of P , and let Λ = (1 − 4t,x,v)

1/2 be a regularizingoperator.

Proposition 7.4.1. For any l ≥ 1, there exists constant Cl > 0 such that, for anyu ∈ C∞

0 (R7), we have

(7.4.9) ‖Wl∇vu‖L2(R7) ≤ Cl

|Re (Pu,W2l u)|+ ‖Wl+1u‖2

L2(R7)

,

and

(7.4.10) ‖X0u‖H−1/3(R7) ≤ C

2∑

j=1

|Re (Pu,Aju)|+ ‖W5u‖2L2(R7)

,

where A1 = W4A1, A2 = W2A2Λ−1/3X0 with A1, A2 ∈ Op(S0

1,0(R7)).

Proof: Even though the proof is similar to the corresponding one in [85], we give themain estimates in the following to make this section to be self-contained.

For u ∈ C∞0 (R7), integration by parts gives

‖∇vu‖2L2(R7) ≤ Re (Pu, u) = Re (a(µ)∇vu,∇vu) ,

and

‖Wl∇vu‖2L2(R7) ≤ ‖∇vWl u‖2

L2(R7) + C‖Wl−1 u‖2L2(R7)

= Re (a(µ)∇v(Wl u),∇v(Wl u)) = Re (P (Wl u), (Wl u))

= Re (Pu, (W2l u)) + Re ([P,Wl] u, (Wl u)).

By direct calculation, (7.4.5) gives

[P,Wl] u = Wl+1 ∇vu+ Wl, u

with |Wl+1| ≤ CWl+1 and |Wl| ≤ CWl. Thus (7.4.9) is proved.By letting w = Λ−2/3X0u, we have

‖Λ−1/3X0u‖2L2 = Re (Pu,w)− Re (a(µ)∇vu,∇vw).

Since (a(µ)∇v · , ∇v · ) is a positive operator on L2, it follows that

2 |Re (a(µ)∇vu,∇vw)| ≤ Re (a(µ)∇vu,∇vu) + Re (a(µ)∇vw,∇vw),

and

(7.4.11) ‖Λ−1/3X0u‖2L2 ≤ |Re (Pu,w)|+ |Re (Pu, u)|+ |Re (Pw,w)|.

We note that

Re (Pw,w) = Re (Pu,X0Λ−4/3X0u) + Re ([Λ−2/3, v] · ∂x(∂t + v · ∂x)u,w)

+ Re ([∇va(µ) · ∇v, Λ−2/3X0]u,w).


Since

[Λ−2/3, v] = −2

3Λ−2/3−2∂v,

we have|Re ([Λ−2/3, v] · ∂x(∂t + v · ∂x)u,w)| ≤ C‖w‖2

L2 ,

and

‖w‖2L2 = ((∂t + v · ∇x)u, Λ−4/3X0u)

≤ |Re (Pu,Λ−4/3X0u)|+ |(∇va(µ) · ∇vu,Λ−4/3X0u)|

≤ |Re (Pu,Λ−4/3X0u)|+ C‖a(µ) · ∇vu‖2L2 + 1

16‖Λ−1/3X0u‖2

L2 .

By using (7.4.4) and (7.4.9), we have

‖a(µ) · ∇vu‖2L2 ≤ ‖W2∇vu‖L2(R7) ≤ C

|Re (Pu,W4u)|+ ‖W5u‖2

L2(R7)

.

For the other commutator terms, since aij(µ) is a polynomial of order 2, one has

(7.4.12) [∇va(µ) · ∇v, Λ−2/3X0] = Λ1/3W1 + Λ1/3W2∇v.

Then

|Re ([∇va(µ) · ∇v, Λ−2/3X0]u,w)| ≤ C (‖W1 u‖L2 + ‖W2∇vu‖L2) ‖Λ−1/3X0u‖L2 ,

which gives (7.4.10) and this completes the proof of the proposition.

Remark 7.4.1. The Proposition 7.4.1 is also true for any u ∈ H2(R7) such thatW2l u ∈ H2(R7), l > 2.

We now proceed to study the interior regularity of the weak solution f to theCauchy problem (7.4.8). Let φ = φ(t) ∈ C∞

0 (]0, T [) be a smooth cutoff function. Thenφ(t)f satisfies

(7.4.13) P (φ(t)f) = G(φ(t)f)− φtf, (t, x, v) ∈ R7.

For any l ∈ N, Theorem 7.4.1 implies the following partial regularity estimate

(7.4.14) ||∇vWl(f)||2L2(R7) ≤ C||Wlf ||2L2(]0,T [×R6) < +∞ ,

where f = φ(t)f .For the higher order regularity, we firstly have the following subelliptic estimate.

Proposition 7.4.2. If f is a weak solution of the equation (7.4.8), then

(7.4.15) ‖(∂t + v · ∂x)(f)‖H−1/3(R7) ≤ C‖W5f‖L2(]0,T [×R6).

Proof: By using the mollifier

Mδ(Dt, Dx, Dv) = (1 + δ(|Dt|2 + |Dx|2 + |Dv|2))−1 ,

one hasuδ = Mδ(Dt, Dx, Dv)f ∈ H2(R7),

and also Wl uδ ∈ H2(R7) for any l ∈ N. Applying (7.4.10) yields

(7.4.16) ‖X0uδ‖H−1/3(R7) ≤ C

2∑

j=1

|Re (Puδ, Ajuδ)|+ ‖W5f‖2L2(R7)

.


Since

[P,Mδ(Dt, Dx, Dv)] = Λ0δMδ(Dt, Dx, Dv) + [∇va(µ)∇v, Mδ(Dt, Dx, Dv)],

the same calculation as the one for (7.4.12) gives

[∇va(µ) · ∇v, Mδ(Dt, Dx, Dv)] = Λ0δW1 + Λ0

δW2∇v,

where Λ0δ is uniformly bounded on L2. Then the equation (7.4.13) and the estimate

(7.4.7) imply

‖X0uδ‖H−1/3(R7) ≤ C‖W2 P (f)‖2

L2 + ‖W5f‖2L2(R7)

≤ C

‖W2G(f)‖2

L2 + ‖W5 f‖2L2(]0,T [×R6)

≤ C‖W5 f‖2

L2(]0,T [×R6) .

The Proposition 7.4.2 is then obtained by letting δ → 0.

The next step, that is the regularity w.r.t. other variables, can be obtained fromthe uncertainty principle.

Proposition 7.4.3. Assume that f is a weak solution of the equation (7.4.8), suchthat Wlf ∈ L2(]0, T [×R6) for any l ∈ N. Then, we have

(7.4.17) Λk/4t,x Wl(f) ∈ L2(R7), ∀k, l > 11

2.

Proof: Since we already know that∇vWl(f), Λ−1/3X0(f) ∈ L2(R7), the Corollary 6.3.1of the uncertainty principle with s = 1, s′ = 1/3, δ = 1/6, yields, for any l > 11/2,

Λ1/4t,x Wl (f) ∈ L2(R7).

It follows that f1 = Λ1/4t,x f satifies

Wlf1, Λ1/4t,x Wl (φtf) ∈ L2(R7), ∀l > 5,

and it is a solution (in the weak sense) of the following linearized Landau type equation,

(7.4.18)∂f1

∂t+ v · ∂xf1 = Lf1 − Λ

1/4t,x (φtf).

Therefore, by iterating the above argument, we can prove the Proposition 7.4.3.

End of the proof for Theorem 7.4.2. By fixing p, l ∈ N, let

f2 = Λpt, x(φ(t)f) and Λm

δ = (1 + |Dv|2)m2 (1 + δ|Dv|2)−N0 ,

for 0 < δ < 1,m ∈ R+ with a large number N0 > 7/2 +m. Then

Wl f2 and Wl Λpx(φtf) ∈ L2(R7) , ∀ l ∈ N.

Taking Λmδ W

2l Λm

δ f2 ∈ H2(R7) as the test function in the weak formulation of thefollowing equation

∂f2

∂t+ v · ∂x(f2) = L(f2)− g,


where g = Λpt,x(φtf), one has(

Wl [v, Λmδ ]∂xf2, Wl Λm

δ f2

)=(Wl Λm

δ L0f2, Wl Λmδ f2

)+(G(f2)− g, Λm

δ W2l Λm

δ f2

).

As above, we find that for any l > 7/2,

||ΛvWl Λmδ f2||2L2 ≤ C

||Wl+2 Λm

δ f2||2L2 + ||Wl Λmδ g||2L2 + ||Wl Λm−1

δ Λxf2||2L2

,

where the constant C > 0 is independent of δ. Since the function g plays the samerole as f2 only with a different cutoff function because φt ∈ C∞

0 (]0, T [), by using aninduction argument on m ∈ N, it follows that

Wl Λmv Λp

t, x(φ(t)f) ∈ L2(R7),

for any p, m ∈ N and any cutoff function φ ∈ C∞0 (]0, T [). This completes the proof of

the Theorem .

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Index

Absolute Maxwellian distribution, 34Adjoint operator, 25, 90Asymptotic expansion, 22

Blanchrel Formula, 4Bobylev’s identity, 37Boltzmann equation, 31Borel Lemma, 22

Cancelation Lemma, 40Classical symbol, 22Coercivity estimates, 39Collision kernel, 31Composition of operators, 26Conservations laws, 33Continuity in L2, 27Continuity in Sobolev spaces, 29Coulomb potential, 32

Debye-Yukawa type potential, 32, 35Density Theorem, 13Dyadic covering, 80

Elliptic pseudo differential operator, 26Energy estimate, 90Entropy dissipation, 33Existence of weak solutions, 90

Formula of commutators, 77Fractional derivative formulas, 4Functional estimate of collision operators, 45

Gevrey hypoellipticity, 76Gevrey regularity, 57Grad’s cutoff assumption, 32Garding inequality, 29

Hardy’s inequality , 13Homogeneous Boltzmann equation, 51Homogeneous Sobolev space, 6Hypoellipticity of kinetic equations, 83

Interpolation inequality, 14Inverse Fourier transformation, 3

Kac’s equation, 34

Landau equation, 34Linearized Boltzmann equation, 58Linearized collision operator, 58Linearized Landau Equation, 98Logarithmic estimate, 44Logarithmic Sobolev inequality, 14Logarithmic Sobolev space, 8

Mathematical Maxwellian case, 32, 52, 89Maxwellian molecule, 32, 34Model of kinetic equations, 50, 65

Non cutoff cross section, 32, 37

Oscillatory integral, 26

Poincare inequality, 16, 17Poisson bracket, 26Principal symbol, 22Pseudo-differential operators, 24

Regularity of weak solution, 70, 95Rellich theorem, 14

Semi-linear equations, 77Sobolev embedding theorem, 10Sobolev inequality, 12, 13Sobolev space on Ω, 19Sub-elliptic estimate, 66, 68, 76Symbol class, 21

Uncertainty principle, 79Unicity of weak solution, 90

Weak formulations for Boltzmann’s kernel, 33Weak solution, 51, 89Weighted interpolation inequality, 18

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