Averaging Lemmas and Dispersion Estimates for kinetic equations

Averaging

LemmasandDispersion

forkineticequations

Pierre-EmmanuelJabin

email:jabin@unice.fr

LaboratoireJ-ADieudonne´

Universite´deNice

ParcValrose,06108NiceCedex02

Estimates

2

Introduction

Kineticequationsareaparticularcaseoftransportequationinphasespace,
i.e.
onfunctions
f
(
x,v
)ofphysical
and
velocityvariableslike

∂
t
f
+
v
∙r
x
f
=
g,t
≥
0
,x,v
∈
R
d
.

Asasolutiontoahyperbolicequation,thesolutioncannotbemoreregular
thantheinitialdataortherighthand-side.Howeveraspecificfeatureof
kineticequationsisthattheaveragesinvelocity,like
Zρ
(
t,x
)=
f
(
t,x,v
)
φ
(
v
)
dv,φ
∈
C
c
∞
(
R
d
)
,
dRareindeedmoreregular.Thisphenomenoniscalledvelocityaveraging.
Itwasfirstobservedin[24]andthenin[23]ina
L
2
framework.Thefinal
L
p
estimatewasobtainedin[17](andslightyrefinedin[3]togetaSobolev
spaceinsteadofBesov).Thecaseofafullderivative
g
=
r
x
∙
h
wastreated
in[45]andalthoughitisinmanywaysalimitcase,itisimportantforsome
applicationsasitcanreplacecompensatedcompactnessarguments.
Inadditiontotheseworks,thiscoursepresentsandsometimesreformu-
latessomeoftheresultsof[6],[17],[22],[31],[32],[36],[37],[45].
Thereareofcoursemanyotherinterestingcontributionsinvestigating
averaginglemmasthatareonlybrieflymentionedthroughthetext.

3

4

Chapter1

Kineticequations:Basictools

1.1Ashortintroductiontokineticequations
Foramorecompleteintroductiontokineticequationsandthebasictheory,
wereferto[6]or[21].Manyproofsareomittedherebutaregenerallywell
knownandnotdifficult.

1.1.1Thebasicequations
Duringmostofthiscourse,wewilldealwiththesimplestequations
∂
t
f
+
α
(
v
)
∙r
x
f
=
g
(
t,x,v
)
,t
∈
R
+
,x
∈
R
d
,v
∈
ω,
(1.1.1)
where
ω
isoften
R
d
(butmightonlybeasubdomain);Orwiththestationary
version
α
(
v
)
∙r
x
f
=
g
(
x,v
)
,t
∈
R
+
,x
∈
O,v
∈
ω,
(1.1.2)
where
O
isanopen,regularsubsetof
R
d
and
ω
isusuallyratherthesphere
S
d
−
1
.Thetransportcoefficient
α
willalwaysberegular,typicallyLipschitz
althoughhereboundedwouldbeenough.
Ofcourse(1.1.1)isreallyasubcaseof(2.1.1)indimension
d
+1andwith
α
0
(
v
)=(1
,v
),
O
=
R
∗
+
×
R
d
,
ω
=
R
d
.
Neither(1.1.1)nor(2.1.1)haveauniquesolutionastherearemanyso-
lutionsto
∂
t
f
+
α
(
v
)
∙r
x
f
=0
,
forinstance.Indeedfor(1.1.1)aninitialdatamustbeprovided
f
(
t
=0
,x,v
)=
f
0
(
x,v
)
,

5

(1.1.3)

6

CHAPTER1.KINETICEQUATIONS:BASICTOOLS

andfor(2.1.1)theincomingvalueof
f
ontheboundarymustbespecified
f
(
x,v
)=
f
in
(
x,v
)
,x
∈
∂O,α
(
v
)
∙
ν
(
x
)
≤
0
,
(1.1.4)
where
ν
(
x
)istheoutwardnormalto
O
at
x
.
Itisthenpossibletohaveexistenceanduniquenessinthespaceofdis-
tributions
Theorem1.1.1
Let
f
0
∈D
0
(
R
d
×
ω
)
and
g
∈
L
l
1
oc
(
R
+
,
D
0
(
R
d
×
ω
))
.Then
thereisauniquesolutionin
L
l
1
oc
(
R
+
,
D
0
(
R
d
×
ω
))
to
(1.1.1)
with
(1.1.3)
in
thesenseofdistributiongivenby
tZf
(
t,x,v
)=
f
0
(
x
−
α
(
v
)
t,v
)+
g
(
t
−
s,x
−
α
(
v
)
s,v
)
ds.
(1.1.5)
0Notethatif
f
solves(1.1.1)thenforany
φ
∈
C
c
∞
(
R
d
×
ω
)
Zdf
(
t,x,v
)
φ
(
x,v
)
∈
L
l
1
oc
(
R
+
)
,
dt
R
d
×
ω
so
f
hasatraceat
t
=0intheweaksenseand(1.1.3)perfectlymakessense.
Proof.
Itiseasytocheckthat(1.1.5)indeedgivesasolution.If
f
isanother
solutionthendefine
tZf
¯=
f
−
f
0
(
x
−
α
(
v
)
t,v
)
−
g
(
t
−
s,x
−
α
(
v
)
s,v
)
ds.
0Remarkthat
∂
t
f
¯+
α
(
v
)
∙r
x
f
¯=0
,
andhence
∂
t
(
f
¯(
t,x
+
α
(
v
)
t,v
)=0sothat
f
¯=0.
Anequivalentresultmaybeprovedfor(2.1.1)withtheconditionthat
thesupportofthesingularpart(in
x
)ofthedistribution
g
doesnotextend
totheboundary
∂O
.
Ontheotherhand,themodifiedequation,whichwewillfrequentlyuse,
α
(
v
)
∙r
x
f
+
f
=
g,x
∈
R
d
,v
∈
ω,
(1.1.6)
iswellposedinthewhole
R
d
withouttheneedforanyboundarycondition
Theorem1.1.2
Let
g
∈S
0
(
R
d
×
ω
)
,thereexistsaunique
f
in
S
0
(
R
d
×
ω
)
solutionto
(1.1.6)
.Itisgivenby
∞Zf
(
x,v
)=
g
(
x
−
α
(
v
)
t,v
)
e
−
t
dt.
(1.1.7)
0

1.1.ASHORTINTRODUCTIONTOKINETICEQUATIONS
7

1.1.2Liouvilleequation
Theequationreads
∂
t
f
+
α
(
v
)
∙r
x
f
+
F
(
t,x,v
)
∙r
v
f
=0
,t
≥
0
,x
∈
R
d
,v
∈
R
d
,
(1.1.8)
where
F
isagivenforcefield.Inmanyapplications,liketheVlasov-Maxwell
system1.2,
F
isinfactcomputedfromthesolution
f
.
Eq.(1.1.8)describesthedynamicsofparticlessubmittedtotheforce
F
andassuchisconnectedtothesolutionoftheODE
dX
(
t,s,x,v
)=
α
(
V
(
t,s,x,v
))
,dV
(
t,s,x,v
)=
F
(
t,X,V
)
,
dtdt
(1.1.9)
X
(
s,s,x,v
)=
x,V
(
s,s,x,v
)=
v,
whichrepresentsthetrajectoryofaparticlestartingwithpositionandve-
locity(
x,v
)attime
t
=
s
.
TheODE(1.1.9)iswellposedforinstanceif
1
,
∞
1
,
∞
α
(
v
)
∈
W
loc
(
R
d
)
,F
∈
W
loc
(
R
+
×
R
2
d
)
,
(1.1.10)
|
α
|
+
|
F
|≤
C
(
t
)(1+
|
x
|
+
|
v
|
)
,
thankstoCauchy-LipschitzTheorem.Weakerassumptionsarehowever
enough,
W
l
1
o,c
1
andboundeddivergencein[16]oreven
BV
loc
in[1],butwill
notberequiredhere.
Under(1.1.10),(1.1.8)isalsowellposed
Theorem1.1.3
Assume
(1.1.10)
and
r
v
∙
F
∈
L
∞
(
R
+
×
R
2
d
)
,foranymea-
surevaluedinitialdata
f
0
∈
M
1
(
R
2
d
)
,thereexistsaunique
f
includedin
L
∞
([0
,T
]
,M
1
(
R
d
))
solutionto
(1.1.8)
inthesenseofdistributionandsat-
isfying
(1.1.3)
.Itisgivenby
f
(
t,x,v
)=
f
0
(
X
(0
,t,x,v
)
,V
(0
,t,x,v
))
.

If
F
and
α
areregularenough(
C
∞
),thesametheoremholdsfor
f
0
a
distribution.
Thistheoremimpliesmanypropertieson
f
,forexample

8

CHAPTER1.KINETICEQUATIONS:BASICTOOLS

Proposition1.1.1
(
i
)
f
≥
0
ifandonlyif
f
≥
0
.
(
ii
)
If
f
0
∈
L
∞
(
R
2
d
)
then
f
∈
L
∞
(
R
+