ON THE TREND TO GLOBAL EQUILIBRIUM FOR SPATIALLY INHOMOGENEOUS KINETIC SYSTEMS

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ON THE TREND TO GLOBAL EQUILIBRIUM FOR SPATIALLY INHOMOGENEOUS KINETIC SYSTEMS: THE BOLTZMANN EQUATION L. DESVILLETTES AND C. VILLANI Abstract. As part of our study of convergence to equilibrium for spatially in- homogeneous kinetic equations, started in [21], we derive estimates on the rate of convergence to equilibrium for solutions of the Boltzmann equation, like O(t?∞). Our results hold conditionally to some strong but natural estimates of smoothness, decay at large velocities and strict positivity, which at the moment have only been established in certain particular cases. Among the most important steps in our proof are 1) quantitative variants of Boltzmann's H-theorem, as proven in [52, 60], based on symmetry features, hypercontractivity and information-theoretical tools; 2) a new, quantitative version of the instability of the hydrodynamic description for non-small Knudsen number; 3) some functional inequalities with geometrical content, in particular the Korn-type inequality which we established in [22]; and 4) the study of a system of coupled differential inequalities of second order, by a treatment inspired from [21]. We also briefly point out the particular role of conformal velocity fields, when they are allowed by the geometry of the problem. Contents I. Introduction and main results 2 I.1. The Boltzmann equation 3 I.

• boltzmann collision


• production bounds via differential inequalities


• main results


• bridge between


• boltzmann's famous


• periodic boundary


• kinetic systems


• homogeneous kinetic

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 28
• Langue: English

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ONTHETRENDTOGLOBALEQUILIBRIUMFORSPATIALLY
INHOMOGENEOUSKINETICSYSTEMS:
THEBOLTZMANNEQUATION
L.DESVILLETTESANDC.VILLANI
Abstract.
Aspartofourstudyofconvergencetoequilibriumforspatiallyin-
homogeneouskineticequations,startedin[21],wederiveestimatesontherateof
convergencetoequilibriumforsolutionsoftheBoltzmannequation,like
O
(
t
∞
).
Ourresultsholdconditionallytosomestrongbutnaturalestimatesofsmoothness,
decayatlargevelocitiesandstrictpositivity,whichatthemomenthaveonlybeen
establishedincertainparticularcases.Amongthemostimportantstepsinour
proofare1)quantitativevariantsofBoltzmann’s
H
-theorem,asprovenin[52,60],
basedonsymmetryfeatures,hypercontractivityandinformation-theoreticaltools;
2)anew,quantitativeversionoftheinstabilityofthehydrodynamicdescription
fornon-smallKnudsennumber;3)somefunctionalinequalitieswithgeometrical
content,inparticulartheKorn-typeinequalitywhichweestablishedin[22];and
4)thestudyofasystemofcoupleddierentialinequalitiesofsecondorder,by
atreatmentinspiredfrom[21].Wealsobrieypointouttheparticularroleof
conformalvelocityelds,whentheyareallowedbythegeometryoftheproblem.

Contents
I.Introductionandmainresults
I.1.TheBoltzmannequation
I.2.Boltzmann’s
H
theorem
I.3.Statementoftheproblemandmainresult
I.4.Ingredients
I.5.Rangeofapplication
I.6.Non-smoothinitialdata
II.Thestrategy
II.1.Quantitative
H
theorem
II.2.Instabilityofthehydrodynamicregime
II.3.Puttingbothfeaturestogether
II.4.Thesystem

1

237014151718181025282

2

L.DESVILLETTESANDC.VILLANI

III.Instabilityofthehydrodynamicdescription
III.1.Equationsforhydrodynamicalelds
III.2.Estimatesatlocalequilibrium
III.3.Thegeneralcase
IV.Somegeometricalinequalities
IV.1.Korn-typeinequalities
IV.2.Poincare-typeinequalities
V.Dampingofhydrodynamicoscillations
VI.Averageentropyproductionboundsviadierential
inequalities
VI.1.Strategy
VI.2.Thesubdivision
VI.3.Estimatesofaverageentropyproduction
VII.Furthercomments
VII.1.Qualitativebehaviorofthegas
VII.2.Quasi-equilibriaandconformalmappings
VII.3.CommentsontheproofofTheorem2
VII.4.Bibliographicalnotes
References

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I.
Introductionandmainresults
Thisworkisthesequelofourprogramstartedin[21]aboutthetrendtother-
modynamicalequilibriumforspatiallyinhomogeneouskineticequations.Inthe
presentpaper,weshallderiveestimatesontherateofconvergencetoequilibrium
forsmoothsolutionsoftheBoltzmannequation,providingarstquantitativebasis
forthemaximumentropyprincipleinthiscontext.
ThereareseveralreasonsforgivingtheBoltzmannequationacentralrolein
thisprogram.First,theproblemofconvergencetoequilibriumforthisequationis
famousforhistoricalreasons,since(togetherwiththe
H
theorem,thatweshallrecall
below)itwasoneofthemainelementsofthecontroversybetweenBoltzmannand
hispeers,andoneofthemostspectacularpredictionsofBoltzmann’sapproach.At
thelevelofpartialdierentialequations,theproblemswhichonehastoovercome
whenstudyingtheBoltzmannequationaretypicalofthoseassociatedwiththe
combinationoftransportphenomenaandcollisions—andforveryfewmodelsdo

TRENDTOGLOBALEQUILIBRIUMFORTHEBOLTZMANNEQUATION3

theseproblemsarisewithsuchintensityasinthecaseoftheBoltzmannequation.
Finally,theBoltzmannequationestablishesabeautifulbridgebetweenstatistical
mechanicsanduidmechanics,apropertywhichwillbecentralinourtreatment.
Inthisintroductorysection,weshallrstintroducebrieythemodel,thenrecall
Boltzmann’sfamous
H
theorem,andnallystateourmainresult,theproofofwhich
willbetheobjectoftherestofthepaper.
I.1.
TheBoltzmannequation.
Let

x
beapositionspaceforparticlesinagas
obeyingthelawsofclassicalmechanics.Forsimplicityweshallassumethat

x
is
eitherasmooth(say
C
1
)boundedconnectedopensubsetof
R
N
(
N

2)orthe
N
-dimensionaltorus
T
N
.Thelattercaseisnotsorelevantfromthephysicalpoint
ofview,butithastheadvantagetoavoidthesubtleproblemscausedbyboundaries,
andisthereforecommonlyusedintheoreticalandnumericalstudies.Withoutloss
ofgeneralityweshallassumethat

x
hasunitLebesguemeasure:
(1)
|

x
|
=1
.
TheunknowninBoltzmann’sdescriptionisatime-dependentprobabilitydensity
(
f
t
)
t

0
onthephasespace

x

R
N
(tothinkofasatangentbundle);itwillbe
denotedeither
f
(
t,x,v
)or
f
t
(
x,v
)andstandsforthedensityofthegasinphase
space.Ifoneassumesthatthegasisdilute,thatparticlesinteractviabinary,elastic,
microscopicallyreversiblecollisions,andthattherearenocorrelationsbetweenpar-
ticleswhicharejustabouttocollide(Boltzmann’schaosassumption),thenonecan
argue,andinsomecases“prove”,thatitisreasonabletouseBoltzmann’sevolution
equation,
f∂(2)+
v
r
x
f
=
Q
(
f,f
)
.
t∂Here
r
x
standsforthegradientoperatorwithrespecttothepositionvariable
x
,
and,accordingly,
v
r
x
istheclassicaltransportoperator,while
Q
isthequadratic
Boltzmanncollisionoperator,
ZZ(3)
Q
(
f,f
)=(
f
0
f
0

ff

)
B
(
v

v

,
)
ddv

.
R
N
S
N

1
Theaboveformulagivesthevalueofthefunction
Q
(
f,f
)at(
t,x,v
),theparameters
v

and

livein
R
N
and
S
N

1
respectively,weusedthecommonshorthands
f
=
f
(
t,x,v
),
f

=
f
(
t,x,v

),
f
0
=
f
(
t,x,v
0
),
f
0
=
f
(
t,x,v
0
),and(
v
0
,v
0
)standfor
thepre-collisionalvelocitiesoftwoparticleswhichinteractandwillhavevelocities
(
v,v

)asaresultoftheinteraction:
(4)
v
0
=
v
+
v

+
|
v

v

|
,v
0
=
v
+
v

|
v

v

|
.
2222

4

L.DESVILLETTESANDC.VILLANI

Thenonnegativefunction
B
=
B
(
v

v

,
),whichwecallBoltzmann’scollision
kernel,onlydependsonthemodulusoftherelativecollisionvelocity,
|
v

v

|
,and
onthecosineofthedeviationangle

,i.e.
vvcos

=
|
v

v
|
,.
Itislinkedtothecross-sectionbytheformula
B
=
|
v

v

|
.Itisnotour
purposeheretodescribethecollisionkernelprecisely;see[59,Chapter1,Section3]
forsomeelementsofclassicationinamathematicalperspective.Ouronlyexplicit
restrictionson
B
willbethatitisstrictlypositive,inthesense