Generalized kinetic Maxwell type models of granular gases

0.3 MB·English

by  A. V. Bobylev

Generalized kinetic Maxwell type models of
granular gases
A.V.Bobylev,C.CercignaniandI.M.Gamba
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4 Abstract
2
Key words: Statistical and kinetic transport models, Dissipative Boltzmann Equa-

tions of Maxwell type interactions, Self-similar solutions and asymptotics, Non-
]
h equilibriumstatisticalstationarystates,Powerlaws.
p
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a 1 Introduction
m
[
Ithasbeennoticedinrecentyearsthatasignificantnon-trivialphysicalphenomena

1 in granular gases can be described mathematically by dissipative Boltzmann type
v
equations, as can be seen in [17] for a review in the area. As motivated by this
4
particularphenomenaofenergydissipationatthekineticlevel,weconsiderinthis
6
8 chapter the Boltzmann equation for non-linear interactions of Maxwell type and
3 somegeneralizationsofsuchmodels.
. The classical conservative(elastic) Boltzmannequationwith the Maxwell-type
1
0 interactions is well-studied in the literature (see [5,14] and references therein).
9 Roughlyspeaking,thisisamathematicalmodelofararefiedgaswithbinarycolli-
0 sionssuchthatthecollisionfrequencyisindependentofthevelocitiesofcolliding
:
v particles,andeventhoughtheintermolecularpotentialsarenotofthosecorrespond-
i ing to hardsphereinteractions,still these modelsprovidea veryrichinside to the
X
understandingofkineticevolutionofgases.
r
a
A.V.Bobylev
Department of Mathematics, Karlstad University, Karlstad, SE-65188 Sweden., e-mail:
[email protected]
C.Cercignani
PolitecnicodiMilano,Milano,Italye-mail:[email protected]
I.M.Gamba
DepartmentofMathematics,TheUniversityofTexasatAustin,Austin,TX78712-1082U.S.A.,
e-mail:[email protected]
1
2 A.V.Bobylev,C.CercignaniandI.M.Gamba
Recently,BoltzmannequationsofMaxwelltypewereintroducedformodelsof
granulargaseswereintroducedin[7]inthreedimensions,andabitearlierin[3]for
in onedimensioncase. Soon after that, these modelsbecame verypopularamong
the community studying granular gases (see, for example, the book [13] and ref-
erences therein). There are two obvious reasons for such studies The first one is
thattheinelasticMaxwell-Boltzmannequationcanbeessentiallysimplifiedbythe
Fourier transform similarly as done for the elastic case, where its study becomes
moretransparent[6,7].Thesecondreasonis motivatedbythe specialphenomena
associated with homogeneous cooling behavior, i.e. solutions to the spatially ho-
mogeneous inelastic Maxwell-Boltzmann equation have a non-trivial self-similar
asymptotics, and in addition, the correspondingself-similar solution has a power-
like tail for large velocities. The latter property was conjectured in [16] and later
proved in [9,11]. This is a rather surprising fact, since the Boltzmannn equation
forhardspheresinelasticinteractionshasbeenshowntohaveselfsimilarsolutions
withallmomentsboundedandlargeenergytailsdecayingexponentially.Thecon-
jectureofself-similar(orhomogeneouscooling)statesforsuchmodelofMaxwell
typeinteractionswasinitiallybasedonanexact1dsolutionconstructedin[1].Itis
remarkablethatsuchanasymptoticsisabsentintheelasticcase(astheelasticBoltz-
mannequationhastoomanyconservationlaws).Later,theself-similarasymptotics
was provedin the elastic case for initialdata with infinite energy[8] byusing an-
othermathematicaltoolscomparedto[9]and[12].
Surprisingly,the recently published exact self-similar solutions [12] for elastic
Maxwelltypemodelforaslowdownprocess,derivedasaformalasymptoticlimit
of a mixture, also is shown to have power-like tails. This fact definitely suggests
thatself-similarasymptoticsarerelatedtototalenergydissipationratherthanlocal
dissipativeinteractions.Asanillustrationtothisfact,wementionsomerecentpub-
lications[2,15],where1d Maxwell-typemodelswereintroducedfornon-standard
applicationssuch asmodelsineconomicsandsocialinteractions,wherealso self-
similarasymptoticsandpower-liketailasymptoticstateswerefound.
Thus all the above discussed models describe qualitatively different processes
in physics or even in economics, however their solutions have a lot in common
frommathematicalpointof view. It is also clear thatsome furthergeneralizations
arepossible:onecan,forexample,includeinthe modelmultiple(notjustbinary)
interactions still assuming the constant (Maxwell-type) rate of interactions. Will
themulti-linearmodelshavesimilarproperties?Theanswerisyes,asweshallsee
below.
Thus,itbecomesclearthattheremustbesomegeneralmathematicalproperties
ofMaxwellmodels,which,inturn,canexplainpropertiesofanyparticularmodel.
That is to say there must be just one maintheorem, from which one can deduce
allabovediscussedfactsandtheirpossiblegeneralizations.Ourgoalistoconsider
Maxwellmodelsfromverygeneralpointofviewandtoestablishtheirkeyproper-
tiesthatleadtotheself-similarasymptotics.
All the results presentedin this chapter are mathematicallyrigorous.Their full
proofscanbefoundin[10].
Generalizedkineticmodelsofgranulargases 3
After this introduction,we introducein Section 2 three specific Maxwell mod-
els ofthe Boltzmannequation:(A)classical (elastic)Boltzmannequation;(B) the
model (A) in the presence of thermostat; (C) inelastic Boltzmann equation for
Maxwell type interactions. Then, in Section 3, we perform the Fourier transform
andintroducean equationthatincludesallthe three modelsas particularcases. A
furthergeneralizationisdoneinSection4,wheretheconceptofgeneralizedmulti-
linear Maxwell model(in the Fourier space) is introduced.Such modelsand their
generalizationsarestudiedindetailinSections5and 6.Themostimportantforour
approachconceptofL-LipschitznonlinearoperatorisexplainedinSection4.Itis
shown(Theorem4.2)thatallmulti-linearMaxwellmodelssatisfy theL-Lipschitz
condition.Thispropertyofthemodelsconstitutesabasisforthegeneraltheory.
Theexistenceand uniquenessof solutionsto the initial valueproblemis stated
inSubsection5.1(Theorem5.2).Then,inSubsection5.2,wepresentandstudythe
large time asymptotics under very generalconditionsthat are fulfilled, in particu-
lar, for all our models. It is shown that L-Lipschitz condition leads to self-similar
asymptotics,providedthecorrespondingself-similarsolutiondoesexist.Theexis-
tenceanduniquenessofsuchself-similarsolutionsisstatedinSubsection5.3(The-
orem5.12).Thistheoremcan be considered,to some extent,as themaintheorem
for general Maxwell-type models. Then, in Subsection 5.4, we go back to multi-
linear modelsof Section 4 and study more specific propertiesof their self-similar
solutions.
We explain in Section 6 how to use our theory for applicationsto any specific
model: it is shown that the results can be expressed in terms of just one function
m (p),p>0,thatdependsonspectralpropertiesofthespecificmodel.Generalprop-
erties(positivity,power-liketails,etc.)self-similarsolutionsarestudiedinSubsec-
tions6.1and6.2.Itincludesalsothecaseof1dmodels,wheretheLaplace(instead
ofFourier)transformisused.InSubsection6.3,weformulate,intheunifiedstate-
ment (Theorem11.1), the main propertiesof Maxwell models(A),(B) and (C) of
the Boltzmann equation. This result is, in particular, an essential improvementof
earlierresultsof[7]forthemodel(A) andquitenewforthemodel(B).
Applicationstoonedimensionalmodelsarealsobrieflydiscussedattheendof
Subsection6.3.
2 Maxwellmodels ofthe Boltzmann equation
We considera spatiallyhomogeneousrarefiedd-dimensionalgas(d =2,3,...) of
particleshavingaunitmass.Let f(v,t),wherev∈Rdandt∈R denoterespectively
+
the velocity and time variables, be a one-particle distribution function with usual
normalization
dvf(v,t)=1. (1)
ZRd
Then f(v,t) has an obvious meaning of a time-dependent probability density in
Rd. We assume thatthe collisionfrequencyis independentof thevelocitiesofthe
4 A.V.Bobylev,C.CercignaniandI.M.Gamba
collidingparticles(Maxwell-typeinteractions).Wediscussthreedifferentphysical
models(A),(B)and(C).
(A) Classical Maxwell gas (elastic collisions). In this case f(v,t) satisfies the
usualBoltzmannequation
u·w
f =Q(f,f)= dwdw g( )[f(v′)f(w′)−f(v)f(w)], (2)
t
ZRd×Sd−1 |u|
wheretheexchangeofvelocitiesafteracollisionaregivenby
1 1
v′= (v+w+|u|w ), and w′= (v+w−|u|w )
2 2
whereu=v−w is therelativevelocityandW ∈Sd−1. Forthe sake of brevitywe
shallconsiderbelowthemodelnon-negativecollisionkernelsg(s)suchthatg(s)is
integrableon[−1,1].Theargumenttof f(v,t)andsimilarfunctionsisoftenomitted
below(asinEq.(2)).
(B)ElasticmodelwithathermostatThiscasecorrespondstomodel(A) inthe
presenceofathermostatthatconsistsofMaxwellparticleswithmassm>0having
theMaxwelliandistribution
2p T m|v|2
M(v)=( )−d/2exp(− ) (3)
m 2T
withaconstanttemperatureT >0.Thentheevolutionequationfor f(x,t)becomes
u·w
f =Q(f,f)+q dwdw g( )[f(v′)M(w′)−f(v)M(w)], (4)
t
Z |u|
whereq >0isacouplingconstant,andtheexchangeofvelocitiesisnow
v+m(w+|u|w ) v+mw−|u|w
v′= , and w′= ,
1+m 1+m
withu=v−wtherelativevelocityand,w ∈Sd−1.
Equation(4)wasderivedin[12]asacertainlimitingcaseofabinarymixtureof
weaklyinteractingMaxwellgases.
(C)Maxwellmodelforinelasticparticles.Weconsiderthismodelintheform
givenin[9].ThentheinelasticBoltzmannequationintheweakformreads
¶ |u·w |
(f,y )= dvdwdw f(v)f(w) [y (v′)−y (v)], (5)
¶ t ZRd×Rd×Sd−1 |u|
wherey (v)isaboundedandcontinuoustestfunction,
1+e
(f,y )= dvf(v,t)y (v), u=v−w, w ∈Sd−1, v′=v− (u·w )w , (6)
ZRd 2
Generalizedkineticmodelsofgranulargases 5
the constant parameter 0<e≤1 denotes the restitution coefficient. Note that the
model(Cwithe=1isequivalenttothemodel(A)withsomekernelg(s).
All three models can be simplified (in the mathematical sense) by taking the
Fouriertransform.
Wedenote
fˆ(k,t)=F[f]=(f,e−ik·v), k∈Rd , (7)
andobtain(byusingthesametrickasin[6]forthemodel(A))forallthreemodels
thefollowingequations:
k·w
(A) fˆ =Q(fˆ,fˆ)= dw g( )[fˆ(k )fˆ(k )−fˆ(k)fˆ(0)],
t + −
ZSd−1 |k|
b (8)
wherek = 1(k±|k|w ),w ∈Sd−1, fˆ(0)=1.
± 2
k·w
(B) fˆ =Q(fˆ,fˆ)+q dw g( )[fˆ(k )M(k )−fˆ(k)M(0)],
t + −
ZSd−1 |k|
b b b (9)
whereM(k)=e−T2|km|2,k+= k+1m+|mk|w ,k−=k−k+,w ∈Sd−1, fˆ(0)=1.
b |k·w |
(C) fˆ = dw [fˆ(k )fˆ(k )−fˆ(k)fˆ(0)],
t + −
ZSd−1 |k|
(10)
where fˆ(0)=1, k = 1+e(k·w )w , k =k−k , with w ∈Sd−1 is the direction
+ 2 − +
containingthetwocentersoftheparticlesatthetimeoftheinteraction.Equivalently,
onemayalternativewritek = 1+e(k−|k|nw˜),andk =k−k ,wherenoww˜ ∈
− 4 + −
Sd−1isthedirectionofthepostcollisionalrelativevelocity,andtheterm |k·w |dwis
|k|
replacedbyafunctiong(k·w˜)dw˜.
|k|
Case(B)canbesimplifiedbythesubstitution
T|k|2
fˆ(k,t)= f(k,t)exp[− ] , (11)
2
eb
leading,omittingtildes,totheequation
k·w k+m|k|w
(B′) fˆ =Q(fˆ,fˆ)+q dw g( )[fˆ( )−fˆ(k)],
t
ZSd−1 |k| 1+m
b (12)
i.e.,themodelfor(B)withT =0,orequivalentlyalinearcollisionaltermtheback-
groundsingulardistribution.Therefore,weshallconsiderbelowjustthecase(B′),
assumingneverthelessthat fˆ(k,t)inEq.(12)istheFouriertransform(7)ofaprob-
abilitydensity f(v,t).
6 A.V.Bobylev,C.CercignaniandI.M.Gamba
3 IsotropicMaxwellmodel inthe Fourierrepresentation
Weshallseethatthesethreemodels(A),(B)and(C)admitaclassofisotropicso-
lutionswithdistributionfunctions f = f(|v|,t).Indeed,accordingto(7)welookfor
solutions fˆ= fˆ(|k|,t)tothecorrespondingisotropicFouriertransformedproblem,
givenby
x=|k|2 , j (x,t)= fˆ(|k|,t)=F[f(|v|,t)], (13)
wherej (x,t)solvesthefollowinginitialvalueproblem
1
j = dsG(s){j [a(s)x]j [b(s)x]−j (x)}+
t
Z0
1 (14)
+ dsH(s){j [c(s)x]−j (x)} ,
Z0
j =j (x), j (0,t)=1,
t=0 0
wherea(s),b(s),c(s)arenon-negativecontinuousfunctionson[0,1],whereasG(s)
andH(s)aregeneralizednon-negativefunctionssuchthat
1 1
dsG(s)<¥ , dsH(s)<¥ . (15)
Z0 Z0
Thus,wedonotexcludesuchfunctionsasG=d (s−s ),0<s <1,etc.Weshall
0 0
seebelowthat,forisotropicsolutions(13),eachofthethreeequations(8),(10),(12)
isaparticularcaseofEq.(14).
LetusfirstconsiderEq.(8)with fˆ(k,t)=j (x,t)inthenotation(13).Inthatcase
1±(w ·w ) k
|k |2=|k|2 0 , w = ∈Sd−1, d=2,...,
± 0
2 |k|
andtheintegralinEq.(8)reads
1+w ·w 1−w ·w
dw g(w ·w )j x 0 j x 0 . (16)
0
ZSd−1 (cid:20) 2 (cid:21) (cid:20) 2 (cid:21)
Itiseasytoverifytheidentity
1
dw F(w ·w 0)=|Sd−2| dzF(z)(1−z2)d−23 , (17)
ZSd−1 Z−1
where|Sd−2|denotesthe“area”oftheunitsphereinRd−1 ford≥3and|S0|=2.
Theidentity(17)holdsforanyfunctionF(z)providedtheintegralasdefinedinthe
righthandsideof(17)exists.
Theintegral(16)nowreads
Generalizedkineticmodelsofgranulargases 7
|Sd−2| 1 dzg(z)(1−z2)d−23j (x1+z)j (x1−z)=
Z−1 2 2
1
= dsG(s)j (sx)j [(1−s)x],
Z0
where
G(s)=2d−2|Sd−2|g(1−2s)[s(1−s)]d−23 , d=2,3,... . (18)
Hence,inthiscaseweobtainEq.(14),where
(A) a(s)=s, b(s)=1−s, H(s)=0,
(19)
G(s)isgiveninEq.(18).
Twoothermodels(B′)and(C),describedbyeqs.(12),(10)respectively,canbe
consideredquitesimilarly.InbothcasesweobtainEq.(14),where
4m
(B′) a(s)=s, b(s)=1−s, c(s)=1− s,
(1+m)2
H(s)=q G(s),
(20)
G(s)isgiveninEq.(18):
(1+e)2 (1+e)(3−e)
(C) a(s)= s, b(s)=1− s,
4 4
H(s)=0, G(s)=|Sd−2|(1−s)d−23 .
(21)
Hence,allthreemodelsaredescribedbyEq.(14)where0<a(s),b(s),c(s)≤1
are non-negative linear functions. One can also find in recent publications some
otherusefulequationsthatcanbereducedafterFourierorLaplacetransformations
to Eq. (14) (see, for example,[2], [15] thatcorrespondto the case G=d (s−s ),
0
H=0).
Theequation(14)withH(s)=0firstappearedinitsgeneralformin[9]incon-
nectionwithmodels(A)and(C).Theconsiderationoftheproblemofself-similar
asymptotics for Eq. (14) in that paper made it quite clear that the most important
properties of “physical” solutions depend very weakly on the specific functions
G(s),a(s)andb(s).
4 Models withmultipleinteractions
We present now a general framework to study solutions to the type of problems
introducedintheprevioussection.
8 A.V.Bobylev,C.CercignaniandI.M.Gamba
We assume, without loss of generality, (scaling transformations t˜= a t, a =
const.)that
1
ds[G(s)+H(s)]=1 (22)
Z0
in Eq. (14). Then Eq. (14) can be consideredas a particular case of the following
equationforafunctionu(x,t)
u +u=G (u), x≥0, t≥0, (23)
t
where
N N
G (u)= (cid:229) a G (n)(u), (cid:229) a =1, a ≥0,
n n n
n=1 n=1 (24)
¥ ¥ n
G (n)(u)= da ... da A (a ,...,a )(cid:213) u(a x), n=1,...,N .
1 n n 1 n k
Z0 Z0 k=1
Weassumethat
¥ ¥
A (a)=A (a ,...,a )≥0, da ... da A(a ,...,a )=1, (25)
n n 1 n 1 n 1 n
Z0 Z0
where A (a)=A (a ,...,a ) is a generalized density of a probability measure in
n n 1 n
Rn foranyn=1,...,N.WealsoassumethatallA (a)haveacompactsupport,i.e.,
+ n
n
A (a ,...,a )≡0 if (cid:229) a2>R2, n=1,...,N , (26)
n 1 n k
k=1
forsufficientlylarge0<R<¥ .
Eq.(14)isaparticularcaseofEq.(23)with
1 1
N=2, a = dsH(s), a = dsG(s)
1 2
Z0 Z0
1 1
A (a )= dsH(s)d [a −c(s)] (27)
1 1 a 1Z0 1
1 1
A (a ,a )= dsG(s)d [a −a(s)]d [a −b(s)].
2 1 2 a 2Z0 1 2
Itis clear that Eq.(23) can be consideredas a generalizedFouriertransformed
isotropic Maxwell modelwith multiple interactionsprovidedu(0,t)=1, the case
N=¥ inEqs.(24)canbetreatedinthesameway.
Generalizedkineticmodelsofgranulargases 9
4.1 Statement ofthegeneral problem
The general problem we consider below can be formulated in the following way.
Westudytheinitialvalueproblem
u +u=G (u), u =u (x), x≥0, t≥0, (28)
t |t=0 0
intheBanachspaceB=C(R )ofcontinuousfunctionsu(x)withthenorm
+
kuk=sup|u(x)|. (29)
x≥0
It is usually assumed that ku k≤1 and that the operator G is given by Eqs.
0
(24). On the other hand, there are just a few properties of G (u) that are essential
for existence, uniqueness and large time asymptotics of the solution u(x,t) of the
problem(28).Therefore,inmanycasestheresultscanbeappliedtomoregeneral
classesofoperatorsG inEqs.(28)andmoregeneralfunctionalspace,forexample
B=C(Rd)(anisotropicmodels).Thatiswhywestudybelowtheclass(24)ofoper-
atorsG asthemostimportantexample,butsimultaneouslyindicatewhichproperties
ofG are relevantin eachcase. In particular,mostof the resultsof Section 4–6do
notuseaspecificform(24)ofG and,infact,arevalidforamoregeneralclassof
operators.
Followingthiswayofstudy,wefirstconsidertheproblem(28)withG givenby
Eqs.(24)andpointoutthemostimportantpropertiesofG .
Wesimplifynotationsandomitinmostofthecasesbelowtheargumentxofthe
functionu(x,t).Thenotationu(t)(insteadofu(x,t))meansthenthefunctionofthe
realvariablet≥0withvaluesinthespaceB=C(R ).
+
Remark1.We shallomitbelowthe argumentx∈R offunctionsu(x), v(x), etc.,
+
inallcaseswhenthisdoesnotcauseamisunderstanding.Inparticular,inequalities
ofthekind|u|≤|v|shouldbeunderstoodasapoint-wisecontrolinabsolutevalue,
i.e.“|u(x)|≤|v(x)|foranyx≥0”andsoon.
Wefirststartbygivingthefollowinggeneraldefinitionforoperatorsactingona
unitballofaBanachspaceBdenotedby
U ={u∈B:kuk≤1} (30)
Definition1. TheoperatorG =G (u)iscalledanL-Lipschitzoperatorifthereexists
alinearboundedoperatorL:B→Bsuchthattheinequality
|G (u )−G (u )|≤L(|u −u |) (31)
1 2 1 2
holdsforanypairoffunctionsu inU.
1,2
Remark2.NotethattheL-Lipschitzcondition(31)holds,bydefinition,atanypoint
x∈R (or x∈Rd if B=C(Rd)). Thus, condition(31) is much stronger than the
+
10 A.V.Bobylev,C.CercignaniandI.M.Gamba
classicalLipschitzcondition
kG (u )−G (u )k<Cku −u k if u ∈U (32)
1 2 1 2 1,2
which obviously follows from (31) with the constantC=kLk , the norm of the
B
operator L in the space of boundedoperatorsacting in B. In other words, the ter-
minology “L-Lipschitz condition” means the point-wise Lipschitz condition with
respecttoanspecificlinearoperatorL.
Weassume,withoutlossofgenerality,thatthekernelsA (a ,...,a )inEqs.(24)
n 1 n
are symmetric with respect to any permutation of the arguments(a ,...,a ), n=
1 n
2,3,...,N.
ThenextlemmastatesthattheoperatorG (u)definedinEqs.(24), whichsatis-
fiesG (1)=1 (mass conservation)and mapsU into itself, satisfies an L-Lipschitz
condition,wherethelinearoperatorListheonegivenbythelinearizationofG near
theunity.See[10]foritsproof.
Theorem1.TheoperatorG (u)definedinEqs.(24)mapsU intoitselfandsatisfies
theL-Lipschitzcondition(31),wherethelinearoperatorLisgivenby
¥
Lu= daK(a)u(ax), (33)
Z0
with
N
K(a)= (cid:229) na K (a),
n n
n=1
(34)
¥ ¥ N
where K (a)= da ... da A (a,a ,...,a ) and (cid:229) a =1.
n 2 n n 2 n n
Z0 Z0 n=1
forsymmetrickernelsA (a,a ,...,a ),n=2,...N .
n 2 n
Andthefollowingcorollaryholds.
Corollary1.The Lipschitz condition (32) is fulfilled for G (u) given in Eqs. (24)
withtheconstant
N ¥
C=kLk= (cid:229) na , (cid:229) a =1, (35)
n n
n=1 n=1
wherekLkisthenormofLinB.
ItcanalsobeshownthattheL-LipschitzconditionholdsinB=C(Rd)for“gain-
operators”inFouriertransformedBoltzmannequations (8),(9)and(10).