La lecture à portée de main
Découvre YouScribe en t'inscrivant gratuitement
Je m'inscrisDécouvre YouScribe en t'inscrivant gratuitement
Je m'inscrisDescription
Sujets
Informations
Publié par | profil-zyak-2012 |
Nombre de lectures | 22 |
Langue | English |
Extrait
(
A
wi
R
ll
M
be
A
in
m
se
a
rt
n
e
u
d
sc
b
r
y
ip
th
t
e
N
ed
o
i
.
tor)
N-particlesapproximationoftheVlasov
equationswithsingularpotential
MaximeHauray
CEREMADE
PlaceduMarechaldeLattredeTassigny
75775ParisCedex16
Pierre-EmmanuelJabin
DMA-ENSParis
45,Rued’Ulm
75005Paris
Abstract.
Weprovetheconvergenceinanytimeintervalofapoint-particle
approximationoftheVlasovequationbyparticlesinitiallyequallyseparated
foraforcein1
/
|
x
|
,with
1.Weintroducediscreteversionsofthe
L
∞
normandtimeaveragesoftheforceeld.Thecoreoftheproofisto
showthatthesequantitiesareboundedandthatconsequentlytheminimal
distancebetweenparticlesinthephasespaceisboundedfrombelow.
Keywords.
Derivationofkineticequations.Particlemethods.Vlasovequa-
tions.
1.Introduction
Weareinterestedherebythevalidityofthemodelingofacontinuousmedia
byakineticequation,withadensityofpresenceinspaceandvelocity.In
otherwords,dothetrajectoriesofmanyinteractingparticlesfollowthe
evolutiongivenbythecontinuousmediaiftheirnumberissucientlylarge?
2
M.HaurayandP.-E.Jabin
Thisisaverygeneralquestionandthispaperclaimstogivea(partial)
answeronlyforthemeaneldapproach.
Letusbemoreprecise.Westudytheevolutionof
N
particles,centered
at(
X
1
,...,X
N
)in
R
d
withvelocities(
V
1
,...,V
N
)andinteractingwitha
centralforce
F
(
x
).Thepositionsandvelocitiessatisfythefollowingsystem
ofODEs
X
˙
i
=
V
i
,
Xj
6
=
i
m
i
V
˙
i
=
E
(
X
i
)=
ij
F
(
X
i
X
j
)
,
(1.1)
wheretheinitialconditions(
X
10
,V
10
,...,X
n
0
,V
n
0
)aregiven.Theprimeex-
amplefor(1.1)consistsinchargedparticleswithcharges
i
andmasses
m
i
,
inwhichcase
F
(
x
)=
x/
|
x
|
3
indimension
d
=3.
Toeasilyderivefrom(1.1)akineticequation(atleastformally),itisvery
convenienttoassumethattheparticlesareidentical,whichmeans
i
=
j
.Moreoverwewillrescalesystem(1.1)intimeandspacetoworkwith
quantitiesoforderone,whichmeansthatwemayassumethat
1ji=
,
∀
i,j.
(1.2)
NmiWenowwritetheVlasovequationmodellingtheevolutionofadensity
f
of
particlesinteractingwitharadialforcein
F
(
x
).Thisisakineticequation
inthesensethatthedensitydependsonthepositionandonthevelocity
(andofcourseonthetime)
)3.1(
∂
t
f
+
v
r
x
f
+
E
(
x
)
r
v
f
=0
,t
∈
R
+
,x
∈
R
d
,v
∈
R
d
,
ZE
(
x
)=
d
(
t,y
)
F
(
x
y
)
dy,
R
(
t,x
)=
f
(
t,x,v
)
dv.
ZvHere
isthespatialdensityandtheinitialdensity
f
0
isgiven.
Whenthenumber
N
ofparticlesislarge,itisobviouslyeasiertostudy
(orsolvenumerically)(1.3)than(1.1).Thereforeitisacrucialpointto
determinewhether(1.3)canbeseenasalimitof(1.1).
ParticlesapproximationofsingularVlasovequations3
Remarkthatif(
X
1
,...,X
N
,V
1
,...,V
N
)isasolutionof(1.1),thenthe
measure
n1N
(
t
)=
(
x
X
i
(
t
))
(
v
V
i
(
t
))
XN1=iisasolutionoftheVlasovequationinthesenseofdistributions.Andthe
questioniswhetheraweaklimit
f
of
N
solves(1.3)ornot.If
F
is
C
1
with
compactsupport,thenitisindeedthecase(itisprovedinthebookby
Spohn[23]forexample).Thepurposeofthispaperistojustifythislimitif
|
F
(
x
)
||
xC
|
,
|r
F
(
x
)
||
x
|
1
C
+
|r
2
F
(
x
)
||
x
|
2
C
+
,
∀
x
6
=0
,
(1.4)
for
<
1,whichistherstrigorousproofofthelimitinacasewhere
F
is
notnecessarilybounded.
Beforebeingmorepreciseconcerningourresult,letusexplainwhatisthe
meaningof(1.1)inviewofthesingularityin
F
.Hereweassumeeither
thatwerestrictourselvestotheinitialcongurationsforwhichthereare
nocollisionsbetweenparticlesoveratimeinterval[0
,T
]withaxed
T
,
independentof
N
,orweassumethat
F
isregularorregularizedbutthat
thenorm
k
F
k
W
1
,
∞
maydependon
N
.Thisprocedureiswellpresentedin
[1]anditistheusualoneinnumericalsimulations(see[24]and[25]).In
bothcases,wehaveclassicalsolutionsto(1.1)buttheonlyboundwemay
useis(1.4).
Otherpossibleapproacheswouldconsistinjustifyingthatthesetofinitial
congurations
X
1
(0)
,...,X
N
(0)
,V
1
(0)
,...,V
N
(0)forwhichthereisatleast
onecollision,isnegligibleorthatitispossibletodeneasolution(unique
ornot)tothedynamicsevenwithcollisions.
Finallynoticethatthecondition
<
1isnotunphysical.Indeedif
F
derivesfromapotential,
=1isthecriticalexponentforwhichrepulsive
andattractiveforcesseemverydierent.Inotherwords,thisisthepoint
wherethebehavioroftheforcewhentwoparticlesareveryclosetakesall
itsimportance.