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Niveau: Supérieur, Doctorat, Bac+8
ARMA manuscript No. (will be inserted by the editor) N-particles approximation of the Vlasov equations with singular potential Maxime Hauray CEREMADE Place du Marechal de Lattre de Tassigny 75775 Paris Cedex 16 Pierre-Emmanuel Jabin DMA - ENS Paris 45, Rue d'Ulm 75005 Paris Abstract. We prove the convergence in any time interval of a point-particle approximation of the Vlasov equation by particles initially equally separated for a force in 1/|x|?, with ? ≤ 1. We introduce discrete versions of the L∞ norm and time averages of the force field. The core of the proof is to show that these quantities are bounded and that consequently the minimal distance between particles in the phase space is bounded from below. Key words. Derivation of kinetic equations. Particle methods. Vlasov equa- tions. 1. Introduction We are interested here by the validity of the modeling of a continuous media by a kinetic equation, with a density of presence in space and velocity. In other words, do the trajectories of many interacting particles follow the evolution given by the continuous media if their number is sufficiently large?

  • many interacting

  • particles approximation

  • distance between

  • over

  • vlasov equations

  • ?i ?j

  • discrete scale

  • l∞ norm

  • any time interval


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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.