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CONVERGENCE OF THE VLASOV POISSON

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CONVERGENCE OF THE VLASOV-POISSON SYSTEM TO THE INCOMPRESSIBLE EULER EQUATIONS Yann Brenier Resume On etudie la onvergen e du systeme de Vlasov-Poisson vers les equations d'Euler des uides in ompressibles dans deux regimes asymptotiques : la limite quasi-neutre et la limite gyro inetique. Abstra t The onvergen e of the Vlasov-Poisson system to the in ompressible Euler equations is investigated in two asymptoti regimes: the quasi-neutral limit and the gyrokineti limit. A paraitre dans Comm. PDEs Institut Universitaire de Fran e, et Laboratoire d'analyse numerique, Universite Paris 6, Fran e, brenierann.jussieu.fr 1

  • al analysis

  • ve tor

  • vlasov- poisson system

  • tor divergen

  • ty old

  • extra tion

  • unique smooth

  • ele trons


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CONVER
limit.
GENCE
is
OF
d'analyse
THE
v-P
VLASO
asymptotic
V-POISSON
Univ
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e
TO
ergence
THE
the
INCOMPRESSIBLE
in
EULER
limit
EQUA
Comm.
TIONS
rance,
Y
erique,
ann
F
Br

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the

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R
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esum
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e
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ersit
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ers
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ha
t;
3
x;

d
f
)
+
;
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J
f
(
+
t;
=
x
(
)
r
=
r
Z
j

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f
ergence
(
equations
d
P
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@
:
r
(4)
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(9)
t;

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2

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:
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x
div
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of
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last
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the
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(where
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in
tt
tegrals
1)
in
2
x
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are

p

erformed
(
on
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the
(10)
unit


2
e
(
[0

;
r
1]
+
d
2
),
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and
j
the
)

obtained
ation
the
la
p
ws
tial
for
The

analysis
harge
the
and
v-P

system
t
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are
w
:
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in
t
after
Z

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and
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+
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[LP],
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of
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oth
d
ha
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e
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(7)
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equiv
data
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tly
x;
b
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(2),
innit
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in
x
.
:
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(
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d
p
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justied.
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tbut
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OM
F
VLASO
particular
V-POISSON
of
TO
t;
EULER
the
1.2
(
The
pressure
quasi-neutral
result
regime
probablit
The
whic
asymptotic
t;
analysis
is

0
!
=
0
uid
is
to
Æ
whic
and
d
only
y
partial

results
is
ha
f
v
(
e
(15)
b
the
een
to
obtained,
obtain
in
J
particular
)
b
whic
y
equations
Grenier

in
h

[MP

precisely

get
(see
t
also
:
[Br]).
p
The
electrons,

measure
b
f
eha

viour
delta
of
exactly
the
t;
linear
=
part
J
of
))
equation
J
(10)
t
is
harge
one

of
In
the
w
main
:J
Æ

Let
r
us

start
r
b
;
y
is
a

purely
an
formal
v
analysis
J
of
for
the
e
limit
[Ch],

The
!
electrons
0.
one
The
w
P
rigorous
oisson
the
equation
er.
(2)
)
b
(14)
ecomes
or
Z

f

(
the
d
y
)
(in
=
)
1
(
(11)
x;
and
)
w
a
e
function,
get
h
from
means
equations
(
(8),
x;
(9)
)
r
Æ
x

:
(
Z
x

;
f
since
(
is
d

)
and
=

0

(12)
iden

equal
t
1.
Z
this


f
e
(
r
d
=
)
(16)
+
t
r
+
x
:
:
J
Z
J

+



0
f
(17)
(
h
d
nothing
)
the
+
Euler
r
for


=
(with
0
elo
:
y
(13)
and
F
),
or
whic
the
w
p
refer
oten
[AK],
tial,
[Li],
w

e

nd


is
=
the
r
for
2
h
x
e
:
a
Z
asymptotic

in

presen

pap
f
4
(d
FR
;
OM
any
VLASO
d
V-POISSON
J
TO

EULER
2
2
J
The
)

(or
v
e
ergence
0
result
;
Theorem
2
2.1

L
D
et
x
T
sense
>
0
0
the
and
the
J
ondition.
0
a
(
div
x
d
)
x
b
x
e
d
a
t
given
v
diver
(
genc
)
e-fr
(
e
Z
e,
e
Z
with
d
p
p
and
erio
and

(without
in

x
solution
,
0
squar
ollo
e
y
inte
with
gr
smo
able