CONVERGENCE OF THE VLASOV POISSON

Yann Brenier

Description

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

Informations

• Publié par: pefav
• Nombre de lectures: 6
• Langue: English

Extrait

CONVER
limit.
GENCE
is
OF
d'analyse
THE
v-P
VLASO
asymptotic
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Univ
SYSTEM
e
TO
ergence
THE
the
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in
EULER
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EQUA
Comm.
TIONS
rance,
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erique,
ann
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esum
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ersit
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syst
The

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eme
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oisson
v-P
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oisson

v
equations
ers
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les
estigated

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equations
o
d'Euler
regimes:
des
quasi-neutral
uides
and


dans
A
deux
dans
r
PDEs

Institut
egimes
ersitaire
asymptotiques
F
:
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limite
n
quasi-neutre

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la

limite
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gyro
6,

rance,

1
etique.e
FR
r
OM
℄
VLASO
,
V-POISSON
Vlaso
TO
energy
EULER
R
W
oisson
e
asymptotic

t
the
gy-

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t
d
of
;
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the
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y
een
the
for
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example)
of



harge

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the
neutralizing
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kground
erna-Ma
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to
mo

ving
is
ions.
y
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eld
equations
b
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t-Ra
giv
w
en
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b
and
y
ormal
the
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v-P
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oisson
x
system,
;
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)
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with
t
f

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and

b
2
(1)

o
)
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and
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(constan
in
t)
still

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p
as
erio
of
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oisson
of
when
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forced
electrons.
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In
ternal
the
has
so-called
v
quasi-neutral
Grenier
regime,
and
namely
℄
as
quasi-neutral

justify
!
limit
0,
the
the
e

dulated
t
1
is
1.1
exp
oisson
ected
normalizations,
to
oisson

[BR]
v

erge

to
f
a
:
solution
=
of
Z
the
(

1
Eu-
where
ler
)
equations,
d
at
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least
v
in
=
the
or

t;
of

a
densit
v
t;
anishing
R
initial
oten
temp
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erature.

This
w
result
v
is
the
pro
(2).
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ed
f
b

y
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adapting
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an
2
argumen
dissipativ
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solutions
used
Dip
b
jda's
y
solutions
P
the
.-L.
equations.
Lions

[Li
teresting
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regime,
to
leading
pro
the
v
equations,
e
wn
the
the

limit
v
the
ergence
v-P
of
system,
the
obtained
Lera
the
y
are
solutions
b
of
a
the

3d
ex-
Na
magnetic
vier-Stok
and
es
b
equation
in
to
estigated
the
y
so-called
[Gr3
dissipativ
Golse
e
Sain
solutions
ymond
of
As
the
the
Euler
limit,
equations.
e
F
the
or

this
b
purp
using
ose,

the
dissipativ
total
solutions
energy
mo
of
total
the
.
system
F
is
analysis
mo
The
dulated
v-P
b
system
y
suitable
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the

v-P
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system
alterna-
(see
tiv
for
e
:
pro
t
of
+
is
:
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x
en,
r
based

on
r
the
f

0
of
(1)
measure-v
R
alued
f
(
d
mv
=
)

solutions
(2)
in
(
tro


2
b
2
y
is
DiP
p
erna
elo
and
y
Ma
ariable,
jda
d
[DM
1
℄
2
and
3,
already
(
used
x;
b
)
y
0
Brenier
electronic
and
y
Grenier
(
[BG
x
℄
2
[Gr2
the
℄
p
for
tial
the

asymptotic
0
analysis

of
t
the
et
Vlaso
een
v-P
Vlaso
oisson
equation
sys-
and
tem
P
in
equation
the
T
quasi-neutral

regime.
system,
Through

this
(0
analysis,
x;
a
)
link
f
is
(
established

b

et
(3)
w
eenthe
FR
y
OM
oisson
VLASO
d
V-POISSON
:
TO
existence
EULER
Z
and

Z
x
d
is
p
tributions
erio
smo

erformed
y
x
in
r
x
)
are
equation,

d
ed.

Up
.
to
wn,
a
and

v
hange

of

sign,
;
w
d
e

harge
jr
and
the

using
t
+
the

t
=
w

o

rst
electric
momen
of
ts
w

the
(
Rein
t;
Pfaelmoser
x
smo
)
pro
=
f
Z

f
Then,
(
ha
t;
3
x;

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

By
f
ergence
(
equations
d
P
)
@
:
r
(4)
Z
Electrons
f
are
=

r

r
d
)
electrons
(
when
2
the
for
temp
oten
erature,
mathematical
prop
Vlaso
ortional
is
to
ell
Z
particular
j
t

Batt
J
℄

erthame
j

2
uniqueness
f
solutions
(
b
t;
ed
x;
initial
d
(
)
Æ
;
at
(5)

v
the
anishes.
w
The
e

fully
ation
)
of
(8)
total
t
energy

reads
(
Z
)
1
r
2
:
j



j
(
2
)
f
r
(
(9)
t;

dx;
:
d
r
)

+
)
Z
2

(
2

jr
2
(
:
t;

x
div
)
of
j
last
2
and
dx
the
(6)
oisson
(where
(
in
tt
tegrals
1)
in
2
x
:
are

p

erformed
(
on
)
the
(10)
unit


2
e
(
[0

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

obtained
ation
the
la
p
ws
tial
for
The

analysis
harge
the
and
v-P

system
t
no
are
w
:
kno

in
t
after
Z

f

(
of
d
and
)
[BR
+
Lions
r
P
x
[LP],
:
[Pf],
Z
Global

and
f
of
(
oth
d
ha
)
e
=
een
0
v
(7)
for
(or,
oth
equiv
data
alen
0
tly
x;
b
),
ecause
tly
of
ying
(2),
innit
r
in
x
.
:
all
Z
formal

tations
f
e
(
v
d
p
)
are
=
justied.
@
tbut
FR

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]).