CONVERGENCE OF THE VLASOV POISSON
19
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe et accède à tout notre catalogue !
Découvre YouScribe et accède à tout notre catalogue !
19
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
- al analysis
- ve tor
- vlasov- poisson system
- tor divergen
- ty old
- extra tion
- unique smooth
- ele trons
Publié par
Langue
English
CONVER
limit.
GENCE
is
OF
d'analyse
THE
v-P
VLASO
asymptotic
V-POISSON
Univ
SYSTEM
e
TO
ergence
THE
the
INCOMPRESSIBLE
in
EULER
limit
EQUA
Comm.
TIONS
rance,
Y
erique,
ann
F
Br
enier
the
system
R
Euler
v
esum
w
the
e
the
On
paraitre
etudie
de
la
Lab
um
v
ersit
ergence
aris
du
brenier@ann.jussieu.fr
syst
The
v
eme
of
de
Vlaso
Vlaso
oisson
v-P
to
oisson
v
equations
ers
in
les
estigated
t
equations
o
d'Euler
regimes:
des
quasi-neutral
uides
and
dans
A
deux
dans
r
PDEs
Institut
egimes
ersitaire
asymptotiques
F
:
et
la
oratoire
limite
n
quasi-neutre
et
Univ
la
limite
P
gyro
6,
rance,
1
etique.e
FR
r
OM
℄
VLASO
,
V-POISSON
Vlaso
TO
energy
EULER
R
W
oisson
e
asymptotic
t
the
gy-
Vlaso
t
d
of
;
an
the
electronic
=
of
generated
kno
b
electrons
y
een
the
for
lo
of
Vlaso
dierence
example)
of
harge
with
a
electric
uniform
the
neutralizing
initial
0
kground
erna-Ma
of
A
non-
to
mo
ving
is
ions.
y
The
eld
equations
b
are
t-Ra
giv
w
en
y
b
and
y
ormal
the
After
Vlaso
reads
v-P
f
oisson
x
system,
;
with
)
a
x;
the
with
t
f
the
=
)
(
and
b
2
(1)
o
)
;
2
x;
where
Lions'
and
is
mv
the
Euler
(constan
in
t)
still
Euler
p
as
erio
of
d
oisson
of
when
the
forced
electrons.
strong
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
osition/v
least
v
in
=
the
or
t;
of
a
densit
v
t;
anishing
R
initial
oten
temp
>
erature.
This
w
result
v
is
the
pro
(2).
v
this
ed
f
b
y
0
adapting
)
an
2
argumen
dissipativ
t
solutions
used
Dip
b
jda's
y
solutions
P
the
.-L.
equations.
Lions
[Li
teresting
℄
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
a
the
v-P
An
system
alterna-
(see
tiv
for
e
:
pro
t
of
+
is
:
giv
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]).
