SYSTEM TO THE INCOMPRESSIBLE EULER
EQUATIONS
Yann Brenier
Resume
Onetudie la onvergene du systeme de Vlasov-Poisson vers les equations
d'Euler des uides inompressibles dans deux regimes asymptotiques : la
limitequasi-neutreetla limitegyroinetique.
Abstrat
The onvergene of the Vlasov-Poisson system to theinompressible Euler
equationsisinvestigatedintwoasymptotiregimes: thequasi-neutrallimit
andthe gyrokinetilimit.
A paraitre dans Comm. PDEs
InstitutUniversitairedeFrane,etLaboratoired'analysenumerique,UniversiteParis
6,Frane,brenierann.jussieu.fr
We onsider the displaement of an eletroni loud generated by the
loal dierene of harge with a uniform neutralizing bakground of non-
moving ions. The equations are given by the Vlasov-Poisson system, with
a oupling onstant = (
2 )
2
where is the (onstant) osillation period
of the eletrons. In the so-alled quasi-neutral regime, namely as ! 0,
theurrent is expeted to onverge to a solutionof theinompressible Eu-
ler equations, at least in the ase of a vanishinginitial temperature. This
result isproved byadaptingan argument usedbyP.-L. Lions[Li℄ to prove
theonvergene of the Leray solutions of the3d Navier-Stokes equation to
theso-alleddissipativesolutionsoftheEulerequations. Forthispurpose,
thetotal energyofthesystem ismodulatedbyatest-funtion. Analterna-
tive proof is given, based on the onept of measure-valued (mv) solutions
introdued by DiPerna and Majda [DM ℄ and already used by Brenier and
Grenier [BG ℄, [Gr2 ℄ for the asymptoti analysis of the Vlasov-Poisson sys-
teminthequasi-neutralregime. Throughthisanalysis,alinkisestablished
between Lions' dissipative solutions and Diperna-Majda's mv solutions of
the Euler equations. A seond interesting asymptoti regime, still leading
totheEulerequations,knownasthegyrokinetilimitoftheVlasov-Poisson
system, is obtained when the eletrons arefored by a strong onstant ex-
ternalmagnetieldand hasbeeninvestigatedbyGrenier[Gr3 ℄,Golseand
Saint-Raymond [GSR℄. As for the quasi-neutral limit, we justify the gy-
rokinetilimitbyusingtheoneptsof dissipativesolutionsand modulated
total energy.
1 Formal analysis
1.1 The Vlasov-Poisson system
Aftersuitablenormalizations,theVlasov-Poissonsystemreads(see[BR℄for
example):
t
f+:r
x
f r
x :r
f =0; (1)
Z
R d
f(d)=1 (2)
where (x;) 2 R 2d
is the position/veloity variable, with d = 1;2 or 3,
f(t;x;) 0 the eletroni density, (t;x) 2 R the eletri potential and
>0theouplingonstantbetweentheVlasovequation(1)andthePoisson
equation (2). Toompletethissystem, initialonditions
f(0;x;)=f
0
(x;)0 (3)
and Z d
periodiityin x are presribed. Up to a hange of sign, we all
harge and urrent thetwo rst moments
(t;x)= Z
f(t;x;d); J(t;x)= Z
f(d): (4)
Eletronsarealled old eletronswhen thetemperature, proportionalto
Z
j J
j
2
f(t;x;d); (5)
vanishes.
The onservation of total energyreads
Z
1
2 jj
2
f(t;dx;d)+ Z
2
jr(t;x)j 2
dx (6)
(where integrals in x are performed on the unit ube [0;1℄
d
), and the
onservation laws forhargeand urrent are:
t Z
f(d)+r
x :
Z
f(d)=0 (7)
(or, equivalentlybeause of(2),
r
x :
Z
f(d)=
t
); (8)
t Z
f(d)+r
x :
Z
f(d)+r (9)
=r:(rr)
2
r(jrj 2
):
ByomputingthedivergeneofthelastequationsandusingthePoisson
equation,
(
tt
+1) r
2
x :
Z
f(d)= (10)
= r
2
:(rr)+
2
(jrj 2
)
isobtainedfor theeletripotential.
The mathematial analysis of the Vlasov-Poisson system is now well
known, in partiular after the reent ontributions of Batt and Rein [BR ℄,
Lions and Perthame [LP℄, Pfaelmoser [Pf℄, et... Global existene and
uniqueness of smooth solutions have been proved for smooth initial data
f
0
(x;), suÆientlydeayingat innityin . Then, all the formal ompu-
tationswehave performedare fullyjustied.
1.2 The quasi-neutral regime
Theasymptotianalysis!0isdiÆultand onlypartialresultshave been
obtained,inpartiularbyGrenierin[Gr1℄, [Gr2℄,[Gr3℄(see also[Br℄). The
osillatory behaviour of thelinear part of equation (10) is one of themain
diÆulties.
Letusstartbyapurely formalanalysisof thelimit!0. ThePoisson
equation (2)beomes
Z
f(d)=1 (11)
and we getfrom equations(8), (9)
r
x :
Z
f(d)=0 (12)
t Z
f(d)+r
x :
Z
f(d)+r=0: (13)
For thepotential,wend
=r
2
x :
Z
f(d): (14)
For perfetyold eletrons, the probablity measure (in ) f(t;x;) is a
delta funtion,whihexatlymeans
f(t;x;)=Æ( J(t;x)); (15)
sine J is the urrent and the harge is identially equal to 1. In this
partiularase, we obtain
r:J =0 (16)
t
J+r:(JJ)+r=0; (17)
whih is nothing but the lassial Euler equations for an inompressible
uid(withveloityJ andpressure),forwhihwereferto[AK℄,[Ch℄,[Li℄,
[MP ℄...
Theaseofoldeletronsispreiselytheoneforwhihwegetarigorous
asymptotiresultinthe present paper.
2 The onvergene result
Theorem 2.1 Let T >0 and J
0
(x) bea given divergene-free, Z d
periodi
in x, square integrable vetor eld. Assume the initial data f
0
(x;) 0 to
be smooth, Z d
periodi in x, niely deaying as ! 1, with total mass 1.
In addition, we assume
Z
f
0
(x;)d=1+o(
1=2
); !0; (18)
in the strong senseof the spae H 1
(R d
=Z d
) and
Z
j v
0 (x)j
2
f
0
(x;)dxd! Z
jJ
0 v
0 (x)j
2
dx; (19)
for all squareintegrable,divergene-free,Z d
periodi, vetor eld v
0 .
Then,uptotheextrationofa sequene
n
!0, thedivergene-freeom-
ponent of the urrent J
onverges in C 0
([0;T℄;D 0
(R d
=Z d
)) to a dissipative
solution J 2C 0
([0;T℄;L 2
(R d
=Z d
) w) of the Euler equations, in the sense
of Lions [Li℄, with initial ondition J
0
. In partiular, if J
0
is smooth and
d =2 (or d =3 and T small), the entirefamily (without extration of any
subsequene)onvergestothe uniquesmooth solutionof the Eulerequations
withJ
0
as initial ondition.
Remark 1
Following Lions [Li℄, we say that J is a dissipative solution with initial
onditionJ
0
if,forallsmoothvetordivergene-freevetoreldsvon[0;T℄
R d
=Z d
,almost every t2[0;T℄,
Z
jJ(t;x) v(t;x)j 2
dx Z
jJ
0
(x) v(0;x)j 2
dxexp(
Z
t
0
2jjd(v())jj)d (20)
+2 Z
t
0 exp(
Z
t
s
2jjd(v())jjd)(
Z
A(v)(s;x):(v J)(s;x))dsdx;
whered(v) is thesymmetri partof Dv=((Dv)
ij )=(
j v
i )
d
ij (v)=
1
2 (
xi v
j +
xj v
i
); (21)
jjd(v(t))jjisthesupremuminxofthespetralradiusofd(v)(t;x), andA(v)
istheaeleration operator
A(v)=
t
v+(v:r)v: (22)
Notie a slight hange of denitionwith respet to [Li℄, sine here we use
thespetralradius oftheentirematrix d(v),not onlyits negative part.
Remark 2
Thequasi-neutralityassumption (18)exatly means,beause of(2),
Z
jr
(0;x)j 2
dx!0: (23)
Assumption(19) means that the eletrons are old and the initial urrent
onverges to J
0
. Indeed, we have(take v
0
=J
0 and v
0
=0)
Z
j J
0 (x)j
2
f
0
(x;)dxd !0; (24)
Z
jj 2
f
0
(x;)dxd! Z
jJ
0 (x)j
2
dx (25)
3 Proofs
The proof is a simple adaptation of the way that Lions follows in [Li℄ to
show the onvergene of Leray solutions of the Navier-Stokes equations to
the so-alled dissipative solutions of the Euler equations. To do that, the
total energyof thesystemis modulatedbyatest-funtion.
3.1 Control of the modulated total energy
LetusomputethetimederivativeofthetotalenergyoftheVlasov-Poisson
system,modulatedbyatestfuntion(t;x)!v(t;x);Z d
periodi,divergene-
freeinx,
H
v (t)=
Z
1
2
j v(t;x)j 2
f
(t;x;)dxd+ Z
2 jr
(t;x)j 2
dx: (26)
Letustemporarily dropthe index. Beause of thetotal energy onserva-
tion,we have,fortheharge and theurrent J,
d
dt H
v (t)=
d
dt Z
1
2
jv(t;x)j 2
(t;x)dx Z
t
(J(t;x):v(t;x))dx: (27)
Elementaryalulationsleadto
d
dt H
v (t)=
Z
d(v)(t;x):( v(t;x))( v(t;x))f(t;x;)dxd (28)
+ Z
d(v)(t;x):r(t;x)r(t;x)dx
+ Z
A(v)(t;x):((t;x)v(t;x) J(t;x))dx
whered(v)is thesymmetrizedgradient ofvdenedby(21)andA(v) isthe
aeleration operator(22). Thus, we get, afterrisingindex,
d
dt H
v
(t)2jjd(v(t))jjH
v (t)+
Z
A(v)(
v J
)dx; (29)
where H
v
is dened by (26) and jjd(v(t))jj is the supremum in x of the
spetralradiusof d(v)(t;x). We dedue,after integrating (29) int,
H
v
(t)H
v
(0)exp ( Z
t
0
2jjd(v())jj)d (30)
+ Z
t
0 exp(
Z
t
s
2jjd(v())jjd)(
Z
A(v)(s;x):(
v J
)(s;x))dsdx:
Inpartiular,inthe asev=0, we reover thetotal energybound
H
0 (t)=
Z
1
2 jj
2
f
(t;x;)dxd+ Z
2 jr
(t;x)j 2
dxH
0
(0): (31)
Remark
Here we use the spetral radius of the entirematrix d(v) and notonly its
negative part (as in Lions' denition for dissipative solutions of the Euler
equations). Indeed, in theright-hand side of (28), the rst and theseond
termsinvolved(v) withoppositesigns!
3.2 A priori bounds
The assumptions on the initial onditions and equations (18), (7), imply
that
Z
f
(t;x;)dxd= Z
0
(x)dx=1; (32)
Z
jj 2
f
0
(x;)dxd+ Z
jr
(0;x)j 2
dx! Z
jJ
0 (x)j
2
dx: (33)
From (31), wededue that
Z
jj 2
f
(t;x;)dxd+ Z
jr
(t;x)j 2
dxC : (34)
ThusJ
is boundedinL 1
([0;T℄;L 1
(R d
=Z d
))sine
( Z
jJ
(t;x)jdx) 2
Z
jj 2
f
(t;x;)ddx Z
f
(t;x;)ddxC :
Up to the extration of a sequene (
n
), we an assume that J
has a
vague limit J, in the sens of (Radon) measures on [0;T℄R d
=Z d
. Simi-
larly, from (32), (7) and (34), we get that
(t;x) 0 onverges to 1 in
C 0
([0;T℄;D 0
(R d
=Z d
))and therefore in thevague senseof measures. Let us
now onsidertheonvexfuntionalof (Radon) measures
K(;m)=sup
b Z
1
2
jb(t;x)j 2
(dtdx)+b(t;x):m(dtdx);
where b spans the spae of all ontinuous funtions from [0;T℄R d
=Z d
to
R d
and,mrespetivelydenotenonnegativeandvetor-valuedmeasureson
[0;T℄R d
=Z d
. When(t;x)=1(the Lebesguemeasure), wesimplyobtain
2K(;m)= Z
jm(t;x)j 2
dtdx;
ifmis asquare integrable funtionand +1otherwise. FuntionalK isls
withrespetto thevagueonvergeneof measures. Sine, foreahnonneg-
ative funtionz2C 0
([0;T℄),
2K(z
;zJ
)= Z
jJ
(t;x)j 2
(t;x)
z(t)dtdx
Z
jj 2
f
(t;x;)z(t)dtdxd C Z
T
0
z(t)dt;
we deduethat
2K(z;zJ)C Z
T
0
z(t)dt;
whih exatly means that J belongs to L 1
([0;T℄;L 2
(R d
=Z d
)). From (8),
we get thatJ is divergene-freein x and, from (9), that
t
J is boundedin
L 1
([0;T℄;D 0
(R d
=Z d
)), sine J is divergene-free(whih allows usto ignore
r
in (9), although this term ould be of size O(
1=2
)). It follows that
thevaguelimit J(t;x)of J
(t;x) isa divergene-free vetor eld belonging
to C 0
([0;T℄;L 2
(R d
=Z d
) w). Forthesame reasons, thedivergene-free (or
solenoidal) part of J
onverges toward J, not only in the vague sense of
measures, butalso inC 0
([0;T℄;D 0
(R d
=Z d
)).
3.3 Convergene
We an rewrite(29) inweak form
Z
H
v (t)z
0
(t)dt z(0)H
v (0)
Z
2jjd(v(t))jjH
v
(t)z(t)dt (35)
+ Z
A(v)(
v J
)(t;x)z(t)dtdx;
forall test funtionz0 inD 0
([0;T[), where H
v
(t)isdened by (26). Let
usintrodue
h
v (t)=
Z
jJ
(t;x) v(t;x)
(t;x)j 2
2
(t;x)
dx (36)
=sup
b Z
[ 1
2 jb(x)j
2
(t;x)+b(x):(J
v
)(t;x)℄dx;
wherebspansthespaeofallontinuousfuntionsfromR d
=Z d
toR d
,whih
is, for eah xed t, a onvex fution of J
(t;:) and
(t;:). (It is a just a
modulated version of funtional K, with a test funtion v.) By Cauhy-
Shwarz inequality,we have
h
v (t)
Z
1
2
j v(t;x)j 2
f
(t;x;)dxd Z
H
v (t):
The a priori bound previously obtained show that, for xed v, H
v
(t) and
h
v
(t) are bounded funtions in L 1
([0;T℄) and, up to the extration of a
sequene (
n
), respetively onverge, in the weak-* sense, to some limits
H
v
(t)andh
v
(t),withH
v h
v
. Sine
!1andJ
!J inthevaguesense
ofmeasures, byonvexity ofthefuntionaldened by(36), weget
Z
jJ(t;x) v(t;x)j 2
dx2h
v
(t): (37)
Theassumptionson theinitialonditions mean
2H
v (0)=
Z
j v(0;x)j 2
f
0
(x;)dxd+ Z
jr
(0;x)j 2
dx!2H
0;v (38)
wherewe set
H
0;v
= 1
2 Z
jJ
0
(x) v(0;x)j 2
dx: (39)
Then,we an pass to thelimitin(35) to get
Z
H
v (t)z
0
(t)dt z(0)H
0;v
Z
2jjd(v(t))jjH
v
(t)z(t)dt (40)
+ Z
A(v)(v J)(t;x)z(t)dtdx:
Byintegrating int, we get
H
v
(t)H
0;v exp(
Z
t
0
2jjd(v())jj)d (41)
+ Z
t
0 exp(
Z
t
s
2jjd(v())jjd)(
Z
A(v)(s;x):(v J)(s;x))dsdx:
Thus
h
v
(t)H
0;v exp(
Z
t
0
2jjd(v())jj)d (42)
+ Z
t
0 exp(
Z
t
s
2jjd(v())jjd)(
Z
A(v)(s;x):(v J)(s;x))dsdx
and,therefore,(20) holdstrue,whih onludesthe proof.
4 An alternative proof
Letusskethanalternativeproof,whihanbeseenasanaturalextension
of the analysis made in [BG ℄ (stationary ase) and [Gr2 ℄ (general ase) to
studythedefetmeasuresoftheVlasov-Poissonsysteminthequasi-neutral
regime.
After adaptatingtheproof(whih requiresanaprioriL 1
boundforf
,
whihisnotaeptableintheframeworkofthepresentpaper),weanshow
1)theexisteneoff(t;x;),anonnegativemeasuref in(x;)2R d
=Z d
R d
,
measurableint,asthevaguelimitoff
,withenoughtightnessin toallow
thezero and rst order momentsin (namelythe harge and the urrent)
to pass to the limit; 2) the existene of
K
(t;x;) and
E
(t;x;), two de-
fet measures in (x;) 2 R d
=Z d
S d 1
, measurable in t, that orrespond
respetivelyto the defetof kinetiand potentialenergies; 3) the existene
of two defet eletri elds E
+
(t;x) and E (t;x) 2 L 1
([0;T℄;L 2
(R d
=Z d
)),
taking into aount the temporal osillationsof the eletri eld generated
by(10);4) theonvergeneofthesolenoidalpartofJ
towardJ = R
f(d)
in C 0
([0;T℄;D 0
(R d
=Z d
)). This is enough to enfore 1) the onservation in
timeof thetotal energywithdefets
2H(t)= Z
jj 2
f(t;dx;d)+ Z
(
K +
E
)(t;dx;d) (43)
+ Z
(jE
+ (t;x)j
2
+jE (t;x)j 2
)dx;
2) thefollowingproperties fortheurrent J(t;x)= R
f(t;x;d) :
r:J =0; (44)
t
J+r:Q=0; (45)
where
Q= Z
f(d)+ Z
(
K
E
)(d) (46)
E
+ E
+
E E :
(Notethehangeofsignbetween
K +
E and
K
E
whenweswithfrom
theenergyonservation to theurrent onservation.) From these relations,
we deduethattheweak-* L 1
limitofthemodulatedtotal energyH
v (t)is
givenby
2H
v (t)=
Z
j v(t;x)j 2
f(t;dx;d)+ Z
(
K +
E
)(t;dx;d) (47)
+ Z
(jE
+ (t;x)j
2
+jE (t;x)j 2
)dx:
Thus, we diretlyget
d
dt H
v (t)=
Z
d(v)(t;x):( v(t;x))( v(t;x))f(t;dx;d) (48)
Z
d(v)(t;x):(
K
E
)(t;dx;d)
+ Z
d(v)(t;x):(E
+ E
+
+E E )(t;x)dx
+ Z
A(v)(t;x):(v(t;x) J(t;x))dx
2jjd(v(t))jjH
v (t)+
Z
A(v)(t;x):(v(t;x) J(t;x))dx; (49)
and we onludeasintherst proof.
5 Comparison of dissipative and mv solutions to the Euler
equations
OuranalysismakesalinkbetweenLions'oneptofdissipativesolutions[Li℄
and Diperna-Majda's onept of measure-valued solutions (\mv solutions)
[DM ℄,bothintroduedtodesribethevanishingvisositylimitoftheNavier-
Stokesequations[Li℄. Ifwegetbakto[DM ℄,weobtain,asbefore,twolimits
f,J = R
f(d), and a kineti defet measure
K
(the only relevent defet
measurewhenapproahingtheEulerequationsfrom theNavier-Stokesside
andnot fromtheVlasov-Poisson side). We getforJ (44) and(45) with,
Q= Z
f(d)+ Z
K
(d): (50)
Inaddition,thetotal kinetienergy,inluding defets,namely :
= Z
jj 2
f(t;dx;d)+ Z
K
(t;dx;d) (51)
isdeayingintime. Thus, after thesame kindof manipulationswealready
used,we see that J is a dissipative solution of the Eulerequations. Thus,
themv solutions are notas dierent from thedissipative solutions asthey
look. Anyway, the onept of dissipative solutions larify the relationship
between mv solutions and lassial solutions, whih was not disussed in
[DM ℄.
6 The gyrokineti limit
There is a seond asymptoti regime of the Vlasov-Poisson system leading
to the Euler equations, the so-alled gyrokineti limit. We onsider, as in
[Gr3 ℄(seealsotheinludedreferenes)orin[GSR℄(withadierentsaling),
theeetof alargeexternalmagnetield. Ifthismagneti eldisparallel
to thethirdoordinate x
3
,wegetthefollowingtwo-dimensional(in bothx
and ) Vlasov-Poisson system
t f
+:r
x f
+ 1
( r
+
?
):r
f
=0; (52)
=1
; Z
(t;x)dx =1; (53)
wherex 2R 2
=Z 2
, 2R 2
, and
?
=(
2
;
1
) is the additionalterm dueto
theexternal magneti eld. We assume thetotal mass of
to be equalto
oneattime0toenfore globalneutrality. Thetotalenergyisstillonserved
and,here,denedby
Z
1
2 jj
2
f
(t;dx;d)+ Z
1
2 jr
(t;x)j 2
dx (54)
(notie thatthemagneti eld isnotinvolved). Inaddition, we get
t
+r:J
=0; (55)
t J
+r
x :
Z
f
(d) (56)
= 1
(
r
+
?
J
):
Byombining(56) and(55), we alsoget
t (
?
r:J
)+
?
r:(
r
) (57)
=
?
r:(r
x :
Z
f
(d)):
Formally, as goes to zero, we expet for the limits , J and , the
self-onsistent system:
t
+r:J =0; (58)
r+
?
J =0; =1 ; (59)
whih is nothing but the Eulerequations written inthe so-alled vortiity
formulation, with 1 standing for the vortiity and for the stream-
funtion. The limit ! 0 has been suessfully investigated in [Gr3 ℄ for
monokinetidataand smalltime, aswellasin[GSR℄fora dierentsaling
andglobalweak solutionsoftheEulerequation inDelort's sense(see [De ℄).
We an performthesame kindofanalysis asforthequasi-neutrallimit,
and show:
Theorem 6.1 Let T > 0 and J
0 (x) =
?
r
0
be a given divergene-free,
Z 2
periodi in x, square integrable vetor eld. Assume the initial data
f
0
(x;)0 tobe smooth, Z 2
periodi in x, nielydeaying as !1, with
totalmass 1. In addition, we assume
Z
jj 2
f
0
(x;)ddx!0; (60)
Z
jr
(0;:)
?
J
0 (x)j
2
dx!0: (61)
Then, up to the extration of a sequene
n
! 0,
?
r
onverges in
C 0
([0;T℄;L 2
(R 2
=Z 2
) w) to a dissipative solution J of the Euler equations
with initial ondition J
0
. In partiular, if J
0
is smooth, the entire family
onverges to the unique smooth solution of the Euler equations with J
0 as
initial ondition.
To provethisresult,we usethesame tehnique asforthequasi-neutral
limitbyintroduingamodulatedtotalenergy,denedinthefollowingway.
Given a smooth divergene-freevetor eld v(t;x)=
?
r (t;x), we set
H
v (t)=
Z
2
j v(t;x)j 2
f
(t;x;)dxd+ Z
1
2 jr(
)(t;x)j 2
dx: (62)
A straightforward but lengthy alulation (using (56) in a ruial way, see
thedetailsintheappendix),leadsto
d
dt H
v
(t)= Z
d(v)(t;x):( v(t;x))( v(t;x))f
(t;x;)dxd (63)
+ Z
d(v)(t;x):r(
)(t;x)r(
)(t;x)dx
+ Z
A(v)(t;x):(
(t;x)v(t;x) J
(t;x))dx
+ Z
A(v)(t;x):(v(t;x)+
?
r
(t;x))dx
whered(v), A(v) arestilldenedby(21), (22).
We also getthefollowingbounds: r
is boundedin
L 1
([0;T℄;L 2
(R 2
=Z 2
));
and 1=2
J
areboundedin
L 1
([0;T℄;L 1
(R 2
=Z 2
))
(beauseoftheonservationofhargeandenergy). Next,
r
isbounded
in
L 1
([0;T℄;D 0
(R 2
=Z 2
)):
Indeed,forall smoothvetor eld g(x), beause of(2),
Z
g(x):
(t;x)r
(t;x)dx= Z
g:r
+ Z
( 1
2 jr
j 2
r:g+(r
:r)g:r
)dxCjjgjj
C 1
(R 2
=Z 2
) :
Then,beauseof (57),
?
r:J
isompat in
C 0
([0;T℄;D 0
(R 2
=Z 2
)):
Sine
?
r:J
= 0(
1=2
) in L 1
([0;T℄;D 0
(R 2
=Z 2
)), we dedue that
, and
thereforer
,arealsoompatinC 0
([0;T℄;D 0
(R 2
=Z 2
)). Thus,weonlude
that, up to the extration of a sequene
n
! 0, H
v
and r
onverge to
some limitsH
v
and r,respetivelyinL 1
([0;T℄)weak-* and
C 0
([0;T℄;L 2
(R 2
=Z 2
) w):
Then,we an pass to thelimitin(62) and(63) to get
Z
jr( )(t;x)j 2
dxH
v
(t); (64)
Z
H
v (t)z
0
(t)dt z(0)H
0;v
Z
2jjd(v(t))jjH
v
(t)z(t)dt (65)
+ Z
A(v):(v+
?
r)(t;x)z(t)dtdx;
forall smoothnonnegative z(t) ompatlysupportedin0t<T,where
H
0;v
= Z
jr(
0
(0;:))(x)j 2
dx (66)
is,byassumption,the limitof H
v
(0). By integrating int, we get
H
v
(t)H
0;v exp(
Z
t
0
2jjd(v())jj)d (67)
+ Z
t
0 exp(
Z
t
s
2jjd(v())jjd)(
Z
A(v)(s;x):(v +
?
r)(s;x))dsdx;
and,nally,byusing(64), we onludethat
?
risadissipativesolution
oftheEulerequationswithinitialonditionJ
0
=
?
r
0
,whihonludes
theproof.
7 Appendix
Inthisappendix, we prove theruialidentity(63). Beause ofthe onser-
vation of energy,weget fromdenition(62) :
d
dt H
v
=I
1 +I
2 +I
3 +I
4 +I
5 +I
6 +I
7
;
whereindexhasbeendroppedand
2I
1
= Z
jvj 2
t
; 2I
2
= Z
t jvj
2
;
I
3
=
Z
v:
t J; I
4
=
Z
J:
t v;
2I
5
= Z
t jvj
2
; I
6
= Z
r:
t
r ; I
7
= Z
r :
t r:
We have
I
7
= Z
t
= Z
t
= Z
r:J = Z
J:r
I
3
= Z
v:(r: Z
f)+ Z
v:r Z
v:
?
J
=
Z
d(v): Z
f
Z
v:r+
Z
J:r
=
Z
d(v): Z
f + Z
d(v):rr I
7 :
Thus
I
3 +I
7
=Q
1 +Q
2 +Q
3 +Q
4 +Q
5 +Q
6 +Q
7
where
Q
1
=
Z
d(v): Z
( v)( v)f + Z
d(v):r( )r( );
Q
2
=
Z
Dv:Jv; Q
3
=
Z
Dv:vJ
Q
4
= Z
Dv:r r; Q
5
= Z
Dv:rr
Q
6
= Z
Dv:vv; Q
7
= Z
Dv:r r :
Then,
Q
3
=
Z
j v
i J
j v
i
= 1
2
Z
r:Jjvj 2
= 1
2
Z
t jvj
2
= I
1 :
Next,weobserve that
I
2 +I
4 +Q
2 +Q
6
= Z
A(v):(v J)
and
I
5 +I
6 +Q
4 +Q
5 +Q
7
= Z
A(v):(v+
?
r)+R ;
where
R=R
1 +R
2 +R
3 +R
4
;
R
1
= Z
Dv:( vv)= Z
v
i v
j
j v
i
=0;
R
2
= Z
Dv:r( )r ;
R
3
= Z
Dv:(r r
?
rv):
Sinev=
?
r and(
?
) 2
= 1,,we get
R
3
= Z
(rv):( v
?
r+r
?
v)
= Z
(r
?
v):(vr rv)= Z
(rr ):(vr rv)
= Z
(rr) :(vr rv)=0:
Similarly,aftersetting = ,
R
2
= Z
(rv):r r
= Z
( r
?
r ):r r= Z
(rr ):r
?
r
= Z
(rr) :r
?
r= Z
r(
1
2 jr j
2
):
?
r=0:
Thus, R=0 andwe nallyget
d
dt H
v
=Q
1 +
Z
A(v):[v+
?
r+(v J)℄;
whihis thedesired result.
Aknowledgments
The author thanks Franois Golse and Emmanuel Grenier for many stim-
ulatingdisussionsonerningthegyrokinetilimit. Thisworkhasbeensup-
portedbytheprogramonChargedPartileKinetisattheErwinShroedinger
Institute.
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