Table Of ContentVLASOV-MAXWELL-BOLTZMANN DIFFUSIVE LIMIT
6
0 JUHIJANG
0
2
Abstract. WestudythediffusiveexpansionforsolutionsaroundMaxwellian
n
equilibrium and in a periodic box to the Vlasov-Maxwell-Boltzmann system,
a
the most fundamental model for an ensemble of charged particles. Such an
J
expansion yieldsasetofdissipativenewmacroscopicPDE’s,theincompress-
0 ible Vlasov-Navier-Stokes-Fourier system and its higher order corrections for
3 describing a charged fluid, where the self-consistent electromagnetic field is
present. The uniform estimate on the remainders is established via a unified
] nonlinearenergymethodanditguarantees theglobalintimevalidityofsuch
P
anexpansionuptoanyorder.
A
.
h
t
a
1. Introduction and Formulation
m
[
The dynamics of charged dilute particles can be described by the celebrated
1
Vlasov-Maxwell-Boltzmannsystem:
v
0 ∂ F +v F +(E+v B) F =Q(F ,F )+Q(F ,F ),
t + x + v + + + +
4 (1.1) ·∇ × ·∇ −
7 ∂tF +v xF (E+v B) vF =Q(F ,F+)+Q(F ,F ),
− ·∇ −− × ·∇ − − − −
1
with initial data F (0,x,v)=F (x,v). For notational simplicity we have set all
0 0,
± ±
6 physicalconstantstobeunity,see[8]formorebackground. HereF (t,x,v) 0are
± ≥
0 the spatially periodic number density functions for the ions (+) and electrons (-)
/ respectively, at time t 0, position x = (x ,x ,x ) [ π,π]3 = T3 and velocity
th v = (v1,v2,v3) R3.≥The collision betwe1en2par3tic∈les−is given by the standard
a ∈
Boltzmann collision operator Q(G ,G ) with hard-sphere interaction:
m 1 2
v: (1.2) Q(G1,G2)= (u v) ω G1(v′)G2(u′) G1(v)G2(u) dudω,
i ZR3×S2| − · |{ − }
X
where v =v [(v u) ω]ω and u =u+[(v u) ω]ω.
′ ′
r The self-con−siste−nt, sp·atially periodic electro−magn·etic field [E(t,x),B(t,x)] in
a
(1.1) is coupled with F (t,x,v) through the Maxwell system:
±
∂ E B = v(F F )dv, B =0,
t +
(1.3) −∇× −ZR3 − − ∇·
∂ B+ E =0, E = (F F )dv,
t +
∇× ∇· ZR3 − −
with initial data E(0,x)=E (x), B(0,x)=B (x).
0 0
It turns out that it is convenient to consider the sum and difference of F and
+
F as proposed in [4]. Defining
−
(1.4) F F +F and G F F ,
+ +
≡ − ≡ − −
Date:January27,2006.
1
2 JUHIJANG
(1.1) and (1.3) can be rewritten as following:
∂ F +v F +(E+v B) G=Q(F,F),
t x v
·∇ × ·∇
∂ G+v G+(E+v B) F =Q(G,F),
t x v
·∇ × ·∇
(1.5) ∂ E B = v Gdv, B =0,
t
−∇× −ZR3 ∇·
∂ B+ E =0, E = Gdv,
t
∇× ∇· ZR3
with initial data F(0,x,v) = F (x,v), G(0,x,v) = G (x,v), E(0,x) = E (x) and
0 0 0
B(0,x)=B (x).
0
Now we introduce the diffusive scaling to (1.5): for any ε>0,
1
ε∂ Fε+v Fε+(Eε+v Bε) Gε = Q(Fε,Fε),
t x v
·∇ × ·∇ ε
1
ε∂ Gε+v Gε+(Eε+v Bε) Fε = Q(Gε,Fε),
t x v
·∇ × ·∇ ε
(1.6)
ε∂ Eε Bε = v Gε dv, Bε =0,
t
−∇× −ZR3 ∇·
ε∂ Bε+ Eε =0, Eε = Gε dv.
t
∇× ∇· ZR3
For notational simplicity, we normalize the global Maxwellian as
1
(1.7) µ(v)= e v2/2.
−| |
(2π)3/2
We consider the following formal expansion in ε around the equilibrium state
[F,G,E,B]=[µ,0,0,0]: for any n 1,
≥
Fε(t,x,v)=µ+√µ εf (t,x,v)+ε2f (t,x,v)+...+εnfε(t,x,v) ,
{ 1 2 n }
Gε(t,x,v)=√µ εg (t,x,v)+ε2g (t,x,v)+...+εngε(t,x,v) ,
(1.8) { 1 2 n }
Eε(t,x)= εE (t,x)+ε2E (t,x)+...+εnEε(t,x) ,
{ 1 2 n }
Bε(t,x)= εB (t,x)+ε2B (t,x)+...+εnBε(t,x) .
{ 1 2 n }
To determine the coefficients f (t,x,v),...,f (t,x,v); g (t,x,v),...,g (t,x,v);
1 n 1 1 n 1
− −
E (t,x),...,E (t,x); B (t,x),...,B (t,x), we plug the formal diffusive expan-
1 n 1 1 n 1
− −
sion (1.8) into the rescaled equations (1.6):
(1.9)
(ε∂ +v ) εf +...+εnfε
t ·∇x { 1 n}
1
+ ε(E +v B )+...+εn(Eε +v Bε) [√µ εg +...+εngε ]
√µ{ 1 × 1 n × n }·∇v { 1 n}
1
= Q(µ+√µ εf +...+εnfε , µ+√µ εf +...+εnfε ),
ε√µ { 1 n} { 1 n}
(ε∂ +v ) εg +...+εngε
t ·∇x { 1 n}
1
+ ε(E +v B )+...+εn(Eε +v Bε) [µ+√µ εf +...+εnfε ]
√µ{ 1 × 1 n × n }·∇v { 1 n}
1
= Q(√µ εg +...+εngε , µ+√µ εf +...+εnfε ),
ε√µ { 1 n} { 1 n}
VLASOV-MAXWELL-BOLTZMANN DIFFUSIVE LIMIT 3
ε∂ εE +...+εnEε εB +...+εnBε = v√µ εg +...εngε dv,
t{ 1 n}−∇×{ 1 n} −ZR3 { 1 n}
ε∂ εB +...+εnBε + εE +...+εnEε =0,
t{ 1 n} ∇×{ 1 n}
εE +...+εnEε = √µ εg +...εngε dv,
∇·{ 1 n} ZR3 { 1 n}
εB +...+εnBε =0.
∇·{ 1 n}
ToexpandtherighthandsideQintheabove,wedefineLthewell-knownlinearized
collision operator and another linearized operater as
L
1
(1.10) Lf Q(µ,√µf)+Q(√µf,µ) ,
≡−√µ{ }
1
(1.11) g Q(√µg,µ),
L ≡−√µ
and the nonlinear collision operator Γ as (non-symmetric)
1
(1.12) Γ(f,g) Q(√µf,√µg).
≡ √µ
Note that Lf and g can be written as following in terms of Γ:
L
(1.13) Lf = Γ(√µ,f)+Γ(f,√µ) , g = Γ(g,√µ).
−{ } L −
Now we equate the coefficients on both sides of the equation (1.9) in front of
different powers of the parameter ε. Let
f =f 0, g =g 0, E 0, B 0
1 0 1 0 0 0
− ≡ − ≡ ≡ ≡
to obtain
1
∂ f +v f + (E +v B ) (√µg )
t m x m+1 i i v j
·∇ √µ × ·∇
i+jX=m+1
i,j 1
≥
= Lf + Γ(f ,f ),
m+2 i j
−
i+jX=m+2
i,j 1
(1.14) ≥
1
∂ g +v g + (E +v B ) (√µf ) E v√µ
t m x m+1 i i v j m+1
·∇ √µ × ·∇ − ·
i+jX=m+1
i,j 1
≥
= g + Γ(g ,f ),
m+2 i j
−L
i+jX=m+2
i,j 1
≥
for 1 m n 3 as well as
− ≤ ≤ −
∂ E B = vg √µdv, B =0,
t m m+1 m+1 m+1
−∇× −Z ∇·
(1.15)
∂ B + E =0, E = g √µdv,
t m m+1 m+1 m+1
∇× ∇· Z
for0 m n 2. Moreover,wecancollecttermsleftin(1.9)withpowersεn 1 or
−
≤ ≤ −
higher and divide by εn 1 to get the equations for the remainders fε,gε,Eε,Bε.
− n n n n
4 JUHIJANG
Notethatalltheεm+1-thordertermsvanishform n 3becauseof(1.14). First,
≤ −
we write equations for fε and gε:
n n
(1.16)
ε2∂ fε+εv fε+Lfε =
t n ·∇x n n
1
∂ f v f (E +v B ) (√µg )+ Γ(f ,f )
t n 2 x n 1 i i v j i j
{− − − ·∇ − − √µi+jX=n 1 × ·∇ i+Xj=n }
i,j 1− i,j 1
≥ ≥
1
+ε ∂ f (E +v B ) (√µg )+ Γ(f ,f )
t n 1 i i v j i j
{− − − √µi+Xj=n × ·∇ i+jX=n+1 }
i,j 1 i,j 1
≥ ≥
n 1
−
+εnΓ(fε,fε)+ εi Γ(fε,f )+Γ(f ,fε) + εi+j nΓ(f ,f )
n n { n i i n } − i j
Xi=1 i+jXn+2
≥
εn+1
(Eε +v Bε) (√µgε)
− √µ n × n ·∇v n
n 1
1 −
εi+1 (E +v B ) (√µgε)+(Eε +v Bε) (√µg )
− √µ { i × i ·∇v n n × n ·∇v i }
Xi=1
1
εi+j−n+1(Ei+v Bi) v(√µgj);
− √µ × ·∇
i+jXn+1
≥
ε2∂ gε +εv gε εEε v√µ+ gε =
t n ·∇x n− n· L n
1
∂ g v g (E +v B ) (√µf )+E v√µ
t n 2 x n 1 i i v j n 1
{− − − ·∇ − − √µ × ·∇ − ·
i+jX=n 1
i,j 1−
≥
1
+ Γ(g ,f ) +ε ∂ g (E +v B ) (√µf )
i j t n 1 i i v j
} {− − − √µ × ·∇
i+Xj=n i+Xj=n
i,j 1 i,j 1
≥ ≥
n 1
−
+ Γ(g ,f ) +εnΓ(gε,fε)+ εi Γ(gε,f )+Γ(g ,fε)
i j } n n { n i i n }
i+jX=n+1 Xi=1
i,j 1
≥
εn+1
+ εi+j−nΓ(gi,fj)− √µ (Enε +v×Bnε)·∇v(√µfnε)
i+jXn+2
≥
n 1
1 −
εi+1 (E +v B ) (√µfε)+(Eε +v Bε) (√µf )
− √µ { i × i ·∇v n n × n ·∇v i }
Xi=1
1
εi+j−n+1(Ei+v Bi) v(√µfj).
− √µ × ·∇
i+jXn+1
≥
Similarly, by (1.15), we get the remainders for Eε,Bε:
n n
ε∂ Eε Bε = ∂ E vgε√µdv, Bε =0,
(1.17) t n−∇× n − t n−1−ZR3 n ∇· n
ε∂ Bε + Eε = ∂ B , Eε = gε√µdv.
t n ∇× n − t n−1 ∇· n ZR3 n
VLASOV-MAXWELL-BOLTZMANN DIFFUSIVE LIMIT 5
The fluid equations can be obtained through the conditions (1.14) and (1.15).
WefirstrecallthattheoperatorL 0,andforanyfixed(t,x),thenullspaceofLis
≥
generatedby [õ,võ, v 2õ]. Foranyfunctionf(t,x,v) wethus candecompose
| |
f =P1f + I P1 f
{ − }
where P1f (hydrodynamic part) is the L2v projection on the null space for L for
given (t,x). We can further denote
v 2 3
(1.18) P1f = ρf(t,x)+v uf(t,x)+(| | )θf(t,x) √µ.
{ · 2 − 2 }
Here we define the hydrodynamic field of f to be
[ρ (t,x),u (t,x),θ (t,x)]
f f f
which representsthe density, velocity and temperature fluctuations physically. For
the velocity field u (t,x), we further define its divergent-free part as
f
P u = the divergent-free projection of u
0 f f
so that
P u 0.
0 f
∇·{ }≡
Similarly, one can show that 0 and for any fixed (t,x), the null space of
L ≥ L
is one dimensional vector space generated by [õ]. Likewise, any g(t,x,v) can be
decomposed into
g =P2g+ I P2 g
{ − }
where P2g (hydrodynamic part) is the L2v projection on the null space for L for
given (t,x). We can further denote
(1.19) P2g =σg(t,x)√µ.
Here σ (t,x), the hydrodynamic field of g, can be interpreted as the concentration
g
difference. Formoredetailsabout andP2,werefer[4]. Beforegoingon,westate
L
thecoercivityofLand whichwillbeoftenusedinthesubsequentsections: there
L
exists a δ >0 such that
(1.20) hLf,fi≥δ|(I−P1)f|2ν, hLg,gi≥δ|(I−P2)g|2ν.
SeeLemma1in[8]foritsproof. NotethattheoperatorLdefinedin[8]isequivalent
to [L, ] in our case.
L
Now define [ρ ,u ,θ ,σ ] to be the corresponding hydrodynamic field of the
m m m m
m-th coefficients f and g . As for the first coefficients f (t,x,v) and g (t,x,v),
m m 1 1
from (1.14)
(1.21) I P1 f1 =0 and I P2 g1 =0
{ − } { − }
which immediately yield that
(1.22) B =0 and E = φ
1 1 1
∇
up to constant and for some function φ (t,x) satisfying φ = σ ; in particular,
1 1 1
△
B may be assumed to be zero physically in a sense that nonzero constants B do
1 1
not cause the hydrodynamic equations (1.25)-(1.28) to change. It will be shown in
Lemma 3.1 that its velocity fluctuation u (t,x) is incompressible:
1
(1.23) u 0 or u =P u ,
1 1 0 1
∇· ≡
6 JUHIJANG
anditsdensityandtemperaturefluctuationsρ (t,x)andθ (t,x)satisfytheBoussi-
1 1
nesq relation:
(1.24) ρ +θ 0.
1 1
≡
Moreover, [u ,θ ,σ ] satisfies the nonlinear incompressible Vlasov-Navier-Stokes-
1 1 1
Fourier equations:
(1.25) ∂ u +u u + p =η∆u +σ φ ,
t 1 1 1 1 1 1 1
·∇ ∇ ∇
(1.26) ∂ σ +u σ =α∆σ ασ ,
t 1 1 1 1 1
·∇ −
(1.27) ∆φ =σ ,
1 1
(1.28) ∂ θ +u θ =κ∆θ ,
t 1 1 1 1
·∇
where p (t,x) is the pressure and η, κ, α>0 are physical constants.
1
Asforthecoefficientsf (t,x,v), g (t,x,v)form 2,by(1.14),themicroscopic
m m
≥
part of f and g is determined by:
m m
I P1 fm =L−1 ∂tfm 2 v xfm 1+ Γ(fi,fj)
{ − } {− − − ·∇ − i+Xj=m
i,j 1
≥
1
(1.29) (E +v B ) (õg ) ,
i i v j
−√µ × ·∇ }
i+jX=m 1
i,j 1−
≥
I P2 gm = −1 ∂tgm 2 v xgm 1+ Γ(gi,fj)
{ − } L {− − − ·∇ − i+Xj=m
i,j 1
≥
1
(1.30) (E +v B ) (õf )+E võ .
i i v j m 1
−√µi+jX=m 1 × ·∇ − · }
i,j 1−
≥
On the other hand, for the hydrodynamic field of f and g :
m m
v 2 3
P1fm = ρm(t,x)+v um(t,x)+ | | θm(t,x) √µ,
{ · { 2 − 2} }
P2gm = σm(t,x)√µ,
we can deduce an m-th order incompressibility condition
(1.31) I P u = ∂ ρ ,
0 m t m 1
∇·{ − } − −
an m-th order Boussinesq relation
(1.32)
ρ +θ =∆ 1 u (P u ) P u u +E σ +E σ
m m − 1 0 m 1 0 m 1 1 1 m 1 m 1 1
∇·{− ·∇ − − − ·∇ − −
v 2õ 5
+Rmu−1}+h| |3 ,L−1({I−P1}v·∇xP1fm−1)i− 2θ1θm−1−um−1·u1,
and an m-th order linear Vlasov-Navier-Stokes-Fouriersystem for [P u ,θ ,σ ]:
0 m m m
(1.33) (∂ +u η∆)P u +P u u + p (E σ +E σ )=Ru,
t 1·∇− 0 m 0 m·∇ 1 ∇ m− 1 m m 1 m
(1.34) (∂ +u α∆+α)σ +P u σ =Rσ,
t 1·∇− m 0 m·∇ 1 m
(1.35) (∂ +u κ∆)θ +P u θ =Rθ ,
t 1·∇− m 0 m·∇ 1 m
VLASOV-MAXWELL-BOLTZMANN DIFFUSIVE LIMIT 7
(1.36) E = ∂ B , E =σ ,
m t m 1 m m
∇× − − ∇·
(1.37) Bm = I P2 gmv√µdv+∂tEm 1, Bm =0,
∇× ZR3{ − } − ∇·
with compatibility conditions coming from conservation laws
d
(1.38) E dx= α E dx+ℓ , B dx=0.
m m m 1 m
dtZT3 − ZT3 − ZT3
HereRu,Rσ,Rθ andℓ ,definedpreciselyin(3.2),(3.3),(3.4)and(3.20),essen-
m m m m 1
−
tiallydependonlyonfj, gj, Ej, Bj forj m 1,since I P1 fm, I P2 gm,
≤ − { − } { − }
I P u , as well as ρ +θ have been determined.
0 m m m
{ − }
In order to state our results precisely in the next section, we introduce the
following norms and notations. We use , to denote the standard L2 inner
product in R3, while we use ( , ) to denoht·e ·Li2 inner product either in T3 R3
or in T3 withvcorresponding L·2 ·norm . We use the standard notation H×s to
||·||
denote the Sobolev space Ws,2. For the Boltzmann collision operator (1.2), define
the collision frequency to be
(1.39) ν(v) v v µ(v )dv ,
′ ′ ′
≡ZR3| − |
which behaves like v as v . It is naturalto define the following weighted L2
| | | |→∞
norm to characterize the dissipation rate.
g 2 g2(v)ν(v)dv, g 2 g2(x,v)ν(v)dvdx.
| |ν ≡ZR3 || ||ν ≡ZT3 R3
×
Observe that for hard sphere interaction,
(1.40) (1+ v )21g C g ν.
|| | | ||≤ || ||
In order to be consistent with the hydrodynamic equations, we define
(1.41) ∂β =∂γ1∂γ2∂γ3∂β1∂β2∂β3
γ x1 x2 x3 v1 v2 v3
where γ = [γ ,γ ,γ ] is related to the space derivatives, while β = [β ,β ,β ] is
1 2 3 1 2 3
related to the velocity derivatives.
We now define instant energy functionals and the dissipation rate.
Definition 1 (Instant Energy) For N 8, for some constant C > 0, an
≥
instant energy functional (f,g,E,B)(t) (t) satisfies:
N N
E ≡E
1
(1.42) (t) [∂βf, ∂βg] 2(t)+ [∂ E, ∂ B] 2(t) C (t).
CEN ≤ || γ γ || || γ γ || ≤ EN
X X
β+γ N γ N
| | | |≤ | |≤
Definition 2 (Dissipation Rate) For N 8, the dissipation rate (f,g)(t)
N
≥ D
is defined as
N(f,g)(t)= [∂γP1f, ∂γP2g] 2(t)
D || ||
X
γ N
(1.43) | |≤
1
+ε2 ||[∂γβ{I−P1}f, ∂γβ{I−P2}g]||2ν(t).
X
β+γ N
| | | |≤
Weremarkthatboththeinstantenergyandthedissipationratearecarefullyde-
signedtocapturethestructureoftherescaledVlasov-Maxwell-Boltzmannequation
8 JUHIJANG
(1.6). First of all, the electromagnetic field [E,B] is included only in the instant
energy, which prevents the exponential decay on unlike the pure Boltzmann
N
E
case for hard potentials. See [9] and [16]. Notice that there is no 1 factor in front
ε2
of the hydrodynamic part [P1f,P2g] in the dissipation rate N(f,g), since only
D
themicroscopicpart[ I P1 f, I P2 g]shouldvanishasε 0. Fornotational
{ − } { − } →
simplicity,theEinstein’s summationconvention isusedforGreekletteruptoorder
N 8 throughout the paper, unless otherwise specified. We denote = and
x
≥ ∇ ∇
use C to denote a constant independent of ε. We also use U() to denote a general
·
positive polynomial with U(0)=0.
2. Main Results
The first result is to determine the coefficients f ,f ,...f ; g ,g ,...g ; E ,E ,
1 2 m 1 2 m 1 2
...E ; B ,B ,...B in a diffusive approximation (1.8).
m 1 2 m
Theorem2.1. Letmdivergent-freevector-valuedfunctions[u0(x),u0(x),...,u0 (x)],
1 2 m
2m scalar functions [θ0(x),θ0(x)...θ0 (x);σ0(x),σ0(x),...σ0 (x)] be given such that
1 2 m 1 2 m
u0 + θ0 + σ0 M
|| 1||H2 || 1||H2 || 1||H2 ≤
and
σ0(x)dx=0, u0(x)dx= E0(x) B0(x)dx,
ZT3 r ZT3 r −i+Xj=rZT3 i × j
i,j 1
≥
3 1
θ0(x)dx= E0(x) E0(x)+B0(x) B0(x)dx,
ZT3 2 r −2i+Xj=rZT3 i · j i · j
i,j 1
≥
for 1 r m. HereE0,B0 are defined by E (0,x),B (0,x) which have been induc-
≤ ≤ i j i j
tively determined at the precedents since i,j < r, starting with the average condi-
tions B0dx =0 and E0dx =0. Then for sufficiently small M and given m
T3 1 T3 1
real veRctors e1(=0),e2,..R.,em, there exist unique functions f1(t,x,v),f2(t,x,v),...,
f (t,x,v); g (t,x,v),g (t,x,v),...,g (t,x,v); E (t,x),E (t,x),...,E (t,x) and
m 1 2 m 1 2 m
B (t,x),B (t,x),...,B (t,x) with
1 2 m
σ (t,x)dx=0, P u (t,x)dx= E (t,x) B (t,x)dx,
r 0 r i j
ZT3 ZT3 −i+Xj=rZT3 ×
i,j 1
(2.1) ≥
3 1
θ (t,x)dx= E (t,x) E (t,x)+B (t,x) B (t,x)dx,
r i j i j
ZT3 2 −2i+Xj=rZT3 · ·
i,j 1
≥
such that initially P u (0,x) = u0(x), θ (0,x) = θ0(x), σ (0,x) = σ0(x) and
0 r r r r r r
E (0,x)dx=e , andf (t,x,v),g (t,x,v),E (t,x),B (t,x)satisfy(1.21)-(1.28)
T3 r r 1 1 1 1
Rand fr(t,x,v),gr(t,x,v),Er(t,x),Br(t,x) satisfy (1.29)-(1.38) for 2 r m.
≤ ≤
VLASOV-MAXWELL-BOLTZMANN DIFFUSIVE LIMIT 9
Moreover, for 1 r m, for any β, and for all s 3, there exists a polyno-
≤ ≤ ≥
mial U with U (0)=0 such that
r,β,s r,β,s
[∂βf , ∂βg (t)+ [∂ E , ∂ B ] (t)
{|| τ r τ r||ν || τ r τ r || }
X
τ s
(2.2) | |≤
≤e−λtUr,β,s( {||u0j||H2s+4(r−j) +||θj0||H2s+4(r−j) +||σj0||H2s+4(r−j)}),
1Xj r
≤ ≤
where space-time derivatives
∂ =∂τ0∂τ1∂τ2∂τ3
τ t x1 x2 x3
and λ can be chosen as 1min η,κ,α for sufficiently small M.
4 { }
We now turn to the most important questionabout the remainder estimates for
fε, gε, Eε and Bε. We first study the classical case for the first order remainders
n n n n
fε fε, gε gε, Eε Eε, Bε Bε
≡ 1 ≡ 1 ≡ 1 ≡ 1
which satisfy the nonlinear Boltzmann type equations:
1 1 1 1
(2.3) ∂ fε+ v fε+ Lfε = Γ(fε,fε) (Eε+v Bε) (√µgε),
t ε ·∇x ε2 ε − √µ × ·∇v
1 1 1 1
∂ gε+ v ( gε √µEε)+ gε = Γ(gε,fε) (Eε+v Bε) (√µfε),
t ε · ∇x − ε2L ε − √µ × ·∇v
ε∂ Eε Bε = gεv√µdv, Bε =0,
t
−∇× −Z ∇·
(2.4)
ε∂ Bε+ Eε =0, Eε = gε√µdv.
t
∇× ∇· Z
Theorem 2.2. Let N 8. Let fε(0,x,v) = fε(x,v), gε(0,x,v) = gε(x,v) and
≥ 0 0
Eε(0,x) = Eε(x), Bε(0,x) = Bε(x) satisfy the mass, momentum and energy con-
0 0
servation laws:
(fε,√µ)=0, (gε,√µ)=0,
0 0
(fε,v√µ)+ε Eε Bεdx=0,
(2.5) 0 ZT3 0 × 0
(fε, v 2√µ)+ε Eε 2+ Bε 2dx=0.
0 | | ZT3| 0| | 0|
Then there exists an instant energy functional (fε,gε,Eε,Bε)(t) such that
N
E
if (fε,gε,Eε,Bε)(0) is sufficiently small, then
N
E
d
(2.6) (fε,gε,Eε,Bε)(t)+ (fε,gε)(t) 0.
N N
dtE D ≤
In particular,
(2.7) sup (fε,gε,Eε,Bε)(t) (fε,gε,Eε,Bε)(0).
N N
E ≤E
0 t
≤ ≤∞
Moreover, for k 1 there exists C >0 such that
N,k
≥
k
(2.8) (fε,gε,Eε,Bε)(t) C (fε,gε,Eε,Bε)(0) 1+ t − .
N N,k N+k
E ≤ E (cid:26) k(cid:27)
10 JUHIJANG
We remark that our initial data fε,gε,Eε,Bε are general and can contain ini-
0 0 0 0
tial layer for the Vlasov-Navier-Stokes-Fourier limit. We can easily include (one)
temporal derivative ∂ in our definition of instant energy and dissipation rate, and
t
obtain the same uniform bound for such a new norm. With such a modification,
the boundedness of ∂ fε, ∂ gε, ∂ Eε, ∂ Bε automatically removes the formation
t 0 t 0 t 0 t 0
of any initial layer.
For higher order remainders fε, gε, Eε, Bε with n 2, we have
n n n n ≥
Theorem 2.3. Let N 8. Given f f ...f ; g g ...g ; E E ...E ; B B ...B
1, 2, n 1, 2, n 1, 2, n 1, 2, n
≥
constructed in Theorem 2.1 and let
[u0,θ0,σ0] (t)
||| n n n |||N
(2.9) ≡ {||u0i||H2N+10+4(n−j) +||θi0||H2N+10+4(n−j) +||σi0||H2N+10+4(n−j)}.
1Xj n
≤ ≤
And let
Fε(0,x,v)≡µ+√µ{εf1(0,x,v)+...+εn−1fn−1(0,x,v)+εnfnε(0,x,v)},
(2.10) Gε(0,x,v)≡√µ{εg1(0,x,v)+...+εn−1gn−1(0,x,v)+εngnε(0,x,v)},
Eε(0,x) εE (0,x)+...+εn 1E (0,x)+εnEε(0,x),
≡ 1 − n−1 n
Bε(0,x) εB (0,x)+...+εn 1B (0,x)+εnBε(0,x)
≡ 1 − n−1 n
be given initial data satisfying the following conservation laws:
Fε(0,x,v) µ(v) dvdx=0, Gε(0,x,v)dvdx=0,
ZT3ZR3{ − } ZT3ZR3
(2.11) vFε(0,x,v)dvdx+ Eε(0,x) Bε(0,x)dx=0,
ZT3ZR3 ZT3 ×
v 2 Fε(0,x,v) µ(v) dvdx+ Eε(0,x)2+ Bε(0,x)2dx=0.
ZT3ZR3| | { − } ZT3| | | |
Then there exist an instant energy functional and a positive polynomial U with
N
E
U(0)=0 such that if both ε and
(fε f ,gε g ,Eε E ,Bε B )(0)
EN n− n n− n n− n n− n
are sufficiently small, then
sup (fε f ,gε g ,Eε E ,Bε B )(t)
EN n− n n− n n− n n− n
0 t
≤ ≤∞
(2.12) ≤{eU(|||[u0n,θn0,σn0]|||N)EN(fnε−fn,gnε −gn,Enε −En,Bnε −Bn)(0)
+ε2U( [u0,θ0,σ0] ) .
||| n n n |||N }
Moreover, for k 1 there exists C >0 such that
N,k
≥
(fε f ,gε g ,Eε E ,Bε B )(t)
EN n− n n− n n− n n− n
(2.13) ≤CN,k{eU(|||[u0n,θn0,σn0]|||N)EN+k(fnε−fn,gnε −gn,Enε −En,Bnε −Bn)(0)
k
t −
+ε2U( [u0,θ0,σ0] ) 1+ .
||| n n n |||N }(cid:26) k(cid:27)
Note that the conservation laws (2.11) in Theorem 2.3 imply the conservation
laws for fε, gε, Eε, Bε due to (2.1). See (7.5) for the precise description. In
n n n n
addition, we remark that the positivity for the initial data (Fε Gε)(0,x,v) 0
± ≥
