Hydrodynamic limit of the Boltzmann-Monge-Ampere system

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by  Fethi Ben Belgacem

HYDRODYNAMIC LIMIT OF THE
BOLTZMANN-MONGE-AMPERE SYSTEM
7
1 FETHI BEN BELGACEM
0
2
Abstract. In this paper we investigate the hydrodynamic limit of the
n
Boltzmann-Monge-Ampere system in the so-called quasineutral regime.
a
WeprovetheconvergenceoftheBoltzmann-Monge-Amperesystemtothe
J
Euler equation by using therelative entropy method.
9

]
A
N 1. Introduction and main results
.
h ThegoalofthisarticleistostudythehydrodynamicallimitoftheBoltzman-
t
a Monge-Ampere system (BMA)
m
(1.1) ∂ fε+ξ.∇ fε+∇ ϕε.∇ fε =Q(fε,fε),
[ t x x ξ
(1.2) det I +ε2D2ϕε =ρε,
1 d
v where I is the identity matrix a(cid:0)nd (cid:1)
3 d
3
3 (1.3) ρε(t,x) = fε(t,x,ξ)dξ
ˆ
2 Rd
0 andfε(t,x,ξ) ≥ 0isthe electronic density attime t ≥ 0point x ∈ [0,1]d = Td,
.
1 andwithavelocityξ ∈Rd.Theperiodicelectricpotentialϕε iscoupledwithfε
0
throughthenonlinearMonge-Ampereequation(1.2). Thequantitiesε> 0and
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1 Q(fε,fε) denote respectively the vacum electric permitivity and the Boltzman
: collision integral. This latter, is given by (see [3,9])
v
i
arX Q(fε,fε)(t,x,ξ) = ˆ ˆS+d−1×Rd(cid:0)(fǫ)′(f1ε)′−fεf1ε(cid:1)b(ξ−ξ1,σ)dσdξ1,
wherethetermsf1ε,(fε)′and(f1ε)′defines,respectivelythevaluesfε(t,x,ξ1),
fε(t,x,ξ′) and fε(t,x,ξ1′) with ξ′ and ξ1′ given in terms of ξ, ξ1 ∈ Rd, and
σ ∈ S+d−1 = σ ∈ Sd−1/σ.ξ ≥ σ.ξ1 by
(cid:8) (cid:9)
′ ξ+ξ1 ξ−ξ1 ′ ξ+ξ1 ξ−ξ1
ξ = + σ, ξ = − σ.
1
2 2 2 2
By linearising the determinant about the identity matrix I , one get
d
det I +ε2D2ϕε = 1+ε2△ϕε+O ε4 .
d
(cid:0) (cid:1) (cid:0) (cid:1)
2010 Mathematics Subject Classification. 35F20, 35B40, 82D10.
Keywordsandphrases. Boltzmanequation,Monge-Ampèreequation,Eulerequationsof
theincompressible fluid.
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HYDRODYNAMIC LIMIT OF THE BOLTZMANN-MONGE-AMPERE SYSTEM 2
It follows that the BMA system is a fully nonlinear version of the Vlasov-
Poisson-Boltzman (VPB) system defined by
(1.4) ∂ fε+ξ.∇ fε+∇ ϕε.∇ fε = Q(fε,fε)
t x x ξ
(1.5) ε2△ϕε = ρε−1
This latter, has been interested many authors. In [5] DiPerna and Lions
showedtheexistenceofrenormalizedsolution. DesvilletesandDolbeault[7]are
interested to the long-time behavior of the weak solutions of the VPB system
for the initial boundary problem. In [10] Guo established the global existence
of smooth solutions to the VPB system in periodic boundary condition case.
For more references for this subject, Boltzmann equation or Vlasov–Poisson
system, one can see [1-7, 9–14].
In [11] L. Hsiao and al. studied the convergence of the VPB system to the
Incompressible Euler Equations. If one consider the case Q(fε,fε) = 0, we
obtain the Vlasov-Monge-Ampère(VMA). This problem, was been considered
by Y. Bernier and Grégoire[2]. They showed that weak solution of VMA con-
verge to a solution of the incompressible Euler equations when the parameter
ε goes to 0.
This work aims to extend these efforts to study such systems.
First, Note that
Q(fε,fε)dξ = ξ Q(fε,fε)dξ = |ξ|2Q(fε,fε)dξ = 0, i = 1,2,...,d.
ˆ ˆ i ˆ
Rd Rd Rd
and the conservation of total energy
1 ε
(1.6) |ξ|2fε(t,x,ξ)dxdξ+ |∇φε(t,x)|2dx = E0
2 ˆRdˆTd 2 ˆTd
where
Jε(t,x) = ξfε(t,x,ξ)dξ.
ˆ
Rd
From the conservation of mass and momentum, it follows that
(1.7) ∂ ρε+∇.Jε = 0
t
and
ε
(1.8) ∂ Jε+∇ . (ξ ⊗ξ)fεdξ+∇φε+ ∇ |∇φε|2 −ε∇.(∇φε⊗∇φε)= 0.
t x ˆRd 2 (cid:16) (cid:17)
Let us consider the periodic boundary problem of Euler equations to the
incompressible fluid
(1.9) ∇.u= 0, t > 0, x ∈Td
(1.10) ∂ u+(u.∇)u+∇p = 0t > 0, x ∈ Td
t
(1.11) u(0,x) = u0(x) ∈ Hs,
where the function space Hs is given by Hs = u∈ Hs Td , ∇.u= 0 .
We have the following result. (cid:8) (cid:0) (cid:1) (cid:9)
HYDRODYNAMIC LIMIT OF THE BOLTZMANN-MONGE-AMPERE SYSTEM 3
Theorem. Let 0 < T < T∗ and u0 in Hs s > 1+ d2 , Zd periodic in x.
Assume that fε(x,ξ) ≥ 0 to be smooth, Zd pe(cid:0)riodic in x(cid:1), and fε decays fast
0 0
as ξ → ∞. In addition, we assume that
f0ε(x,ξ)dξ = 1+o ε21 , as ε → 0,
ˆ
Rd (cid:16) (cid:17)
in the strong sense of the space H−1 Td and
(cid:0) (cid:1)
1 |ξ−u0(x)|2f0ε(x,ξ)dxdξ + ε ∇φ2(0,x) 2dx → 0 as ε → 0.
2 ˆRdˆTd 2 ˆTd
(cid:12) (cid:12)
(cid:12) (cid:12)
Let fε be any nonnegative smooth solution of (1.1)-(1.2). Then, up to the
extraction of a subsequence, the current Jε converges weakly to the unique
solution u(x,t) of the Euler equations (1.9)-(1.10)-(1.11). Moreover, the di-
vergence free part of f converges to u in L∞ [0,T],L2 Td .
(cid:0) (cid:0) (cid:1)(cid:1)
2. Proof of the theorem
First introduce the modulated energy functional
Hε(t)= 1 |ξ −u(x)|2fε(t,x,ξ)dxdξ+ ε ∇φ2(t,x) 2dx.
2 ˆRdˆTd 2 ˆTd
(cid:12) (cid:12)
(cid:12) (cid:12)
In the squel we need the following two Lemmas
Lemma 1. Under the hypothesis of the above theorem ,we have up to the
extraction of a sequence, ρε converges to 1 in C0 [0,T],D′ Td , the current
Jε converges to J in L∞ [0,T],D′ Td , J ∈ L(cid:0)∞ [0,T],(cid:0)L2 (cid:1)T(cid:1)d , and the
divergence free parts of Jε(cid:0)converges(cid:0)to(cid:1)J(cid:1)in C0 [0,T(cid:0)],D′ Td(cid:0) . (cid:1)(cid:1)
(cid:0) (cid:0) (cid:1)(cid:1)
Proof. we take d= 2, and we notice that
det I +εD2φε = 1+ε△φε+ε2detD2φε.
We first show that ρ(cid:0)ε → 1 in C(cid:1)0 [0,T],D′ Td . In fact, for η ∈ C∞ Td ,
0
we get (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1)
(ρε(t,x)−1)η(x)dx = det I +εD2φε −1 η(x)dx
ˆ ˆ
(cid:0) (cid:0) (cid:1) (cid:1)
= ε△φε +ε2detD2φε η(x)dx.
ˆ
(cid:0) (cid:1)
But
1 1
detD2φε = tr cofD2φε D2φε = div cofD2φε ∇φε ,
2 2
(cid:0)(cid:0) (cid:1) (cid:1) (cid:0)(cid:0) (cid:1) (cid:1)
it follows by integrating by parts that
ε2
(ρε(t,x)−1)η(x)dx = ε ∇φε∇η(x)dx+ div cofD2φε ∇φε η(x)dx
ˆ ˆ 2 ˆ
(cid:0)(cid:0) (cid:1) (cid:1)
ε2
= ε ∇φε∇η(x)dx− cofD2φε ∇φε.∇η(x)dx.
ˆ 2 ˆ
(cid:0) (cid:1)
HYDRODYNAMIC LIMIT OF THE BOLTZMANN-MONGE-AMPERE SYSTEM 4
Thus, by the Hölder inequality one has
1/2 1/2
(ρε(t,x)−1) η(x)dx ≤ ε1/2 ε |∇φε|2 |∇η|2 +
(cid:12)ˆ (cid:12) (cid:18) ˆ (cid:19) (cid:18)ˆ (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12) ε2
(cid:12) (cid:12) + cofD2φε k∇φεk k∇ηk .
2 L2 L2 L2
(cid:13) (cid:13)
Recall that from regularity result of M(cid:13)onge-Am(cid:13)père equation we have [8]
cofD2φε . ε−21,
L2
(cid:13) (cid:13)
So, by the conservation of th(cid:13)e energy,(cid:13)one deduce that
(cid:12)ˆ (ρε(t,x)−1)η(x)dx(cid:12) ≤C0ε1/2k∇ηkL2 +C.ε3/2k∇φεkL2k∇ηkL2.
(cid:12) (cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) ≤C0ε1/2k∇ηkL2 +C.εk∇ηkL2
≤ε1/2 C0+Cε1/2 k∇ηkL2
(cid:16) (cid:17)
(cid:3)
By the total energy equality (1.6) we have
(2.1)
1/2 1/2
|Jε(t,x)|dx ≤ |ξ|2fε(t,x,ξ)dxdξ fε(t,x,ξ)dxdξ ≤ C.
ˆ (cid:18)ˆ ˆ (cid:19) (cid:18)ˆ ˆ (cid:19)
Thus Jε is bounded in L∞ [0,T],L1 Td . Up to extracting a subsequence,
we can assume that Jε ha(cid:0)s a limit J(cid:0) in(cid:1)(cid:1)the sens of (Radon) measures on
[0,T] × Rd/Zd = Td. Let us define as in [11], for each non-negative function
z(t) ∈ C0([0,T]), the convex functional of a (Radon) measure
T |Jε(t,x)|2
K(ρε,Jε)= z(t)dxdt
ˆ 2ρε(t,x)
0
T 1
= sup − |b(t,x)|2ρε(t,x)+b(t,x)Jε(t,x) z(t)dt.
ˆ (cid:26) 2 (cid:27)
b 0
where b belongs to the space of all continuous functions from [0,T]×Td to Rd.
From (2.1) and since the functional K is lower semi-continuous with respect
to the convergence of measure, it follows that
T T
2
z(t) |J(t,x)| dx dt ≤ C z(t)dt,
ˆ (cid:18)ˆ (cid:19) ˆ
0 0
which means that J ∈ L∞ [0,T],L2 Td .
From (1.7) and (1.8) one(cid:0)write (cid:0) (cid:1)(cid:1)
∂ ρε = ∂ det I+ε2D2ϕε = −∇Jε,
t t
(cid:0) (cid:1)
thus
∇Jε = −ε∂ △φε−ε2∂ detD2φε.
t t
HYDRODYNAMIC LIMIT OF THE BOLTZMANN-MONGE-AMPERE SYSTEM 5
For η ∈ C∞ Td , we have
0
(cid:0) (cid:1)
∇Jεη(x)dx = −ε ∂ (△φεη)dx−ε2 ∂ detD2φεηdx,
ˆ ˆ t ˆ t
thus J is divergence free in x in the sense of distribution.
By (1.8), we deduce that ∂ J is bounded in L∞ [0,T],D′ Td . So, we
t
obtainthatuptotheexractionofasubsequance,J ∈ C(cid:0) 0 [0,T],(cid:0)L2 (cid:1)T(cid:1)d −w .
Inthesameway ,we canshow thatthedivergence -fre(cid:0)epartofJ(cid:0)ε co(cid:1)nverg(cid:1)es
to J in C0 [0,T],D′ Td . Since Jε converges to J, it remains to show that
J = u in L∞(cid:0) [0,T],L(cid:0)2 T(cid:1)(cid:1)d . For this, it suffies to use the next Lemma.
(cid:0) (cid:0) (cid:1)(cid:1)
Lemma 2. [11]Let u be the unique solution of the Euler equations (1.9)-(1.10)
with initial datum and u0 and the hypotheses of theorem 1 hold. Then, for any
t ∈ (0,T], Hε(t)→ 0 as ε → 0.
To end the proof of the Theorem, we define a new functional
|Jε(t,x)−ρε(t,x)u(t,x)|2
(2.2) hε(t) = dx.
ˆ 2ρε(t,x))
With b belongs to the space of all continuous functions from Td to Rd. By the
Cauchy-Shwarz inequality, one get
1
hε(t) ≤ |ξ −u(t,x)|2fε(t,x,ξ)dxdξ ≤ Hε(t).
2ˆ
Since ρε → 1, Jε → J and from the convexity of the functional defined by
(2.2), we obtain
|J(t,x)−u(t,x)|2dx ≤ 2limhε(t)≤ 2limHε(t)= 0.
ˆ ε→0 ε→0
This finish the proof of Theorem 1.2.
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LR03ES04 Equations aux dérivées partielles et applications, Université de
Tunis El Manar, Faculté des Sciences de Tunis, 2092 Tunis, Tunisie.
E-mail address: [email protected]