convergence of the vlasov–poisson–boltzmann system to the incompressible euler equations

Acta Mathematica Sinica, English Series

Apr., 2007, Vol. 23, No. 4, pp. 761–768

Published online: May 5, 2006

DOI: 10.1007/s10114-005-0774-3

Http://www.ActaMath.com

Convergence of the Vlasov–Poisson–Boltzmann System tothe Incompressible Euler Equations

Ling HSIAOAcademy of Mathematics and Systems Science, Chinese Academy of Sciences,

Beijing 100080, P. R. China

E-mail: [email protected]

Fu Cai LIDepartment of Mathematics, Nanjing University, Nanjing 210093, P. R. China


Shu WANGCollege of Applied Sciences, Beijing University of Technology, Beijing 100022, P. R. China


Abstract In this paper we establish the convergence of the Vlasov–Poisson–Boltzmann system to

the incompressible Euler equations in the so-called quasi-neutral regime. The convergence is rigorously

proved for time intervals on which the smooth solution of the Euler equations of the incompressible

fluid exists. The proof relies on the relative-entropy method.

Keywords Vlasov–Poisson–Boltzmann system, Euler equations of the incompressible fluid, Relative-

entropy method

MR(2000) Subject Classification 35F20, 35B40, 82D10

1 Introduction and Main ResultsIn this paper, we investigate the hydrodynamical limit of the Vlasov–Poisson–Boltzmann (VPB)system, which describes the behavior of dilute charged particles when the magnetic forces areneglected. The precise system takes the form

∂tfε + ξ · ∇xf ε −∇xΦε · ∇ξf

ε = Q(f ε, f ε), (1.1)∫Rd

f ε(t, x, ξ)dξ = 1 − εΔΦε, (1.2)

f ε(0, x, ξ) = f ε0(x, ξ) ≥ 0, (1.3)

where f ε(t, x, ξ) is the spatially periodic distribution function for particles, which expresses theprobability of finding a particle at time t ≥ 0 in a position x ∈ [0, 1]d ≡ T

d, and with a velocityξ ∈ R

d, d = 1, 2, 3. The spatially periodic electric potential Φε is coupled with f ε(t, x, ξ) throughthe Poisson equation (1.2) and ε > 0 is the (rescaled) vacuum electric permittivity. Q(f ε, f ε) isthe standard Boltzmann collision integral. This collision integral acts only on the ξ-argumentof the distribution function f ε and is given by the expression (see [1, 2])

Q(f ε, f ε)(t, x, ξ) =∫∫

Sd−1+ ×Rd

((f ε)′(f ε

1)′ − f εf ε1

)b(ξ − ξ1, σ)dσdξ1, (1.4)

Received July 8, 2004, Accepted February 17, 2005L. Hsiao is partially supported by the Special Funds of State Major Basic Research Projects (Grant 1999075107),

by The Grant of NSAF (No. 10276036), and by NSFC (Grant 10431060)

F. Li is partially supported by Tianyuan Youth Funds of China (Grant 10426030), by NSFC (Grant 10501047),

and by Nanjing University Talent Development Foundation.

S. Wang is supported by NSFC (Grant 10471009)

762 Hsiao L., et al.

where the terms f ε1 , (f ε)′ and (f ε

1)′ designate, respectively, the values f ε(t, x, ξ1), f ε(t, x, ξ′) andf ε(t, x, ξ′1) with ξ′ and ξ′1 given in terms of ξ, ξ1 ∈ R

d, and σ ∈ Sd−1+ ≡ {σ ∈ Sd−1|σ · ξ ≥ σ · ξ1}

by the formulas

ξ′ =ξ + ξ1

2+

ξ − ξ1

2σ, ξ′1 =

ξ + ξ1

2− ξ − ξ1

2σ. (1.5)

The collision kernel b = b(z, σ) is, in general, an a.e. positive function defined on Rd × Sd−1

+

and depends only on |z| and |z · σ|. For simplicity, we consider a hard sphere kernel, i.e.b(z, σ) = |z · σ|. The solutions of the function equation Q(f ε, f ε) = 0 are Maxwellians, whichhave the form

Mε = M(ρε,uε,Kε)(ξ) =

ρε

(2πKε)d/2exp

{− |ξ − uε|2

2Kε

}, (1.6)

for some ρε > 0, Kε > 0 and uε ∈ Rd.

The VPB system has been studied by many authors. DiPerna and Lions [3] establishedthe global-in-time renormalized solution with arbitrary amplitude. Desvilletes and Dolbeault[4] studied the long-time behavior of the weak solutions with an extra regularity assumptionto the VPB system for the initial boundary problem. Guo [5] obtained the global existence ofsmooth solutions to the VPB system with periodic boundary condition when the initial datumis a perturbation of a global Maxwellian. For the Cauchy problem of the VPB system, Guo[6] established global smooth solutions for certain soft potential when the initial datum is aperturbation of vacuum; by using the so-called Macro–Micro decomposition of the distributionfunction of Boltzmann equations originally introduced in [7], Yang et al. [8] obtained the globalexistence of smooth solutions when the initial datum is a perturbation of a global Maxwellian.For more references on either the pure Boltzmann equation, Vlasov–Poisson system or thesemiconductor Vlasov–Poisson–Boltzmann system, see [1, 2, 7, 9–12].

The purpose of this paper is to investigate the hydrodynamic limit of the problem (1.1)–(1.3) in the so-called quasineutral regime. It is well known that, for the Boltzmann operator,we have ∫

Rd

Q(f ε, f ε)dξ =∫

Rd

ξi Q(f ε, f ε)dξ =∫

Rd

|ξ|2 Q(f ε, f ε)dξ = 0, i = 1, 2, 3, (1.7)

and the conservation of total energy12

∫Rd

∫Td

|ξ|2f ε(t, x, ξ)dxdξ +ε

2

∫Td

|∇Φε(t, x)|2dx = E0, (1.8)

where E0 = 12

∫Rd

∫Td |ξ|2f ε

0(x, ξ)dxdξ + ε2

∫Td |∇Φε(0, x)|2dx. Denote

ρε(t, x) =∫

Rd

f ε(t, x, ξ)dξ, Jε(t, x) = ρεuε(t, x) =∫

Rd

ξ f ε(t, x, ξ)dξ. (1.9)

Then the conservation of mass and of momentum reads∂tρ

ε + ∇ · Jε = 0, (1.10)

∂tJε + ∇x ·

∫Rd

(ξ ⊗ ξ)f εdξ + ∇Φε +ε

2∇(|∇Φε|2) − ε∇ · (∇Φε ⊗∇Φε) = 0 (1.11)

with−εΔΦε = ρε − 1, (1.12)

where the symbol “⊗” denotes the tensor product of vectors.To understand the fluid structure of the term

∫Rd(ξ ⊗ ξ)f εdξ, we rewrite the distribution

function f ε in the problem (1.1)–(1.3) asf ε = M

ε + εGε, (1.13)where M

ε, the local Maxwellian given by (1.6), is the macroscopic fluid part of f ε and Gε(x, t, ξ)is the microscopic non-fluid part of f ε. Then, a direct computation gives∫

Rd

(ξ ⊗ ξ)f εdξ =Jε

⊗Jε

ρε+ ρεKε

Id + O(ε), (1.14)

Convergence of the Vlasov–Poisson–Boltzmann System 763

where Id is an identity matrix, Kε denotes the temperature, and

ρε(3

2Kε +

12|uε|2

)(t, x) =

12

∫Rd

|ξ|2f ε(t, x, ξ)dξ.

Let us start with a purely formal analysis on the limit ε → 0. First, it follows from the Poissonequation that ρε → 1. Then setting ε = 0, we obtain from (1.2), (1.10), (1.11) and (1.14) thatthere holds, for perfectly cold electrons, i.e. the case where the temperature Kε vanishes, that

∇ · J = 0, (1.15)∂tJ + ∇ · (J ⊗ J) + ∇Φ = 0, (1.16)

which is the classical Euler equations to the incompressible fluid, where J is the limit of Jε (ifit exists) in some sense, and Φ is a pressure function.

The main aim of this paper is to rigorously establish the above hydrodynamical limit.Before stating the main result of this paper, we recall the existence result on the incom-

pressible Euler equations. Consider the periodic boundary problem of Euler equations to theincompressible fluid

∇ · u = 0, t > 0, x ∈ Td, (1.17)

∂tu + (u · ∇)u + ∇p = 0, t > 0, x ∈ Td, (1.18)

u(0, x) = u0(x) ∈ H s, (1.19)where the function space H s is given by H s = {u ∈ Hs(Td) | ∇ · u = 0}. We haveProposition 1.1 ([13, 14, 15]) For each u0 ∈ H s(s > 1 + d/2), there exist a unique T ∗ ∈(0,∞] (T ∗ = +∞ if d = 2) and a unique solution u ∈ L∞

loc([0, T ∗), Hs) satisfying∫ T∗

0

|| curl u ||L∞(Td) dt = +∞, if d = 3,

and for any T < T ∗,sup

0≤t≤T

(‖u‖Hs + ‖∂tu‖Hs−1 + ‖∇p‖Hs + ‖∂t∇p‖Hs−1

) ≤ C(T ), (1.20)

for some positive constant C(T ), depending only upon T .Now we state the main result.

Theorem 1.2 Let 0 < T < T ∗ and u0 be a given vector in H s(s > 1 + d/2), Zd periodic in

x. Assume that f ε0(x, ξ) ≥ 0 to be smooth, Z

d is periodic in x, and f ε0 decays fast as ξ → ∞.

In addition, we assume that∫Rd

f ε0(x, ξ)dξ = 1 + o(ε1/2) as ε → 0, (1.21)

in the strong sense of the space H−1(Td) and12

∫Rd

∫Td

|ξ − u0(x)|2f ε0(x, ξ)dxdξ +

ε

2

∫Td

|∇Φε(0, x)|2dx → 0 as ε → 0. (1.22)

Let f ε be any nonnegative smooth solution of the problem (1.1)–(1.3). Then, up to the extractionof a subsequence, the current Jε converges weakly to the unique solution u(x, t) of the Eulerequations (1.17)–(1.18) with initial datum u0. Moreover, the divergence-free part of Jε convergesto u in L∞(0, T ; L2(Td)).Remark 1.1 The assumption on the initial datum in Theorem 1.2 can be guaranteed, forexample, by taking f ε

0 as

f ε0(x, ξ) =

1(2πεα)d/2

exp{− |ξ − u0|2

2εα

},

for some α > 0.The proof of our result is based on the so-called relative-entropy method, going back to

Dafermos [16] and introduced by Brenier [17] to prove the convergence of the Vlasov–Poissonsystem to the Euler equations of the incompressible fluid. This method seems to be a powerfultool in the argument for the hydrodynamical limit problem. The method has also been applied


to show the convergence of the Schrodinger–Poisson system to Euler equations, see [18–20].Introduce the modulated energy functional

Hε(t) =12

∫Rd

∫Td

|ξ − u(t, x)|2f ε(t, x, ξ) dxdξ +ε

2

∫Td

|∇Φε(t, x)|2 dx, (1.23)

where u(t, x) is the smooth solution to equations (1.17)–(1.18). By a combination of the fluiddynamic formulation (1.10)–(1.11) and the equations (1.2), (1.17)–(1.18), we can obtain that

d

dtHε(t) ≤ CHε(t) + Cε1/2, (1.24)

for some constant C > 0. The condition (1.22) and inequality (1.24) imply that Hε(t) → 0 asε → 0, which gives our desired result.

For pure Bolztmann equations, the Euler limit and Navier–Stokes limit of the renormalizedsolution have been studied by Golse and Saint–Raymond, see [21, 22] for details. For thesemiconductor VPB system, the convergence regime is very different from that of the presentpaper due to the different collision operator, see Golse and Poupaud [23], where the authorsproved the convergence of the bipolar semiconductor VPB system to a nonlinear parabolicsystem.

When Q(f ε, f ε) ≡ 0, the VPB system becomes the Vlasov–Poisson (VP) system. Brenier[17] proved that, for the VP system in the so-called quasi-neutral regime with cold electrons,the current converges to the dissipative solution to the incompressible Euler equations. Thisresult was extended by Masmoudi [24], in which he shows that the current converges to thestrong solution of the Euler equations for the case of an “ ill-prepared” initial datum. Lighted bythese ideas, it is possible to prove that the divergence-free part of Jε converges to the dissipativesolution of the Euler equations of the incompressible fluid for the case of cold elections. Namely,we haveTheorem 1.3 Let 0 < T < T ∗ and J0(x) be a given vector which is divergence-free, Z

d beperiodic in x, and square integrable. Assume that f ε

0(x, ξ) ≥ 0 to be smooth, Zd is periodic in

x, and f ε0 decays fast as ξ → ∞. In addition, we assume that (1.21) holds and

12

∫Rd

∫Td

|ξ − v0(x)|2f ε0(x, ξ)dxdξ +

ε

2

∫Td

|∇Φε(0, x)|2dx

→∫

Td

|J0 − v0(x)|2dx as ε → 0 (1.25)

for any divergence-free and Zd periodic vector field v0(x). Let f ε be any nonnegative smooth

solution to the problem (1.1)–(1.3). Then, up to the extraction of a subsequence, the divergence-free part of Jε converges in C0([0, T ], D ′(Rd)) to a dissipative solution J ∈ C0([0, T ], L2(Td)−w)for the incompressible Euler equations (1.15)–(1.16), in the sense of Lions [14], with initialdatum J0.

2 Proofs of Theorem 1.2 and Theorem 1.3We shall prove our main theorems by using the relative-entropy method. In the following, weomit the integration domain in the integral symbols for convenience, and denote C for a genericconstant.Proof of Theorem 1.2 To prove this theorem we establish the following lemmas first:Lemma 2.1 Under the hypotheses of Theorem 1.2 or Theorem 1.3, we have, up to theextraction of a subsequence, ρε converges to 1 in C0([0, T ], D ′(Td)), the current Jε converges toJ in L∞([0, T ], D ′(Td)), J ∈ L∞([0, T ], L2(Td)), and the divergence-free parts of Jε convergesto J in C0([0, T ], D ′(Td)).Proof We first show that ρε → 1 in C0([0, T ], D ′(Td)). In fact, for any test function η(x) ∈C∞

0 (Td), using the Poisson equation (1.2) and the total energy equality (1.8), we get∣∣∣∫

(ρε(t, x) − 1)η(x)dx∣∣∣ = ε1/2

(ε

∫|∇Φε(t, x)|2dx

)1/2( ∫|∇η(x)|2dx

)1/2


≤ Cε1/2( ∫

|∇η(x)|2dx)1/2

.

By the total energy equality (1.8) we have∫|Jε(t, x)|dx ≤

( ∫∫|ξ|2f ε(t, x, ξ)dxdξ

)1/2( ∫∫f ε(t, x, ξ)dxdξ

)1/2

≤ C, (2.1)

thus, Jε is bounded in L∞([0, T ], L1(Td)). Hence, up to extracting a subsequence, we can claimthat Jε converges weakly to a Radon measure J on [0, T ] × T

d. For each nonnegative functionz(t) ∈ C0([0, T ]), we consider the convex functional of a (Radon) measure

K(ρε, Jε)�=

∫ T

0

∫ |Jε(t, x)|22ρε(t, x)

z(t) dxdt

= supb

∫ T

0

∫ {− 1

2|b(t, x)|2ρε(t, x) + b(t, x) · Jε(t, x)

}z(t)dxdt,

where b spans the space of all continuous functions from [0, T ] × Td to R

d. Noticing theinequality (2.1) and the fact that the functional K is lower semi-continuous with respect to theconvergence of measure, we get∫ T

0

z(t)(∫

|J(t, x)|2dx

)dt ≤ C

∫ T

0

z(t)dt,

which means that J ∈ L∞([0, T ], L2(Td)). From the Poisson equation (1.2) and the conservationof mass (1.10), we get ∇x · Jε = ε∂tΔΦε, thus, J is divergence-free in x in the sense of distri-butions. From the equation (1.11), we have that ∂tJ is bounded in L∞([0, T ], D ′(Td)). UsingAubin’s lemma, we obtain that, up to the extraction of a subsequence, J ∈ C0([0, T ], L2(Td)−w).Similarly, we can show that the divergence-free part of Jε converges to J in C0([0, T ], D ′(Td)).

Since Jε converges to J , we need to show that J = u in L∞([0, T ], L2(Td)). To this end, weprove the convergence of Hε(t) as ε → 0 next.

Lemma 2.2 Let u be the unique solution of the Euler equations (1.17)–(1.18) with initialdatum u0 and the hypotheses of Theorem 1.2 hold. Then, for any t ∈ (0, T ], Hε(t) → 0 asε → 0.

Proof By the definition of Hε(t) (Eq. (1.23)) and the conservation of total energy, we haved

dtHε(t) =

12

∫ρε

t|u|2 dx +12

∫ρε∂t|u|2 dx −

∫u · ∂tJ

ε dx −∫

Jε · ∂tu dx

≡ I1 + I2 + I3 + I4.

We first compute the third term I3. Using the equations (1.10) and (1.11), one gets

I3 = −∫

u · ∂tJε dx

=∫

u ·(∇x ·

∫(ξ ⊗ ξ)f ε dξ

)dx +

∫u · ∇Φε dx

− ε

∫u · (∇ · (∇Φε ⊗∇Φε)

)dx +

ε

2

∫u · ∇(|∇Φε|2) dx

= ε

∫d(u) : ∇Φε ⊗∇Φε dx −

∫d(u) : (ξ − u) ⊗ (ξ − u)f ε dξdx

−∫

Du : Jε ⊗ u dx −∫

Du : u ⊗ Jε dx +∫

ρεDu : u ⊗ u dx,

where Du denotes the matrix (∂uj/∂xk)jk, 1 ≤ j, k ≤ d, d(u) is the symmetric part of Du, andthe symbol “ : ” means summation over both matrix indices.

Notice

−∫

Du : Jε ⊗ u dx =12

∫∇ · Jε|u|2 dx = −

∫12∂tρ

ε|u|2 dx = −I1,


and

I2 + I4 −∫

Du : u ⊗ Jε dx +∫

ρεDu : u ⊗ u dx

=∫

[∂tu + (u · ∇)u] · (ρεu − Jε) dx = −∫

∇p · (ρεu − Jε) dx.

Thus, one can obtaind

dtHε(t) ≤ 2‖d(u)‖L∞Hε(t) −

∫∇p · (ρεu − Jε) dx. (2.2)

Now we deal with the second term on the right-hand side of (2.2). We have

−∫

∇p · (ρεu − Jε) dx = −∫

∇p · (ρε − 1)u dx −∫

∇p · (u − Jε) dx

= ε

∫ΔΦε(∇p · u) dx +

∫p∂tρ

ε dx. (2.3)

Applying the Young inequality we obtain

ε∣∣∣∫

ΔΦε(∇p · u) dx∣∣∣ ≤ ε

(√ε||∇Φε||2L2 +

1√ε||∇(∇p · u)||2L2

). (2.4)

A direct calculation gives∫p ∂tρ

ε dx =∫

p ∂t(ρε−1) dx = −ε

∫p ∂t(ΔΦε) dx = −ε

d

dt

∫pΔΦε dx+ε

∫ΔΦεpt dx. (2.5)

Applying the Young inequality again, we get

ε∣∣∣∫

ΔΦεptdx∣∣∣ = ε

∣∣∣∫

∇Φε∇pt dx∣∣∣ ≤ ε

(√ε||∇Φε||2L2 +

1√ε||∇pt||2L2

). (2.6)

Combining (2.2)–(2.6) together, we getd

dtHε(t) ≤ CHε(t) − ε

d

dt

∫pΔΦεdx + Cε1/2. (2.7)

To apply Gronwall’s inequality, we rewrite (2.7) asd

dt

{Hε(t) + ε

∫pΔΦεdx

}≤ C

(Hε(t) + ε

∫pΔΦεdx

)− Cε

∫pΔΦεdx + Cε1/2

≤ C(Hε(t) + ε

∫pΔΦεdx

)+ Cε1/2,

where we have used the estimate

ε∣∣∣∫

pΔΦε dx∣∣∣ ≤ ε

(√ε||∇Φε||2L2 +

1√ε||∇p||2L2

).

By condition (1.22) and Gronwall’s inequality, we obtainlimε→0

Hε(t) = 0. (2.8)

To complete the proof of Theorem 1.2, we introduce a new functional

hε(t)�=

∫ |Jε(t, x) − ρε(t, x)u(t, x)|22ρε(t, x)

dx

= supb

∫ {− 1

2|b(x)|2ρε(t, x) + b(x) · (Jε − ρε(t, x)u(t, x)

)}dx, (2.9)

where b spans the space of all continuous functions from Td to R

d. By the Cauchy–Schwarzinequality, we have

hε(t) ≤ 12

∫∫|ξ − u(t, x)|2f ε(t, x, ξ) dxdξ ≤ Hε(t).

Since ρε → 1, Jε → J , in view of the convexity of the functional defined by (2.9), we get∫|J(t, x) − u(t, x)|2 dx ≤ 2 lim

ε→0hε(t) ≤ 2 lim

ε→0Hε(t) = 0.

This completes the proof of Theorem 1.2.Proof of Theorem 1.3 To prove our conclusion, we first give a weak formula of Gronwall’sinequality.


Lemma 2.3 Let a(t), f(t), u(t) be functions well defined in [0, T ] with a(t) ∈ L1([0, T ]),a(t) ≥ 0, and f ∈ C0([0, T ]). Suppose that the inequlity

−∫ T

0

u(t)z′(t)dt − u(0)z(0) ≤∫ T

0

(a(t)u(t) + f(t))z(t)dt (2.10)

holds for any z(t) ∈ C∞([0, T ]) satisfying z(t) ≥ 0, z′(t) + a(t)z(t) ≤ 0, and z(T ) = 0. Then

u(t) ≤ u(0)e� t0 a(s)ds +

∫ t

0

e� t

sa(τ)dτf(s)ds, a.e. t ∈ [0, T ]. (2.11)

Proof Denote y(t) = z(t)e� t0 a(τ)dτ , v(t) = u(t)e−

� t0 a(τ)dτ . Then one gets, from (2.10), that

−∫ T

0

v(t)y′(t)dt − v(0)y(0)≤∫ T

0

f(t)e−� t0 a(τ)dτy(t)dt=−

∫ T

0

(∫ t

0

f(s)e−� s0 a(τ)dτds

)y′(t)dt,

i.e.∫ T

0{v(t) − v(0) − ∫ t

0f(s)e−

� s0 a(τ)dτds}(−y′(t))dt ≤ 0, which gives the result (2.11).

Now we prove Theorem 1.3. Since the proof is similar to that of Theorem 1.2, we onlysketch it here. First, we introduce the modulated energy functional

Hεv(t) =

12

∫Rd

∫Td

|ξ − v(t, x)|2f ε(t, x, ξ) dxdξ +ε

2

∫Td

|∇Φε(t, x)|2 dx,

where v(t, x) is a test function on [0, T ]×Td, which is Z

d periodic and divergence free in x. Asin the proof of Theorem 1.2, we can get

d

dtHε

v(t) ≤ 2‖d(v)‖L∞Hεv(t) +

∫(∂tv + v · ∇v) · (ρεv − Jε) dx. (2.12)

Now we define the functional

hεv(t)

�=

∫ |Jε(t, x) − ρε(t, x)v(t, x)|22ρε(t, x)

dx

= supb

∫ {− 1

2|b(x)|2ρε(t, x) + b(x) · (Jε − ρε(t, x)v(t, x)

)}dx, (2.13)

where b spans the space of all continuous functions from Td to R

d. By the Cauchy–Schwarzinequality, we have

hεv(t) ≤ 1

2

∫∫|ξ − v(t, x)|2f ε(t, x, ξ) dxdξ ≤ Hε

v(t).

Thus, for fixed v, Hεv(t) and hε

v(t) are bounded functions in L∞([0, T ]) and, up to the extractionof a subsequence, respectively converge, in the weak-∗ sense, to some functions Hv(t) and hv(t)with Hv ≥ hv. Since ρε → 1, Jε → J as ε → 0, in view of the convexity of the functional definedby (2.13), we get

∫ |J(t, x) − v(t, x)|2 dx ≤ 2hv(t). The assumptions on the initial conditionsgive

Hεv(0) =

12

∫Rd

∫Td

|ξ − v(0, x)|2f ε0(x, ξ) dxdξ +

ε

2

∫Td

|∇Φε(0, x)|2 dx → H0,v,

where H0,v = 12

∫ |J0 − v0|2dx.Then we rewrite (2.12) in the weak form

−∫ T

0

Hεv(t)z′(t)dt − z(0)Hε

v(0) ≤∫ T

0

2‖d(v)‖L∞Hεv(t)z(t)dt

+∫ T

0

∫(∂tv + v · ∇v) · (ρεv − Jε)z(t) dxdt, (2.14)

for all test functions z ≥ 0 in D ′([0, T ]). Thus, we can pass to the limit in (2.14) to get

−∫ T

0

Hv(t)z′(t)dt−z(0)H0,v ≤∫ T

0

2‖d(v)‖L∞Hv(t)z(t)dt+∫ T

0

∫(∂tv+v·∇v)·(v−J)(t, x)z(t)dxdt,

which, together with Lemma 2.3, gives∫|J(t, x) − v(t, x)|2dx ≤

∫|J0(x) − v(0, x)|2dx e

� t0 2‖d(v(τ))‖L∞dτ

+ 2∫ t

0

e� t

s2‖d(v(τ))‖L∞dτ

∫(∂tv + v · ∇v) · (v − J)(s, x)dxds.


This completes our proof of Theorem 1.3.

References[1] Cercignani, C.: The Boltzmann equation and its applications, Applied Mathematical Sciences, 67. Springer,

New York, 1988

[2] Glassey, R. T.: The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics

(SIAM), Philadelphia, 1996

[3] DiPerna, R. J., Lions, P. L.: On the Cauchy problem for Boltzmann equations: global existence and weak

stability. Ann. of Math., 130, 321–366 (1989)

[4] Desvillettes, L., Dolbeault, J.: On long time asymptotics of the Vlasov–Poisson–Boltzmann equation.

Comm. Partial Differential Equations, 16, 451–489 (1991)

[5] Guo, Y.: The Vlasov–Poisson–Boltzmann system near Maxwellians. Comm. Pure Appl. Math., 55, 1104–

1135 (2002)

[6] Guo, Y.: The Vlasov–Poisson–Boltzmann system near vacuum. Comm. Math. Phys., 218, 293–313 (2001)

[7] Liu, T. P., Yang, T., Yu, S. H.: Energy method for Boltzmann equation. Physica D, 188, 178–192 (2004)

[8] Yang, T., Yu, H. J., Zhao, H. J.: Cauchy problem to Vlasov–Poisson–Boltzmann system, Preprint, 2004

http://www.math.ntnu.no/conservation/2004/027.html

[9] Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases, Applied Mathematical

Sciences, 106, Springer, New York, 1994

[10] DiPerna, R. J., Lions, P. L.: Global weak solution of Vlasov-Maxwell systems. Comm. Pure Appl. Math.,

42, 729–757 (1989)

[11] Dolbeault, J.: On long time asymptotics of the Vlasov–Poisson–Boltzmann system. Nonlinear kinetic theory

and mathematical aspects of hyperbolic systems (Rapallo, 1992), 115–123, Ser. Adv. Math. Appl. Sci., 9,

World Sci. Publishing, River Edge, NJ, 1992

[12] Sone, Y.: Kinetic theory and fluid dyamics, Birkhauer, Boston, 2002

[13] Beale, J. T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler

equations. Comm. Math. Phys., 94, 61–66 (1984)

[14] Lions, P. L.: Mathematical topics in fluid mechanics, Vol. 1. Incompressible models, Oxford Lecture

Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford

University Press, New York, 1996

[15] McGrath, F. J.: Nonstationary plane flow of viscous and ideal fluids. Arch. Rational Mech. Anal., 27,

229–348 (1968)

[16] Dafermos, C.: Hyperbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wis-

senschaften, 325, Springer-Verlag, Berlin, 2000

[17] Brenier, Y.: Convergence of the Vlasov–Poisson system to the incompressible Euler equations. Comm.

Partial Differential Equations, 25, 737–754 (2000)

[18] Jungel, A., Wang, S.: Convergence of nonlinear Schrodinger-Poisson systems to the compressible Euler

equations. Comm. Partial Differential Equations, 28, 1005–1022 (2003)

[19] Puel, M.: Convergence of the Schrodinger-Poisson system to the incompressible Euler equations. Comm.

Partial Differential Equations 27, 2311–2331 (2002)

[20] Puel, M.: Convergence of the Schrodinger-Poisson system to the Euler equations under the influence of a

large magnetic field. M 2AN Math. Model. Numer. Anal., 36, 1071–1090 (2002)

[21] Golse, F., Saint-Raymond, L.: The Navier–Stokes limit of the Boltzmann equation for bounded collision

kernels. Invent. Math., 155, 81–161 (2004)

[22] Saint-Raymond, L.: Convergence of solutions to the Boltzmann equation in the incompressible Euler limit.

Arch. Ration. Mech. Anal., 166, 47–80 (2003)

[23] Golse, F., Poupaud, F.: Fluid limit of the Vlasov–Poisson–Boltzmann equation of semiconductors. BAIL

V (Shanghai, 1988), 133–136, Boole Press Conf. Ser., 12, Boole, Dun Laoghaire, 1988

[24] Masmoudi, N.: From Vlasov–Poisson system to the incompressible Euler system. Comm. Partial Differen-

tial Equations, 26. 1913–1928 (2001)

convergence of the vlasov–poisson–boltzmann system to the incompressible euler equations

Documents