Quasineutral Limits of the Vlasov- Introduction VPFP system - - PowerPoint PPT Presentation

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Quasineutral Limits of the Vlasov- Poisson/Maxwell-Fokker-Planck System

FUCAI LI Department of Mathematics, Nanjing University Email: fli@nju.edu.cn ( A joint work with Ling Hsiao and Shu Wang )

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1. Introduction

Kinetic equations are mathematical models used to describe the “dilute parti- cle gases” at an intermediate scale between the microscopic and the macro- scopic level. They appear in a variety of sciences such as plasma, astrophysics, aerospace engineering, nuclear engineering, particle-fluid interactions, semi- conductor technology, social sciences and biologies, see

• B. Perthame. Bull. Amer. Math. Soc. (N.S.), 41(2):205–244 2004.
• N. Bellomo, A. Palczewski, and G. Toscani. Mathematical topics in non-

linear kinetic theory,1988.

• L. Arlotti, N. Bellomo, and E. De Angelis. Math. Models Methods Appl.

Sci., 12(4):567–591, 2002.

• A. Decoster, P. A. Markowich, and B. Perthame. Modeling of collisions,

1998.

• N. Bellomo and M. Pulvirenti, editors.

Modeling in applied sciences, A kinetic theory approach, 2000.

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Now the study on kinetic equations is one of the most active areas in applied

• mathematics. The quasineutral limit problem plays an important role in the

theory of kinetic equations since it gives the relations between the microscopic mathematical models and the macroscopic ones. We discuss the quasineutral limits of some kinetic equations including

• Vlasov-Poisson-Fokker-Planck (VPFP) system
• Vlasov-Maxwell-Fokker-Planck (VMFP) system

We shall show that the solution of VPFP system converges to the solution of incompressible Euler equations with or without damping, and the renormalized solution of VMFP system converges to the solution of so-called electron mag- netohydrodynamics (e-MHD) equations.

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2. The Vlasov-Poisson-Fokker-Planck (VPFP) system

The VPFP system takes the form ∂tf ǫ + ξ · ∇xf ǫ − ∇xΦǫ · ∇ξf ǫ = 1 τ divξ

• ξf ǫ + σ∇ξf ǫ

, (1)

• RN f ǫ(t, x, ξ)dξ = 1 − ǫ∆Φǫ

(2) with initial datum f ǫ(0, x, ξ) = f ǫ

0(x, ξ) ≥ 0.

(3) Here f(t, x, ξ) denotes the distribution function for particles, which expresses the probability of finding a particle at time t ≥ 0 in a position x ∈ [0, 1]N ≡ TN, and with a velocity ξ ∈ RN, N = 1, 2, 3. The spatially periodic electric potential Φǫ is coupled with f ǫ(t, x, ξ) through the Poisson equation (2).

• ǫ > 0:

The vacuum electric permittivity

• σ > 0:

The thermal diffusion coefficient

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• τ > 0:

The relaxation time due to collisions of the particles with the thermal bath. For more physical background and the derivation of the VPFP system, see

• S. Chandrasekhar.

Stochastic problems in physics and astronomy. Rev. Modern Physics, 15:1–89, 1943.

• S. Chandrasekhar.

Brownian motion, dynamical friction, and stellar dy-

• namics. Rev. Modern Physics, 21:383–388, 1949.

Some know results of VPFP system:

• 1. The existence and uniqueness of smooth solutions of VPFP system:
• F. Bouchut. J. Funct. Anal.,1993.;
• G. Rein and J. Weckler. J. Differential Equations, 1992;
• Jr. H. D. Victory and B. P. O’Dwyer. Indiana Univ. Math. J., 1990.

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• 2. Different kind of limits problem on VPFP system.

Some results on different kind of limits problem on VPFP system under different scalings and different assumptions have been obtained:

• Parabolic limit. F. Poupaud and J. Soler [ Math. Models Methods Appl.

Sci., 2000 ] consider the form ǫ2∂tfǫ + ǫ((v · ∇x)fǫ − (∇xΦǫ · ∇v)fǫ) = ∆vfǫ + divv(vfǫ), ∆xΦǫ =

• R3 fǫdv

They showed the above VPFP sysmtem converges to a parabolic equation as ǫ → 0.

• High-field limit. T. Goudon, J. Nieto, F. Poupaud, and J. Soler [ JDE, 2005]

consider the form ǫ(∂tfǫ + v · ∇xfǫ) − ∇xΦǫ · ∇vfǫ = ∆vfǫ + divv(vfǫ), ∆xΦǫ = nǫ − Dǫ, nǫ(t, x) =

• R3 fǫdv

They showed the above VPFP system converges to a nonlinear systems of PDEs related to Ohm’s Law as ǫ → 0.

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• Low-field limit. Using the Hilbert expansion, A. Arnold, J. A. Carrillo,
• I. Gamba, and C.-W. Shu [Transport Theory Statist. Phys., 2001] obtain

the convergence of VPFP system to drift-diffusion equations. We are interested in the quasineutral limits of VPFP system (1)-(2), i.e., ∂tf ǫ + ξ · ∇xf ǫ − ∇xΦǫ · ∇ξf ǫ = 1 τ divξ

• ξf ǫ + σ∇ξf ǫ

,

• RN f ǫ(t, x, ξ)dξ = 1 − ǫ∆Φǫ.

. We discuss two cases:

• ǫ → 0, τ > 0 is fixed, and σ → 0.

In this case the VPFP system converges to incompressible Euler Equations with damping.

• ǫ → 0, τ → +∞, and σ > 0 is fixed.

In this case the VPFP system converges to incompressible Euler Equations without damping. Remark: 1. The case ǫ, τ, → 0 is trivial because we obtain J = 0 formally.

• 2. The case for which τ → 0, ǫ > 0 and σ > 0 are fixed corresponds to the

hyperbolic limit conjectured by F. Poupaud and J. Soler in the paper [ Math. Models Methods Appl. Sci., 2000], but we don’t know whether or not they have

• btained the result.

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3. The convergence to incompressible Euler equations with damping

We study the quasi-neutral limits of VPFP system for the first case, i.e. ǫ → 0, τ, σ > 0 are fixed. We set τ = 1 for convenience. Now the VPFP system reads ∂tf ǫ + ξ · ∇xf ǫ − ∇xΦǫ · ∇ξf ǫ = divξ

• ξf ǫ + σ∇ξf ǫ

, (4)

• RN f ǫ(t, x, ξ)dξ = 1 − ǫ∆Φǫ

(5) with initial datum f ǫ(0, x, ξ) = f ǫ

0(x, ξ) ≥ 0.

(6)

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Denote ρǫ(t, x) =

• RN f ǫ(t, x, ξ)dξ,

Jǫ(t, x) = ρǫuǫ(t, x) =

• RN ξ f ǫ(t, x, ξ)dξ. (7)

Multiplying the equation (4) by 1 and ξ respectively, and integrating the results with respect to ξ, we have ∂tρǫ + ∇ · Jǫ = 0, (8) ∂tJǫ + ∇x ·

• RN(ξ ⊗ ξ)f ǫdξ + ∇Φǫ

+ ǫ 2∇(|∇Φǫ|2) − ǫ∇ · (∇Φǫ ⊗ ∇Φǫ) + Jǫ = 0 (9) with −ǫ∆Φǫ = ρǫ − 1, (10) where we have used the identity ∇Φǫ(∆Φǫ) = 1 2∇(|∇Φǫ|2) − ∇ · (∇Φǫ ⊗ ∇Φǫ) and the symbol “⊗” which denotes the tensor product of vectors.

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To understand the fluid structure of the term

• RN(ξ ⊗ ξ)f ǫdξ, we rewrite the

distribution function f ǫ in the system (4)-(5) as f ǫ = ρǫ (2πKǫ)N/2 exp

• −|ξ − uǫ|2

2Kǫ

• +ǫGǫ(t, x, ξ) ≡ M(ρǫ,uǫ,Kǫ)(ξ)+ǫGǫ, (11)

where M(ρǫ,uǫ,Kǫ)(ξ) is a local Maxwellian denoting the macroscopic fluid part of f ǫ, Gǫ is the microscopic non-fluid part of f ǫ, and Kǫ denotes the temperature. Then, a direct computation gives

• RN(ξ ⊗ ξ)f ǫdξ = Jǫ Jǫ

ρǫ + ρǫKǫ IN + O(ǫ), (12) where IN is an identity matrix, and 1 2

• RN |ξ|2f ǫ(t, x, ξ)dξ = ρǫ

3 2Kǫ + 1 2|uǫ|2

• (t, x) + O(ǫ).

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We start with a purely formal analysis on the limit process as ǫ → 0 First, it follows from the Poisson equation (10) that ρǫ → 1. Then setting ǫ = 0, we

• btain from (8), (9), and (10) that it holds gives

∇ · J = 0, (13) ∂tJ + ∇ · (J ⊗ J) + ∇Φ + J = 0 (14) for the case where the temperature Kǫ vanishes, which is the case for perfectly cold electrons. (13)-(14) are the incompressible Euler equations with the damp- ing, where J is the limit of Jǫ (if it exists) in some sense, and Φ is a pressure function. We will establish the above limit rigorously. Before stating the main result, we give the following existence result on the incompressible Euler equations with

• damping. Consider the periodic boundary problem of the incompressible Euler

equations with damping ∇ · u = 0, t > 0, x ∈ TN, (15) ∂tu + (u · ∇)u + ∇p + u = 0, t > 0, x ∈ TN, (16) u(0, x) = u0(x) ∈ Hs, (17) where the function space Hs is given by

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We have

• Theorem. 1 For each u0 ∈ Hs(s > 1 + N/2), there exist a T ∗ ∈ (0, ∞) and a

unique solution u ∈ L∞

loc([0, T ∗), Hs) satisfying, for any T < T ∗, that

sup

0≤t≤T

• uHs + ∂tuHs−1 + ∇pHs + ∂t∇pHs−1

≤ C(T) (18) for some positive constant C(T), depending only upon T. The proof of this theorem is similar to that of Theorem 3.4 in

• Y. J. Peng and Y. G. Wang.

Convergence of compressible Euler-Poisson equations to incompressible type Euler equations. Asymptot. Anal. 41 (2005), 141–160. The proof of above theorem is based on the iteration method and compactness argument.

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• Theorem. 2 Let 0 < T < T ∗ and u0 be a given vector in Hs(s > 1+N/2), ZN

periodic in x. Assume that f ǫ

0(x, ξ) ≥ 0 is smooth, ZN periodic in x with total

mass 1, and decays fast as ξ → ∞. In addition, we assume that there exists a constant C0 > 0, independent of ǫ, such that

• RN×TN
• 1 + |x|2 + |ξ|2 + | ln f ǫ

0(x, ξ)|

• f ǫ

0(x, ξ)dxdξ ≤ C0,

(19) furthermore,

• RN f ǫ

0(x, ξ)dξ = 1 + o(ǫ1/2)

as ǫ → 0 (20) in the strong sense of the space H−1(TN), and 1 2

• RN×TN |ξ − u0(x)|2f ǫ

0(x, ξ)dxdξ + σ

• RN×TN f ǫ

0(x, ξ)| ln f ǫ 0(x, ξ)|dxdξ

+ ǫ 2

• TN |∇Φǫ(0, x)|2dx → 0

as ǫ, σ → 0. (21) Let f ǫ be any nonnegative smooth solution of the problem (4)-(6). Then, up to the extraction of a subsequence, the current Jǫ converges weakly to the unique solution u(x, t) of the incompressible Euler equations (15)-(16) with initial data u0. Moreover, the divergence-free part of Jǫ converges to u in L∞(0, T; L2(TN)).

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Remark: The assumption on initial datum in Theorem 2 can be guaranteed, for example, by taking f ǫ

0 as

f ǫ

0(x, ξ) =

1 (2πǫα)N/2 exp

• − |ξ − u0|2

2ǫα

• for some α > 0.

The proof of Theorem 2 is based on the compactness argument and the so- called modulated energy method (also known as relative-entropy method). The idea of this method is to modulate the energy of the system by test functions, and to obtain a stability inequality when these test functions are solutions of the limiting system. We must point out the fact that the energy of the VPFP system is dissipative due to the effects of the friction forces and heat diffusions, which brings new diffi- culties to use the modulated energy mothod. To overcome it, we modified the method for the part energy of VPFP system and require σ → 0 in our argument.

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Outline of the proof of Theorem 2:

• Step 1:

Obtain basic estimates for the solution f ǫ;

• Step 2:

Analyze the compactness of ρǫ, Jǫ;

• Step 3:

Show that Jǫ → J in some sense and obtain that the limit J = u in L2. This is a crucial step in the proof.

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We first give some basic properties of the solution to the VPFP system and then prove Theorem 2. In the sequel, we omit the integration domain in the integral symbols for convenience, and denote C for a generic constant, only depending upon T and u. Lemma 3 Assume that f ǫ

0 satisfies the conditions in Theorem 2. Let f ǫ be the

classical solution to the problem (4)-(6), then the following equalities hold:

• f ǫ(t, x, ξ)dxdξ =
• f ǫ

0dxdξ,

(22) d dt |ξ|2 2 + σ ln f ǫ fdxdξ + ǫ 2

• |∇Φǫ(t, x)|2dx
• = −
• |ξ
• f ǫ + 2σ∇ξ
• f ǫ|2dxdξ.

(23)

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Denote Eǫ(t) = |ξ|2 2 + σ ln f ǫ f ǫdxdξ + ǫ 2

• |∇Φǫ(t, x)|2dx.

(24) We know from (23) that Eǫ(t) ≤ Eǫ(0). (25) We shall show that Eǫ(t) is also bounded from below. First, we have Lemma 4 Assume that f ǫ

0 satisfies the conditions in Theorem 2. Let f ǫ be the

classical solution to the problem (4)-(6). Then the following quantities are bounded for any t ∈ [0, T],

• |ξ|2f ǫdxdξ ≤ CT,

(26)

• |x|2f ǫdxdξ ≤ CT,

(27)

• f ǫ| ln f ǫ|dxdξ ≤ CT,

(28) where CT is independent of ǫ, σ and t.

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Lemma 5 Under the hypotheses of Theorem 2, we have, up to the extraction of a subsequence, ρǫ converges to 1 in C0([0, T], D′(TN)), the current Jǫ converges to J in L∞([0, T], D′(TN)), J ∈ L∞([0, T], L2(TN)), and the divergence-free parts of Jǫ converges to J in C0([0, T], D′(TN)). With these estimates, we prove Theorem 2 next. By Lemma 5, we get that Jǫ converges to J in L∞([0, T], D′(TN)) and J ∈ L∞([0, T], L2(TN)). To complete the proof, we only need to show that J = u in L∞([0, T], L2(TN)). Denote Hǫ(t) = 1 2

• |ξ|2f ǫ(t, x, ξ)dxdξ + ǫ

2

• |∇Φǫ(t, x)|2dx.

(29) We have Eǫ(t) = Hǫ(t) + σ

• f ǫ ln f ǫdxdξ.

(30) Introduce the modulated energy Eǫ

u(t) = Hǫ u(t) + σ

• f ǫ ln f ǫdxdξ,

(31) where Hǫ

u(t) =

|ξ − u|2 2 f ǫdxdξ + ǫ 2

• |∇Φǫ(t, x)|2dx.

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We have the following conclusion on Hǫ

u(t).

Lemma 6 Let u be the unique solution of the incompressible Euler equations (15)-(16) with initial datum u0 and the hypotheses of Theorem 2 hold. Then, for any t ∈ (0, T], Hǫ

u(t) → 0

as ǫ, σ → 0. Proof By the definition of Eǫ

u(t) (Eq. (31)) and the total energy equation (23),

we have d dtEǫ

u(t) = 1

2

• ρǫ

t|u|2 dx + 1

2

• ρǫ∂t|u|2 dx −
• u · ∂tJǫ dx

−

• Jǫ · ∂tu dx −
• |ξ
• f ǫ + 2σ∇ξ
• f ǫ|2dxdξ.

By a careful calculation with the help of using Yang’s inequality, Gronwall’s inequality, the hypothesis condition on initial datum, and the definition of Hǫ

u(t),

we are able to get Hǫ

u(t) ≤ CTǫ1/2 + CTǫ3/2 + 2CTσ + C

t Hǫ

u(s)ds.

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By the condition (21) and Gronwall’s inequality, we obtain lim

ǫ,σ→0 Hǫ u(t) = 0.

(32) To complete the proof of Theorem 2, we only have to prove J ∈ L∞([0, T], L2(TN)) now. We introduce a new functional hǫ(t)

△

= |Jǫ(t, x) − ρǫ(t, x)u(t, x)|2 2ρǫ(t, x) dx = sup

b

− 1 2|b(x)|2ρǫ(t, x) + b(x) ·

• Jǫ − ρǫ(t, x)u(t, x)
• dx, (33)

where b spans the space of all continuous functions from TN to RN. By Cauchy- Schwarz inequality, we have hǫ(t) ≤ 1 2

• |ξ − u(t, x)|2f ǫ(t, x, ξ) dxdξ ≤ Hǫ

u(t).

Since ρǫ → 1, Jǫ → J as ǫ → 0, and the convexity of the functional defined by (33), we get, by Lemma 6,

• |J(t, x) − u(t, x)|2 dx ≤ 2 lim

ǫ,σ→0 hǫ(t) ≤ 2 lim ǫ,σ→0 Hǫ u(t) = 0.

This completes the proof of Theorem 2.

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4. The convergence to incompressible Euler equations without damping

We study the quasi-neutral limits of VPFP system for the second case, i.e. ǫ → 0, σ is fixed and τ → +∞. For convenience, we set σ = 1 and then denote 1

τ = γ. Now the VPFP system

reads ∂tf ǫ + ξ · ∇xf ǫ − ∇xΦǫ · ∇ξf ǫ = γdivξ

• ξf ǫ + ∇ξf ǫ

, (34)

• RN f ǫ(t, x, ξ)dξ = 1 − ǫ∆Φǫ

(35) with initial datum f ǫ(0, x, ξ) = f ǫ

0(x, ξ) ≥ 0.

(36) x ∈ TN, velocity ξ ∈ RN, N = 1, 2, 3. Φǫ is coupled with f ǫ(t, x, ξ) through the Poisson equation (35).

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We denote ρǫ(t, x) =

• RN f ǫ(t, x, ξ)dξ, Jǫ(t, x) = ρǫuǫ(t, x) =
• RN ξ f ǫ(t, x, ξ)dξ. (37)

Multiplying the equation (34) by 1 and ξ respectively, and integrating the results with respect to ξ, we have ∂tρǫ + ∇ · Jǫ = 0, (38) ∂tJǫ + ∇x ·

• RN(ξ ⊗ ξ)f ǫdξ + ∇Φǫ

+ ǫ 2∇(|∇Φǫ|2) − ǫ∇ · (∇Φǫ ⊗ ∇Φǫ) + γJǫ = 0 (39) with −ǫ∆Φǫ = ρǫ − 1. The similar formal analysis on limit process as ǫ, γ → 0 shows that ∇ · J = 0, (40) ∂tJ + ∇ · (J ⊗ J) + ∇Φ = 0, (41) for the case where the temperature Kǫ vanishes, which is the case for perfectly cold electrons. (40)-(41) are nothing else but the Euler equations to the incom- pressible fluid, where J is the limit of Jǫ (if it exists) in some sense, and Φ is a pressure function.

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Consider the periodic boundary problem of Euler equations to the incompress- ible fluid ∇ · u = 0, t > 0, x ∈ TN, (42) ∂tu + (u · ∇)u + ∇p = 0, t > 0, x ∈ TN, (43) u(0, x) = u0(x) ∈ Hs, (44) where the function space Hs is given by Hs = {u ∈ Hs(TN) | ∇ · u = 0}. We have

• Theorem. 7 (Beale-Kato-Majda, CMP, 1984.) For each u0 ∈ Hs(s > 1 +

N/2), there exist a T ∗ ∈ (0, ∞](T ∗ = +∞ if N = 2) and a unique solution u ∈ L∞

loc([0, T ∗), Hs) satisfying, for any T < T ∗, that

sup

0≤t≤T

• uHs + ∂tuHs−1 + ∇pHs + ∂t∇pHs−1

≤ C(T) (45) for some positive constant C(T), depending only upon T.

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Now we state the main result.

• Theorem. 8 Let 0 < T < T ∗ and u0 be a given vector in Hs(s > 1+N/2), ZN

periodic in x. Assume that f ǫ

0(x, ξ) ≥ 0 is smooth, ZN periodic in x, and decays

fast as ξ → ∞. In addition, we assume that there exists a constant C0 > 0, independent of ǫ, such that

• RN×TN
• 1 + |x|2 + |ξ|2

f ǫ

0(x, ξ)dxdξ ≤ C0,

(46) furthermore,

• RN f ǫ

0(x, ξ)dξ = 1 + o(ǫ1/2) as ǫ → 0

(47) in the strong sense of the space H−1(TN), and 1 2

• RN×TN |ξ − u0(x)|2f ǫ

0(x, ξ)dxdξ + ǫ

2

• TN |∇Φǫ(0, x)|2dx → 0

as ǫ, γ → 0. (48) Let f ǫ be any nonnegative smooth solution of the problem (34)-(36). Then, up to the extraction of a subsequence, the current Jǫ converges weakly to the unique solution u(x, t) of the Euler equations (42)-(43) with initial data u0. Moreover, the divergence-free part of Jǫ converges to u in L∞(0, T; L2(TN)).

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Using the definition of dissipative solution to incompressible Euler equations due to P. L. Lions [Mathematical topics in fluid mechanics. Vol.1, 1996], we are able to prove that the divergence-free part of Jǫ converges to the dissipative solution of the Euler equations to the incompressible fluid for the case of cold

• electrons. Namely, we have
• Theorem. 9 Let 0 < T < T ∗ and J0(x) be a given vector which is divergence-

free, ZN periodic in x, and square integrable. Assume that f ǫ

0(x, ξ) ≥ 0 to be

smooth, ZN periodic in x, and f ǫ

0 decays fast as ξ → ∞. In addition, we assume

that (46), (47) hold, and 1 2

• RN×TN |ξ − v0(x)|2f ǫ

0(x, ξ)dxdξ + ǫ

2

• TN |∇Φǫ(0, x)|2dx

→ 1 2

• TN |J0(x) − v0(x)|2dx

as ǫ, γ → 0 (49) for any divergence-free and ZN periodic vector field v0(x). Let f ǫ be any non- negative smooth solution to the problem (34)-(36). Then, up to the extraction of a subsequence, the divergence-free part of Jǫ converges in C0([0, T], D′(RN)) to a dissipative solution J ∈ C0([0, T], L2(TN) − w) for the Euler equations (40)-(41), in the sense of P. L. Lions, with initial datum J0.

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Since both the proof of Theorem 8 and Theorem 9 are very similar to that of Theorem 2, we only give a sketch on the proof of Theorem 9. Before proving the Theorem 9, we first give a weak formula for Gronwall’s inequality. Lemma 10 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 u(t)z′(t)dt − u(0)z(0) ≤ T (a(t)u(t) + f(t))z(t)dt (50) 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

t

0 a(s)ds +

t e

t

s a(τ)dτf(s)ds,

a.e. t ∈ [0, T]. (51)

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Now we sketch the proof of Theorem 9. First, the results obtained by lemmas 3-5 are also true for the problem (34)-(36), which give ρǫ → 1, Jǫ → J as ǫ → 0. Then, to show that Jǫ converges in C0([0, T], D′(RN)) to a dissipative solution J ∈ C0([0, T], L2(TN) − w) for the Euler equations (40)-(41), We introduce the modulated energy functional Hǫ

v(t) =

|ξ − v(t, x)|2 2 f ǫdxdξ + ǫ 2

• |∇Φǫ(t, x)|2dx,

where v(t, x) is a test function on [0, T] × TN, which is ZN periodic and divergence-free in x. As in the proof of Theorem 2, we can get d dtHǫ

v(t) ≤ 2d(v)L∞Hǫ v(t)+

• (∂tv +v ·∇v)·(ρǫv −Jǫ) dx+CTγǫ1/2 +Cγ.

(52) Now we define the functional hǫ

v(t) △

= |Jǫ(t, x) − ρǫ(t, x)v(t, x)|2 2ρǫ(t, x) dx = sup

b

− 1 2|b(x)|2ρǫ(t, x) + b(x) ·

• Jǫ − ρǫ(t, x)v(t, x)
• dx, (53)

where b spans the space of all continuous functions from TN to RN.

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By Cauchy-Schwarz inequality, we have hǫ

v(t) ≤ 1

2

• |ξ − v(t, x)|2f ǫ(t, x, ξ) dxdξ ≤ Hǫ

v(t).

Thus, it is easy to know that for fixed v, Hǫ

v(t) and hǫ v(t) are bounded functions

in L∞([0, T]) and, up to the extraction of a subsequence, respectively converge, in the weak-∗ sense, to some functions Hv(t) and hv(t) with Hv ≥ hv. Since ρǫ → 1, Jǫ → J as ǫ → 0, and the convexity of the functional defined by (53), we get

• |J(t, x) − v(t, x)|2 dx ≤ 2hv(t).

The assumptions on the initial conditions give Hǫ

v(0) =1

2

• RN
• TN |ξ − v(0, x)|2f ǫ

0(x, ξ) dxdξ

+ ǫ 2

• TN |∇Φǫ(0, x)|2 dx → H0,v,

as ǫ → 0, where H0,v = 1

2

• |J0 − v0|2dx.

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Then we rewrite (52) in a weak form − T Hǫ

v(t)z′(t)dt − z(0)Hǫ v(0) ≤

T 2d(v)L∞Hǫ

v(t)z(t)dt

+ T

• (∂tv + v · ∇v) · (ρǫv − Jǫ)z(t) dxdt + Cγǫ1/2 + CTγ

(54) for all test function z ≥ 0 in D′([0, T]). Thus, by (46), we can pass to the limit (ǫ, γ → 0) in (54) to get − T Hv(t)z′(t)dt − z(0)H0,v ≤ T 2d(v)L∞Hv(t)z(t)dt + T

• (∂tv + v · ∇v) · (v − J)(t, x)z(t) dxdt,

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which, together with Lemma 10, gives

• |J(t, x) − v(t, x)|2dx

≤

• |J0(x) − v(0, x)|2dx e

t

0 2d(v(τ))L∞dτ

+ 2 t e

t

s 2d(v(τ))L∞dτ

• (∂tv + v · ∇v) · (v − J)(s, x)dxds.

By the definition of dissipative solution for the Euler equations given by P. L. Lions in his book [ Mathematical topics in fluid mechanics, Vol. 1 Incompress- ible models, Oxford University Press, New York (1996)], we know that J is a dissipative solution for the Euler equations (40)-(41). Thus, we complete the proof of Theorem 9.

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5. The Vlasov-Maxwell-Fokker-Planck System

The Vlasov-Maxwell-Fokker-Planck (VMFP) system is a mathematical model in the kinetic theory of plasmas. It provides a statistic description of plasma in the terms of charged particle density f ǫ(t, x, ξ) depending on the time t ≥ 0, the position x = (x1, x2, x3) ∈ [0, 1]3 ≡ T3, and the velocity ξ = (ξ1, ξ2, ξ3) ∈ R3. The (rescaled) VMFP system takes the form ∂tf ǫ + ξ · ∇xf ǫ + (Eǫ + αξ × Bǫ) · ∇ξf ǫ = divξ

• βξf ǫ + σ∇ξf ǫ

, (55) ǫ2α∂tEǫ − curlxBǫ = −α

• R3 ξf ǫdξ, −ǫ2divxEǫ = 1 −
• R3 f ǫdξ,

(56) α∂tBǫ + curlxEǫ = 0, divxBǫ = 0, (57) where ǫ, α, β, σ are positive parameters. More explanation on the system and the parameters involved can be found in

• R. J. DiPerna and P.-L. Lions, Comm. Pure Appl. Math., 42(6):729–757,

1989;

• R. Lai. Math. Methods Appl. Sci., 18(13):1013–1040, 1995.

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Some known results on the VMFP system:

• Wollman [CPAM,84] obtained the local existence and the uniqueness of

solution to the system (55)-(57) in the case of β = σ = 0.

• DiPerna and Lions [CPAM, 89] obtained the renormalized solution of the

VMFP system (55)-(57) with β = 0 for general initial data.

• Lai[M2AS,98,95] studied the relativistic VMFP system in one and one-half

dimension.

• Yan Guo [ARMA,95; CMP,93] studied the initial boundary problem of

Vlasov-Maxwell system.

• More results on the relativistic Vlasov-Maxwell system can be found in the

book by Glassey [The Cauchy problem in kinetic theory]. In this talk, we discuss the quasineutral limit of the VMFP system (55)-(57). The quasineutral limit problem plays an important role in the theory of kinetic equa- tions since it gives the relations between the microscopic mathematical models and the macroscopic ones.

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Denote ρǫ(t, x) =

• R3 f ǫ(t, x, ξ)dξ,

Jǫ(t, x) = ρǫuǫ(t, x) =

• R3 ξf ǫ(t, x, ξ)dξ. (58)

Formally, multiplying the equation (55) by 1 and ξ respectively, and integrating the results with respect to ξ, we have ∂tρǫ + divxJǫ = 0, (59) ∂tJǫ + divx

• R3(ξ ⊗ ξ)f ǫdξ − ρǫEǫ − αJǫ × Bǫ + βJǫ = 0,

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

• R3(ξ ⊗ ξ)f ǫdξ, we rewrite the

distribution function f ǫ in the system (55)-(57) as f ǫ = ρǫ (2πKǫ)3/2 exp

• −|ξ − uǫ|2

2Kǫ

• +ǫGǫ(t, x, ξ) ≡ M(ρǫ,uǫ,Kǫ)(ξ)+ǫGǫ, (61)

where M(ρǫ,uǫ,Kǫ)(ξ) is a local Maxwellian denoting the macroscopic fluid part of f ǫ, Gǫ is the microscopic non-fluid part of f ǫ, and Kǫ denotes the temperature.

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Then, a direct computation gives

• R3(ξ ⊗ ξ)f ǫdξ = Jǫ Jǫ

ρǫ + ρǫKǫI3 + O(ǫ), (62) where I3 is an identity matrix, and 1 2

• R3 |ξ|2f ǫ(t, x, ξ)dξ = ρǫ

3 2Kǫ + 1 2|uǫ|2

• (t, x) + O(ǫ).

Let us start with a purely formal analysis on the limit process as ǫ → 0. (The condition σ → 0 comes from the proof of our result stated below.) First, it follows from the second equation in (56) that ρǫ → 1. Then setting ǫ = 0, we

• btain from (59), (60), (56) and (62) that it holds

∂tJ + divx(J ⊗ J) − E − αJ × B + βJ = 0, divxJ = 0 (63) for the case where the temperature Kǫ vanishes, which is the case for perfectly cold electrons. Here J, E, and B denote the limit of Jǫ, Eǫ, and Bǫ (if they exist) in some sense, respectively. From the Maxwell equations (56)-(57), we get α∂tB + curlxE = 0, (64) αJ − curlxB = 0, divxB = 0. (65) The system (63)-(65) is the so-called so-called electron magnetohydrodynamics equations.

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We will establish the above limit rigorously in the present talk. Before stating

• ur result, we consider the periodic boundary problem to the following limiting

equations ∂tv + divx(v ⊗ v) − e − αv × b + βv = 0, (66) α∂tb + curlxe = 0, divxv = 0 (67) αv − curlxb = 0, divxb = 0, (68) v(0, x) = v0(x) ∈ Hs, b(0, x) = b0(x) ∈ Hs, (69) where the function space Hs is given by Hs = {φ(x) ∈ Hs(T3) | ∇ · φ(x) = 0}. We have Lemma 11 Let v0, b0 ∈ Hs(s > 1 + 3/2) satisfying αv0 − curlxb0 = 0. Then there exist a T ∗ ∈ (0, ∞) and a unique solution (v, b, e) ∈ C([0, T ∗), Hs) which solves the problem (66)-(69). The proof of this lemma is similar to that of Lemma 5.1 in [M. Puel, Asymptot. Anal., 04 ].

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Now we state our result.

• Theorem. 12 Let (v0, b0) be two given vectors in Hs(s > 2 + 3/2), Z3 periodic

in x. Consider the periodic boundary problem for the system (55)-(57) with initial data f ǫ(0, x, ξ) = f ǫ

0(x, ξ),

Eǫ(0, x) = Eǫ

0(x),

Bǫ(0, x) = Bǫ

0(x).

(70) Assume that the initial data f ǫ

0(x, ξ) ≥ 0, Eǫ 0(x), Bǫ 0(x) are Z3 periodic in x,

and satisfy the standard compatibility conditions −ǫ2divxEǫ

0 = 1 −

• R3 f ǫ

0dξ,

divxBǫ

0 = 0.

(71) In addition, we assume that there exists a constant C0 > 0, independent of ǫ, such that

• R3×T3
• 1 + |x|2 + |ξ|2 + | ln f ǫ

0(x, ξ)|

• f ǫ

0(x, ξ)dxdξ

+

• T3(ǫ|Eǫ

0(x)|2 + |Bǫ 0(x)|2)dx ≤ C0,

(72) furthermore,

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β 2

• R3×T3 |ξ − v0(x)|2f ǫ

0(x, ξ)dxdξ + σ

• R3×T3 f ǫ

0(x, ξ)| ln f ǫ 0(x, ξ)|dxdξ

+ βǫ2 2

• T3 |Eǫ

0(x)|2dx + β

2

• T3 |Bǫ

0(x) − b0(x)|2dx → 0

as ǫ, σ → 0. (73) Let (f ǫ, Eǫ, Bǫ) be the smooth solutions of the VMFP system (55)-(57) with initial data (70). Then, for all 0 < T < T ∗(defined in Lemma 1), the current Jǫ converges weakly to v in L∞([0, T], L1(T3)), the scaled electric field and the magnetic field converge strongly ǫEǫ → 0 and Bǫ → b in L∞([0, T], L2(T3)) as ǫ, σ → 0, where (v, b, e) be the local strong solutions of the limiting equations (66)-(68) constructed in Lemma 11. The proof of Theorem 12 is based on the compactness argument and the so- called modulated energy method. Since it is similar to that of above theorem for VPFP system, we omit it here.