High-electric-field limit for the Vlasov-Maxwell-Fokker-Planck system

Mihai Bostan^{∗}, Thierry Goudon^{†}

(November 21, 2005)

Abstract In this paper we derive the high-electric-field limit of the three dimensional Vlasov-Maxwell-Fokker-Planck system. We use the modulated energy method which requires the smoothness of the solution of the limit problem. We obtain convergences of charge and current densities in the space of bounded measures.

Keywords: High-field limit, Vlasov-Maxwell-Fokker-Planck system, Modulated energy.

AMS classification: 35Q99, 35B40.

1 Introduction

We consider a plasma in which the dilute charged particles interact both through collisions and through the action of their self-consistent electro-magnetic field. Actually, we are concerned with the evolution of the negative particles which are described in terms of a distribution function in phase space while the charge and current of the positive particles are given functions. Up to a dimensional analysis (postponed to the Appendix) the evolution of the plasma is governed by the following equations

ε(∂_{t}f_{ε}+v·∇_{x}f_{ε})−(E_{ε}+αε(v∧B_{ε}))·∇_{v}f_{ε} = div_{v}(vf_{ε}+∇_{v}f_{ε}), (t, x, v) ∈]0, T [×ℝ^{3}×ℝ^{3}, (1)

∂_{t}E_{ε} − curl_{x}B_{ε} = −(J − j_{ε}) = ∫_{ℝ^{3}} vf_{ε} dv − J(t, x), (t, x) ∈]0, T [×ℝ^{3}, (2)

∗Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex France. E-mail : mbostan@math.univ-fcomte.fr

†Team SIMPAF–INRIA Futurs & Laboratoire Paul Painlevé, UMR CNRS 8524, Université des Sciences et Technologies Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex France. E-mail : Thierry.Goudon@math.univ-lille1.fr

αε∂_{t}B_{ε} + curl_{x}E_{ε} = 0, (t, x) ∈]0, T [×ℝ^{3},  (3)

div_{x}E_{ε} = D(t, x) − ρ_{ε}(t, x) = D(t, x) − ∫_{ℝ^{3}} f_{ε} dv, div_{x}B_{ε} = 0, (t, x) ∈]0, T [×ℝ^{3}.

(4)

The system (1), (2), (3), (4) is refered to as the Vlasov-Maxwell-Fokker-Planck (VMFP) system. Here f_{ε}(t, x, v) ≥ 0 is the distribution function of the negative particles, E_{ε}, B_{ε} stand for the electric and magnetic fields respectively while D(t, x), J(t, x) are the (given) charge and current densities of positive particles. They are supposed to satisfy the conservation law

∂_{t}D + div_{x}J = 0, (t, x) ∈]0, T [×ℝ^{3}.  (5)

The dimensionless parameter ε = (\frac{l}{Λ})^{2} is the square of the ratio between the mean

free path and the Debye length and α = (\frac{Λ}{τ·c_{0}} )^{2} is the square of the ratio between the Debye length and the distance travelled by the light during the relaxation time due to collisions. We are interested in the asymptotic regime

0 < ε ≪ 1, α = O(1).

Notice that in (1) the nonlinear term E_{ε}·∇_{v}f_{ε} is of the same order of magnitude that the diffusion Fokker-Planck term (which is due to the hypothesis that the mean free path l is much smaller than the Debye length Λ). We call this asymptotic regime for ε ↘ 0 the high-electric-field limit. The system is completed by prescribing initial conditions for the distribution function f_{ε} and the electro-magnetic field (E_{ε}, B_{ε})

f_{ε}(0, x, v) = f_{ε}^{0}(x, v), (x, v) ∈ ℝ^{3} × ℝ^{3},  (6)
E_{ε}(0, x) = E_{ε}^{0}(x), B_{ε}(0, x) = B_{ε}^{0}(x), x ∈ ℝ^{3}.  (7)

We suppose that initially the plasma is globally neutral i.e.,

∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx = ∫_{ℝ^{3}} D(0, x) dx,  (8)

and also that the initial conditions satisfy

div_{x}E_{ε}^{0} = D(0, x) − ∫_{ℝ^{3}} f_{ε}^{0}(x, v) dv, div_{x}B_{ε}^{0} = 0, x ∈ ℝ^{3}.  (9)

After integration of (1) with respect to v ∈ ℝ^{3} we deduce that the charge and current densities of the negative particles

ρ_{ε}(t, x) = ∫_{ℝ^{3}} f_{ε} dv, j_{ε}(t, x) = ∫_{ℝ^{3}} vf_{ε} dv

verify the conservation law

∂_{t}ρ_{ε} + div_{x}j_{ε} = 0, (t, x) ∈]0, T [×ℝ^{3}.  (10)

By using (10), (5) and by taking the divergence with respect to x of equations (2),

(3) we deduce that (4) are consequences of (9).

Now, note that the Fokker-Planck operator can be written as follows

L_{F P} (f) := div_{v}(vf + ∇_{v}f) = div_{v} (e^{− \frac{|v|^{2}}{2}} ∇_{v} (fe^{\frac{|v|^{2}}{2}} )) ,  (11)

and therefore the kinetic equation (1) becomes

∂_{t}f_{ε} + v · ∇_{x}f_{ε} − α(v ∧ B_{ε}) · ∇_{v}f_{ε} = \frac{1}{ε}div_{v} (e^{− \frac{|v+Eε(t,x)|^{2}}{2}} ∇_{v} (f_{ε}e^{\frac{|v+Eε(t,x)|^{2}}{2}} )) . (12)

From (12) we can expect that when ε ↘ 0, the distribution function f_{ε} converges to

f_{ε} ≈ \frac{ρ(t, x)}{(2π)^{\frac{3}{2}}}e^{− \frac{|v+E(t,x)|^{2}}{2}} ,  (13)

and therefore we can guess that

j_{ε}(t, x) = ∫_{ℝ^{3}} vf_{ε} dv ≈ −ρ(t, x)E(t, x).

Using the charge conservation law (10) together with (2), (3), (4), we are thus formally led to the following limit system

\{ \begin{matrix}∂_{t}ρ − div_{x}(ρE) = 0, (t, x) ∈]0, T [×ℝ^{3}, \\ div_{x}E = D(t, x) − ρ(t, x), curl_{x}E = 0, (t, x) ∈]0, T [×ℝ^{3}, \\ ∂_{t}E − curl_{x}B = −J(t, x) − ρ(t, x)E(t, x), div_{x}B = 0, (t, x) ∈]0, T [×ℝ^{3}. \end{matrix} (14)

The problem is motivated from plasma physics, as for instance in the theory of semiconductors, the evolution of laser-produced plasmas or the description of tokamaks. High-field asymptotics has been first analyzed in the kinetic theory of semiconductors in [30], see also [14]. Results for different physical models have been obtained in [1], [4], [17], [27]. Of course, the main difficulty relies on the non linear acceleration term E_{ε} · ∇_{v}f_{ε}. The problem slightly simplifies in the electrostatic case where the electric field is simply defined through the Poisson equation (complete (1) by E_{ε} = −∇_{x}Φ_{ε}, Δ_{x}Φ_{ε} = ρ_{ε} − D and B_{ε} = 0). This actually means that the electric field E_{ε} is defined by a convolution with ρ_{ε} − D. The resulting Vlasov-Poisson-Fokker-Planck (VPFP) can be seen, at least formally, as an asymptotic limit of the VMFP model in a physical regime where the light speed is large compared to the thermal velocity. The high-field limit of the VPFP system can be addressed by appealing to usual compactness methods; however, constraints on the space dimension appear, due to the singularity of the convolution kernel. It turns out that the strategy works in dimension 1 [28] and dimension 2 [21]. Another approach uses modulated energy (or relative entropy) methods, as introduced in [34]. With such an approach, we try to evaluate how far the solution is from the expected limit. This method has been used to treat various asymptotic questions in plasma physics [10], [11], [20], gas dynamics [32], [5], fluid-particles interaction [23]... Concerning the VPFP system, it allows to justify the L^{2} strong convergence for the electric field and we can pass to the limit for any space dimension [21]. However, this method requires some smoothness on the solutions of the limit system.

The aim of this paper is therefore to analyze the high-electric-field limit of the three dimensional VMFP system by using the modulated energy method. This extension is interesting both from the viewpoint of physics: we are dealing with a more realistic and complete model; and those of mathematics: replacing the Poisson equation by the Maxwell system we cannot expect regularizing effects from the coupling, and this also shows how robust the modulated energy method is. We consider only smooth solutions of the VMFP system. Unfortunately, to our knowledge, there are no mathematical results concerning the existence/uniqueness of strong solution for the VMFP system. Indeed, the theory for VPFP is well established: existence of weak solutions can be found in [12], [33] while for existence and uniqueness results of strong solution we refer to [6], [7], [16], [29]; but coupling with the Maxwell equations leads to much more difficult analysis. Weak solutions for VMFP are dealt with in [18]. The collisionless case has been further investigated and global existence of classical solutions relies on the behavior of the tip of the support of the solution, as shown by different approaches in [19], [8], [25]. It is also worth mentioning the recent results when considering data close to equilibrium [24], [35] or reduced model [13]. Eventually, we point out that a low-field regime, where diffusion dominates the transport terms, can also be considered: for the VPFP system, we refer to [31], [22]. Here, our main result states as follows (see Section 4 for the definition of tight convergence).

Theorem 1.1 Let ρ^{0} ≥ 0 and D ≥ 0 such that ρ^{0} ∈ W^{1,1}(ℝ^{3}) ∩ W^{1,∞}(ℝ^{3}), and D ∈ L^{∞}(]0, T [; W^{1,1}(ℝ^{3})) ∩ W^{1,∞}(]0, T [×ℝ^{3}), ∂_{t}D ∈ L^{∞}(]0, T [; L^{1}(ℝ^{3})). Let J ∈ L^{2}(]0, T [; L^{2}(ℝ^{3}))^{3}, ∂_{t}J ∈ L^{2}(]0, T [; H^{−1}(ℝ^{3}))^{3} such that ∂_{t}D + div_{x}J = 0. Consider (ρ, E, B) the unique solution of (14) with the initial condition ρ^{0}. Let f_{ε}^{0} ≥ 0 be a sequence of smooth distribution functions verifying

∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx = ∫_{ℝ^{3}}D(0, x) dx, sup_{ε>0} ∫_{ℝ^{3}}∫_{ℝ^{3}} |x|f_{ε}^{0} dv dx < +∞,  (15)

lim_{ε↘0} ∫_{ℝ^{3}}∫_{ℝ^{3}} ε (\frac{|v|^{2}}{2} + | ln f_{ε}^{0}|) f_{ε}^{0} dv dx = 0,  (16)
lim_{ε↘0} ∫_{ℝ^{3}}{|E_{ε}^{0} − E^{0}|^{2} + αε|B_{ε}^{0}|^{2}} dx = 0,  (17)

where E^{0} is the solution of div_{x}E^{0} = D(0, x) − ρ^{0}(x), curl_{x}E^{0} = 0, x ∈ ℝ^{3}. We assume that (f_{ε}, E_{ε}, B_{ε})_{ε>0} are strong solutions of the VMFP system (1), (2), (3), (4), (6), (7). Then (E_{ε})_{ε>0} converges to E in L^{∞}(]0, T [; L^{2}(ℝ^{3}))^{3}, (\sqrt{ε}B_{ε})_{ε>0} converges to 0 in L^{∞}(]0, T [; L^{2}(ℝ^{3}))^{3}, (B_{ε})_{ε>0} converges to B in D′(]0, T [×ℝ^{3})^{3}, (ρ_{ε})_{ε>0} converges to ρ in C^{0}([0, T], M_{+}^{1}(ℝ^{3}) − tight ) and (j_{ε})_{ε>0} converges to −ρE in M^{1}([0, T] × ℝ^{3})^{3}- tight.

The paper is organized as follows. In Section 2 we establish some priori estimates satisfied by smooth solutions (f_{ε}, E_{ε}, B_{ε}) of the three dimensional VMFP system. In the next section, we introduce the modulated energy and calculate its time evolution. There, we also analyze the well-posedness of the limit equation. In Section 4 we detail the passage to the limit. The dimension analysis of the equations and the physical meaning of the different parameters are detailed in the Appendix.

2 A priori estimates

In this section we establish a priori estimates for the smooth solutions (f_{ε}, E_{ε}, B_{ε}) of VMFP, uniformly with respect to ε > 0.

Proposition 2.1 Let (f_{ε}, E_{ε}, B_{ε}) be a smooth solution of the problem (1), (2), (3), (4), (6), (7) where the initial conditions satisfy f_{ε}^{0} ≥ 0, and

M_{ε}^{0} := ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx < +∞,

W_{ε}^{0} := ε ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε}^{0} dv dx + \frac{1}{2} ∫_{ℝ^{3}}|E_{ε}^{0}|^{2} dx + \frac{αε}{2} ∫_{ℝ^{3}}|B_{ε}^{0}|^{2} dx < +∞,

L_{ε}^{0} := ∫_{ℝ^{3}}∫_{ℝ^{3}} |x|f_{ε}^{0} dv dx < +∞,
H_{ε}^{0} := ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0}| ln f_{ε}^{0}| dv dx < +∞.

We assume also that J ∈ L^{1}(]0, T [; L^{2}(ℝ^{3}))^{3}. Then, we have for any 0 < t < T < ∞

∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(t, x, v) dv dx = ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx < +∞,

sup_{0≤t≤T} \{ε ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε}(t, x, v) dv dx + \frac{1}{2} ∫_{ℝ^{3}}|E_{ε}(t, x)|^{2} dx + \frac{αε}{2} ∫_{ℝ^{3}}|B_{ε}(t, x)|^{2} dx\}
+ ∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε} dv dx dt
≤ ((2W_{ε}^{0} + 6T M_{ε}^{0})^{\frac{1}{2}} + \sqrt{2}∥J∥_{L^{1}(]0,T [;L^{2}(ℝ^{3}))})^{2} ,

sup_{0≤t≤T} ∫_{ℝ^{3}}∫_{ℝ^{3}} |x|f_{ε}(t, x, v) dv dx ≤ C_{T} (M_{ε}^{0} + W_{ε}^{0} + L_{ε}^{0} + ∥J∥_{L^{1}(]0,T [;L^{2}(ℝ^{3}))}^{2}),

sup_{0≤t≤T} ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}| ln f_{ε}|(t, x, v) dv dx + ∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} |∇_{v}\sqrt{f_{ε}}|^{2} dv dx dt ≤ C_{T} (ε + M_{ε}^{0}
+ W_{ε}^{0} + εL_{ε}^{0} + H_{ε}^{0} + ∥J∥_{L^{1}(]0,T [;L^{2}(ℝ^{3}))}^{2}).

Before starting our computations let us state the following lemma, based on classical arguments due to Carleman.

Lemma 2.1 Assume that f = f(x, v) satisfies f ≥ 0, (|x| + |v|^{2} + | ln f|)f ∈ L^{1}(ℝ^{N} × ℝ^{N}). Then for all k > 0 we have

f| ln f| ≤ f ln f + 2k(|x| + |v|^{2})f + 2Ce^{− \frac{k}{2} (|x|+|v|^{2})}, with C = sup_{0<y<1}\{−\sqrt{y} ln y\},

and

∫_{ℝ^{N}} ∫_{ℝ^{N}} f| ln f| dv dx ≤ ∫_{ℝ^{N}} ∫_{ℝ^{N}} f ln f dv dx + 2k ∫_{ℝ^{N}} ∫_{ℝ^{N}}(|x| + |v|^{2})f dv dx + C_{k},

with C_{k} = 2C ∫_{ℝ^{N}} ∫_{ℝ^{N}} e^{− \frac{k}{2} (|x|+|v|^{2})} dv dx.

Proof. Since f| ln f| = f ln f + 2f(ln f)_{−}, it is sufficient to estimate f(ln f)_{−}. Take k > 0 and let C = sup_{0<y<1}\{−\sqrt{y} ln y\} < +∞. We have

f(ln f)_{−} = −f ln f · 1_{{0<f<e^{−k(|x|+|v|^{2})}}} − f ln f · 1_{{e^{−k(|x|+|v|^{2})}≤f<1}}
≤ Ce^{− \frac{k}{2} (|x|+|v|^{2})} + k(|x| + |v|^{2})f, ∀ (x, v) ∈ ℝ^{N} × ℝ^{N}.

Therefore

∫_{ℝ^{N}} ∫_{ℝ^{N}} f(ln f)_{−} dv dx ≤ k ∫_{ℝ^{N}} ∫_{ℝ^{N}}(|x|+|v|^{2})f dv dx+C ∫_{ℝ^{N}} ∫_{ℝ^{N}} e^{− \frac{k}{2} (|x|+|v|^{2})} dv dx,

and the conclusion follows easily.

Proof of Proposition 2.1. Integrating (1) with respect to (x, v) ∈ ℝ^{3} × ℝ^{3} yields

\frac{d}{dt} ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(t, x, v) dv dx = 0, t ∈]0, T [,

which implies that

∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(t, x, v) dv dx = ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx = M_{ε}^{0}, t ∈]0, T [.  (18)

Note that integrating (5) with respect to x implies \frac{d}{dt} ∫_{ℝ^{3}}D(t, x) dx = 0 and therefore we deduce that if initially the plasma is globally neutral, i.e., ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx = ∫_{ℝ^{3}}D(0, x) dx, then it remains globally neutral for all t ∈]0, T [

∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(t, x, v) dv dx = ∫_{ℝ^{3}}D(t, x) dx.

Multiplying (1) by \frac{|v|^{2}}{2} and integrating with respect to (x, v) implies

ε\frac{d}{dt} ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε} dv dx+∫_{ℝ^{3}}∫_{ℝ^{3}} E_{ε}·vf_{ε} dv dx = − ∫_{ℝ^{3}}∫_{ℝ^{3}} |v|^{2}f_{ε} dv dx+3 ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} dv dx.

(19)

Multiplying (2), (3) by E_{ε}, resp. B_{ε} and integrating with respect to x yields

\frac{1}{2} \frac{d}{dt} ∫_{ℝ^{3}}(|E_{ε}|^{2} + αε|B_{ε}|^{2}) dx = − ∫_{ℝ^{3}}E_{ε} · (J − j_{ε}) dx.  (20)

By combining (19), (20) we obtain

∫_{ℝ^{3}}∫_{ℝ^{3}} |v|^{2}f_{ε} dv dx + \frac{d}{dt} (ε ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε} dv dx + \frac{1}{2} ∫_{ℝ^{3}}(|E_{ε}|^{2} + αε|B_{ε}|^{2}) dx)
= − ∫_{ℝ^{3}}E_{ε} · J dx + 3 ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx,

and therefore we have

∫_{0}^{t}∫_{ℝ^{3}}∫_{ℝ^{3}} |v|^{2}f_{ε} dv dx ds + ε ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε} dv dx + \frac{1}{2} ∫_{ℝ^{3}}(|E_{ε}|^{2} + αε|B_{ε}|^{2}) dx
≤ ε ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε}^{0} dv dx + \frac{1}{2} ∫_{ℝ^{3}}(|E_{ε}^{0}|^{2} + αε|B_{ε}^{0}|^{2}) dx
+ ∫_{0}^{t} (∫_{ℝ^{3}}|E_{ε}(s, x)|^{2} dx)^{\frac{1}{2}} · (∫_{ℝ^{3}}|J(s, x)|^{2} dx)^{\frac{1}{2}} ds
+ 3t ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx.

By using Bellman’s lemma we obtain for all 0 ≤ t ≤ T

(∫_{ℝ^{3}}|E_{ε}(t, x)|^{2} dx)^{\frac{1}{2}} ≤ (∫_{ℝ^{3}}∫_{ℝ^{3}}(ε|v|^{2} + 6T)f_{ε}^{0} dv dx + ∫_{ℝ^{3}}(|E_{ε}^{0}|^{2} + αε|B_{ε}^{0}|^{2}) dx)^{\frac{1}{2}}
+ ∫_{0}^{T} (∫_{ℝ^{3}}|J(s, x)|^{2} dx)^{\frac{1}{2}} ds
= (2W_{ε}^{0} + 6T M_{ε}^{0})^{\frac{1}{2}} + ∥J∥_{L^{1}(]0,T [;L^{2}(ℝ^{3}))}.

Finally we get

∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} |v|^{2}f_{ε} dv dx dt + sup_{0≤t≤T} \{ε ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε} dv dx + \frac{1}{2} ∫_{ℝ^{3}}(|E_{ε}|^{2} + αε|B_{ε}|^{2}) dx\}
≤ ((2W_{ε}^{0} + 6T M_{ε}^{0})^{\frac{1}{2}} + \sqrt{2}∥J∥_{L^{1}(]0,T [;L^{2}(ℝ^{3}))})^{2} .  (21)

We multiply now (1) by |x| and we obtain after integration with respect to (x, v)

\frac{d}{dt} ∫_{ℝ^{3}}∫_{ℝ^{3}} |x|f_{ε} dv dx − ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{(v · x)}{|x|}f_{ε} dv dx = 0.

We deduce that

sup_{0≤t≤T} ∫_{ℝ^{3}}∫_{ℝ^{3}} |x|f_{ε} dv dx ≤ ∫_{ℝ^{3}}∫_{ℝ^{3}} |x|f_{ε}^{0} dv dx + ∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} |v|f_{ε} dv dx dt
≤ ∫_{ℝ^{3}}∫_{ℝ^{3}} |x|f_{ε}^{0} dv dx + ∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{1}{2}(|v|^{2} + 1)f_{ε} dv dx dt
≤ C_{T} (M_{ε}^{0} + W_{ε}^{0} + L_{ε}^{0} + ∥J∥_{L^{1}(]0,T [;L^{2}(ℝ^{3}))}^{2}).  (22)

We multiply now (1) by (1 + ln f_{ε}) and after integration with respect to (x, v) we get

ε\frac{d}{dt} ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} ln f_{ε} dv dx = − ∫_{ℝ^{3}}∫_{ℝ^{3}}(vf_{ε} + ∇_{v}f_{ε})\frac{∇_{v}f_{ε}}{f_{ε}} dv dx
= 3 ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} dv dx − 4 ∫_{ℝ^{3}}∫_{ℝ^{3}} |∇_{v}\sqrt{f_{ε}}|^{2} dv dx.

Finally we deduce that

ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} ln f_{ε} dv dx + 4 ∫_{0}^{t}∫_{ℝ^{3}}∫_{ℝ^{3}} |∇_{v}\sqrt{f_{ε}}|^{2} dv dx ds = ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} ln f_{ε}^{0} dv dx
+ 3t ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx. (23)

Combining (21), (22), (23) and Lemma 2.1 with k = 1 yields

sup_{0≤t≤T} ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}| ln f_{ε}| dv dx + ∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} |∇_{v}\sqrt{f_{ε}}|^{2} dv dx dt ≤ C_{T} (ε + M_{ε}^{0} + W_{ε}^{0}
+ εL_{ε}^{0} + H_{ε}^{0} + ∥J∥_{L^{1}(]0,T [;L^{2}(ℝ^{3}))}^{2}).

3 The modulated energy method

In this section we introduce the modulated energy (see [10], [34]) which allows us to show that the electro-magnetic field (E_{ε}, \sqrt{αε}B_{ε}) converges strongly in L^{2}(ℝ^{3})^{6}, uniformly in time. The proof requires some regularity properties of the limit solutions (ρ, E, B) of (14) as well as the convergence of the initial electro-magnetic field

lim_{ε↘0} (∫_{ℝ^{3}}|E_{ε}^{0}(x) − E(0, x)|^{2} dx + αε ∫_{ℝ^{3}}|B_{ε}^{0}(x)|^{2} dx) = 0.

Let us define the modulated energy H_{ε} and the positive modulated energy H_{ε,p} as follows

H_{ε}(t) = ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(ln f_{ε} + \frac{1}{2}|v + E|^{2}) dv dx + \frac{1}{2} ∫_{ℝ^{3}}(|E_{ε} − E|^{2} + αε|B_{ε} − B|^{2}) dx,

and

H_{ε,p}(t) = ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(| ln f_{ε}| + \frac{1}{2}|v + E|^{2}) dv dx + \frac{1}{2} ∫_{ℝ^{3}}(|E_{ε} − E|^{2} + αε|B_{ε} − B|^{2}) dx,

where (f_{ε}, E_{ε}, B_{ε})_{ε>0} are smooth solutions of (1), (2), (3), (4), (6), (7) and (ρ, E, B) is a smooth solution of (14). According to the derivation of the high-field limit of the VPFP system in [21], the idea is to study the time variation of H_{ε}(t) (which is also called relative entropy), which in turn will give information on H_{ε,p}.

3.1 Analysis of the limit system

We start with the analysis of the system (14). Note that this system can be split into two problems. First solve for (ρ, E)

\{ \begin{matrix}∂_{t}ρ − div_{x}(ρE) = 0, (t, x) ∈]0, T [×ℝ^{3}, \\ div_{x}E = D(t, x) − ρ(t, x), (t, x) ∈]0, T [×ℝ^{3}, \\ curl_{x}E = 0, (t, x) ∈]0, T [×ℝ^{3}, \\ ρ(0, x) = ρ_{0}(x), x ∈ ℝ^{3}, \end{matrix}  (24)

and secondly find B solution of

\{ \begin{matrix}∂_{t}E − curl_{x}B = −J(t, x) − ρ(t, x)E(t, x), (t, x) ∈]0, T [×ℝ^{3}, \\ div_{x}B = 0, (t, x) ∈]0, T [×ℝ^{3}, \end{matrix}  (25)

where the charge and current densities D, J are given functions satisfying ∂_{t}D + div_{x}J = 0. We give here an existence result for (24) which is a direct consequence of the existence result obtained in [28], see also [21].

Proposition 3.1 Let ρ^{0} ∈ W^{1,1}(ℝ^{3}) ∩ W^{1,∞}(ℝ^{3}), D ∈ L^{∞}(]0, T [; W^{1,1}(ℝ^{3})) ∩ W^{1,∞}(]0, T [×ℝ^{3}), ∂_{t}D ∈ L^{∞}(]0, T [; L^{1}(ℝ^{3})). Then there is a unique solution for (24) satisfying

ρ ∈ W^{1,∞}(]0, T [×ℝ^{3}), E ∈ W^{1,∞}(]0, T [×ℝ^{3})^{3}.

Proof. We introduce the exterior electric field E_{0} given by div_{x}E_{0} = D, curl_{x}E_{0} = 0, so that div_{x}(E − E_{0}) = −ρ, curl_{x}(E − E_{0}) = 0. The hypotheses imply that E_{0} ∈ W^{1,∞}(]0, T [×ℝ^{3})^{3}. Following the arguments of Theorem 3 and Lemma 8 of [28] we deduce that there is a unique strong solution (ρ, E) for (24) verifying ρ ∈ L^{∞}(]0, T [; W^{1,∞}(ℝ^{3})), D − ρ ∈ L^{∞}(]0, T [; W^{1,∞}(ℝ^{3})), E ∈ L^{∞}(]0, T [; W^{1,∞}(ℝ^{3}))^{3}. By differentiating the first equation of (24) with respect to x we check that ∇_{x}ρ ∈ L^{∞}(]0, T [; L^{1}(ℝ^{3}))^{3} and since ∂_{t}ρ = E · ∇_{x}ρ + ρ(D − ρ) ∈ L^{∞}(]0, T [; L^{1}(ℝ^{3})) ∩ L^{∞}(]0, T [; L^{∞}(ℝ^{3})) we deduce that ∂_{t}E ∈ L^{∞}(]0, T [; W^{1,p}(ℝ^{3}))^{3} ⊂ L^{∞}(]0, T [; L^{∞}(ℝ^{3}))^{3} for all p > 3. Finally we obtain that ρ ∈ W^{1,∞}(]0, T [×ℝ^{3}) and E ∈ W^{1,∞}(]0, T [×ℝ^{3})^{3}. In fact, since D − ρ ∈ L^{∞}(]0, T [; W^{1,1}(ℝ^{3})) ∩ L^{∞}(]0, T [; W^{1,∞}(ℝ^{3})) we have E ∈ L^{∞}(]0, T [; W^{2,p}(ℝ^{3}))^{3} for all 1 < p < +∞. On the other hand since ∂_{t}D − ∂_{t}ρ ∈ L^{∞}(]0, T [; L^{1}(ℝ^{3})) ∩ L^{∞}(]0, T [; L^{∞}(ℝ^{3})) we have ∂_{t}E ∈ L^{∞}(]0, T [; W^{1,p}(ℝ^{3}))^{3} for all 1 < p < +∞. In particular we obtain that E, ∂_{t}E ∈ L^{∞}(]0, T [; L^{2}(ℝ^{3}))^{3}.

Once we find (ρ, E) it is easy to solve (25).

Proposition 3.2 Under the hypotheses of Proposition 3.1 assume also that J, ∂_{t}J ∈ L^{2}(]0, T [; H^{−1}(ℝ3))^{3} and ∂_{t}D + div_{x}J = 0 in D′(]0, T [×ℝ^{3}). Then there is a unique solution B for (25) verifying B, ∂_{t}B ∈ L^{2}(]0, T [; L^{2}(ℝ^{3}))^{3}.

Proof. Observe that we have div_{x}(∂_{t}E + ρE + J) = 0 and that ∂_{t}E + ρE + J ∈ L^{2}(]0, T [; H^{−1}(ℝ^{3}))^{3}. Therefore there is a unique B ∈ L^{2}(]0, T [; L^{2}(ℝ^{3}))^{3} such that ∂_{t}E + ρE + J = curl_{x}B, div_{x}B = 0. In order to estimate ∂_{t}B in L^{2}(]0, T [; L^{2}(ℝ^{3}))^{3} it is sufficient to estimate ∂_{t}(∂_{t}E + ρE + J) in L^{2}(]0, T [; H^{−1}(ℝ^{3}))^{3}. We have

div_{x}(∂_{t}^{2}E) = ∂_{t}^{2}(D − ρ) = ∂_{t}(−div_{x}J − div_{x}(ρE)),

and thus

∥∂_{t}^{2}E∥_{L^{2}(]0,T [;H^{−1}(ℝ^{3}))} ≤ C · ∥∂_{t}(J + ρE)∥_{L^{2}(]0,T [;H^{−1}(ℝ^{3}))}
≤ C · {∥∂_{t}J∥_{L^{2}(]0,T [;H^{−1}(ℝ^{3}))} + ∥ρ∥_{L^{∞}} · ∥∂_{t}E∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}
+ ∥∂_{t}ρ∥_{L^{∞}} · ∥E∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}},

and the conclusion follows.

3.2 Evolution of the modulated energy

This section is devoted to the study of the evolution of the modulated energy. Let us multiply (1) by (1+ln f_{ε} + \frac{|v+E|^{2}}{2} ) where (ρ, E, B) is solution of (14). It is convenient to rewrite (1) as follows

ε(∂_{t}f_{ε} + v · ∇_{x}f_{ε}) − div_{v}(f_{ε}(v + E) + ∇_{v}f_{ε}) = div_{v}{(E_{ε} − E + αε(v ∧ B_{ε}))f_{ε}}. (26)

We perform our computations in three steps.

Lemma 3.1 Assume that E ∈ W^{1,∞}(]0, T [×ℝ^{3})^{3}. Let us set

Q_{1}(t) := ε ∫_{ℝ^{3}}∫_{ℝ^{3}}(∂_{t}f_{ε} + v · ∇_{x}f_{ε}) · (1 + ln f_{ε} + \frac{|v + E|^{2}}{2} ) dv dx.

Then we have

Q_{1}(t) = ε\frac{d}{dt} ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} (ln f_{ε} + \frac{|v + E|^{2}}{2} ) dv dx
−ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(v + E) · (∂_{t}E + (D_{x}E)v) dv dx,

where D_{x}E stands for the jacobian matrix of E. Moreover, there is a constant C = C(∥E∥_{W^{1,∞}(]0,T [×ℝ^{3})}) such that

|∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(v + E) · (∂_{t}E + (D_{x}E)v) dv dx| ≤ C ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} (1 + \frac{|v + E|^{2}}{2} ) dv dx.

Proof. We can write

(∂_{t}f_{ε} + v · ∇_{x}f_{ε}) · (1 + ln f_{ε} + \frac{|v + E|^{2}}{2} ) = ∂_{t}(f_{ε} ln f_{ε}) + v · ∇_{x}(f_{ε} ln f_{ε})
+ ∂_{t} (f_{ε} \frac{|v + E|^{2}}{2} ) + v · ∇_{x} (f_{ε} \frac{|v + E|^{2}}{2} ) − f_{ε}(v + E) · (∂_{t}E + (D_{x}E)v),

where D_{x}E = (\frac{∂E_{i}}{∂x_{j}} )_{1≤i,j≤3}. After integration with respect to (x, v) we get

Q_{1}(t) = ε\frac{d}{dt} ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} (ln f_{ε} + \frac{|v + E|^{2}}{2} ) dv dx
−ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(v + E) · (∂_{t}E + (D_{x}E)v) dv dx.

The last term in the above equation can be estimated by

|∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(v + E) · ∂_{t}E dv dx| ≤ \frac{1}{2} ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}|v + E|^{2} dv dx + \frac{1}{2} ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}|∂_{t}E|^{2} dv dx
≤ C ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} (1 + \frac{|v + E|^{2}}{2} ) dv dx,

and

|∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(v + E) · (D_{x}E)v dv dx| ≤ ∥D_{x}E∥_{L^{∞}} ∫_{ℝ^{3}}∫_{ℝ^{3}} (\frac{3}{2}|v + E|^{2} + \frac{1}{2}|E|^{2}) f_{ε} dv dx
≤ C ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} (1 + \frac{|v + E|^{2}}{2} ) dv dx.

Lemma 3.2 Let us set

Q_{2}(t) := − ∫_{ℝ^{3}}∫_{ℝ^{3}} div_{v}(f_{ε}(v + E) + ∇_{v}f_{ε}) (1 + ln f_{ε} + \frac{|v + E|^{2}}{2} ) dv dx.

Then, we have

Q_{2}(t) = ∫_{ℝ^{3}}∫_{ℝ^{3}} |\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx.

Proof. By using the formula

div_{v}(f_{ε}(v + E) + ∇_{v}f_{ε}) = div_{v} \{e^{− \frac{|v+E|^{2}}{2}} ∇_{v} (f_{ε}e^{\frac{|v+E|^{2}}{2}} )\} ,

we deduce that

Q_{2}(t) = − ∫_{ℝ^{3}}∫_{ℝ^{3}} div_{v} \{e^{− \frac{|v+E|^{2}}{2}} ∇_{v} (f_{ε}e^{\frac{|v+E|^{2}}{2}} )\} ln (f_{ε}e^{\frac{|v+E|^{2}}{2}} ) dv dx
= ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{e^{−|v+E|^{2}}}{f_{ε}} |∇_{v} (f_{ε}e^{\frac{|v+E|^{2}}{2}} )|^{2} dv dx
= ∫_{ℝ^{3}}∫_{ℝ^{3}} |\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx.

Lemma 3.3 Let (ρ, E, B) be a solution of (14) satisfying E ∈ W^{1,∞}(]0, T [×ℝ^{3})^{3}, B, ∂_{t}B, J ∈ L^{2}(]0, T [; L^{2}(ℝ^{3}))^{3}. Let us set

Q_{3}(t) := − ∫_{ℝ^{3}}∫_{ℝ^{3}} div_{v}((E_{ε} − E + αε(v ∧ B_{ε}))f_{ε}) (1 + ln f_{ε} + \frac{|v + E|^{2}}{2} ) dv dx.

Then, we have

Q_{3}(t) = \frac{d}{dt} \{\frac{1}{2} ∫_{ℝ^{3}}|E_{ε} − E|^{2} dx + \frac{αε}{2} ∫_{ℝ^{3}}|B_{ε} − B|^{2} dx\}
− ∫_{ℝ^{3}}{(E_{ε} − E)div_{x}(E_{ε} − E) − (E_{ε} − E) ∧ curl_{x}(E_{ε} − E)} · E dx
− αε ∫_{ℝ^{3}}{B_{ε}div_{x}B_{ε} − B_{ε} ∧ curl_{x}B_{ε}} · E dx + αε ∫_{ℝ^{3}}∂_{t}B · (B_{ε} − B) dx
+ αε ∫_{ℝ^{3}}(J ∧ B_{ε}) · E dx + αε ∫_{ℝ^{3}}∂_{t}(E_{ε} ∧ B_{ε}) · E dx.

Moreover, there is a constant C = C(∥E∥_{W^{1,∞}}) such that |∫_{ℝ^{3}}{(E_{ε} − E)div_{x}(E_{ε} − E) − (E_{ε} − E) ∧ curl_{x}(E_{ε} − E)} · E dx| ≤ C ∫_{ℝ^{3}}|E_{ε}−E|^{2} dx, |∫_{ℝ^{3}}{B_{ε}div_{x}B_{ε} − B_{ε} ∧ curl_{x}B_{ε}} · E dx| ≤ C ∫_{ℝ^{3}}|B_{ε}|^{2} dx ≤ C ∫_{ℝ^{3}}{|B_{ε}−B|^{2}+|B|^{2}} dx. Proof. We can write

div_{v}\{f_{ε}(E_{ε} − E + αε(v ∧ B_{ε}))\} (1 + ln f_{ε} + \frac{|v + E|^{2}}{2} )
= div_{v}{f_{ε} ln f_{ε}(E_{ε} − E + αε(v ∧ B_{ε}))}

+div_{v}\{f_{ε}(E_{ε} − E + αε(v ∧ B_{ε}))\} \frac{|v + E|^{2}}{2} .

After integration with respect to (x, v) we get

Q_{3}(t) = ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(E_{ε} − E + αε(v ∧ B_{ε})) · (v + E) dv dx
= ∫_{ℝ^{3}}(E_{ε} − E) · j_{ε} dx + ∫_{ℝ^{3}}ρ_{ε}(E_{ε} − E) · E dx + αε ∫_{ℝ^{3}}(j_{ε} ∧ B_{ε}) · E dx
= ∫_{ℝ^{3}}(E_{ε} − E) · (j_{ε} + ρE) dx + ∫_{ℝ^{3}}(ρ_{ε} − ρ)(E_{ε} − E) · E dx
+αε ∫_{ℝ^{3}}(j_{ε} ∧ B_{ε}) · E dx
= I_{1} + I_{2} + I_{3}.

From (2), (3), (14) we have

∂_{t}(E_{ε} − E) − curl_{x}(B_{ε} − B) = j_{ε} + ρE,  (27)
αε∂_{t}(B_{ε} − B) + curl_{x}(E_{ε} − E) = −αε∂_{t}B.  (28)

By multiplying (27), (28) by E_{ε}−E, and B_{ε}−B respectively, we find after integration with respect to x

\frac{1}{2} \frac{d}{dt} ∫_{ℝ^{3}}(|E_{ε} − E|^{2} + αε|B_{ε} − B|^{2}) dx = ∫_{ℝ^{3}}(j_{ε} + ρE) · (E_{ε} − E) dx
− αε ∫_{ℝ^{3}}∂_{t}B · (B_{ε} − B) dx
= I_{1} − αε ∫_{ℝ^{3}}∂_{t}B · (B_{ε} − B) dx. (29)

From (4), (14) we have div_{x}(E_{ε} − E) = −(ρ_{ε} − ρ) and thus

I_{2} = ∫_{ℝ^{3}}(ρ_{ε} − ρ)(E_{ε} − E) · E dx = − ∫_{ℝ^{3}}div_{x}(E_{ε} − E)(E_{ε} − E) · E dx. (30)

Now by using (2), (3) we deduce

I_{3} = αε ∫_{ℝ^{3}}(j_{ε} ∧ B_{ε}) · E dx = αε ∫_{ℝ^{3}}((∂_{t}E_{ε} − curl_{x}B_{ε} + J) ∧ B_{ε}) · E dx
= αε ∫_{ℝ^{3}}(B_{ε} ∧ curl_{x}B_{ε}) · E dx + αε ∫_{ℝ^{3}}(J ∧ B_{ε}) · E dx
+ αε ∫_{ℝ^{3}}∂_{t}(E_{ε} ∧ B_{ε}) · E dx − αε ∫_{ℝ^{3}}(E_{ε} ∧ ∂_{t}B_{ε}) · E dx
= αε ∫_{ℝ^{3}}(B_{ε} ∧ curl_{x}B_{ε}) · E dx + αε ∫_{ℝ^{3}}(J ∧ B_{ε}) · E dx
+ αε ∫_{ℝ^{3}}∂_{t}(E_{ε} ∧ B_{ε}) · E dx + ∫_{ℝ^{3}}(E_{ε} ∧ curl_{x}E_{ε}) · E dx.  (31)

Note also that since curl_{x}E = 0 we have

∫_{ℝ^{3}}(E_{ε} ∧ curl_{x}E_{ε}) · E dx = ∫_{ℝ^{3}}((E_{ε} − E) ∧ curl_{x}(E_{ε} − E)) · E dx.

The equalities (30), (31) imply

I_{2} + I_{3} = − ∫_{ℝ^{3}}{(E_{ε} − E)div_{x}(E_{ε} − E) − (E_{ε} − E) ∧ curl_{x}(E_{ε} − E)} · E dx
− αε ∫_{ℝ^{3}}{B_{ε}div_{x}B_{ε} − B_{ε} ∧ curl_{x}B_{ε}} · E dx
+ αε ∫_{ℝ^{3}}∂_{t}(E_{ε} ∧ B_{ε}) · E dx + αε ∫_{ℝ^{3}}(J ∧ B_{ε}) · E dx.

Finally we get

Q_{3}(t) = \frac{1}{2} \frac{d}{dt} ∫_{ℝ^{3}}\{|E_{ε} − E|^{2} + αε|B_{ε} − B|^{2}\} dx + αε ∫_{ℝ^{3}}∂_{t}B · (B_{ε} − B) dx
− ∫_{ℝ^{3}}{(E_{ε} − E)div_{x}(E_{ε} − E) − (E_{ε} − E) ∧ curl_{x}(E_{ε} − E)} · E dx
− αε ∫_{ℝ^{3}}{B_{ε}div_{x}B_{ε} − B_{ε} ∧ curl_{x}B_{ε}} · E dx
+ αε ∫_{ℝ^{3}}∂_{t}(E_{ε} ∧ B_{ε}) · E dx + αε ∫_{ℝ^{3}}(J ∧ B_{ε}) · E dx.

Now, using the formula

A_{i}(u, u) := (udiv_{x}u − u ∧ curl_{x}u)_{i} = ∑_{j=1}^{3} \frac{∂}{∂x_{j}} (u_{i}u_{j}) − \frac{1}{2} \frac{∂}{∂x_{i}} |u|^{2}, 1 ≤ i ≤ 3,

one gets easily after integration by parts that

|∫_{ℝ^{3}}A(E_{ε} − E, E_{ε} − E) · E dx| ≤ C(∥E∥_{W^{1,∞}}) ∫_{ℝ^{3}}|E_{ε} − E|^{2} dx,

and similarly

|∫_{ℝ^{3}}A(B_{ε}, B_{ε}) · E dx| ≤ C(∥E∥_{W^{1,∞}}) ∫_{ℝ^{3}}|B_{ε}|^{2} dx
≤ C(∥E∥_{W^{1,∞}}) (∫_{ℝ^{3}}|B_{ε} − B|^{2} dx + ∫_{ℝ^{3}}|B|^{2} dx) .

From (26) we know that Q_{1}(t) + Q_{2}(t) + Q_{3}(t) = 0 for all 0 ≤ t ≤ T. Therefore, combining Lemmas 3.1, 3.2, 3.3 characterizes the evolution of the modulated energy as follows.

Proposition 3.3 Let f_{ε}^{0} ≥ 0 verify the assumptions of Proposition 2.1. Let D ≥ 0, D ∈ L^{∞}(]0, T [; L^{1}(ℝ^{3})) and J ∈ L^{2}(]0, T [; L^{2}(ℝ^{3}))^{3} verify ∂_{t}D + div_{x}J = 0. Let (f_{ε}, E_{ε}, B_{ε}) be a smooth solution of the VMFP system (1), (2), (3), (4) with the initial conditions f_{ε}^{0}, E_{ε}^{0}, B_{ε}^{0} satisfying (9). Assume that the solution (ρ, E, B) of (14) verifies E ∈ W^{1,∞}(]0, T [×ℝ^{3})^{3}, E, B, ∂_{t}E, ∂_{t}B ∈ L^{2}(]0, T [; L^{2}(ℝ^{3}))^{3}. Then the balance of the modulated energy is given by

\frac{d}{dt}H_{ε}(t) + ∫_{ℝ^{3}}∫_{ℝ^{3}} |\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx
= ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(v + E) · (∂_{t}E + (D_{x}E)v) dv dx
+ ∫_{ℝ^{3}}A(E_{ε} − E, E_{ε} − E) · E dx + αε ∫_{ℝ^{3}}A(B_{ε}, B_{ε}) · E dx (32)
−αε ∫_{ℝ^{3}} ∂_{t}B · (B_{ε} − B) dx − αε ∫_{ℝ^{3}} (J ∧ B_{ε}) · E dx
−αε ∫_{ℝ^{3}} ∂_{t}(E_{ε} ∧ B_{ε}) · E dx,

where, for a given u : ℝ^{3} → ℝ^{3}, A(u, u) denotes the vector u div_{x}u − u ∧ curl_{x}u.

In turn, we are able to establish a useful estimate for the positive modulated energy.

Corollary 3.1 Let us denote E^{0} := E(0, ·), B^{0} := B(0, ·). Then, for all 0 ≤ t ≤ T the following inequality holds

H_{ε,p}(t) + ∫_{0}^{t}∫_{ℝ^{3}}∫_{ℝ^{3}}|\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx ds≤C∫_{ℝ^{3}}\{|E_{ε}^{0} − E^{0}|^{2} + αε|B_{ε}^{0}|^{2}\}
+ Cε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} (| ln f_{ε}^{0}| + 1 + |x| + \frac{|v|^{2}}{2} ) dv dx
+ C (ε + (ε + αε + \sqrt{αε})∥J∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2} + \sqrt{αε} ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx)
+ (ε + \sqrt{αε}) (\frac{1}{2} ∫_{ℝ^{3}}(|E_{ε}^{0}|^{2} + αε|B_{ε}^{0}|^{2}) dx + ε ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε}^{0} dv dx)
+ Cαε (∥B∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2} + ∥∂_{t}B∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2}) + C ∫_{0}^{t} H_{ε,p}(s) ds. (33)

Proof. We use the estimates proved in Lemmas 3.1 and 3.3

|∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}(v + E) · (∂_{t}E + (D_{x}E)v) dv dx| ≤ C ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} (1 + \frac{|v + E|^{2}}{2} ) dv dx,

|∫_{ℝ^{3}}A(E_{ε} − E, E_{ε} − E) · E dx| + αε |∫_{ℝ^{3}}A(B_{ε}, B_{ε}) · E dx| ≤ C ∫_{ℝ^{3}}|E_{ε} − E|^{2} dx
+ Cαε (∫_{ℝ^{3}}|B_{ε} − B|^{2} dx + ∫_{ℝ^{3}}|B|^{2} dx) ,

and the trivial inequalities

αε |∫_{ℝ^{3}}∂_{t}B · (B_{ε} − B) dx| ≤ \sqrt{αε} (∫_{ℝ^{3}}|∂_{t}B|^{2} dx)^{\frac{1}{2}} (∫_{ℝ^{3}}αε|B_{ε} − B|^{2} dx)^{\frac{1}{2}} ,

αε |∫_{ℝ^{3}}(J ∧ B_{ε}) · E dx| ≤ C\sqrt{αε} (∫_{ℝ^{3}}|J|^{2} dx)^{\frac{1}{2}} (∫_{ℝ^{3}}αε|B_{ε} − B|^{2} dx)^{\frac{1}{2}}

+Cαε (∫_{ℝ^{3}}|J|^{2} dx)^{\frac{1}{2}} (∫_{ℝ^{3}}|B|^{2} dx)^{\frac{1}{2}} ,

where C = C(∥E∥_{W^{1,∞}}). After integration of (32) with respect to t ∈]0, T [ we obtain

H_{ε}(t) + ∫_{0}^{t}∫_{ℝ^{3}}∫_{ℝ^{3}} |\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx ds ≤ H_{ε}(0)
+ Cε ∫_{0}^{t}∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} (1 + \frac{|v + E|^{2}}{2} ) dv dx ds
+ C ∫_{0}^{t} ∫_{ℝ^{3}}(\frac{1}{2}|E_{ε} − E|^{2} + \frac{αε}{2}|B_{ε} − B|^{2}) dxds
+ Cαε(∥B∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2} + ∥∂_{t}B∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2} + ∥J∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2})
− αε ∫_{0}^{t} ∫_{ℝ^{3}} ∂_{t}(E_{ε} ∧ B_{ε}) · E dx ds.  (34)

Note that we have

H_{ε}(0) ≤ Cε ∫_{0}^{t}∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} (| ln f_{ε}^{0}| + 1 + \frac{|v|^{2}}{2} ) dv dx ds
+ \frac{1}{2} ∫_{ℝ^{3}}\{|E_{ε}^{0}(x) − E(0, x)|^{2} + αε|B_{ε}^{0}(x) − B(0, x)|^{2}\} dx.  (35)

By using Proposition 2.1 we deduce also that

|αε ∫_{0}^{t}∫_{ℝ^{3}} ∂_{t}(E_{ε} ∧ B_{ε}) · E dx ds| ≤ αε |∫_{ℝ^{3}}(E_{ε} ∧ B_{ε}) · E dx|_{0}^{t} |
+ αε |∫_{0}^{t}∫_{ℝ^{3}}(E_{ε} ∧ B_{ε}) · ∂_{t}E dx ds|
≤ C\sqrt{αε}(M_{ε}^{0} + W_{ε}^{0} + ∥J∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2}).(36)

Finally, combining (34), (35), (36) yields for all 0 ≤ t ≤ T

H_{ε}(t) + ∫_{0}^{t}∫_{ℝ^{3}}∫_{ℝ^{3}} |\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx ds ≤ Cε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx

+Cε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} (| ln f_{ε}^{0}| + \frac{|v|^{2}}{2} ) dv dx + \frac{1}{2} ∫_{ℝ^{3}}\{|E_{ε}^{0} − E^{0}|^{2} + αε|B_{ε}^{0} − B^{0}|^{2}\}

+ Cαε(∥B∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2} + ∥∂_{t}B∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2} + ∥J∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2})
+ C\sqrt{αε}(M_{ε}^{0} + W_{ε}^{0} + ∥J∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2}) + C ∫_{0}^{t} H_{ε,p}(s) ds.  (37)

Now we can show exactly as in the proof of Lemma 2.1 that

f_{ε}| ln f_{ε}| ≤ f_{ε} ln f_{ε} + \frac{1}{2}(|x| + |v + E|^{2})f_{ε}
+ 2Ce^{− \frac{1}{8} (|x|+|v+E|^{2})}, ∀ (t, x, v) ∈]0, T [×ℝ^{3} × ℝ^{3}.  (38)

After integration with respect to (x, v) we deduce

ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}| ln f_{ε}| dv dx ≤ ε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} (ln f_{ε} + \frac{|v + E|^{2}}{2} ) dv dx  (39)
+ \frac{ε}{2} ∫_{ℝ^{3}}∫_{ℝ^{3}} |x|f_{ε} dv dx + 2εC ∫_{ℝ^{3}}∫_{ℝ^{3}} e^{− \frac{1}{8} (|x|+|v|^{2})} dv dx.

Using Proposition 2.1 and (39) we obtain the inequality

H_{ε,p}(t) ≤ H_{ε}(t) + Cε(1 + M_{ε}^{0} + W_{ε}^{0} + L_{ε}^{0} + ∥J∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2}).  (40)

Therefore, from (37), (40) we deduce

H_{ε,p}(t) + ∫_{0}^{t}∫_{ℝ^{3}}∫_{ℝ^{3}}|\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx ds≤C∫_{ℝ^{3}}\{|E_{ε}^{0} − E^{0}|^{2} + αε|B_{ε}^{0}|^{2}\}
+ Cε ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} (| ln f_{ε}^{0}| + 1 + |x| + \frac{|v|^{2}}{2} ) dv dx
+ C (ε + (ε + αε + \sqrt{αε})∥J∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2} + \sqrt{αε} ∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx)
+ (ε + \sqrt{αε}) (\frac{1}{2} ∫_{ℝ^{3}}(|E_{ε}^{0}|^{2} + αε|B_{ε}^{0}|^{2}) dx + ε ∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε}^{0} dv dx)
+ Cαε (∥B∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2} + ∥∂_{t}B∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}^{2} + ∥B^{0}∥_{L^{2}(ℝ^{3})})
+ C ∫_{0}^{t} H_{ε,p}(s) ds, ∀ 0 ≤ t ≤ T.  (41)

The conclusion follows easily by observing that

∥B∥_{L^{∞}(]0,T [;L^{2}(ℝ^{3}))} ≤ C{∥B∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))} + ∥∂_{t}B∥_{L^{2}(]0,T [;L^{2}(ℝ^{3}))}}.

4 Asymptotics

In this section we analyze the asymptotic behavior of smooth solutions (f_{ε}, E_{ε}, B_{ε})_{ε>0} of the VMFP system (1), (2), (3), (4), (6), (7) when the parameter ε ↘ 0. We will prove that (E_{ε})_{ε>0} converges to E strongly in L^{∞}(]0, T [; L^{2}(ℝ^{3}))^{3}, (B_{ε})_{ε>0} converges to B in D′(]0, T [×ℝ^{3})^{3} and ρ_{ε} → ρ, j_{ε} → −ρE in a sense that will be precised, where (ρ, E, B) is the unique strong solution of (14). Throughout the paper we denote by M^{1}(ℝ^{3}) the set of bounded Radon measures on ℝ^{3}, while M_{+}^{1}(ℝ^{3}) stands for its positive cone. We recall some definitions and compactness properties in measure spaces (see [9] for more details).

Definition 4.1 Let (ρ_{n})_{n∈ℕ} be a sequence in M^{1}(ℝ^{3}). We say that

1) (ρ_{n})_{n∈ℕ} converges vaguely to ρ iff

lim_{n→+∞} ∫_{ℝ^{3}} φ dρ_{n} = ∫_{ℝ^{3}} φ dρ,  (42)

for any continuous function with compact support φ ∈ C_{c}^{0}(ℝ^{3}) (actually the convergence holds for any continuous function φ vanishing at infinity i.e., lim_{|x|→+∞} φ(x) = 0) ;

2) (ρ_{n})_{n∈ℕ} converges tightly to ρ iff (42) holds for any continuous and bounded function φ ∈ C^{0}(ℝ^{3}) ∩ L^{∞}(ℝ^{3}).

We have the following classical results.

Proposition 4.1 1) Let (ρ_{n})_{n∈ℕ} be a sequence in M_{+}^{1}(ℝ^{3}) which converges vaguely to ρ. Assume also that lim_{n→+∞}ρ_{n}(ℝ^{3}) = ρ(ℝ^{3}). Then (ρ_{n})_{n∈ℕ} converges to ρ tightly.

2) Let (ρ_{n})_{n∈ℕ} be a sequence in M^{1}(ℝ^{3}) verifying sup_{n} |ρ_{n}|(ℝ^{3}) < +∞ and such that for any η > 0 there exists a compact set K_{η} ⊂ ℝ^{3} satisfying sup_{n} |ρ_{n}|(ℝ^{3} − K_{η}) ≤ η. Then (ρ_{n})_{n∈ℕ} is relatively compact for the tight topology.

We recall also the following compactness result, cf. [21].

Proposition 4.2 Assume that (ρ_{ε})_{ε>0}, (j_{ε})_{ε>0} satisfy ρ_{ε} ≥ 0, ∂_{t}ρ_{ε} + div_{x}j_{ε} = 0 in D′(]0, T [×ℝ^{3}), ∀ε > 0 and

sup_{ε>0} sup_{t∈[0,T]} ∫_{ℝ^{3}}ρ_{ε}(t, x)(1 + |x|) dx < +∞,

sup_{ε>0} ∫_{0}^{T} (∫_{ℝ^{3}} |j_{ε}(t, x)| dx)^{2} dt < +∞,

sup_{ε>0} ∫_{0}^{T} ∫_{ℝ^{3}}(1 + \sqrt{|x|})|j_{ε}(t, x)| dxdt < +∞.

Then (ρ_{ε})_{ε>0} is relatively compact in C^{0}([0, T], M_{+}^{1}(ℝ^{3}) − tight ) and (j_{ε})_{ε>0} is relatively compact in M^{1}([0, T] × ℝ^{3})^{3}- tight.

Now we are ready to end the proof of Theorem 1.1.

Proof of Theorem 1.1. From Proposition 3.3 and the hypotheses (15), (16), (17) we deduce that for 0 ≤ t ≤ T we have

H_{ε,p}(t) + ∫_{0}^{t}∫_{ℝ^{3}}∫_{ℝ^{3}} |\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx ds ≤ r(ε) + C ∫_{0}^{t} H_{ε,p}(s) ds,

with lim_{ε↘0} r(ε) = 0. By Gronwall’s Lemma we obtain

H_{ε,p}(t) ≤ r(ε)e^{Ct}, 0 ≤ t ≤ T, ε > 0,

and

∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} |\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx dt ≤ r(ε)e^{CT} , ∀ ε > 0.

In particular we have

lim_{ε↘0} ∫_{ℝ^{3}}{|E_{ε}(t, x) − E(t, x)|^{2} + αε|B_{ε}(t, x)|^{2}} dx = 0, uniformly for t ∈ [0, T], (43)

and

lim_{ε↘0} ∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} |\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx dt = 0.  (44)

By Proposition 2.1 and the hypotheses (15), (16), (17) we have

sup_{ε>0} ∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} \frac{|v|^{2}}{2} f_{ε} dv dx dt < +∞,  sup_{ε>0} sup_{t∈[0,T]} ∫_{ℝ^{3}}(1 + |x|)ρ_{ε}(t, x) dx < +∞.

Moreover we have the inequalities

∫_{0}^{T} ∥j_{ε}(t)∥_{L^{1}(ℝ^{3})}^{2} dt ≤ ∫_{0}^{T}(∫_{ℝ^{3}}∫_{ℝ^{3}} |v|f_{ε} dv dx)^{2} dt
≤ ∫_{0}^{T}(∫_{ℝ^{3}}∫_{ℝ^{3}} |v|^{2}f_{ε} dv dx) · (∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} dv dx) dt
≤ (∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} |v|^{2}f_{ε} dv dx dt) · (∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε}^{0} dv dx)
≤ C,  ∀ ε > 0,

and

∫_{0}^{T} ∫_{ℝ^{3}}(1 + \sqrt{|x|})|j_{ε}(t, x)| dxdt ≤ ∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}}(1 + \sqrt{|x|})|v|f_{ε} dv dx dt
≤ \frac{1}{2} ∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}}\{(1 + |v|^{2}) + (|x| + |v|^{2})\}f_{ε} dv dx dt
≤ C,  ∀ ε > 0.

Therefore, by using Proposition 4.2 we deduce that (ρ_{ε})_{ε>0} is relatively compact in C^{0}([0, T], M_{+}^{1}(ℝ^{3})−tight) and (j_{ε})_{ε>0} is relatively compact in M^{1}([0, T]×ℝ^{3})^{3}-tight. Since div_{x}(E_{ε}−E) = (D−ρ_{ε})−(D−ρ) we obtain that ρ_{ε} → ρ in C^{0}([0, T], M_{+}^{1}(ℝ^{3})− tight). Using now (44) yields

∫_{0}^{T}∫_{ℝ^{3}}|j_{ε} + ρ_{ε}E| dx dt ≤ ∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} \sqrt{f_{ε}}|\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}| dv dx dt  (45)

≤ (∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} f_{ε} dv dx dt)^{\frac{1}{2}} (∫_{0}^{T}∫_{ℝ^{3}}∫_{ℝ^{3}} |\sqrt{f_{ε}}(v + E) + 2∇_{v}\sqrt{f_{ε}}|^{2} dv dx dt)^{\frac{1}{2}} ,

and thus we have for all continuous bounded function θ

|∫_{0}^{T}∫_{ℝ^{3}}(j_{ε} + ρE)θ dx dt| ≤ |∫_{0}^{T}∫_{ℝ^{3}}(j_{ε} + ρ_{ε}E)θ dx dt| + |∫_{0}^{T}∫_{ℝ^{3}}(ρ − ρ_{ε})Eθ dx dt|
≤ ∥θ∥_{L^{∞}} ∫_{0}^{T} ∫_{ℝ^{3}}|j_{ε} + ρ_{ε}E| dx dt
+ |∫_{0}^{T} ∫_{ℝ^{3}}(ρ_{ε} − ρ)Eθ dx dt| .  (46)

Since Eθ is bounded and continuous we have lim_{ε↘0} ∫_{0}^{T} ∫_{ℝ^{3}}(ρ_{ε} − ρ)Eθ dxdt = 0 and thus (44), (45), (46) imply that j_{ε} → −ρE in M^{1}([0, T] × ℝ^{3})^{3}-tight.

In order to prove that (B_{ε})_{ε>0} converges to B in D′(]0, T [×ℝ^{3})^{3} we will show that

lim_{ε↘0} ∫_{0}^{T}∫_{ℝ^{3}}(B_{ε} − B) · φ dx dt = 0,

for all function φ ∈ C_{c}^{2}(]0, T [×ℝ^{3})^{3}. Take φ such a function and observe that in particular we have φ, ∂_{t}φ ∈ L^{2}(]0, T [; H^{1}(ℝ^{3}))^{3}. By using the decomposition

φ = ∇_{x}φ_{1} + curl_{x}φ_{2},

with φ_{1}, ∂_{t}φ_{1} ∈ L^{2}(]0, T [; H^{2}(ℝ^{3})) and φ_{2}, ∂_{t}φ_{2} ∈ L^{2}(]0, T [; H^{2}(ℝ^{3}))^{3}, it is sufficient to prove that

lim_{ε↘0} ∫_{0}^{T}∫_{ℝ^{3}}(B_{ε} − B) · ∇_{x}φ_{1} dx dt = 0,  (47)

and

lim_{ε↘0} ∫_{0}^{T}∫_{ℝ^{3}}(B_{ε} − B) · curl_{x}φ_{2} dx dt = 0.  (48)

The convergence (47) follows easily since div_{x}B_{ε} = div_{x}B = 0. To justify (48) we use the equations

∂_{t}E_{ε} − curl_{x}B_{ε} = j_{ε} − J, ∂_{t}E − curl_{x}B = −ρE − J.

After multiplication by the test function φ_{2} we find

− ∫_{0}^{T}∫_{ℝ^{3}}(E_{ε}−E)·∂_{t}φ_{2} dx dt−∫_{0}^{T}∫_{ℝ^{3}}(B_{ε}−B)·curl_{x}φ_{2} dx dt = ∫_{0}^{T}∫_{ℝ^{3}}(j_{ε}+ρE)·φ_{2} dx dt.

Since ∂_{t}φ_{2} ∈ L^{2}(]0, T [; L^{2}(ℝ^{3}))^{3} and lim_{ε↘0} E_{ε} = E in L^{2}(]0, T [; L^{2}(ℝ^{3}))^{3} the first integral in the left hand side vanishes as ε ↘ 0. To deal with the right hand side, observe that φ_{2} is a continuous bounded function since φ_{2}, ∂_{t}φ_{2} ∈ L^{2}(]0, T [; H^{2}(ℝ^{3}))^{3} imply φ_{2} ∈ C^{0}([0, T]; H^{2}(ℝ^{3}))^{3} ⊂ C^{0}([0, T] × ℝ^{3})^{3} ∩ L^{∞}(]0, T [×ℝ^{3})^{3}. By using the convergence lim_{ε↘0} j_{ε} = −ρE in M^{1}([0, T] × ℝ^{3})^{3}-tight we deduce

lim_{ε↘0} ∫_{0}^{T}∫_{ℝ^{3}}(j_{ε} + ρE) · φ_{2} dx dt = 0,

and therefore (48) holds.

Let us end with the following remark, which makes clear the formal result (13).

Corollary 4.1 Let us set M_{E}(t, x, v) = (2π)^{−3/2}e^{−|v+E(t,x)|^{2}/2}. Under the assumptions of Theorem 1.1, then f_{ε} converges to ρM_{E} in the following sense

lim_{ε↘0} ∫_{0}^{T}∫_{ℝ^{3}} | ∫_{ℝ^{3}} (f_{ε}(t, x, v) − ρ(t, x)M_{E}(t, x, v)) φ(x) dx| dv dt = 0,

for any test function φ ∈ C^{0}(ℝ^{3}) ∩ L^{∞}(ℝ^{3}).

Proof. At first, we expand

f_{ε} − ρM_{E} = (f_{ε} − ρ_{ε}M_{E}) + (ρ_{ε} − ρ)M_{E}.

Consider φ ∈ C^{0}(ℝ^{3}) ∩ L^{∞}(ℝ^{3}). Since E ∈ W^{1,∞}(]0, T [×ℝ^{3})^{3}, for all (t, v) ∈ [0, T] × ℝ^{3} the function x → M_{E}(t, x, v)φ(x) is continuous and bounded. We have already shown that ρ_{ε} − ρ tends to 0 in C^{0}([0, T]; M^{1}(ℝ^{3}) − tight) and thus we have

lim_{ε↘0} ∫_{ℝ^{3}}(ρ_{ε}(t, x) − ρ(t, x))M_{E}(t, x, v)φ(x) dx = 0, ∀(t, v) ∈ [0, T] × ℝ^{3}.

Moreover we have the inequality |v + E(t, x)|^{2} ≥ \frac{1}{2}|v|^{2} − |E(t, x)|^{2} ≥ \frac{1}{2}|v|^{2} − ∥E∥_{L^{∞}}^{2} and thus M_{E}(t, x, v) ≤ C(∥E∥_{L^{∞}})e^{−|v|^{2}/4}, ∀ (t, x, v) ∈ [0, T] × ℝ^{3} × ℝ^{3}. We deduce that

|∫_{ℝ^{3}}(ρ_{ε}(t, x) − ρ(t, x))M_{E}(t, x, v)φ(x) dx| ≤ 2C(∥E∥_{L^{∞}})∥φ∥_{L^{∞}}∫_{ℝ^{3}} D(0, x) dx e^{−|v|^{2}/4},

and by using the dominated convergence theorem we obtain

lim_{ε↘0} ∫_{0}^{T}∫_{ℝ^{3}} |∫_{ℝ^{3}}(ρ_{ε}(t, x) − ρ(t, x))M_{E}(t, x, v)φ(x) dx| dv dt = 0.

It remains to discuss f_{ε} − ρ_{ε}M_{E}. We appeal to the logarithmic Sobolev inequality, see e.g. [3], [2], which yields

\begin{matrix}0 ≤ ∫_{ℝ^{N}} \{\frac{f_{ϵ}}{ρ_{ϵ}M_{E}} ln (\frac{f_{ϵ}}{ρ_{ϵ}M_{E}} ) − \frac{f_{ϵ}}{ρ_{ϵ}M_{E}} + 1\} ρ_{ϵ} M_{E} dv = ∫_{ℝ^{N}} f_{ϵ} ln (\frac{f_{ϵ}}{ρ_{ϵ}M_{E}} ) dv \\ ≤ λ ∫_{ℝ^{N}} |∇_{v}\sqrt{f_{ϵ}/M_{E}}|^{2} M_{E} dv, \end{matrix}

for some λ > 0. By using (44), the integral with respect to time and space of this quantity tends to 0 as ε ↘ 0. Eventually, we conclude by using the Csiszar-Kullback-Pinsker inequality, see [15], [26], which implies that

(∫_{ℝ^{N}}∫_{ℝ^{N}} |f_{ϵ} − ρ_{ϵ}M_{E}| dv dx)^{2} ≤ μ ∫_{ℝ^{N}} f_{ϵ} ln (\frac{f_{ϵ}}{ρ_{ϵ}M_{E}} ) dv dx,

with μ > 0.

5 Appendix

We detail here the dimensional analysis of the equations and the physical meaning of the different parameters introduced previously. Let us write the equations in physical variables. We distinguish the following physical constants

- ε_{0} the vacuum permittivity ;

- μ_{0} the vacuum permeability ;

- c_{0} the vacuum light speed given by ε_{0}μ_{0}c_{0}^{2} = 1 ;

- q the charge of (negative) particles ;

- m the mass of particles ;

- τ the relaxation time which characterizes the interactions of the particles with the thermal bath ;

- K_{B} the Planck constant ;

- T_{th} the temperature of the thermal bath. Let f(t, x, v) denote the particle distribution function, which depends on the time t > 0, space coordinates x ∈ ℝ^{3} and velocity coordinates v ∈ ℝ^{3}. The evolution of f is described by the Fokker-Planck equation

∂_{t}f + v · ∇_{x}f + \frac{F}{m} · ∇_{v}f = L_{F P} (f), (t, x, v) ∈]0, +∞[×ℝ^{3} × ℝ^{3},

where the Fokker-Planck collision operator is given by

L_{F P} (f) = \frac{1}{τ}div_{v} (vf + \frac{K_{B}T_{th}}{m} ∇_{v}f) ,

and F (t, x, v) = q(E(t, x) + v ∧ B(t, x)) represents the Lorentz force. The evolution of the electro-magnetic field (E, B) is given by the Maxwell equations

∂_{t}E − c_{0}^{2} · curl_{x}B = −\frac{j(t, x)}{ε_{0}}, ∂_{t}B + curl_{x}E = 0, (t, x) ∈]0, +∞[×ℝ^{3},

div_{x}E = \frac{ρ(t, x)}{ε_{0}}, div_{x}B = 0, (t, x) ∈]0, +∞[×ℝ^{3},

where ρ = q ∫_{ℝ^{3}} f dv and j = q ∫_{ℝ^{3}} vf dv are respectively the charge and current

densities. The plasma is characterized by the mean free path l = \sqrt{\frac{K_{B}T_{th}}{m}} · τ, which is the average distance travelled by a particle between two successive collisions, and the Debye length Λ = \sqrt{\frac{ε_{0}K_{B}T_{th}L^{3}}{q^{2}N}}, which is the typical length of perturbations of a quasi-neutral plasma. Here N stands for a typical value for the number of particles in the plasma. In this paper we focus our attention on asymptotic regimes where the mean free path is much smaller than the Debye length i.e., l ≪ Λ. We set ε = (\frac{l}{Λ})^{2} which is a small parameter. We introduce time, length and velocity units

T = \frac{τ}{ε} , L = \frac{l}{ε}, V = \sqrt{\frac{K_{B}T_{th}}{m}} .

Observe also that we have L = T V. We define dimensionless variables and unknowns by the relations

t = T t′, x = Lx′, v = V v′,

f(t, x, v) = \frac{N}{L^{3}V^{3}} f′(\frac{t}{T} , \frac{x}{L}, \frac{v}{V} ), E(t, x) = \frac{U_{th}}{Lε} E ′(\frac{t}{T} , \frac{x}{L}), B(t, x) = \frac{V U_{th}}{c_{0}^{2}Lε} B ′(\frac{t}{T} , \frac{x}{L}),

ρ(t, x) = \frac{qN}{L^{3}} ρ ′(\frac{t}{T} , \frac{x}{L}), j(t, x) = \frac{qV N}{L^{3}} j ′(\frac{t}{T} , \frac{x}{L}),

where U_{th} = \frac{K_{B}T_{th}}{q} is the thermal potential. After changing variables and unknowns, we obtain dropping the primes

ε(∂_{t}f + v · ∇_{x}f) − (E(t, x) + \frac{V^{2}}{c_{0}^{2}} v ∧ B(t, x)) · ∇_{v}f = div_{v}(vf + ∇_{v}f),

∂_{t}E − curl_{x}B = −j(t, x), \frac{V^{2}}{c_{0}^{2}} ∂_{t}B + curl_{x}E = 0,

div_{x}E = ρ(t, x), div_{x}B = 0,

where ρ(t, x) = ∫_{ℝ^{3}} f(t, x, v) dv, j(t, x) = ∫_{ℝ^{3}} vf(t, x, v) dv. Notice that we have

\frac{V^{2}}{c_{0}^{2}} = (\frac{l}{τ} )^{2} \frac{1}{c_{0}^{2}} = \frac{Λ^{2}ε}{τ^{2}c_{0}^{2}} = αε,

where α = (\frac{Λ}{τc_{0}} )^{2}, so that we are interested in a regime where the light speed remains large compared to the velocity unit of observation.

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