Asymptotic behavior for the Vlasov-Poisson equations with strong external curved magnetic field. Part II : general initial conditions

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Asymptotic behavior for the Vlasov-Poisson equations with strong external curved magnetic field. Part II :

general initial conditions

Mihai Bostan

To cite this version:

Mihai Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong external curved

magnetic field. Part II : general initial conditions. 2019. �hal-02047472�

Asymptotic behavior for the Vlasov-Poisson

equations with strong external curved magnetic field.

Part II : general initial conditions

Miha¨ı BOSTAN ^∗ (February 4, 2019)

Abstract

We discuss the asymptotic behavior of the Vlasov-Poisson system in the framework of the magnetic confinement, that is, under a strong external mag- netic field. We concentrate on curved three dimensional magnetic fields. We derive second order approximations, when the magnetic field becomes large, for general initial particle densities.

Keywords: Vlasov-Poisson system, averaging, homogenization.

AMS classification: 35Q75, 78A35, 82D10.

1 Introduction

The asymptotic analysis of the transport of charged particles under strong magnetic fields is a very important topic in plasma physics [7, 8, 9, 10, 14, 15, 11, 12, 13, 1, 2, 3, 4, 5]. It is related to real life applications, such that the energy production through magnetic confinement. When the particle velocities are small with respect to the light speed, the evolution of the particle density f = f(t, x, v) is described by the Vlasov-Poisson system

∂

_t

f

^ε

+ v ·∇

_x

f

^ε

+ q

m {E[f

^ε

(t)](x) + v ∧ B

^ε

(x)}·∇

_v

f

^ε

= 0, (t, x, v) ∈ R

+

× R

³

× R

³

. (1) E[f

^ε

(t)] = −∇

x

Φ[f

^ε

(t)], Φ[f

^ε

(t)](x) = q

4π

₀

Z

R³

Z

R³

f

^ε

(t, x

⁰

, v

⁰

)

|x − x

⁰

| dv

⁰

dx

⁰

where ε > 0 is a small parameter, entering the strong external non vanishing magnetic field

B

^ε

(x) = B

^ε

(x)e(x), B

^ε

(x) = B(x)

ε , |e(x)| = 1, x ∈ R

³

.

∗

Aix Marseille Universit´ e, CNRS, Centrale Marseille, I2M, Marseille France, Centre de

Math´ ematiques et Informatique, UMR 7373, 39 rue Fr´ ed´ eric Joliot Curie, 13453 Marseille Cedex

13 France. E-mail : mihai.bostan@univ-amu.fr.

The potential Φ[f

^ε

] satisfies the Poisson equation

−∆

_x

Φ[f

^ε

(t)] = q

0

Z

R³

f

^ε

(t, x, v) dv, (t, x) ∈ R

+

× R

³

whose fundamental solution is z →

_4π|z|¹

, z ∈ R

³

\ {0}. Here

₀

represents the electric permittivity. For any particle density f = f(x, v), the notation E[f] stands for the Poisson electric field

E[f ](x) = q 4π

0

Z

R³

Z

R³

f (x

⁰

, v

⁰

) x − x

⁰

|x − x

⁰

|

³

dv

⁰

dx

⁰

(2) and ρ[f ], j[f] are the charge and current densities respectively

ρ[f ] = q Z

R³

f (·, v) dv, j[f ] = q Z

R³

f (·, v)v dv.

The above system is supplemented by the initial condition f

^ε

(0, x, v) = f

_in

(x, v), (x, v) ∈ R

³

× R

³

.

In [6] a regular reformulation (when ε & 0) of the Vlasov-Poisson system has been derived, in the three dimensional setting, for well prepared initial particle densities. In this work we extend the previous analysis to general initial particle densities. Consider- ing general initial conditions leads to fast oscillations in time. In order to describe the asymptotic behavior (when ε & 0), we need to introduce a fast time variable s = t/ε.

The analysis follows closely that in [6] and the arguments rely on averaging along the flow of a vector field. As a fast time variable has been introduced, we need to consider the extended phase space (s, x, v) for averaging functions and vector fields.

Our paper is organized as follows. The average operators on the extended phase space and main properties are discussed in Section 2. The regular reformulation of the Vlasov-Poisson problem with strong external magnetic field is derived in Section 3 and revisited in the last Section 4.

2 Average operators and main properties

As in [6], we introduce the relative velocity with respect to the electric cross field drift

˜

v = v − ε E

^ε

(t, x) ∧ e(x) B (x) .

Accordingly, at any time t ∈ [0, T ], we consider the new particle density f ˜

^ε

(t, x, v) = ˜ f

^ε

t, x, v ˜ + ε E[f

^ε

(t)](x) ∧ e(x) B(x)

, (x, ˜ v) ∈ R

³

× R

³

. The particle densities f

^ε

, f ˜

^ε

have the same charge density

ρ[ ˜ f

^ε

(t)] = q Z

R³

f ˜

^ε

(t, ·, ˜ v) d˜ v = q Z

R³

f

^ε

(t, ·, v) dv = ρ[f

^ε

(t)], t ∈ [0, T ]

implying that the Poisson electric fields corresponding to f

^ε

, f ˜

^ε

coincide E[f

^ε

(t)] = E[ ˜ f

^ε

(t)], t ∈ [0, T ].

Therefore we can use the same notation E

^ε

(t) for denoting them. We assume that the magnetic field satisfies

B

₀

:= inf

x∈R³

|B(x)| > 0 or equivalently ω

₀

:= inf

x∈R³

|ω

_c

(x)| > 0.

The new particle densities ( ˜ f

^ε

)

_ε>0

verify

∂

_t

f ˜

^ε

+

˜

v + ε E

^ε

∧ e B

· ∇

_x

f ˜

^ε

− ε

∂

_t

E

^ε

∧ e B + ∂

_x

E

^ε

∧ e

B ˜ v + ε E

^ε

∧ e B

· ∇

_˜v

f ˜

^ε

+ h ω

_c

ε v ˜ ∧ e + q

m (E

^ε

· e) e i

· ∇

_˜v

f ˜

^ε

= 0, (t, x, v ˜ ) ∈ [0, T ] × R

³

× R

³

(3) f ˜

^ε

(0, x, ˜ v) = f

_in

x, v ˜ + ε E[f

_in

](x) ∧ e(x) B(x)

, (x, ˜ v) ∈ R

³

× R

³

. As in [6], thanks to the continuity equation

∂

t

ρ[f

^ε

] + div

x

j[f

^ε

] = 0

we obtain the following representation for the time derivative of the electric field E

^ε

, in terms of the particle density ˜ f

^ε

∂

_t

E[f

^ε

] = − 1 4π

₀

div

_x

Z

R³

x − x

⁰

|x − x

⁰

|

³

⊗

j [ ˜ f

^ε

(t)](x

⁰

) + ερ[ ˜ f

^ε

(t)](x

⁰

) E

^ε

(t, x

⁰

) ∧ e(x

⁰

) B(x

⁰

)

dx

⁰

. We introduce the new Larmor center ˜ x = x + ε

^˜v∧e(x)_ω

c(x)

, which is a second order ap- proximation of the Larmor center x + ε

^v∧e(x)_ω

c(x)

. We decompose the transport field in the Vlasov equation in such a way that ˜ x remains invariant with respect to the fast dynamics. We will distinguish between the orthogonal and parallel directions, taking as reference direction the magnetic line passing through the new Larmor center ˜ x, that is e(˜ x) (which is left invariant with respect to the fast dynamics)

˜

v = [˜ v − (˜ v · e(˜ x))e(˜ x)] + (˜ v · e(˜ x))e(˜ x).

Finally the Vlasov equation (3) writes

∂

_t

f ˜

^ε

+c

^ε

[ ˜ f

^ε

(t)]·∇

_x,˜v

f ˜

^ε

+εa

^ε

[ ˜ f

^ε

(t)]·∇

_x,˜v

f ˜

^ε

+ b

^ε

ε ·∇

_x,˜v

f ˜

^ε

= 0, (t, x, ˜ v) ∈ [0, T ]× R

³

× R

³

(4) where the autonomous vector field

^b_ε^ε

· ∇

_x,˜v

is given by

b

^ε

ε · ∇

x,˜v

= [˜ v − (˜ v · e(˜ x)) e(˜ x) + εA

^ε_x

(x, v)] ˜ · ∇

x

+ ω

_c

(x)

ε (˜ v ∧ e(˜ x)) · ∇

v˜

and for any particle density ˜ f , a

^ε

[ ˜ f ] · ∇

_x,˜v

, c

^ε

[ ˜ f] · ∇

_x,˜v

stand for the vector fields a

^ε

[ ˜ f] · ∇

_x,˜v

= E[ ˜ f] ∧ e

B − A

^ε_x

!

· ∇

_x

+

"

−∂

_x

E[ ˜ f ] ∧ e B

!

˜

v + ε E [ ˜ f ] ∧ e B

!

+ 1

4π

₀

B div

x

Z

R³

x − x

⁰

|x − x

⁰

|

³

⊗ j [ ˜ f] + ερ[ ˜ f] E[ ˜ f] ∧ e B

!

(x

⁰

) dx

⁰

∧ e(x)

#

· ∇

v˜

c

^ε

[ ˜ f ] · ∇

x,˜v

= (˜ v · e(˜ x)) e(˜ x) · ∇

x

+

ω

c

(x)˜ v ∧ e(x) − e(˜ x)

ε + q

m (E[ ˜ f] · e(x)) e(x)

· ∇

˜v

= q

m (E[ ˜ f ] · e(x)) e(x) − ω

_c

v ˜ ∧ Z

1

0

∂

_x

e

x + εs ˜ v ∧ e(x) ω

_c

(x)

v ˜ ∧ e(x) ω

_c

(x) ds

· ∇

_v˜

+ (˜ v · e(˜ x)) e(˜ x) · ∇

_x

.

The vector field A

^ε_x

(x, ˜ v) · ∇

_x

will be determined by imposing that the Larmor center

˜

x is left invariant by the fast dynamics b

^ε

· ∇

_x,˜v

x + ε v ˜ ∧ e(x) ω

c

(x)

= 0 that is

I

₃

+ ε∂

_x

˜ v ∧ e ω

_c

A

^ε_x

(x, v) = ˜ −∂

_x

˜ v ∧ e ω

_c

[˜ v−(˜ v·e(˜ x)) e(˜ x)]− e(˜ x) − e(x)

ε ∧(˜ v∧e(˜ x)).

The method employed in [6] applies as well when the initial particle density is not well prepared. In this case we deal with two time scales: the slow time variable t and the fast time variable s = t/ε. We need to average in the extended phase space(s, x, v ˜ ).

We say that a function u = u(s, x, ˜ v) is S = S(x, v) periodic with respect to ˜ s iff u(s + S(x, v), x, ˜ v) = ˜ u(s, x, v), ˜ (s, x, v) ˜ ∈ R × R

³

× R

³

.

Similarly, we say that a function u = u(s, x, v) is ˜ S

^ε

= S

^ε

(x, v) periodic with respect ˜ to s iff

u(s + S

^ε

(x, ˜ v), x, v) = ˜ u(s, x, ˜ v), (s, x, v) ˜ ∈ R × R

³

× R

³

.

With the notations in [6] Propositions 3.1, 3.2, we observe that if u is S periodic with respect to s, therefore the function (s, x, ˜ v) → u(Λ

^ε

(s; x, ˜ v), T

^ε

(x, v)) is ˜ S

^ε

periodic with respect to s. For establishing that, notice that

Λ

^ε

(s + S

^ε

(x, ˜ v); x, v ˜ ) = Λ

^ε

(s; x, ˜ v) + S(T

^ε

(x, v)). ˜ Indeed, we have, thanks to Proposition 3.2 [6]

Λ

^ε

(s + S

^ε

(x, v ˜ ); x, v) = ˜

Z

s+S^ε(x,˜v) 0

λ

^ε

( X

^ε

(σ; x, v), ˜ V ˜

^ε

(σ; x, ˜ v)) dσ

= Z

s

0

λ

^ε

( X

^ε

(σ; x, v), ˜ V ˜

^ε

(σ; x, ˜ v)) dσ +

Z

S^ε(x,˜v) 0

λ

^ε

(( X

^ε

, V ˜

^ε

)(τ ; ( X

^ε

, V ˜

^ε

)(s; x, ˜ v))) dτ

= Λ

^ε

(s; x, v) + ˜

Z

S^ε((X^ε,V˜^ε)(s;x,˜v)) 0

λ

^ε

(( X

^ε

, V ˜

^ε

)(τ; ( X

^ε

, V ˜

^ε

)(s; x, ˜ v))) dτ

= Λ

^ε

(s; x, v) + Λ ˜

^ε

(S

^ε

(( X

^ε

, V ˜

^ε

)(s; x, ˜ v)); ( X

^ε

, V ˜

^ε

)(s; x, ˜ v))

= Λ

^ε

(s; x, v) + ˜ S(T

^ε

(( X

^ε

, V ˜

^ε

)(s; x, ˜ v)))

= Λ

^ε

(s; x, v) + ˜ S(( X , V ˜ )(Λ

^ε

(s; x, v); ˜ T

^ε

(x, ˜ v)))

= Λ

^ε

(s; x, v) + ˜ S(T

^ε

(x, v)). ˜

It is easily seen that

u(Λ

^ε

(s + S

^ε

(x, v ˜ ); x, v), T ˜

^ε

(x, v)) = ˜ u(Λ

^ε

(s; x, ˜ v) + S(T

^ε

(x, v)), T ˜

^ε

(x, ˜ v))

= u(Λ

^ε

(s; x, ˜ v), T

^ε

(x, ˜ v))

saying that the function (s, x, v) ˜ → u(Λ

^ε

(s; x, v ˜ ), T

^ε

(x, v)) is ˜ S

^ε

periodic with respect to s.

Observe that for any (s, x, ˜ v) ∈ R × R

³

× R

³

, the characteristics of ∂

_s

+ b · ∇

_x,˜v

, ∂

_s

+ b

^ε

· ∇

_x,˜v

issued from (s, x, v) are ˜

(s + σ, X (σ; x, ˜ v), V ˜ (σ; x, v ˜ )), (s + σ, X

^ε

(σ; x, v), ˜ V ˜

^ε

(σ; x, v ˜ ))

respectively. We define the average operators for continuous S periodic, S

^ε

periodic functions by

hui (s, x, v) = ˜ 1 S(x, v) ˜

Z

S(x,˜v) 0

u(s + σ, X (σ; x, ˜ v), V ˜ (σ; x, ˜ v)) dσ, (s, x, v ˜ ) ∈ R × R

³

× R

³

hui

_ε

(s, x, ˜ v) = 1

S

^ε

(x, v) ˜

Z

S^ε(x,˜v) 0

u(s+σ, X

^ε

(σ; x, v), ˜ V ˜

^ε

(σ; x, ˜ v)) dσ, (s, x, v) ˜ ∈ R × R

³

× R

³

. Notice that the above operators extend the corresponding average operators defined in Proposition 3.1 [6] for continuous functions, not depending on s. As in Proposition 3.2 [6], we establish a relation between the average operators h·i , h·i

_ε

. We will work under the hypothesis ∇

_x

ω

_c

= 0, implying that S(x, ˜ v) = S

^ε

(x, v ˜ ) = 2π/ω

_c

, λ

^ε

(x, ˜ v) = 1, Λ

^ε

(s; x, ˜ v) = s, (s, x, v) ˜ ∈ R × R

³

× R

³

.

Proposition 2.1

Let u ∈ C( R × R

³

× R

³

) be a S periodic function with respect to s such that supp u ⊂ {(s, x, v) ˜ ∈ R × R

³

× R

³

: |˜ v| ≤ R} for some R > 0. For any ε > 0 satisfying εRk∂

x

ek

L^∞

/|ω

c

| < 1 we have

hu(·, T

^ε

)i

_ε

= hui (·, T

^ε

).

Proof.

It is enough to consider (s, x, ˜ v) ∈ R × R

³

× R

³

such that |˜ v| ≤ R. In that case we have, cf. Proposition 3.2 [6]

hu(·, T

^ε

)i

_ε

(s, x, ˜ v) = 1 S

Z

S 0

u(s + σ, T

^ε

(( X

^ε

, V ˜

^ε

)(σ; x, v))) dσ ˜

= 1 S

Z

S 0

u(s + σ, ( X , V ˜ )(σ; T

^ε

(x, ˜ v))) dσ

= hui (s, T

^ε

(x, v)). ˜

We also need to adapt the result in Proposition 3.3 [6] for S periodic functions.

Proposition 2.2

Let z ∈ C( R × R

³

× R

³

) be a S periodic function of zero average hzi (s, x, ˜ v) = 1

S Z

S

0

z(s + σ, x, V ˜ (σ; x, ˜ v)) dσ = 0, (s, x, v) ˜ ∈ R × R

³

× R

³

.

1. There is a unique continuous S periodic function u of zero average whose deriva- tive along the flow of ∂

_s

+ b · ∇

_x,˜v

is z

(∂

_s

+ b · ∇

_x,˜v

)u = z, hui = 0.

If z is bounded, so is u and

kuk

_C(R×B(Rx)×B(R_v˜))

≤ S

2 kzk

_C(R×B(Rx)×B(R_v˜))

for any R

x

, R

˜v

> 0. If supp z ⊂ R × B (R

x

) × B(R

v˜

), then supp u ⊂ R × B(R

x

) × B(R

_˜v

).

2. If z is of class C

¹

, then so is u and we have for any R

_x

, R

_v˜

> 0 k∇

v˜

uk

C(R×B(Rx)×B(Rv˜))

≤ S √

3k∇

˜v

zk

C(R×B(Rx)×B(R˜v))

k∇

_x

uk

_C(R×B(Rx)×B(R_˜v))

≤ C k∇

_x

zk

_C(R×B(Rx)×B(R_v˜))

+ R

_v˜

k∇

_˜v

zk

_C(R×B(Rx)×B(R_˜v))

k∂

_s

uk

_{C(R×B(Rx)×B(Rv˜))}

≤ kzk

_{C(R×B(Rx)×B(R˜v))}

+ 2 √

3R

_v˜

k∇

_˜v

zk

_{C(R×B(Rx)×B(R˜v))}

for some constant C depending on k∂

_x

ek

_L^∞

and S.

Proof.

1. Take

u(s, x, ˜ v) = 1 S

Z

S 0

(σ − S)z(s + σ, ( X , V ˜ )(σ; x, v)) dσ, ˜ (s, x, v) ˜ ∈ R × R

³

× R

³

. 2. Use the vector fields (c

ⁱ

· ∇

_x,˜v

)

1≤i≤6

, see Proposition 3.1 [6], which are in involution with ∂

_s

+ b · ∇

_x,˜v

, since ∇

_x

ω

_c

= 0.

3 The limit model and convergence result

We are ready to investigate the limit model in (4) as ε & 0. In this case we intend to capture the fast oscillations due to the operator ∂

_t

+

^b_ε^ε

· ∇

_x,˜v

. We are looking for a development whose dominant term belongs to the kernel of ∂

_t

+

^b_ε^ε

· ∇

_x,˜v

. It is easily seen that for any function u ∈ ker(∂

_s

+ b · ∇

_x,˜v

), we have u(·, T

^ε

) ∈ ker(∂

_s

+ b

^ε

· ∇

_x,˜v

), since

u(s + σ, T

^ε

(( X

^ε

, V ˜

^ε

)(σ; x, v))) = ˜ u(s + σ, ( X , V ˜ )(σ; T

^ε

(x, v))) = ˜ u(s, T

^ε

(x, v)). ˜ Similarly, for any S periodic function of zero average hui = 0, the S periodic function u(·, T

^ε

) has zero average

hu(·, T

^ε

)i

_ε

= hui (·, T

^ε

) = 0.

The previous discussion suggests to consider the Ansatz

f ˜

^ε

(t) = ˜ f

_ε

(t, t/ε) ◦ T

^ε

+ ε f ˜

_ε¹

(t, t/ε) ◦ T

^ε

+ ε

²

f ˜

_ε²

(t, t/ε) ◦ T

^ε

+ . . . (5) where (∂

_s

+ b · ∇

_x,˜v

) ˜ f

_ε

= 0, D

f ˜

_ε¹

E

= 0. As in [6], at the leading order, the particle

density ˜ f

^ε

has no fluctuation (with respect to the extended average operator), and the

averages at the orders O(ε

⁰

), O(ε) combine together in ˜ f

ε

(t, t/ε) ◦ T

^ε

. Notice also that the constraint (∂

_s

+b ·∇

_x,˜v

) ˜ f

_ε

= 0 is equivalent to ˜ f

_ε

(t, s, x, v) = ˜ ˜ F

_ε

(t, ( X , V ˜ )(−s; x, ˜ v)), for some function ˜ F

_ε

(t). We are looking for a closure determining ˜ f

_ε

, f ˜

_ε¹

. The error estimate will require to introduce the second order correction ε

²

f ˜

_ε²

(t, t/ε)◦T

^ε

. Plugging the Ansatz (5) in (4) we obtain

1

ε ∂

_s

f ˜

_ε

(t, s) ◦ T

^ε

+ ∂

_t

f ˜

_ε

(t, s) ◦ T

^ε

+ ∂

_s

f ˜

_ε¹

(t, s) ◦ T

^ε

+ ε∂

_t

f ˜

_ε¹

(t, s) ◦ T

^ε

+ ε∂

_s

f ˜

_ε²

(t, s) ◦ T

^ε

+ . . . + c

^ε

[ ˜ f

_ε

(t, s) ◦ T

^ε

+ ε f ˜

_ε¹

(t, s) ◦ T

^ε

+ . . .] · ∇[( ˜ f

_ε

+ ε f ˜

_ε¹

+ . . .)(t, s) ◦ T

^ε

] + εa

^ε

[( ˜ f

_ε

+ . . .)(t, s) ◦ T

^ε

] · ∇[( ˜ f

_ε

+ . . .)(t, s) ◦ T

^ε

]

+ b

^ε

ε · ∇[( ˜ f

_ε

+ ε f ˜

_ε¹

+ ε

²

f ˜

_ε²

+ . . .)(t, s) ◦ T

^ε

] = 0.

By construction we have (∂

_s

+ b

^ε

· ∇)( ˜ f

_ε

◦ T

^ε

) = 0, and therefore we deduce

∂

_t

f ˜

_ε

(t, s) ◦ T

^ε

+ (∂

_s

+ b

^ε

· ∇)(( ˜ f

_ε¹

+ ε f ˜

_ε²

)(t, s) ◦ T

^ε

) + ε∂

_t

f ˜

_ε¹

(t, s) ◦ T

^ε

(6) + c

^ε

[( ˜ f

ε

+ ε f ˜

_ε¹

)(t, s) ◦ T

^ε

] · ∇[( ˜ f

ε

+ ε f ˜

_ε¹

)(t, s) ◦ T

^ε

]

+ εa

^ε

[ ˜ f

_ε

(t, s) ◦ T

^ε

] · ∇[ ˜ f

_ε

(t, s) ◦ T

^ε

] = O(ε

²

).

We will take the (extended) average of (6) by discarding all second order contributions.

Obviously we have

D

∂

_t

f ˜

_ε

(t, ·) ◦ T

^ε

E

ε

= ∂

_t

f ˜

_ε

(t, ·) ◦ T

^ε

D

(∂

_s

+ b

^ε

· ∇

_x,˜v

)( ˜ f

_ε¹

+ ε f ˜

_ε²

)(t, ·) ◦ T

^ε

) E

ε

= 0 D

∂

_t

f ˜

_ε¹

(t, ·) ◦ T

^ε

E

ε

= D

∂

_t

f ˜

_ε¹

(t, ·) E

◦ T

^ε

= ∂

_t

D

f ˜

_ε¹

(t, ·) E

◦ T

^ε

= 0 and

εa

^ε

[ ˜ f

_ε

(t, s) ◦ T

^ε

] · ∇

_x,˜v

[ ˜ f

_ε

(t, s) ◦ T

^ε

] = ε(a[ ˜ f

_ε

(t, s)] · ∇

_x,˜v

f ˜

_ε

(t, s)) ◦ T

^ε

+ O(ε

²

) which implies, cf. Proposition 2.1

D

εa

^ε

[ ˜ f

_ε

(t, ·) ◦ T

^ε

] · ∇

_x,˜v

[ ˜ f

_ε

(t, ·) ◦ T

^ε

] E

ε

= ε D

(a[ ˜ f

_ε

(t, ·)] · ∇

_x,˜v

f ˜

_ε

(t, ·)) ◦ T

^ε

E

ε

+ O(ε

²

)

= ε D

a[ ˜ f

_ε

(t, ·)] · ∇

_x,˜v

f ˜

_ε

(t, ·) E

◦ T

^ε

+ O(ε

²

).

We concentrate now on the term corresponding to the vector field c

^ε

· ∇

_x,˜v

c

^ε

[( ˜ f

_ε

+ ε f ˜

_ε¹

)(t, s) ◦ T

^ε

] · [( ˜ f

_ε

+ ε f ˜

_ε¹

)(t, s) ◦ T

^ε

] = c

^ε

[ ˜ f

_ε

(t, s) ◦ T

^ε

] · ∇( ˜ f

_ε

(t, s) ◦ T

^ε

)

+ ε(c

^ε

[ ˜ f

_ε

(t, s)] · ∇ f ˜

_ε¹

(t, s)) ◦ T

^ε

+ ε q

m [(E[ ˜ f

_ε¹

(t, s)] · e)(e · ∇

_˜v

f ˜

_ε

(t, s)] ◦ T

^ε

+ O(ε

²

)

= (c

₀

[ ˜ f

_ε

(t, s)] · ∇ f ˜

_ε

(t, s)) ◦ T

^ε

+ ε(c

₁

[ ˜ f

_ε

(t, s)] · ∇ f ˜

_ε

(t, s)) ◦ T

^ε

+ ε(c

0

[ ˜ f

ε

(t, s)] · ∇ f ˜

_ε¹

(t, s)) ◦ T

^ε

+ ε q

m [(E[ ˜ f

_ε¹

(t, s)] · e)(e · ∇

v˜

f ˜

ε

)] ◦ T

^ε

+ O(ε

²

) where

c

₀

[ ˜ f ] · ∇

_x,˜v

= (˜ v · e)e · ∇

_x

+ q

m (E[ ˜ f ] · e)e · ∇

_˜v

− [˜ v ∧ ∂

_x

e(˜ v ∧ e)] · ∇

_˜v

.

We claim that the average along the flow of ∂

s

+ b · ∇

x,˜v

of (E[ ˜ f

_ε¹

(t, s)] · e)e · ∇

˜v

f ˜

ε

(t, s) vanishes. Indeed, as e · ∇

_˜v

is in involution with respect to ∂

_s

+ b · ∇

_x,˜v

, we have

(∂

_s

+ b · ∇

_x,˜v

)(e · ∇

_v˜

f ˜

_ε

(t)) = e · ∇

_v˜

((∂

_s

+ b · ∇

_x,˜v

) ˜ f

_ε

(t)) = 0 implying that

D

(E[ ˜ f

_ε¹

(t, ·)] · e)(e · ∇

_˜v

f ˜

_ε

(t)) E

= D

E[ ˜ f

_ε¹

(t)] · e E

e · ∇

_v˜

f ˜

_ε

(t) = 0 because the charge density of ˜ f

_ε¹

(t) has zero average

D

ρ[ ˜ f

_ε¹

(t)] E

(s, x) = q S

Z

S 0

Z

R³

f ˜

_ε¹

(t, s + σ, x, ˜ v) d˜ vdσ

= q S

Z

S 0

Z

R³

f ˜

_ε¹

(t, s + σ, x, V ˜ (σ; x, ˜ v)) d˜ vdσ

= q Z

R³

D f ˜

_ε¹

(t) E

(s, x, v) d˜ ˜ v = 0.

Therefore, thanks to Proposition 2.1 we obtain D

[(E[ ˜ f

_ε¹

(t)] · e)(e · ∇

˜v

f ˜

ε

(t))] ◦ T

^ε

E

ε

= D

(E[ ˜ f

_ε¹

(t)] · e)(e · ∇

˜v

f ˜

ε

(t)) E

◦ T

^ε

= 0 and thus

D

c

^ε

[( ˜ f

_ε

(t, ·) + ε f ˜

_ε¹

(t, ·)) ◦ T

^ε

] · ∇[( ˜ f

_ε

(t, ·) + ε f ˜

_ε¹

(t, ·)) ◦ T

^ε

] E

ε

= D

c

₀

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε

(t, ·) E

◦ T

^ε

+ ε D

c

₁

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε

(t, ·) E

◦ T

^ε

+ ε D

c

₀

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε¹

(t, ·) E

◦ T

^ε

+ O(ε

²

).

The previous computations lead to the following model for the particle density ˜ f

_ε

∂

_t

f ˜

_ε

(t, s) + D

c

₀

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε

(t, ·) E + ε D

(a[ ˜ f

_ε

(t, ·)] + c

₁

[ ˜ f

_ε

(t, ·)]) · ∇ f ˜

_ε

(t, ·) E

(7) + ε D

c

₀

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε¹

(t, ·) E

= 0, (t, s, x, v) ˜ ∈ [0, T ] × R × R

³

× R

³

together with the constraint (∂

_s

+ b · ∇

_x,˜v

) ˜ f

_ε

= 0. The equation for the fluctuation ˜ f

_ε¹

comes by comparing (6) with respect to (7). Indeed, we have

(∂

_s

+ b

^ε

· ∇)[( ˜ f

_ε¹

+ ε f ˜

_ε²

)(t, s) ◦ T

^ε

] = ∂

_s

( ˜ f

_ε¹

+ ε f ˜

_ε²

) ◦ T

^ε

+ ∂T

^ε

b

^ε

| {z }

b◦T^ε

·[∇( ˜ f

_ε¹

+ ε f ˜

_ε²

)] ◦ T

^ε

= [(∂

_s

+ b · ∇)( ˜ f

_ε¹

+ ε f ˜

_ε²

)] ◦ T

^ε

and therefore (6) also writes

∂

_t

f ˜

_ε

(t, s) + (∂

_s

+ b · ∇)( ˜ f

_ε¹

+ ε f ˜

_ε²

) + ε∂

_t

f ˜

_ε¹

+ c

₀

[ ˜ f

_ε

(t, s)] · ∇ f ˜

_ε

(t, s) (8) + ε(a[ ˜ f

_ε

(t, s)] + c

₁

[ ˜ f

_ε

(t, s)]) · ∇ f ˜

_ε

(t, s) + εc

₀

[ ˜ f

_ε

(t, s)] · ∇ f ˜

_ε¹

(t, s)

+ ε q

m (E[ ˜ f

_ε¹

(t, s)]·)(e · ∇

_˜v

f ˜

_ε

(t, s)) = O(ε

²

).

Taking the difference between (8), (7) yields c

₀

[ ˜ f

_ε

(t, s)] · ∇ f ˜

_ε

(t, s) − D

c

₀

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε

(t, ·) E

+ ε(a[ ˜ f

_ε

(t, s)] + c

₁

[ ˜ f

_ε

(t, s)]) · ∇ f ˜

_ε

(t, s)

− ε D

(a[ ˜ f

_ε

(t, ·)] + c

₁

[ ˜ f

_ε

(t, ·)]) · ∇ f ˜

_ε

(t, ·) E + εc

0

[ ˜ f

ε

(t, s)] · ∇ f ˜

_ε¹

(t, s) − ε

D

c

0

[ ˜ f

ε

(t, ·)] · ∇ f ˜

_ε¹

(t, ·) E + ε q

m (E[ ˜ f

_ε¹

(t, s)] · e)(e · ∇

_v˜

f ˜

_ε

(t, s)) + (∂

_s

+ b · ∇)( ˜ f

_ε¹

+ ε f ˜

_ε²

) + ε∂

_t

f ˜

_ε¹

= O(ε

²

).

The above equality suggests to determine the fluctuation ˜ f

_ε¹

by c

₀

[ ˜ f

_ε

(t, s)] · ∇ f ˜

_ε

(t, s) − D

c

₀

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε

(t, ·) E

+ (∂

_s

+ b · ∇) ˜ f

_ε¹

(t, s) = 0, D f ˜

_ε¹

E

= 0 (9) and to consider the corrector ˜ f

_ε²

such that

(a[ ˜ f

_ε

(t, s)] + c

₁

[ ˜ f

_ε

(t, s)]) · ∇ f ˜

_ε

(t, s) − D

(a[ ˜ f

_ε

(t, ·)] + c

₁

[ ˜ f

_ε

(t, ·)]) · ∇ f ˜

_ε

(t, ·) E + c

₀

[ ˜ f

_ε

(t, s)] · ∇ f ˜

_ε¹

(t, s) − D

c

₀

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε¹

(t, ·) E + q

m (E[ ˜ f

_ε¹

(t, s)] · e)(e · ∇

v˜

f ˜

ε

(t, s)) + ∂

t

f ˜

_ε¹

+ (∂

s

+ b · ∇) ˜ f

_ε²

= 0, D f ˜

_ε²

E

= 0.

The well posedness of (7), (9) is stated in Section 4, see Theorem 4.1. As in Theorem 1.2 [6], we can establish the following error estimate. The proof details are left to the reader.

Theorem 3.1

Let B = Be ∈ C

_b⁴

( R

³

) be a smooth magnetic field such that ∇

_x

B = 0, div

_x

B = 0.

Consider a non negative, smooth, compactly supported initial particle density G ˜ ∈ C

_c³

( R

³

× R

³

) and g(s, x, ˜ v) = ˜ ˜ G(x, V ˜ (−s; x, v ˜ )), (s, x, v) ˜ ∈ R × R

³

× R

³

. We denote by (f

^ε

)

_0<ε≤1

the solutions of the Vlasov-Poisson equations with external magnetic field (1), (2) on [0, T ], 0 < T < T (f

_in

), cf. Theorem 2.1 [6], corresponding to the initial condition

f

^ε

(0, x, v) = (˜ g + ε˜ g

¹

) 0, x + ε v ∧ (x)

ω

_c

, v − ε E[ ˜ G] ∧ e(x) B

!

, (x, v) ∈ R

³

× R

³

where

c

0

[˜ g] · ∇˜ g − hc

0

[˜ g] · ∇˜ gi + (∂

s

+ b · ∇)˜ g

¹

= 0,

˜ g

¹

= 0.

For ε ∈]0, ε

T

] small enough, we consider the solution f ˜

ε

= ˜ f

ε

(t, s, x, ˜ v) on [0, T ] of the problem

∂

_t

f ˜

_ε

(t, s) + D

c

₀

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε

(t, ·) E + ε D

(a[ ˜ f

_ε

(t, ·)] + c

₁

[ ˜ f

_ε

(t, ·)]) · ∇ f ˜

_ε

(t, ·) E + ε D

c

₀

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε¹

(t, ·) E

= 0, (t, s, x, v) ˜ ∈ [0, T ] × R × R

³

× R

³

c

₀

[ ˜ f

_ε

(t, s)] · ∇ f ˜

_ε

(t, s) − D

c

₀

[ ˜ f

_ε

(t, ·)] · ∇ f ˜

_ε

(t, ·) E

+ (∂

_s

+ b · ∇) ˜ f

_ε¹

(t, s) = 0, D f ˜

_ε¹

E

= 0

corresponding to the initial condition

f ˜

_ε

(0, s, x, ˜ v) = ˜ G(x, V ˜ (−s; x, v)) = ˜ ˜ g(s, x, ˜ v), (s, x, v) ˜ ∈ R × R

³

× R

³

. Therefore there is a constant C

_T

such that for any 0 < ε ≤ ε

_T

sup

t∈[0,T]





 Z

R³

Z

R³

f

^ε

(t, x, v) − ( ˜ f

ε

+ ε f ˜

_ε¹

) t, t/ε, x + ε v ∧ e

ω

_c

, v − ε E[ ˜ f

ε

(t, t/ε)] ∧ e B

!

2

dvdx







1/2

≤ C

T

ε

²

.

4 Equivalent formulation of the limit model

We determine now the equivalent formulation for (7), (9) by computing the average of the vector fields entering this model. Most of the computations has been performed in [6], where the formulae for hai · ∇, hc

₀

i · ∇, hc

₁

i · ∇ are detailed, and we only need to complete them by treating the extra terms due to the presence of the fast time variable.

Proposition 4.1

Assume that e ∈ C

²

( R

³

), ∇

_x

ω

_c

= 0, div

_x

e = 0 and let us consider f ˜

_ε

∈ C

²

([0, T ] × R × R

³

× R

³

). Then f ˜

_ε

solves

∂

_t

f ˜

_ε

+ D

c

₀

[ ˜ f

_ε

] · ∇ f ˜

_ε

E + ε D

(a[ ˜ f

_ε

] + c

₁

[ ˜ f

_ε

]) · ∇ f ˜

_ε

E + ε D

c

₀

[ ˜ f

_ε

] · ∇ f ˜

_ε¹

E

= 0, (∂

_s

+ b · ∇) ˜ f

_ε

= 0 (10) c

₀

[ ˜ f

_ε

] · ∇ f ˜

_ε

− D

c

₀

[ ˜ f

_ε

] · ∇ f ˜

_ε

E

+ (∂

_s

+ b · ∇) ˜ f

_ε¹

= 0, D f ˜

_ε¹

E

= 0 (11)

iff f ˜

_ε

satisfies f ˜

_ε

(t, s, x, v ˜ ) = ˜ F

_ε

(t, ( X , V ˜ )(−s; x, v)), where ˜

∂

_t

F ˜

_ε

+ D

c

₀

[ ˜ F

_ε

] E

· ∇ F ˜

_ε

+ ε D a[ ˜ F

_ε

] E

+ D

c

₁

[ ˜ F

_ε

] E

· ∇ F ˜

_ε

+ εD[ ˜ F

_ε

] · ∇ F ˜

_ε

= 0 (12)

D[ ˜ F ] · ∇

_X,V˜

=

"

j[ ˜ F ] ∧ e(X) 3

₀

B +

R

R³

N (e(X), X − X

⁰

, e(X

⁰

))j[ ˜ F ](X

⁰

) dX

⁰

8π

₀

B ∧ e(X)

#

· ∇

V˜

+

"

q

m e ⊗ e E

"

(e · rot

X

e) ( ˜ V · e) ω

_c

F ˜

#

− E

"

V ˜ ∧ e

ω

_c

· ∇

X

F ˜

#!

+ d

V˜

(X, V ˜ )

#

· ∇

V˜

d

V˜

(X, V ˜ ) =

*

(c

₀

· ∇

_x,˜v

)

3

X

k=1

[(A

_k

)

_−s

cos(ksω

_c

) + (B

_k

)

_−s

sin(ksω

_c

)]

+

(0, X, V ˜ )

f ˜

_ε¹

(t, s, x, v) = ˜

3

X

k=1

[A

_k

(( X , V ˜ )(−s; x, v)) cos(ksω ˜

_c

) + B

_k

(( X , V ˜ )(−s; x, v)) sin(ksω ˜

_c

)]

· (∇

V˜

F ˜

_ε

(t))(( X , V ˜ )(−s; x, v)) ˜ (13)

where N (e, z, e

⁰

) = (I

3

−e⊗e)K(z)(I

3

−e

⁰

⊗e

⁰

)−M [e]K(z)M [e

⁰

], K (z) = (I

3

− 3z ⊗ z/|z|

²

) /|z|

³

A

_k

(X, V ˜ ) = 1

kπ Z

S

0

F(s, X, V ˜ ) sin(ksω

_c

) ds, B

_k

(X, V ˜ ) = − 1 kπ

Z

S 0

F (s, X, V ˜ ) cos(ksω

_c

) ds for k ∈ {1, 2, 3} and

F (s, X, V ˜ ) = ∂

_x,˜v

V ˜ (−s; ( X , V ˜ )(s; X, V ˜ ))c

₀

(( X , V ˜ )(s; X, V ˜ )) c

₀

(x, ˜ v) · ∇

_x,˜v

= (˜ v · e)e · ∇

_x

− [˜ v ∧ ∂

_x

e(˜ v ∧ e)] · ∇

_˜v

. Proof.

Clearly the constraint (∂

s

+ b · ∇) ˜ f

ε

= 0 writes ˜ f

ε

(t, s, x, v ˜ ) = ˜ F

ε

(t, ( X , V ˜ )(−s; x, v)) for ˜ some function ˜ F

_ε

= ˜ F

_ε

(t, X, V ˜ ) ∈ C

²

([0, T ]× R

³

× R

³

). We need to compute the averages D

c

₀

[ ˜ f

_ε

] · ∇ f ˜

_ε

E , D

(a[ ˜ f

_ε

] + c

₁

[ ˜ f

_ε

]) · ∇ f ˜

_ε

E , D

c

₀

[ ˜ f

_ε

] · ∇ f ˜

_ε¹

E

along the flow of ∂

_s

+b ·∇

_x,˜v

and to invert the operator ∂

_s

+ b · ∇

_x,˜v

on zero average functions, in order to solve (11).

Recall that for any particle density ˜ f , the vector field a[ ˜ f] · ∇

_x,˜v

writes a[ ˜ f] · ∇

_x,˜v

= E[ ˜ f] ∧ e

B − A

_x

(x, ˜ v)

!

· ∇

_x

− ∂

_x

E[ ˜ f] ∧ e B

!

˜ v · ∇

_v˜

+ 1

4π

₀

B

div

_x

Z

R³

x − x

⁰

|x − x

⁰

|

³

⊗ j[ ˜ f](x

⁰

) dx

⁰

∧ e(x)

· ∇

_˜v

.

We have ρ[ ˜ f

_ε

(t, s)] = ρ[ ˜ F

_ε

(t)] implying that E[ ˜ f

_ε

(t, s)] = E[ ˜ F

_ε

(t)] and j[ ˜ f

_ε

(t, s)](x) = R(−sω

_c

, e(x))j [ ˜ F

_ε

(t)](x)

and therefore

a[ ˜ f

_ε

(t, s)] · ∇

_x,˜v

= a[ ˜ F

_ε

(t)] · ∇

_x,˜v

+ a

_s

[ ˜ F

_ε

(t)] · ∇

_v˜

where

a

_s

[ ˜ F

_ε

]·∇

_˜v

= 1 4π

₀

B

div

_x

Z

R³

x − x

⁰

|x − x

⁰

|

³

⊗ [R(−sω

_c

, e(x

⁰

)) − I

₃

] j[ ˜ F

_ε

](x

⁰

) dx

⁰

∧ e(x)

·∇

_˜v

.

Thanks to the equality ˜ f

ε

(t, s, x, v) = ˜ ˜ f

ε

(t, s + σ, ( X , V ˜ )(σ; x, v)) we have ˜

∇

_x,˜v

f ˜

_ε

(t, s, x, v) = ˜

^t

∂( X , V ˜ )(σ; x, v ˜ )(∇ f ˜

_ε

)(t, s + σ, ( X , V ˜ )(σ; x, v)) ˜ implying that

(∇ f ˜

_ε

)(t, s + σ, ( X , V ˜ )(σ; x, ˜ v)) =

^t

∂ ( X , V ˜ )(−σ; ( X , V ˜ )(σ; x, ˜ v))∇ f ˜

_ε

(t, s, x, ˜ v) and

(a[ ˜ f

_ε

] · ∇ f ˜

_ε

)(t, s + σ, ( X , V ˜ )(σ; x, ˜ v))

= ∂( X , V ˜ )(−σ; ( X , V ˜ )(σ; x, v))a[ ˜ ˜ F

_ε

(t)](( X , V ˜ )(σ; x, v)) ˜ · ∇ f ˜

_ε

(t, s, x, v) ˜

+ R(σω

c

, e(x))a

s+σ

[ ˜ F

ε

(t)](x) · ∇

˜v

f ˜

ε

(t, s, x, v). ˜

Averaging with respect to σ, we obtain cf. Proposition 5.1 [6]

D

a[ ˜ f

_ε

] · ∇ f ˜

_ε

E

(t, s, x, ˜ v) = 1 S

Z

S 0

∂( X , V ˜ )(−σ; ( X , V ˜ )(σ; x, v))a[ ˜ ˜ F

_ε

(t)](( X , V ˜ )(σ; x, v)) dσ ˜

· ∇

_x,˜v

f ˜

_ε

(t, s, x, v ˜ ) + 1 S

Z

S 0

R(σω

_c

, e(x))a

_s+σ

[ ˜ F

_ε

(t)](x) dσ · ∇

_˜v

f ˜

_ε

(t, s, x, ˜ v)

= D

a[ ˜ F

_ε

(t)] E

(x, v ˜ ) · ∇

_x,˜v

f ˜

_ε

(t, s, x, ˜ v) + 1 S

Z

S 0

R(σω

_c

, e(x))a

_s+σ

[ ˜ F

_ε

(t)](x) dσ · ∇

_˜v

f ˜

_ε

(t, s, x, ˜ v)

= ∂( X , V ˜ )(−s; x, v) ˜ D

a[ ˜ F

_ε

(t)] E

(x, ˜ v) · (∇

_X,V˜

F ˜

_ε

)(t, ( X , V ˜ )(−s; x, v ˜ )) + 1

S Z

S

0

R((s + σ)ω

c

, e(x))a

s+σ

[ ˜ F

ε

(t)](x) dσ · (∇

V˜

F ˜

ε

)(t, ( X , V ˜ )(−s; x, v)) ˜

= D

a[ ˜ F

_ε

(t)] E

(( X , V ˜ )(−s; x, ˜ v)) · (∇

_X,V˜

F ˜

_ε

)(t, ( X , V ˜ )(−s; x, v)) ˜ + 1

S Z

S

0

R(σω

_c

, e(x))a

_σ

[ ˜ F

_ε

(t)](x) dσ · (∇

V˜

F ˜

_ε

)(t, ( X , V ˜ )(−s; x, ˜ v))

where hai · ∇ has been computed in Proposition 5.4 [6]. Notice that in the last equality we have used the involution of D

a[ ˜ F

_ε

(t)] E

· ∇ with respect to the vector field b · ∇, cf.

Proposition 5.1 [6]. For the last average, observe that 1

S Z

S

0

R(σω

_c

, e(x))a

_σ

[ ˜ F

_ε

(t)](x) dσ

= 1

4π

₀

BS Z

S

0

R(σω

_c

, e(x))

div

_x

Z

R³

x − x

⁰

|x − x

⁰

|

³

⊗ R(−σω

_c

, e(x

⁰

))j [ ˜ F

_ε

(t)](x

⁰

) dx

⁰

∧ e(x)

dσ

= 1

4π

₀

BS Z

S

0

[cos(σω

c

)(I

3

− e(x) ⊗ e(x)) + sin(σω

c

)M[e(x)] + e(x) ⊗ e(x)]

div

_x

Z

R³

x − x

⁰

|x − x

⁰

|

³

⊗ [cos(σω

_c

)(I

₃

− e(x

⁰

) ⊗ e(x

⁰

)) − sin(σω

_c

)M[e(x

⁰

)] + e(x

⁰

) ⊗ e(x

⁰

)]j[ ˜ F

_ε

(t)] dx

⁰

∧ e(x) dσ

= 1

4π

₀

B

I

₃

− e ⊗ e 2

div

x

Z

R³

x − x

⁰

|x − x

⁰

|

³

⊗ [I

3

− e(x

⁰

) ⊗ e(x

⁰

)]j[ ˜ F

ε

(t)](x

⁰

) dx

⁰

∧ e(x)

− 1

4π

₀

B M [e]

2

div

x

Z

R³

x − x

⁰

|x − x

⁰

|

³

⊗ M [e(x

⁰

)]j[ ˜ F

ε

(t)](x

⁰

) dx

⁰

∧ e(x)

= − M [e]

8π

₀

B lim

δ&0

Z

|x−x⁰|>δ

[I

₃

− e(x) ⊗ e(x)](x − x

⁰

)

|x − x

⁰

|

³

div{[I

₃

− e(x

⁰

) ⊗ e(x

⁰

)]j[ ˜ F

_ε

(t)](x

⁰

)} dx

⁰

+ M [e]

8π

₀

B lim

δ&0

Z

|x−x⁰|>δ

M [e(x)](x − x

⁰

)

|x − x

⁰

|

³

div{M [e(x

⁰

)]j[ ˜ F

_ε

(t)](x

⁰

)} dx

⁰

= − M[e(x)]

8π

₀

B lim

δ&0

Z

|x−x⁰|>δ

N (e(x), x − x

⁰

, e(x

⁰

))j[ ˜ F

ε

(t)](x

⁰

) dx

⁰

− M [e(x)]

8π

0

B lim

δ&0

Z

|x−x⁰|=δ

{[I

₃

− e(x) ⊗ e(x)] x − x

⁰

|x − x

⁰

|

³

⊗ [I

₃

− e(x

⁰

) ⊗ e(x

⁰

)] x − x

⁰

|x − x

⁰

| + M [e(x)] x − x

⁰

|x − x

⁰

|

³

⊗ M [e(x

⁰

)] x − x

⁰

|x − x

⁰

| }j[ ˜ F

_ε

(t)](x

⁰

) dσ(x

⁰

)

where K(z) = (I

3

− 3

_|z|^z

⊗

_|z|^z

)/|z|

³

, z ∈ R

³

\ {0}, N (e, z, e

⁰

) = (I

3

− e ⊗ e)K (z)(I

3

− e

⁰

⊗ e

⁰

) − M [e]K(z)M[e

⁰

], e, z, e

⁰

∈ R

³

\ {0}, |e| = |e

⁰

| = 1. Performing the change of variable x

⁰

= x + δz in the last integral yields

Z

|x−x⁰|=δ

{(I

₃

− e(x) ⊗ e(x)) x − x

⁰

|x − x

⁰

|

³

⊗ (I

₃

− e(x

⁰

) ⊗ e(x

⁰

)) x − x

⁰

|x − x

⁰

| + M [e(x)] x − x

⁰

|x − x

⁰

|

³

⊗ M [e(x

⁰

)] x − x

⁰

|x − x

⁰

| }j[ ˜ F

_ε

(t)](x

⁰

)dσ(x

⁰

)

= Z

|z|=1

{(I

₃

− e(x) ⊗ e(x))z ⊗ (I

₃

− e(x + δz) ⊗ e(x + δz))z + M [e(x)]z ⊗ M [e(x + δz)]z}j [ ˜ F

_ε

(t)](x + δz)dσ(z)

δ&0

→ Z

|z|=1

{(I

₃

− e(x) ⊗ e(x))z ⊗ (I

₃

− e(x) ⊗ e(x))z + M [e(x)]z ⊗ M [e(x)]z}dσ(z) j[ ˜ F

_ε

(t)](x)

= Z

|z|=1

|z ∧ e(x)|

²

(I

₃

− e(x) ⊗ e(x))dσ j[ ˜ F

_ε

(t)](x)

= 8π

3 (I

₃

− e(x) ⊗ e(x)) j[ ˜ F

_ε

(t)](x).

Therefore the average of a[ ˜ f

_ε

] · ∇

_x,˜v

f ˜

_ε

writes D

a[ ˜ f

_ε

] · ∇ f ˜

_ε

E

(t, s, x, ˜ v) = D

a[ ˜ F

_ε

(t)] E

· ∇

_X,V˜

F ˜

_ε

(t)

(( X , V ˜ )(−s; x, v)) ˜ (14)

−

e(x)

8π

0

B ∧ lim

δ&0

Z

|x−x⁰|>δ

N (e(x), x − x

⁰

, e(x

⁰

))j[ ˜ F

_ε

(t)](x

⁰

) dx

⁰

· (∇

V˜

F ˜

_ε

(t))(( X , V ˜ )(−s; x, v)) ˜ + j[ ˜ F

_ε

(t)](x) ∧ e(x)

3

₀

B · (∇

V˜

F ˜

_ε

(t))(( X , V ˜ )(−s; x, v)). ˜ We inquire about the convergence, when δ & 0 of Z

|x−x⁰|>δ

N (e(x), x − x

⁰

, e(x

⁰

))j [ ˜ F

_ε

](x

⁰

) dx

⁰

= Z

δ<|x−x⁰|<R

N (e(x), x − x

⁰

, e(x

⁰

))(j [ ˜ F

_ε

](x

⁰

) − j[ ˜ F

_ε

](x)) dx

⁰

+

Z

δ<|x−x⁰|<R

N(e(x), x − x

⁰

, e(x

⁰

)) dx

⁰

j[ ˜ F

_ε

](x) for R large enough. We are done if we establish the convergence of R

1

_{δ<|x−x⁰_|<R}

N (e(x), x−

x

⁰

, e(x

⁰

)) dx

⁰

when δ & 0. But we can write Z

δ<|x−x⁰|<R

N (e(x), x − x

⁰

, e(x

⁰

)) dx

⁰

= Z

δ<|x−x⁰|<R

(N (e(x), x − x

⁰

, e(x

⁰

)) − N (e(x), x − x

⁰

, e(x))) dx

⁰

+

Z

δ<|x−x⁰|<R

N(e(x), x − x

⁰

, e(x)) dx

⁰

and therefore it is enough to prove the convergence, as δ & 0, of R

1

{δ<|x−x⁰|<R}

N (e(x), x−

x

⁰

, e(x)) dx

⁰

. Actually, for any r > 0 we have R

|z|=r

N (e(x), z, e(x)) dσ(z) = 0. This is a consequence of the fact that K (z) has zero trace for any z ∈ R

³

\ {0}. Indeed, for any ξ ∈ R

³

we have

K (z) : (I

₃

− e ⊗ e)ξ ⊗ (I

₃

− e ⊗ e)ξ + K(z) : M[e]ξ ⊗ M [e]ξ + |ξ ∧ e|

²

K (z) : e ⊗ e

= |ξ ∧ e|

²

traceK(z) = 0, z ∈ R

³

\ {0}

implying that

[N (e, z, e) + (K(z)e · e)(I

₃

− e ⊗ e)] : ξ ⊗ ξ = 0, ξ ∈ R

³

.

As the matrix N (e, z, e) + (K (z)e · e)(I

₃

− e ⊗ e) is symmetric, we obtain N (e, z, e) =

−(K(z)e · e)(I

₃

− e ⊗ e). By direct computation we obtain R

|z|=r

(K(z)e · e) dσ(z) = 0, r > 0. Similarly we have

D

c

0

[ ˜ f

ε

] · ∇

x,˜v

f ˜

ε

E

(t, s, x, v) = ˜ D

c

0

[ ˜ F

ε

(t)]

E

· ∇

_X,V˜

F ˜

ε

(t)

(( X , V ˜ )(−s; x, v)) ˜ (15) where hc

₀

i · ∇ has been computed in Proposition 5.5 [6]. In order to treat the average of c

1

[ ˜ f

ε

] · ∇

x,˜v

f ˜

ε

we need to compute E[(˜ v ∧ e) · ∇

x

f ˜

ε

] and therefore the charge density

q Z

R³

(˜ v ∧ e) · ∇

_x

f ˜

_ε

(t, s, x, v) d˜ ˜ v = q div

_x

Z

R³

f ˜

_ε

(t, s, x, v)(˜ ˜ v ∧ e) d˜ v

− q Z

R³

f ˜

_ε

(t, s, x, v)div ˜

_x

(˜ v ∧ e) d˜ v

= div

x

(j[ ˜ f

ε

(t, s)] ∧ e) + j [ ˜ f

ε

(t, s)] · rot

x

e

= e · rot

_x

(R(−sω

_c

, e(x))j[ ˜ F

_ε

(t)]).

We obtain the following formula for the vector field c

₁

· ∇

c

₁

[ ˜ f

_ε

(t, s)] · ∇

_x,˜v

= c

₁

[ ˜ F

_ε

(t)] · ∇

_x,˜v

+ c

_1s

[ ˜ F

_ε

(t)] · ∇

_˜v

where

c

_1s

[ ˜ F

_ε

] · ∇

_˜v

= 1 B

E[e · rot

_x

((R(−sω

_c

, e(x)) − I

₃

)˜ v F ˜

_ε

)] · e e · ∇

_v˜

and therefore

D

c

₁

[ ˜ f

_ε

] · ∇ f ˜

_ε

E

(t, s, x, v) = ˜ D

c

₁

[ ˜ F

_ε

(t)] E

· ∇

_X,V˜

F ˜

_ε

(t)

(( X , V ˜ )(−s; x, v)) ˜ + 1

S Z

S

0

R(σω

c

, e(x))c

1σ

[ ˜ F

ε

(t)](x) dσ · (∇

V˜

F ˜

ε

(t))(( X , V ˜ )(−s; x, v)) ˜ where hc

₁

i · ∇ has been computed in Proposition 5.6 [6]. As before, the last average writes

1 S

Z

S 0

R(σω

_c

, e(x))c

_1σ

[ ˜ F

_ε

(t)](x) dσ = e ⊗ e BS

Z

S 0

E[e · rot

_x

((R(−σω

_c

, e) − I

₃

)˜ v F ˜

_ε

)] dσ

= e ⊗ e

B E[e · rot

_x

((e ⊗ e − I

₃

)˜ v F ˜

_ε

(t))]

= e ⊗ e

B E[(e · rot

_x

e)(˜ v · e) ˜ F

_ε

(t)] − e ⊗ e

B E[(˜ v ∧ e) · ∇

_x

F ˜

_ε

(t)].

Finally the average of c

₁

[ ˜ f

_ε

] · ∇

_x,˜v

f ˜

_ε

is given by D

c

₁

[ ˜ f

_ε

] · ∇

_x,˜v

f ˜

_ε

E

(t, s, x, v) = ˜ D

c

₁

[ ˜ F

_ε

(t)] E

· ∇

_X,V˜

F ˜

_ε

(t)

(( X , V ˜ )(−s; x, v)) ˜ (16) + q

m

E

(e·rot

_x

e) (˜ v ·e) ω

_c

F ˜

_ε

(t)

·e − E v ˜ ∧e

ω

_c

·∇

_X

F ˜

_ε

(t)

·e

e·(∇

V˜

F ˜

_ε

(t))(( X , V ˜ )(−s; x, v)). ˜

We concentrate now on the average of c

₀

[ ˜ f

_ε

] · ∇

_x,˜v

f ˜

_ε¹

. Before ending the proof of

Proposition 4.1, we need to generalize the Proposition 5.2 [6] to the periodic case.

Proposition 4.2

Assume that e ∈ C

²

( R

³

), ∇

_x

ω

_c

= 0, div

_x

e = 0. Let χ · ∇

_s,x,˜v

= χ

_s

∂

_s

+ χ

_x

· ∇

_x

+ χ

_˜v

· ∇

_˜v

be a C

¹

vector field on R × R

³

× R

³

, S periodic with respect to s. There is a continuous vector field ξ · ∇

s,x,˜v

= ξ

s

∂

s

+ ξ

x

· ∇

x

+ ξ

v˜

· ∇

˜v

in involution with respect to ∂

s

+ b · ∇

x,˜v

, such that for any S periodic function u ∈ C

²

( R × R

³

× R

³

) ∩ ker(∂

_s

+ b · ∇

_x,˜v

)

χ · ∇

_s,x,˜v

u

¹

= ξ · ∇

_s,x,˜v

u (17)

where

χ · ∇

_s,x,˜v

u − hχ · ∇

_s,x,˜v

ui + (∂

_s

+ b · ∇

_x,˜v

)u

¹

= 0, u

¹

= 0 and

hχ · ∇

_s,x,˜v

ϕi = ξ

_s

(18)

where

χ

_s

− hχ

_s

i + (∂

_s

+ b · ∇

_x,˜v

)ϕ = 0, hϕi = 0.

Proof.

The vector field ξ is uniquely determined by imposing (17) with ( X , V ˜ )(−s; x, ˜ v) ∈ ker(∂

s

+ b · ∇

x,˜v

) and (18)

χ · ∇

_s,x,˜v

U

¹

= −ξ

_s

b(( X , V ˜ )(−s; x, v)) + ˜ ∂

_x,˜v

( X , V ˜ )(−s; x, v)ξ ˜

_x,˜v

where U

¹

is the unique solution of

χ ·∇

_s,x,˜v

( X , V ˜ )(−s; x, v)− ˜ D

χ · ∇

_s,x,˜v

( X , V ˜ )(−s; x, v) ˜ E

+(∂

_s

+ b ·∇

_x,˜v

)U

¹

= 0, U

¹

= 0 (19) and

hχ · ∇

_s,x,˜v

ϕi = ξ

_s

, χ

_s

− hχ

_s

i + (∂

_s

+ b · ∇

_x,˜v

)ϕ = 0, hϕi = 0.

It remains to check that (17) holds true for any u(s, x, v) = ˜ U(z), z = ( X , V ˜ )(−s; x, v ˜ ), U ∈ C

²

. We use the notation f

_s

(x, ˜ v) = f (( X , V ˜ )(s; x, ˜ v)), for any function f. As χ·∇

_s,x,˜v

u = (∇

z

U )

−s

· (χ · ∇

s,x,˜v

)( X , V ˜ )(−s; x, v), it comes that the solution ˜ u

¹

of

χ · ∇

_s,x,˜v

u − hχ · ∇

_s,x,˜v

ui + (∂

_s

+ b · ∇

_x,˜v

)u

¹

= 0, u

¹

= 0

is given by u

¹

= (∇

_z

U )

−s

· U

¹

, where U

¹

is the unique solution of (19). Therefore we have

χ · ∇

_s,x,˜v

u

¹

= (∇

_z

U)

−s

·

χ · ∇

_s,x,˜v

U

¹

+

(χ · ∇

_s,x,˜v

)(∇

_z

U )

−s

· U

¹

= (∇

_z

U)

_−s

· (ξ · ∇

_s,x,˜v

)( X , V ˜ )(−s; x, v) + ˜

(χ · ∇

_s,x,˜v

)(∇

_z

U )

_−s

· U

¹

= ξ · ∇

_s,x,˜v

u +

(χ · ∇

_s,x,˜v

)(∇

_z

U)

−s

· U

¹

and we are done provided that the last average vanishes. Indeed, as hU

¹

i = 0 we write, thanks to the symmetry of the Hessian matrix (∂

_z²

U )

_−s

(χ · ∇

_s,x,˜v

)(∇

_z

U)

−s

· U

¹

= D

(∂

_z²

U )

−s

(χ · ∇

_s,x,˜v

)( X , V ˜ )(−s; x, v) ˜ · U

¹

E

= D

(∂

_z²

U )

−s

h

χ · ∇

_s,x,˜v

( X , V ˜ )(−s; x, v) ˜ − D

χ · ∇

_s,x,˜v

( X , V ˜ )(−s; x, v) ˜ Ei

· U

¹

E

= − 1 2

(∂

_s

+ b · ∇

_x,˜v

)

(∂

_z²

U )

_−s

U

¹

· U

¹

= 0.