On the three-dimensional finite Larmor radius approximation: the case of electrons in a fixed background of ions

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On the three-dimensional finite Larmor radius approximation: the case of electrons in a fixed

background of ions

Daniel Han-Kwan

To cite this version:

Daniel Han-Kwan. On the three-dimensional finite Larmor radius approximation: the case of electrons

in a fixed background of ions. 2010. �hal-00540385�

On the three-dimensional finite Larmor radius approximation:

the case of electrons in a fixed background of ions

Daniel Han-Kwan

∗

Abstract

This paper is concerned with the analysis of a mathematical model arising in plasma physics, more specifically in fusion research. It directly follows [17], where the tri-dimensional analysis of a Vlasov-Poisson equation with finite Larmor radius scaling was led, corresponding to the case of ions with massless electrons whose density follows a linearized Maxwell-Boltzmann law. We now consider the case of electrons in a background of fixed ions, which was only sketched in [17]. Unfortunately, there is evidence that the formal limit is false in general. Nevertheless, we formally derive a fluid system for particular monokinetic data. We prove the local in time existence of analytic solutions and rigorously study the limit (when the Debye length vanishes) to a new anisotropic fluid system. This is achieved thanks to Cauchy-Kovalevskaya type techniques, as introduced by Caflisch [7] and Grenier [13]. We finally show that this approach fails in Sobolev regularity, due to multi-fluid instabilities.

Keywords: Finite Larmor Radius Approximation - Anisotropic quasineutral limit - Anisotropic hydrodynamic systems - Cauchy-Kovalevskaya theorem - Ill-posedness in Sobolev spaces.

1 Introduction

1.1 Presentation of the problem

The main goal of this paper is to derive some fluid model in order to understand the behaviour of a quasineutral gas of electrons in a neutralizing background of fixed ions and submitted to a strong magnetic field. For simplicity, we consider that the magnetic field has fixed direction and intensity. The density of the electrons is governed by the classical Vlasov-Poisson equation. We first introduce some notations:

Notations. Let (e

₁

, e

₂

, e

_k

) be a fixed orthonormal basis of R

³

.

• The subscript ⊥ stands for the orthogonal projection on the plane (e

1

, e

2

), while the subscript k stands for the projection on e

_k

.

• For any vector X = (X

₁

, X

₂

, X

_k

), we define X

^⊥

as the vector (X

_y

, − X

_x

, 0) = X ∧ e

_k

.

• We define the differential operators ∆

x_k

= ∂

²_xk

and ∆

x_⊥

= ∂

_x²₁

+ ∂

_x²₂

.

∗Ecole Normale Sup´´ erieure, D´epartement de Math´ematiques et Applications, 45 rue d’Ulm 75230 Paris Cedex 05 France, email : hankwan@dma.ens.fr

The scaling we consider (we refer to the appendix for physical explanations) leads to the study of the scaled Vlasov-Poisson system (for t > 0, x ∈ T

³

:= R

³

/ Z

³

, v ∈ R

³

):



 

 

 

 

∂

_t

f

_ǫ

+

^v_ǫ^⊥

. ∇

x

f

_ǫ

+ v

_k

. ∇

x

f

_ǫ

+ (E

_ǫ

+

^v∧_ǫ^ek

). ∇

v

f

_ǫ

= 0 E

_ǫ

= ( −∇

x_⊥

V

_ǫ

, − ǫ ∇

x_k

V

_ǫ

)

− ǫ

²

∆

_xk

V

_ǫ

− ∆

_x⊥

V

_ǫ

= R

f

_ǫ

dv − R

f

_ǫ

dvdx f

_ǫ,t=0

= f

_ǫ,0

≥ 0, R

f

_ǫ,0

dvdx = 1.

(1.1)

The quantity f

_ǫ

(t, x, v) is interpreted as the distribution function of the electrons: this means that f

_ǫ

(t, x, v)dxdv is the probability of finding particles at time t with position x and velocity v; V

_ǫ

(t, x) and E

_ǫ

(t, x) are respectively the electric potential and force.

This corresponds to the so-called finite Larmor radius scaling for the Vlasov-Poisson equation, which was introduced by Fr´enod and Sonnendr¨ ucker in the mathematical liter- ature [9]. The 2D version of the system (obtained when one restricts to the perpendicular dynamics) was studied in [9] and more recently in [3] and [10]. A version of the full 3D system describing ions with massless electrons was studied by the author in [17]. In this work, we considered that the density of electrons follows a linearized Maxwell-Boltzmann law. This means that we studied the following Poisson equation for the electric potential:

V

_ǫ

− ǫ

²

∆

_xk

V

_ǫ

− ∆

_x⊥

V

_ǫ

= Z

f

_ǫ

dv − Z

f

_ǫ

dvdx. (1.2)

In this case it was shown after some filtering that the number density f

_ǫ

weakly-* converges to some solution f to another kinetic system exhibiting the so-called E × B drift in the orthogonal plane, but with trivial dynamics in the parallel direction. This last feature seems somehow disappointing.

We observed in [17] that in the case where the Poisson equation reads:

− ǫ

²

∆

_xk

V

_ǫ

− ∆

_x⊥

V

_ǫ

= Z

f

_ǫ

dv − Z

f

_ǫ

dvdx, (1.3)

we could expect to make a pressure appear in the limit process ǫ → 0, due to the incom- pressibility constraint:

Z

f dvdx

_⊥

= Z

f dvdx.

Unfortunately, we were not able to rigorously derive a kinetic limit or even a fluid limit from (1.1). This is not only due to technical mathematical difficulties. This is related to the existence of instabilities for the Vlasov-Poisson equation, such as the double-humped instabilities (see Guo and Strauss [15]) and their counterpart in the multi-fluid Euler equations, such as the two-stream instabilities (see Cordier, Grenier and Guo [8]). Such instabilities actually take over in the limit ǫ → 0 and the formal limit is false in general, unless f

_ǫ,0

does not depend on parallel variables, which corresponds to the 2D problem studied by Fr´enod and Sonnendr¨ ucker [9].

Actually, we can observe that if on the contrary the initial data f

_ǫ,0

depends only on parallel variables, we obtain the one-dimensional quasineutral system:







∂

_t

f

_ǫ

+ v

_k

∂

_xk

f

_ǫ

− ∂

_xk

V

_ǫ

∂

_vk

f

_ǫ

= 0

− ǫ

²

∂

_x²_k

V

_ǫ

= R

f

_ǫ

dv − R

f

_ǫ

dvdx

_k

f

_ǫ,t=0

= f

_ǫ,0

≥ 0, R

f

_ǫ,0

dvdx

_k

= 1.

(1.4) The formal limit is easily obtained, by taking ǫ = 0:







∂

t

f + v

_k

∂

x_k

f − ∂

x_k

V ∂

v_k

f = 0

− ǫ

²

∂

_x²_k

V = R

f dv − R

f dvdx

_k

f

_t=0

= f

₀

≥ 0, R

f

₀

dvdx

_k

= 1.

(1.5)

In [14], an explicit example of Grenier shows that the formal limit is false in general, because of the double-humped instability:

Theorem 1.1 ([14]). For any N and s in N , and for any ǫ < 1, there exist for i = 1, 2, 3, 4, v

^ǫ_i

(x) ∈ H

^s

( T ) with k v

^ǫ₁

(x) + 1 k

H^s

≤ ǫ

^N

, k v

^ǫ₂

(x) + 1/2 k

H^s

≤ ǫ

^N

, k v

^ǫ₃

(x) − 1/2 k

H^s

≤ ǫ

^N

, k v

^ǫ₄

(x) − 1 k

H^s

≤ ǫ

^N

, such that the solution f

_ǫ

(t, x, v) associated to the initial data defined by:

f

_ǫ,0

(x, v) = 1 for v

₁^ǫ

(x) ≤ v ≤ v

₂^ǫ

(x) and v

₃^ǫ

(x) ≤ v ≤ v

₄^ǫ

(x)

= 0 elsewhere.

We also define f

₀

by:

f

₀

(x, v) = 1 for − 1 ≤ v ≤ − 1/2 and 1/2 ≤ v ≤ 1

= 0 elsewhere.

Then f

_ǫ

does not converge to f

₀

in the following sense:

lim inf

ǫ→0

sup

t≤T

Z

| f

_ǫ

(t, x, v) − f

₀

(v) | v

²

dvdx > 0 (1.6) for any T > 0 and also for T = ǫ

^α

, with α < 1/2.

In order to overcome the effects of these instabilities for the usual quasineutral limit, there are two possibilities:

• One consists in restricting to particular initial profiles chosen in order to be stable (this would imply in particular some monotony conditions on the data, such as the Penrose condition [20]).

• The other one consists in considering data with analytic regularity, in which case the instabilities ( which are essentially of “Sobolev” nature) do not have any effect.

Here the situation is worst: by opposition to the usual quasineutral limit (see [6], [14]), restricting to stable profiles is not sufficient. This is due to the anisotropy of the problem and the dynamics in the perpendicular variables.

In this paper, we illustrate this phenomenon by formally deriving the following fluid system, obtained from the kinetic system (1.1) by considering some physically relevant monokinetic data (we refer to the appendix for the detailed formal derivation).



 

 

 

 

 

 

∂

t

ρ

ǫ

+ ∇

⊥

(E

_ǫ^⊥

ρ

ǫ

) + ∂

_k

(v

_k,ǫ

ρ

ǫ

) = 0

∂

_t

v

_k,ǫ

+ ∇

⊥

(E

^⊥_ǫ

v

_k,ǫ

) + v

_k,ǫ

∂

_k

(v

_k,ǫ

) = − ǫ∂

_k

φ

_ǫ

(t, x) − ∂

_k

V

_ǫ

(t, x

_k

) E

_ǫ^⊥

= −∇

^⊥

φ

ǫ

− ǫ

²

∂

_k²

φ

_ǫ

− ∆

_⊥

φ

_ǫ

= ρ

_ǫ

− R ρ

_ǫ

dx

_⊥

− ǫ∂

_k²

V

_ǫ

= R

ρ

_ǫ

dx

_⊥

− 1,

(1.7)

where:

• ρ

_ǫ

(t, x

_⊥

, x

_k

) : R

⁺

× T

³

→ R

⁺

∗

can be interpreted as a charge density,

• v

_k,ǫ

(t, x

_⊥

, x

_k

) : R

⁺

× T

³

→ R can be interpreted as a “parallel” current density.

• φ

_ǫ

(t, x

_k

) and V

_ǫ

(t, x) are electric potentials.

Although we have considerered monokinetic data, (1.7) is intrinsically a “multi-fluid”

system, because of the dependence on x

_⊥

. Hence, we still have to face the two-stream instabilities ([8]): because of these, the limit is false in Sobolev regularity and we thus decide to study the associated Cauchy problem for analytic data.

We then prove the limit to a new fluid system which is strictly speaking compressible but also somehow “incompressible in average”. This rather unusual feature is due to the anisotropy of the model. The fluid system is the following (obtained formally by taking ǫ = 0):



 



 



∂

_t

ρ + ∇

⊥

(E

^⊥

ρ) + ∂

_k

(v

_k

ρ) = 0

∂

t

v

_k

+ ∇

⊥

(E

^⊥

v

_k

) + v

_k

∂

_k

(v

_k

) = − ∂

_k

p(t, x

_k

) E

^⊥

= ∇

^⊥

∆

⁻_⊥¹

ρ − R

ρdx

_⊥

R ρdx

_⊥

= 1.

(1.8)

We observe that this system can be interpreted as an infinite system of Euler-type equations, coupled together through the “parameter” x

_⊥

. It has some interesting features:

• This system is highly anisotropic in x

_⊥

and x

_k

. The 2D part of the dynamics of the equation for ρ is nothing but the vorticity formulation of 2D incompressible Euler.

Physically speaking, ρ should be interpreted here as a density rather than a vorticity.

The dynamics in the parallel direction is similar to the dynamics of incompressible Euler written in velocity. We finally observe that the pressure p only depends on the parallel variable x

_k

and not on x

_⊥

.

• This does not strictly speaking describe an incompressible fluid, since (E

^⊥

, v

_k

) is not divergence free. Somehow, the fluid is hence compressible. But the constraint R ρdx

_⊥

= 1 can be interpreted as a constraint of “incompressibility in average” which allows one to recover the pressure law from the other unknowns. Indeed, we easily get, thanks to the equation on ρ:

∂

_xk

Z

ρv

_k

dx

_⊥

= 0. (1.9)

So by plugging this constraint in the equation on ρv

_k

:

∂

_t

(ρv

_k

) + ∇

⊥

(E

^⊥

ρ

_k

v

_k

) + ∂

_k

(ρv

_k²

) = − ∂

_k

p(t, x

_k

)ρ,

we get the (one-dimensional !) elliptic equation allowing to recover − ∂

_xk

p:

− ∂

_k²

p(t, x

_k

) = ∂

_k²

Z

ρv

_k²

dx

_⊥

, from which we get:

− ∂

_k

p(t, x

_k

) = ∂

_k

Z

ρv

²_k

dx

_⊥

. (1.10)

• From the point of view of plasma physics, E

^⊥

. ∇

⊥

is the so-called electric drift. By analogy with the so-called drift-kinetic equations [23], we can call this system a drift- fluid equation. To the best of our knowledge, this is the very first time such a model is exhibited in the literature.

From now on, when there is no risk of confusion, we will sometimes write v and v

_ǫ

instead of v

_k

and v

_k,ǫ

.

1.2 Organization of the paper

The outline of this paper is as follows. In Section 2, we will state the main results of this paper that are: the existence of analytic solutions to (1.7) locally in time but uniformly in ǫ (Theorem 2.1), the strong convergence to (1.8) with a complete description of the plasma oscillations (Theorem 2.2) and finally the existence and uniqueness of local analytic solutions to (1.8), in Proposition 2.1.

Section 3 is devoted to the proof of Theorem 2.1. First we recall some elementary fea- tures of the analytic spaces we consider (section 3.1), then we implement an approximation scheme for our Cauchy-Kovalesvkaya type existence theorem. The results are based on a decomposition of the electric field allowing for a good understanding of the so-called plasma waves (section 3.2).

In section 4, we prove Theorem 2.2, by using the uniform in ǫ estimates we have obtained in the previous theorem. The proof relies on another decomposition of the electric field, in order to exhibit the effects of the plasma waves as ǫ goes to 0.

Then, in section 5, we discuss the sharpness of our results:

• In sections 5.1 and 5.2, we discuss the analyticity assumption and explain why we can not lower down the regularity to Sobolev. In section 5.3, we explain why it is not possible to obtain global in time results. We obtain these results by considering some well-chosen initial data and using results of Brenier on multi-fluid Euler systems [5].

• Because of the multi-stream instabilities, studying the limit with the relative entropy method is bound to fail. Nevertheless we found it interesting to try to apply the method and see at which point things get nasty: this is the object of section 5.4, where we study a kinetic toy model which retains the main unstable feature of system (1.7).

The two last sections are respectively a short conclusion and an appendix where we explain the scaling and the formal derivation of system (1.7).

2 Statement of the results

In order to prove both the existence of strong solutions to systems (1.7) and (1.8) and also prove the results of convergence, we follow the construction of Grenier [13], with some modifications adapted to our problem.

In [13], Grenier studies the quasineutral limit of the family of coupled Euler-Poisson systems:



 



 



∂

t

ρ

^ǫ_Θ

+ div(ρ

^ǫ_Θ

v

_Θ^ǫ

) = 0

∂

_t

v

_Θ^ǫ

+ v

^ǫ_Θ

. ∇ (v

^ǫ_Θ

) = E

^ǫ

rot E

^ǫ

= 0

ǫ div E

^ǫ

= R

M

ρ

^ǫ_Θ

µ(dΘ) − 1,

(2.1)

with (M, Θ, µ) a probability space.

Following the proof of the Cauchy-Kovalevskaya theorem given by Caflisch [7], Grenier

proved the local existence of analytic functions (with respect to x) uniformly with respect

to ǫ and then, after filtering the fast oscillations due to the force field, showed the strong

convergence to the system:



 



 



∂

_t

ρ

_Θ

+ div(ρ

_Θ

v

_Θ

) = 0

∂

_t

v

_Θ

+ v

_Θ^ǫ

. ∇ (v

_Θ

) = E rot E = 0

R ρ

_Θ

µ(dΘ) = 1.

(2.2)

We notice that the class of systems studied by Grenier is close to system (1.7), if we take x = x

_k

, Θ = x

_⊥

and (M, µ) = ( T

²

, dx

_⊥

), the main difference being that we have to deal with a dynamics in Θ = x

_⊥

.

Hence, we introduce the same spaces of analytic functions as in [13], but this time depending also on Θ = x

_⊥

.

Definition. Let δ > 1. We define B

_δ

the space of real functions φ on T

³

such that

| φ |

δ

= X

k∈Z³

|F φ(k) | δ

^|k|

< + ∞ , (2.3)

where F φ(k) is the k-th Fourier coefficient of φ defined by:

F φ(k) = Z

T³

φ(x)e

^−i2πk.x

dx.

The first theorem proves the existence of local analytic solutions of (1.7) with a life span uniform in ǫ.

Theorem 2.1. Let δ

0

> 1. Let ρ

ǫ

(0) and v

ǫ

(0) be two bounded families of B

_δ0

such that R ρ

_ǫ

(0)dx = 1 and:

Z

ρ

_ǫ

(0)dx

_⊥

− 1

Bδ0

≤ C √

ǫ, (2.4)

then there exists η > 0 such that for every δ

1

∈ ]1, δ

0

[, for any ǫ > 0, there exists a unique strong solution (ρ

_ǫ

, v

_ǫ

) to (1.7) bounded uniformly in C ([0, η(δ

₀

− δ

₁

)[, B

_δ1

) with initial conditions (ρ

_ǫ

(0), v

_ǫ

(0)). Moreover, √ ǫ∂

_k

V

_ǫ

is uniformly bounded in C ([0, η(δ

₀

− δ

₁

)[, B

_δ1

).

Remark 2.1. • The condition R

ρ

_ǫ

(0)dx

_⊥

− 1

Bδ0

≤ C √

ǫ implies that √

ǫ∂

_k

V

_ǫ

(0) is bounded uniformly in B

_δ0

(this is the correct scale in view of the energy conservation).

• Note that for all t ≥ 0, R

ρ

ǫ

dx = 1. Hence the Poisson equation − ǫ∂

_k²

V

ǫ

= R

ρ

ǫ

dx

_⊥

− 1 can always be solved.

We can then prove the convergence result:

Theorem 2.2. Let (ρ

_ǫ

, v

_ǫ

) be solutions to the system (1.7) for 0 ≤ t ≤ T satisfying for some s > 7/2 the following estimate:

(H) : sup

t≤T,ǫ

k ρ

_ǫ

k

H_x^s_⊥,x

k

+ k v

_ǫ

k

H^s_x⊥,x

k

+ k √

ǫ∂

_xk

V

_ǫ

k

H^s_x

k

< + ∞ . (2.5)

Then we get the following convergences

ρ

_ǫ

→ ρ, v

_ǫ

− 1

i (E

₊

e

^it/√ǫ

− E

₋

e

^−it/√ǫ

) → v,

strongly respectively in C ([0, T ], H

_x^s′_⊥,xk

) and C ([0, T ], H

_x^s′_⊥⁻_,x¹_k

) for all s

^′

< s, and

√ ǫ

− ∂

_xk

V

_ǫ

− (E

₊

e

^it/√ǫ

+ E

₋

e

^−it/√ǫ

)

→ 0,

strongly in C ([0, T ], H

_x^s′_k

) for all s

^′

< s − 1, and where (ρ, v) is solution to the asymptotic system (1.8) on [0, T ] with initial conditions:

ρ(0) = lim

ǫ→0

ρ

_ǫ

(0), v(0) = lim

ǫ→0

v

_ǫ

(0) − Z

ρ

_ǫ

v

_ǫ

dx

_⊥

(0)

and E

₊

(t, x

_k

), E

₋

(t, x

_k

) are gradient correctors which satisfy the transport equations:

∂

_t

E

_±

+ Z

ρvdx

_⊥

∂

_xk

E

_±

= 0, with initial data:

∂

_xk

E

₊

(0) = lim

ǫ→0

1 2 ∂

_xk

− √

ǫ∂

_xk

V

_ǫ

(0) + i Z

ρ

_ǫ

v

_ǫ

dx

_⊥

(0)

,

∂

x_k

E

₋

(0) = lim

ǫ→0

1 2 ∂

x_k

− √

ǫ∂

x_k

V

ǫ

(0) − i Z

ρ

ǫ

v

ǫ

dx

_⊥

(0)

.

As explained in the introduction, due to the two-streams instabilities, we have to restrict to data with analytic regularity: the Sobolev version of these results is false in general (see [8] and the discussion of Section 5).

Remark 2.2. • It is clear that solutions built in Theorem 2.1 satisfy (H).

• If instead of (H) we make the stronger assumption, for δ > 1 (H

^′

) : sup

t≤T,ǫ

k ρ

_ǫ

k

Bδ

+ k v

_ǫ

k

Bδ

+ k √

ǫ∂

_xk

V

_ǫ

k

Bδ

< + ∞ , (2.6)

then we get the same strong convergences in C ([0, T ], B

_δ^′

) for all δ

^′

< δ.

Using Lemma 3.1 (ii), (iv), the proof under assumption (H

^′

) is the same as under assumption (H).

• The “well-prepared” case corresponds to the case when:

lim

ǫ→0

− √

ǫ∂

_x²_k

V

_ǫ

(0) = 0, lim

ǫ→0

∂

_xk

Z

ρ

_ǫ

v

_ǫ

dx

_⊥

(0) = 0.

Then there is no corrector.

With the same method used for Theorem 2.1, we can also prove a theorem of existence

and uniqueness of analytic solutions to system (1.8).

Proposition 2.1. Let δ

₀

> δ

₁

> 1. For initial data ρ(0), v(0) ∈ B

_δ0

satisfying

ρ(0) ≥ 0, (2.7)

Z

ρ(0)dx

_⊥

= 1 (2.8)

and

∂

_k

Z

ρ(0)v(0)dx

_⊥

= 0, (2.9)

there exists η > 0 depending on δ

0

and on the initial conditions only such that there is a unique strong solution (ρ, v

_k

, p) to the system (1.8) with ρ, v ∈ C ([0, η(δ

₀

− δ

₁

)[, B

_δ

) for all δ < δ

₁

.

3 Proof of Theorem 2.1

3.1 Functional analysis on B

_δ

spaces

First we define the time dependent analytic spaces we will work with.

Let β be an arbitrary constant in ]0, 1[ (take for instance β = 1/2 to fix ideas) and η > 0 a parameter to be chosen later.

Definition. Let δ

₀

> 1. We define the space B

_δ^η₀

= { u ∈ C

⁰

([0, η(δ

₀

− 1)], B

_δ0−t/η

) } , endowed with the norm

k u k

δ0

= sup







1 < δ ≤ δ

₀

0 ≤ t ≤ η(δ

₀

− δ)

| u(t) |

δ

+ (δ

₀

− δ − t

η )

^β

|∇ u(t) |

δ

)

,

where the norm | u |

δ

was defined in (2.3):

| u |

δ

= X

k∈Z³

|F u(k) | δ

^|k|

< + ∞ ,

We now gather from [13] a few elementary properties of these spaces, that we recall for the reader’s convenience.

Lemma 3.1. For all δ > 1

(i) The spaces B

_δ

and B

^η_δ

are Banach algebra.

(ii) If δ

^′

< δ then B

_δ

⊂ B

_δ^′

, the embedding being continuous and compact.

(iii) For all s ∈ R , B

_δ

⊂ H

^s

, the embedding being continuous and compact.

(iv) For all 1 < δ

^′

< δ, if φ ∈ B

_δ

,

|∇ φ |

δ^′

≤ δ δ − δ

^′

| φ |

δ

. (v) If u is in B

_δ^η₀

and if δ + t/η < δ

₀

then

| ∂

_x²_i,xj

u(t) |

δ

≤ 2

^1+β

k u k

δ0

δ

₀

(δ

₀

− δ − t

η )

^−β−1

.

For further properties of these spaces we refer to the recent work of Mouhot and Villani [19], in which similar analytic spaces (and more sophisticated versions) are considered.

Proof. We give an elementary proof for (ii) which is not given in [13] . The embedding is obvious. We consider for N ∈ N the map i

_N

defined by:

i

_N

(φ) = X

|k|≤N

F φ(k)e

^ixk

.

We then compute:

| (Id − i

_N

)φ |

δ^′

= X

|k|>N

|F φ(k) | δ

^′|k|

≤ δ

^′

δ

N

X

|k|>N

|F φ(k) | δ

^|k|

≤ δ

^′

δ

N

| φ |

δ

.

So the embedding B

_δ

⊂ B

_δ^′

is compact as the limit of finite rank operators.

For (v), take δ

^′

= δ +

^δ0−δ₂^−t/η

and apply (iv). We refer to [13] for the other proofs.

We will also need the following elementary observation:

Remark 3.1. Let φ ∈ B

_δ

. Then:

Z

φdx

_⊥

δ

≤ | φ |

δ

. Proof. We simply compute:

Z

φdx

_⊥

δ

= X

k⊥=0,k_k∈N

|F (φ)(k

_⊥

, k

_k

) | δ

^|k|

≤ X

k∈N³

|F (φ) | δ

^|k|

= | φ |

δ

.

3.2 Description of plasma oscillations

To simplify notations, we set E

_ǫ,k

= − ∂

_xk

V

_ǫ

(t, x

_k

) (which has nothing to do with E

_ǫ^⊥

). In this paragraph, we want to understand the oscillatory behaviour of E

_ǫ,k

. We will see that the dynamics in x

_⊥

does not interfer too much with the equations on E

_ǫ,k

, so that we get almost the same description of oscillations as in Grenier’s paper [13].

First we differentiate twice with respect to time the Poisson equation satisfied by V

ǫ

: ǫ∂

²_t

∂

x_k

E

_ǫ,k

= ∂

_t²

Z

ρ

ǫ

dx

_⊥

. (3.1)

We use the equation on ρ

_ǫ

to compute the right hand side of (3.1).

∂

_t

Z

ρ

_ǫ

dx

_⊥

= − Z

∇

⊥

(E

_ǫ^⊥

ρ

_ǫ

)dx

_⊥

| {z }

=0

− ∂

_xk

Z

ρ

_ǫ

v

_ǫ

dx

_⊥

. (3.2)

Then we integrate with respect to x

_⊥

the equation satisfied by ρ

_ǫ

v

_ǫ

, that is:

∂

_t

(ρ

_ǫ

v

_ǫ

) + ∇

⊥

(E

_ǫ^⊥

ρ

_ǫ

v

_ǫ

) + ∂

_xk

(v

_ǫ²

ρ

_ǫ

) = − ρ

_ǫ

(ǫ∂

_xk

φ

_ǫ

(t, x) + ∂

_xk

V

_ǫ

(t, x

_k

)) and we get:

− ∂

t

Z

ρ

ǫ

v

ǫ

dx

_⊥

= ∂

x_k

Z

ρ

ǫ

v

_ǫ²

dx

_⊥

+ E

_ǫ,k

Z

ρ

ǫ

dx

_⊥

− Z

ρ

ǫ

(ǫ∂

x_k

φ

ǫ

)dx

_⊥

, (3.3)

so that:

∂

_t²

Z

ρ

_ǫ

dx

_⊥

= ∂

_x²_k

Z

ρ

_ǫ

v

_ǫ²

dx

_⊥

− ∂

_xk

(E

_ǫ,k

Z

ρ

_ǫ

dx

_⊥

) + ∂

_xk

Z

ρ

_ǫ

(ǫ∂

_xk

φ

_ǫ

)dx

_⊥

. Thus it comes:

ǫ∂

_t²

∂

_xk

E

_ǫ,k

+ ∂

_xk

E

_ǫ,k

= ∂

_x²_k

Z

ρ

_ǫ

v

²_ǫ

dx

_⊥

+ ǫ∂

_xk

[E

_ǫ,k

∂

_xk

E

_ǫ,k

] − ∂

_xk

Z

ρ

_ǫ

(ǫ∂

_xk

φ

_ǫ

)dx

_⊥

. (3.4) Equation (3.4) is the wave equation allowing to describe the essential oscillations. At least formally, this equation indicates that there are time oscillations with frequency

√¹ǫ

and magnitude

√¹ǫ

created by the right-hand side of the equation which acts like a source.

We observe here that the source is expected to be of order O (1): indeed, by assumption on the data at t = 0, we can check that this quantity is bounded in a B

_δ

space.

In particular if we want to prove strong convergence results we will have to introduce non-trivial correctors in order to get rid of these oscillations. We notice also that (3.4) is very similar to the wave equation obtained in [13] (the only difference is a new term in the source), so that most of the calculations and estimates on E

_ǫ,k

we will need are done in [13].

3.3 A priori estimates

We have just observed that E

_ǫ,k

roughly behaves like

√¹

ǫ

e

^±it/√ǫ

. Hence if we consider the average in time

G

_ǫ

= Z

_t

0

E

_ǫ,k

(s, x

_k

)ds, (3.5)

we expect that G

_ǫ

is bounded uniformly with respect to ǫ in some functional space. We also introduce the translated current (which corresponds to some filtering of the time oscillations created by the electric field):

w

ǫ

= v

ǫ

− G

ǫ

, (3.6)

so that system (1.7) now writes:

∂

_t

ρ

_ǫ

+ ∇

⊥

(E

_ǫ^⊥

ρ

_ǫ

) + ∂

_k

((w

_ǫ

+ G

_ǫ

)ρ

_ǫ

) = 0

∂

_t

w

_ǫ

+ ∇

⊥

(E

_ǫ^⊥

(w

_ǫ

+ G

_ǫ

)) + (w

_ǫ

+ G

_ǫ

)∂

_k

(v

_ǫ

+ G

_ǫ

)) = − ǫ∂

_k

φ

_ǫ

(t, x

_k

). (3.7) The goal is now to prove some a priori estimates for G

_ǫ

, ρ

_ǫ

and w

_ǫ

. We are also able to get similar estimates on E

_ǫ^⊥

and ǫ∂

_xk

φ

_ǫ

, thanks to the Poisson equation satisfied by φ

_ǫ

. 3.3.1 Estimate on G

_ǫ

and √ ǫE

_ǫ,k

We use Duhamel’s formula for the wave equation (3.4) to get the following identity:

F

_k

G

_ǫ

(t, k

_k

) = Z

_t

0

1 ik

_k

1 − cos( t − s

√ ǫ )

F

_k

g

_ǫ

(s, k

_k

)

ds + F

_k

G

⁰_ǫ

, (3.8) denoting by F

_k

the Fourier transform with respect to the parallel variable only and k

_k

the Fourier variable and where:

g

ǫ

= ∂

_x²_k

Z

ρ

ǫ

v

²_ǫ

dx

_⊥

+ ǫ∂

x_k

[E

_ǫ,k

∂

x_k

E

_ǫ,k

] − ∂

x_k

Z

ρ

ǫ

(ǫ∂

x_k

φ

ǫ

)dx

_⊥

,

G

⁰_ǫ

= √

ǫE

_ǫ,k

(0, x

_k

) sin s

√ ǫ

− ǫ∂

_t

E

_ǫ,k

(0, x

_k

)

cos s

√ ǫ

− 1

. (3.9)

We now estimate k G

_ǫ

k

δ0

.

| G

_ǫ

|

δ

≤ Z

t

0

F

_k⁻¹

1

ik

_k

[1 − cos( t − s

√ ǫ )] F

_k

g

_ǫ

(s, k

_k

)

δ

ds + | G

⁰_ǫ

|

δ

. 1

ik

_k

F

_k

g

_ǫ

= F

_k

∂

_xk

Z

ρ

_ǫ

v

_ǫ²

dx

_⊥

+ ǫ F

_k

E

_ǫ,k

∂

_xk

E

_ǫ,k

. Thanks to Remark 3.1 and Lemma 3.1 , (i):

Z

∂

x_k

(ρ

ǫ

v

²_ǫ

)dx

_⊥

δ

≤

∂

x_k

(ρ

ǫ

v

_ǫ²

)

δ

≤ (δ

0

− δ − s

η )

^−β

k ρ

ǫ

k

δ0

k v

ǫ

k

²δ0

. (3.10) Similarly, we prove:

ǫ

E

_ǫ,k

∂

_xk

E

_ǫ,k

δ

≤ 1

2 (δ

₀

− δ − s η )

^−β

k √

ǫE

_ǫ,k

k

²δ0

,

Z

∂

x_k

ρ

ǫ

(ǫ∂

x_k

φ

ǫ

) dx

_⊥

δ

≤ (δ

0

− δ − s

η )

^−β

k ρ

ǫ

k

δ0

k ǫ∂

x_k

φ

ǫ

k

δ0

. Thus, we have:

| G

_ǫ

|

δ

≤ C Z

t

0

(δ

₀

− δ − s

η )

^(−β)

( k ρ

_ǫ

k

δ0

k v

_ǫ

k

²δ0

+ k √

ǫE

_ǫ,k

k

²δ0

+ k ρ

_ǫ

k

δ0

k ǫ∂

_xk

φ

_ǫ

k

δ0

) + | G

⁰_ǫ

|

δ

. Likewise, one can show (this time we use lemma 3.1, (v)):

| ∂

_xk

G

_ǫ

|

δ

≤ C Z

_t

0

(δ

₀

− δ − s

η )

^(−β−1)

( k ρ

_ǫ

k

δ0

k v

_ǫ

k

²δ0

+ k √

ǫE

_ǫ,k

k

²δ0

+ k ρ

_ǫ

k

δ0

k ǫ∂

_xk

φ

_ǫ

k

δ0

) + | ∂

_xk

G

⁰_ǫ

|

δ

. Hence using the elementary estimates

Z

t 0

ds

(δ

₀

− δ −

^σ_η

)

^β

≤ η 2

1 − β δ

₀^1−β

, Z

t

0

ds

(δ

0

− δ −

^σ_η

)

^β+1

≤ 2η

β (δ

₀

− δ − t η )

^−β

, and k v

_ǫ

k

δ0

≤ k w

_ǫ

k

δ0

+ k G

_ǫ

k

δ0

, we get:

k G

_ǫ

k

δ0

≤ ηC (δ

₀

, β)

( k w

_ǫ

k

δ0

+ k G

_ǫ

k

δ0

)

²

k ρ

_ǫ

k

δ0

+ k √

ǫE

_ǫ,k

k

²δ0

+ k ρ

_ǫ

k

δ0

k ǫ∂

_xk

φ

_ǫ

k

δ0

+ k G

⁰_ǫ

k

δ0

. (3.11) If we compare two solutions (w

⁽¹⁾

, ρ

⁽¹⁾

) and (w

⁽²⁾

, ρ

⁽²⁾

) with the same inital data we obtain:

k G

⁽¹⁾_ǫ

− G

⁽²⁾_ǫ

k

δ0

≤ ηC

( k w

⁽¹⁾_ǫ

− w

⁽²⁾_ǫ

k

δ0

+ k G

⁽¹⁾_ǫ

− G

⁽²⁾_ǫ

k

δ0

)

× ( k w

_ǫ⁽¹⁾

k

δ0

+ k w

_ǫ⁽²⁾

k

δ0

+ k G

⁽¹⁾_ǫ

k

δ0

+ k G

⁽²⁾_ǫ

k

δ0

)( k ρ

⁽¹⁾_ǫ

k

δ0

+ k ρ

⁽²⁾_ǫ

k

δ0

) + ( k w

_ǫ⁽¹⁾

k

²δ0

+ k w

⁽²⁾_ǫ

k

²δ0

+ k G

⁽¹⁾_ǫ

k

²δ0

+ k G

⁽²⁾_ǫ

k

²δ0

)( k ρ

⁽¹⁾_ǫ

− ρ

⁽²⁾_ǫ

k

δ0

) + k ρ

⁽¹⁾_ǫ

− ρ

⁽²⁾_ǫ

k

δ0

( k ǫ∂

_xk

φ

⁽¹⁾_ǫ

k

δ0

+ k ǫ∂

_xk

φ

⁽²⁾_ǫ

k

δ0

)

+ ( k ρ

⁽¹⁾_ǫ

k

δ0

+ k ρ

⁽²⁾_ǫ

k

δ0

) k ǫ∂

_xk

φ

⁽¹⁾_ǫ

− ǫ∂

_xk

φ

⁽²⁾_ǫ

k

δ0

+ k √

ǫE

_ǫ,⁽¹⁾_k

− √

ǫE

_ǫ,⁽²⁾_k

k

δ0

( k √

ǫE

_ǫ,⁽¹⁾_k

k

δ0

+ k √

ǫE

_ǫ,⁽²⁾_k

k

δ0

)

. (3.12)

Likewise we get the same estimates on k √

ǫE

_ǫ,k

k

δ0

since we have the formula:

F

_k

( √

ǫE

_ǫ,k

)(t, k

_k

) = Z

t

0

1

ik

_k

[sin( t − s

√ ǫ )] F

_k

g

_ǫ

(s, k

_k

)

ds + F

_k

( √

ǫE

_ǫ,⁰_k

), (3.13) with

E

_ǫ,⁰_k

= E

_ǫ,k

(0, x) cos( s

√ ǫ ) + √

ǫ∂

t

E

_ǫ,k

(0, x) sin( s

√ ǫ ). (3.14)

3.3.2 Estimate on E

_ǫ^⊥

and ǫ∂

_xk

φ

_ǫ

We now use the scaled Poisson equation satisfied by φ

_ǫ

to get some a priori estimates.

The principle here is to look at the symbols of the operators involved in the Poisson equations. Accordingly, we compute in Fourier variables:

ǫ

²

k

_k²

F φ

_ǫ

+ | k

_⊥

|

²

F φ

_ǫ

= F

ρ

_ǫ

− Z

ρ

_ǫ

dx

_⊥

. (3.15)

Thus it comes:

F φ

_ǫ

= F (ρ

_ǫ

− R

ρ

_ǫ

dx

_⊥

) ǫ

²

k

_k²

+ | k

_⊥

|

²

. Since R

(ρ

_ǫ

− R

ρ

_ǫ

dx

_⊥

)dx

_⊥

= 0, we have for all k

_k

: F

ρ

_ǫ

−

Z

ρ

_ǫ

dx

_⊥

(0, k

_k

) = 0.

Thus it comes, for all k

_⊥

, k

_k

:

|F φ

ǫ

| ≤ |F (ρ

_ǫ

− R

ρ

_ǫ

dx

_⊥

) |

| k

_⊥

|

²

. (3.16)

In particular we easily get, using the relation E

_ǫ^⊥

= −∇

^⊥

φ

_ǫ

:

|F E

^⊥_ǫ

| ≤ |F (ρ

_ǫ

− R

ρ

_ǫ

dx

_⊥

) |

| k

_⊥

| ≤ F

ρ

_ǫ

−

Z

ρ

_ǫ

dx

_⊥

. Hence:

k E

_ǫ^⊥

k

δ0

≤ C k ρ

_ǫ

k

δ0

. (3.17) Likewise, since ab ≤

¹₂

(a

²

+ b

²

) and | k

_⊥

| ≥ 1:

|F (ǫ∂

x_k

φ

ǫ

) | ≤ ǫ | k

_k

||F (ρ

_ǫ

− R

ρ

_ǫ

dx

_⊥

) | ǫ

²

k

²_k

+ | k

_⊥

|

²

≤ 1

2 |F (ρ

ǫ

− Z

ρ

ǫ

dx

_⊥

) | , and consequently:

k ǫ∂

_xk

φ

_ǫ

k

δ0

≤ C k ρ

_ǫ

k

δ0

. (3.18) And if we compare two solutions with the same initial data:

k E

_ǫ^⊥,(1)

− E

_ǫ^⊥,(2)

k

δ0

+ k ǫ∂

_xk

φ

⁽¹⁾_ǫ

− ǫ∂

_xk

φ

⁽²⁾_ǫ

k

δ0

≤ C k ρ

⁽¹⁾_ǫ

− ρ

⁽²⁾_ǫ

k

δ0

. (3.19)

3.3.3 Estimate on ρ

_ǫ

and w

_ǫ

We now use the conservation laws satisfied by ρ

_ǫ

and w

_ǫ

to get the appropriate estimates.

The density ρ

_ǫ

satisfies the equation:

∂

_t

ρ

_ǫ

+ ∇

⊥

(E

_ǫ^⊥

ρ

_ǫ

) + ∂

_k

((w

_ǫ

+ G

_ǫ

)ρ

_ǫ

) = 0.

With the same kind of computations as before and thanks to estimate (3.17) we get:

| ρ

_ǫ

|

δ

≤ Z

_t

0

| ∂

_t

ρ

_ǫ

|

δ

+ | ρ

_ǫ

(0) |

δ

≤ k ρ

ǫ

(0) k

δ0

+ Z

t

0

(δ

0

− δ − s

η )

^−β

k ρ

ǫ

k

δ0

( k ρ

ǫ

k

δ0

+ k w

ǫ

k

δ0

+ k G

ǫ

k

δ0

)ds.

Similarly we estimate | ∂

_xi

ρ

_ǫ

|

δ

by differentiating with respect to x

_i

the equation satisfied by ρ

_ǫ

. Finally we get:

k ρ

ǫ

k

δ0

≤ ηC k ρ

ǫ

k

δ0

( k ρ

ǫ

k

δ0

+ k w

ǫ

k

δ0

+ k G

ǫ

k

δ0

). (3.20) If we compare two solutions with the same initial conditions, we get likewise:

k ρ

⁽¹⁾_ǫ

− ρ

⁽²⁾_ǫ

k

δ0

≤ ηC

( k ρ

⁽¹⁾_ǫ

k

δ0

+ k ρ

⁽²⁾_ǫ

k

δ0

)( k w

⁽¹⁾_ǫ

− w

⁽²⁾_ǫ

k

δ0

+ k G

⁽¹⁾_ǫ

− G

⁽²⁾_ǫ

k

δ0

) + ( k ρ

⁽¹⁾_ǫ

k

δ0

+ k ρ

⁽²⁾_ǫ

k

δ0

+ k w

⁽¹⁾_ǫ

k

δ0

+ k w

⁽²⁾_ǫ

k

δ0

+ k G

⁽¹⁾_ǫ

k

δ0

+ k G

⁽²⁾_ǫ

k

δ0

)

× ( k ρ

⁽¹⁾_ǫ

− ρ

⁽²⁾_ǫ

k

δ0

)

. (3.21)

In the same fashion, we estimate the δ

₀

norm of w

_ǫ

:

k w

ǫ

k

δ0

≤ ηC ( k w

ǫ

k

δ0

+ 1) k ρ

ǫ

k

δ0

+ ( k w

ǫ

k

δ0

+ k G

ǫ

k

δ0

)

²

+ k ǫ∂

_k

φ

ǫ

k

δ0

, (3.22)

and if we compare two solutions with the same initial data:

k w

_ǫ⁽¹⁾

− w

_ǫ⁽²⁾

k

δ0

≤ ηC

( k ρ

⁽¹⁾_ǫ

k

δ0

+ k ρ

⁽²⁾_ǫ

k

δ0

)( k w

_ǫ⁽¹⁾

k

δ0

+ k w

_ǫ⁽²⁾

k

δ0

) + ( k w

_ǫ⁽¹⁾

− w

⁽²⁾_ǫ

k

δ0

+ k ρ

⁽¹⁾_ǫ

− ρ

⁽²⁾_ǫ

k

δ0

)

+ ( k w

_ǫ⁽¹⁾

k

δ0

+ k w

⁽²⁾_ǫ

k

δ0

+ k G

⁽¹⁾_ǫ

k

δ0

+ k G

⁽²⁾_ǫ

k

δ0

)

× ( k w

⁽¹⁾_ǫ

− w

_ǫ⁽²⁾

k

δ0

+ k G

⁽¹⁾_ǫ

− G

⁽²⁾_ǫ

k

δ0

) k ǫ∂

_k

φ

⁽¹⁾_ǫ

− ǫ∂

_k

φ

⁽¹⁾_ǫ

k

δ0

. (3.23)

3.4 Approximation scheme

We use the usual approximation scheme for Cauchy-Kovalevskaya type of results ([7]).

We define ρ

ⁿ_ǫ

, w

_ǫⁿ

, G

ⁿ_ǫ

, V

_ǫⁿ

, φ

ⁿ_ǫ

by recursion:

Initialization For 0 < t < η(δ

₀

− 1), we define:

ρ

⁰_ǫ

(t) = ρ

_ǫ

(0), w

⁰_ǫ

= v

ǫ

(0) − G

⁰_ǫ

,

− ǫ

²

∂

_x²_k

φ

⁰_ǫ

− ∆

x_⊥

φ

⁰_ǫ

= ρ

⁰_ǫ

− Z

ρ

⁰_ǫ

dx

_⊥

,

E

_ǫ^⊥,0

= −∇

^⊥

φ

⁰_ǫ

,

and − ǫ

²

∂

²_xk

V

_ǫ⁰

= ρ

⁰_ǫ

− R

ρ

⁰_ǫ

dx. Finally, G

⁰_ǫ

is given by formula G

⁰_ǫ

= − R

t 0

∂

_k

V

_ǫ⁰

. Recursion We define ρ

ⁿ⁺¹_ǫ

, w

ⁿ⁺¹_ǫ

by the equations:

( ∂

_t

ρ

ⁿ⁺¹_ǫ

+ ∇

⊥

(E

^⊥_ǫ^,n

ρ

ⁿ_ǫ

) + ∂

_k

((w

ⁿ_ǫ

+ G

ⁿ_ǫ

)ρ

ⁿ_ǫ

) = 0

∂

t

w

ⁿ⁺¹_ǫ

+ ∇

⊥

(E

ǫ^⊥,n

(w

ⁿ_ǫ

+ G

ⁿ_ǫ

)) + (w

_ǫⁿ

+ G

ⁿ_ǫ

)∂

_k

(v

ⁿ_ǫ

+ G

ⁿ_ǫ

)) = − ǫ∂

_k

φ

ⁿ_ǫ

(t, x

_k

).

with the initial conditions: ρ

ⁿ⁺¹_ǫ

(0) = ρ

_ǫ

(0) and w

_ǫⁿ⁺¹

= v

_ǫ

(0) − G

⁰_ǫ

. Then we can define φ

ⁿ⁺¹_ǫ

as the solution to the Poisson equation:

− ǫ

²

∂

_x²_k

φ

ⁿ⁺¹_ǫ

− ∆

x_⊥

φ

ⁿ⁺¹_ǫ

= ρ

ⁿ⁺¹_ǫ

− Z

ρ

ⁿ⁺¹_ǫ

dx

_⊥

.

E

_ǫ^⊥,n+1

= −∇

^⊥

φ

ⁿ⁺¹_ǫ

, Similarly,

− ǫ

²

∂

_x²_k

V

_ǫⁿ⁺¹

= ρ

ⁿ⁺¹_ǫ

− Z

ρ

ⁿ⁺¹_ǫ

dx.

Then we can define G

ⁿ⁺¹_ǫ

with the formula: G

ⁿ⁺¹_ǫ

= − R

t

0

∂

_k

V

_ǫⁿ⁺¹

.

Now let C

₁

be a constant larger than k ρ

_ǫ

(0) k

δ0

, k w

_ǫ

(0) k

δ0

, k G

_ǫ

(0) k

δ0

, k √ ǫE

_ǫ

(0) k

δ0

and all the other constants in the previous estimates. It is possible to choose η small enough with respect to C

₁

to propagate the following estimates by recursion (we refer to [13] for details; we use in particular estimates (3.12),(3.21),(3.23)). There exists C

₂

> C

₁

, for all n ≥ 1:

(i) 

 



 



k ρ

ⁿ_ǫ

k

δ0

≤ C

₂

, k w

ⁿ_ǫ

k

δ0

≤ C

₂

, k G

ⁿ_ǫ

k

δ0

≤ C

₂

, k √

ǫE

_ǫ,ⁿ_k

k

δ0

≤ C

₂

.

(ii) 

 

 

 

 

k ρ

ⁿ_ǫ

− ρ

ⁿ_ǫ⁻¹

k

δ0

≤

^C₂ⁿ²

, k w

ⁿ_ǫ

− w

ⁿ_ǫ⁻¹

k

δ0

≤

^C₂ⁿ²

, k G

ⁿ_ǫ

− G

ⁿ_ǫ⁻¹

k

δ0

≤

^C₂ⁿ²

, k √ ǫE

_ǫ,ⁿ_k

− √ ǫE

_ǫ,ⁿ⁻¹

k

k

δ0

≤

^C₂ⁿ²

. This proves that the sequences ρ

ⁿ_ǫ

, w

_ǫⁿ

, G

ⁿ_ǫ

, √

ǫE

_ǫ

, E

^⊥ǫ^,n

, ǫ∂

_xk

φ

ⁿ_ǫ

are Cauchy sequences (with respect to n) in B

_δ^η₀

, and consequently converge strongly in B

_δ^η₀

, the estimates being uniform in ǫ. It is clear that the limit satisfies System (1.7).

The requirement δ

₁

< δ

₀

and the explicit life span in Theorem 2.1 come directly from the definition of the B

_δ^η₀

spaces.

For the uniqueness part, one can simply notice that the estimates we have shown allow us to prove that the application F defined by:

F(ρ

_ǫ

, w

_ǫ

) =

R

t

0

( −∇

⊥

(E

_ǫ^⊥

ρ

_ǫ

) − ∂

_k

((w

_ǫ

+ G

_ǫ

)ρ

_ǫ

)))ds R

t

0

( −∇

⊥

(E

_ǫ^⊥

(w

_ǫ

+ G

_ǫ

)) − (w

_ǫ

+ G

_ǫ

)∂

_k

(v

_ǫ

+ G

_ǫ

)) − ǫ∂

_k

φ

_ǫ

(t, x

_k

))ds

! , is a contraction on the closed subset B of B

_δ0

× B

_δ0

, defined by:

B = { ρ, w ∈ B

_δ0

; k ρ k

δ0

≤ C, k w k

δ0

≤ C } ,

with C large enough, provided that η is chosen small enough. The uniqueness of the

analytic solution then follows.

Proof of Proposition 2.1

We can lead the same analysis as for the proof of Theorem 2.1, but even simpler since here we do not have to deal anymore with the fast oscillations in time. The only slightly different point is to estimate the norm of R

_t

0

− ∂

_k

pds = R

_t

0

∂

_k

R

ρv

²

dx

_⊥

ds, which is straightforward:

Z

t 0

∂

_k

pds

δ0

≤ ηC k ρ k

δ0

k v k

²δ0

.

Then as before, we can use a contraction argument to prove the proposition.

4 Proof of Theorem 2.2

Step 1: Another average in time for E

_ǫ,k

We have observed previously that the wave equation (3.4) describing the time oscillations of E

_ǫ,k

was the same as the one appearing in Grenier’s work, except for a slight change in the source. Therefore the following decomposition taken from [13] identically holds:

Lemma 4.1. Under assumption (H), there exist vector fields E

_ǫ¹

, E

_ǫ²

and W

ǫ

such that E

_ǫ,k

= E

¹_ǫ

+ E

_ǫ²

and a positive constant C independent of ǫ such as:

(i) k √

ǫE

_ǫ¹

k

_L^∞_(Hx^s−1

k )

≤ C.

(ii) ∂

_t

W

_ǫ

= E

_ǫ¹

, k W

_ǫ

k

_L^∞_(Hx^s−1

k )

≤ C and W

_ǫ

⇀ 0 in L

²

. (iii) W

^ǫ

(0) = − ǫ∂

_t

E

_ǫ,k

(0) = R

ρ

_ǫ

(0)v

_ǫ

(0)dx

_⊥

. (iv) k E

_ǫ²

k

_L^∞_(Hx^s−1

k )

≤ C.

(v) R

E

_ǫ¹

dx

_k

= R

E

_ǫ²

dx

_k

= 0.

Idea of the proof. The idea in order to build E

_ǫ²

is to cut off the essential temporal oscil- lations (of frequency

√¹

ǫ

). Hence, we can define E

_ǫ²

defined by its Fourier transform:

F

_k

E

_ǫ²

(t, k

_k

) = 1 2π √

ǫ

Z

t+2π√ǫ

t

F

_k

E

_ǫ,k

(s, k

_k

)ds

and E

_ǫ¹

= E

_ǫ,k

− E

_ǫ²

, so that E

_ǫ¹

gathers the essential information on the oscillations.

Let us refer to [13] for details.

Step 2: Uniform bound on E

_ǫ^⊥

and ∂

_xk

φ

_ǫ

Under hypothesis (H) we clearly get that E

_ǫ^⊥

and ∂

_xk

φ

_ǫ

are bounded in L

^∞_t

(H

^s−1

) uni- formly with respect to ǫ (we do not need any gain of elliptic regularity).

Since Z

(ρ

_ǫ

− Z

ρ

_ǫ

dx

_⊥

)dx

_⊥

= 0, we easily check that:

k φ

_ǫ

k

Hx^s⊥,xk

≤ k ρ − Z

ρdx

_⊥

k

Hx^s⊥,xk

.

Hence the result.