Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates

(1)

Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates

E. Fr´ enod, A. Mouton

October 11, 2010 - Modified by A. Mouton

Abstract

In this paper, we present some new results about the approximation of the Vlasov-Poisson system with a strong external magnetic field by the 2D finite Larmor radius model. The proofs within the present work are built by using two-scale convergence tools, and can be viewed as an improvement of previous works of Frénod & Sonnendrücker and Bostan on the 2D finite Larmor Radius model. In a first part, we recall the physical and mathematical contexts. We also recall two main results from previous papers of Frénod & Sonnendrücker and Bostan. Then, we introduce a set of variables which are so-calledcanonical gyrokinetic coordinates, and we write the Vlasov equation in these new variables. Then, we establish some two-scale convergence and weak-* convergence results.

1 Introduction

Nowadays, domestic energy production by using magnetic confinement fusion (MCF) techniques is a huge technological and human challenge, as it is illustrated by the international scientific collaboration around ITER which is under construction in Cadarache (France).

Since magnetic confinement, needed to reach nuclear fusion reaction, is a very complex physical phenomenon, the mathematical models which are linked with this plasma physics subject need to be rigorously studied from theoretical and numerical points of view. Such a work programme based on rigorous mathematical studies and high precision numerical simulations can bring some additional informations about the behavior of the studied plasma before the launch of real experiments.

The present paper can be viewed as a part of the recent work programme about the mathematical justification of the mathematical models which are used for numerical simulations of MCF experiments. Indeed, the first tokamak plasma models have been proposed by Littlejohn, Leeet al., Dubinet al. or Brizardet al. (see [20], [18, 19], [7], [5, 6]) nevertheless most of these models were established by using formal assumptions. For ten years, many mathematicians have been working on mathematical justification of these models, especially the gyrokinetic approaches like guiding-center approximations and finite Larmor radius approximations: many results in this research field are due to Fr´enod & Sonnendr¨ucker et al.

[9, 11, 12, 13], Golse & Saint-Raymond [14, 15], Bostan [4] or, more recently, Han-Kwan [17]. These mathematical results mostly rely on two-scale convergence theory (see Allaire [3], Nguetseng [24]) or compactness arguments.

In this paper, we are focused on the 2D finite Larmor radius model and its mathematical justification: more precisely, the goal is to make a synthesis of previous mathematical proofs of the convergence of (f,E˜), where

f(x, k, α, t) = ˜f x₁−√

2ksinα, x₂+√

2kcosα,√

2ksinα, t

, (1.1)

(2)

and where ( ˜f,E˜) is the solution of the following 2D Vlasov-Poisson system











∂_tf˜+1

v˜· ∇x˜f˜+ E˜+1

v˜2

−˜v1

· ∇v˜f˜= 0, f˜(˜x,v,˜ 0) = ˜f⁰(˜x,v)˜ ,

− ∇x˜φ˜= ˜E, −∆˜xφ˜= Z

R²

f˜d˜v−n˜e,

(1.2)

towards the couple (f,E) which is the solution of the 2D finite Larmor radius model given in˜ [4] (see also Theorem 2 below). The main results on this model are due to Sonnendr¨ucker, Fr´enod and Bostan, and indicates that (f,E˜) somehow weak-* converges to the solution of the finite Larmor radius model. However the proofs within these articles are based on various assumptions and use various tools. Then, it seems useful to gather these convergence results and to simplify them as more as possible.

The first part of the present paper is devoted to a state-of-the-art about the two- dimensional finite Larmor radius approximation. Firstly, we recall the procedure which allows us to obtain the dimensionless model (1.2) from the complete Vlasov-Poisson model by considering specific assumptions. Then we recall the two-scale convergence theorem of Fr´enod & Sonnendr¨ucker [13] on the one hand, and the weak-* convergence theorem of Bostan [4] on the other hand.

In a second part, we introduce a set of variables which are so-calledcanonical gyrokinetic coordinatesand we reformulate the Vlasov-Poisson system (1.2) in these new variables. Then, we establish a two-scale convergence theorem which only relies on Fr´enod & Sonnendr¨ucker’s assumptions. Finally, we deduce from this result a new justification of the 2D finite Larmor radius model through an almost trivial proof.

2 State-of-the-art

2.1 Scaling of the Vlasov-Poisson model

This paragraph is devoted to the scaling of the following Vlasov-Poisson model:











∂_tf˜+ ˜v· ∇x˜f˜+ e mi

E˜ + ˜v×B˜

· ∇v˜f˜= 0, f˜(˜x,v,˜ 0) = ˜f⁰(˜x,v)˜ ,

−∇˜xφ˜= ˜E, −∆˜xφ˜= e ε0

Z

R³_v_˜

f d˜˜ v− e ε0

˜ n_e,

(2.3)

where ˜x = (˜x₁,x˜₂,x˜₃) ∈ R³˜x is the position variable, ˜v = (˜v₁,v˜₂,˜v₃) ∈ R³v˜ is the velocity variable,t∈R+ is the time variable, ˜f = ˜f(˜x,v, t) is the ion distribution function, ˜˜ ne is the electron density, ˜E = ˜E(˜x, t) is the self-consistent electric field generated by the ions and the electrons, ˜B= ˜B(˜x, t) is the magnetic field which is applied on the considered plasma, φ˜= ˜φ(˜x, t) is the electric potential linked with ˜E, eis the elementary charge and mi is the elementary mass of an ion.

In this model, the external magnetic field ˜Bis assumed to be uniform and carried by the unit vectore₃. We also assume that the electron density ˜n_eis given for any (˜x, t)∈R³x˜×R+. Following the same approach as in Bostan [4], Fr´enodet al. [9, 13], Golse et al. [14, 15] and Han-Kwan [17], we add the following assumptions:

(i) The magnetic field is supposed to be strong,

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(ii) The finite Larmor radius effects are taken into account, (iii) The ion gyroperiod is supposed to be small.

We define the dimensionless variables and unknowns ˜x⁰ = (˜x⁰₁,x˜⁰₂,x˜⁰₃), ˜v⁰ = (˜v₁⁰,˜v⁰₂,v˜₃⁰), t⁰, ˜f⁰, ˜E⁰ and ˜φ⁰ by

˜

x₁=L_⊥˜x⁰₁, x˜₂=L_⊥x˜⁰₂, x˜₃=L_||x˜⁰₃, t=t t⁰, v˜=vv˜⁰, f˜(L⊥x˜⁰₁, L⊥x˜⁰₂, L_||x˜⁰₃, vv˜₁⁰, vv˜₂⁰, v˜v₃⁰, t t⁰) =ff˜⁰(˜x⁰₁,x˜⁰₂,x˜⁰₃,˜v⁰₁,v˜₂⁰,˜v₃⁰, t⁰),

E(L˜ ⊥x˜⁰₁, L⊥x˜⁰₂, L_||x˜⁰₃, t t⁰) =EE˜⁰(˜x⁰₁,x˜⁰₂,x˜⁰₃, t⁰), φ(L˜ _⊥x˜⁰₁, L_⊥x˜⁰₂, L_||x˜⁰₃, t t⁰) =φφ˜⁰(˜x⁰₁,x˜⁰₂,x˜⁰₃, t⁰).

(2.4)

In these definitions, L_⊥ is the characteristic length in the direction perpendicular to the magnetic field,L_|| is the characteristic length in the direction of the magnetic field,vis the characteristic velocity andt is the characteristic time. We also rescale the electron density as follows:

˜

ne(L⊥x˜⁰₁, L⊥x˜⁰₂, L_||x˜⁰₃, t t⁰) =nn˜⁰_e(˜x⁰₁,x˜⁰₂,x˜⁰₃, t⁰). (2.5) Following the assumptions on the magnetic field ˜B, we setB as being such that

B˜ =Be3. (2.6)

Then, we setL_|| as the size of the physical device in game. We also linkf, E andφ with the characteristic Debye lengthλDby

f = n

v³, E= λDe n

ε₀ , φ= λD 2e n

ε₀ , (2.7)

and we takeλ_D as the characteristic length in the direction perpendicular to the magnetic field,i.e.

L_⊥=λ_D. (2.8)

Since we want to take into account the smallness of the gyroperiod and the finite Larmor radius effects, we define the characteristic gyrofrequency ωi and the characteristic Larmor radiusrL as

ωi= e B

m_i, rL = v

ω_i. (2.9)

With these notations, the Vlasov-Poisson system is rescaled as follows:











∂_t⁰f˜⁰+t v







r_L λ_D ˜v₁⁰

rL

λD ˜v₂⁰

r_L L_||˜v⁰₃





· ∇˜x⁰f˜⁰+t λDe²n ε₀m_iv

E˜⁰+t ω_iv˜⁰×e₃

· ∇v˜⁰f˜⁰= 0,

f˜⁰(˜x⁰,v˜⁰,0) = ˜f⁰⁰(˜x⁰,v˜⁰),

E˜⁰=







−∂_x_˜⁰

1

φ˜⁰

−∂x˜⁰₂φ˜⁰

−^λ_L^D

||

∂_˜_x⁰

3

φ˜⁰





 , −∆_(˜_x⁰

1,˜x⁰₂)φ˜⁰−λD 2

L_||²

∂_x²_˜0 3

φ˜⁰= Z

R³

f˜⁰d˜v⁰−n˜⁰_e.

(2.10)

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Taking into account the finite Larmor radius effects consists in considering a regime in which the Larmor radius is of the order of the Debye length. This implies

rL

λD

= 1. (2.11)

Since the magnetic field is assumed to be strong, the Larmor radius is small when compared with the size of the physical domain. Then it is natural to take

rL

L_||= , (2.12)

where >0 is small.

Assumption (iii) can be translated in terms of characteristic scales by t ω_i= 1

. (2.13)

Assumption (i) means that the magnetic force is much stronger than the electric force, so we consider

E e v miωi

= . (2.14)

Then, removing the primes and adding in subscript, the rescaled Vlasov-Poisson model writes











∂_tf˜+1

v˜1

˜ v2

· ∇_(˜_x₁_,˜_x₂₎f˜+ ˜v₃∂_x_˜₃f˜+ E˜+1

v˜×e₃

· ∇v˜f˜= 0, f˜(˜x,v,˜ 0) = ˜f⁰(˜x,v)˜ ,

−∇(˜x₁,˜x₂)φ˜

− ∂˜x₃φ˜

= ˜E, −∆(˜x₁,˜x₂)φ˜−²∂_x²_˜₃φ˜= Z

R³

f˜d˜v−˜ne,

(2.15)

which is the model studied in previous works of Fr´enod & Sonnendr¨ucker [13], Golse &

Saint-Raymond [14, 15], and Bostan [4].

2.2 Previous results

In this paragraph, we recall two main results about the asymptotic behavior of the sequences ( ˜f) >0 and ( ˜E) >0 when goes to 0. The first one is based on the use of two-scale convergence and homogenization techniques developed by Allaire [3] and Nguetseng [24], and was established by Fr´enod and Sonnendr¨ucker in [13]. The second one relies on compactness arguments and was proved by Bostan in [4]. After recalling these two results, we discuss the main differences between them. These differences are the source of the motivation of the present paper.

In order to simplify, we consider that the whole model (2.15) does not depend on ˜x₃nor

˜

v₃, and we assume that ˜f⁰ = ˜f⁰ for all. Then, it is reduced to a singularly perturbed 2D Vlasov-Poisson model of the form











∂tf˜+1

v˜· ∇x˜f˜+ E˜+1

v˜2

−˜v1

· ∇v˜f˜= 0, f˜(˜x,v,˜ 0) = ˜f⁰(˜x,v)˜ ,

− ∇x˜φ˜= ˜E, −∆˜xφ˜= Z

R²

f˜d˜v−n˜e,

(2.16)

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where ˜x= (˜x₁,x˜₂)∈R²and ˜v= (˜v₁,v˜₂)∈R².

The following theorem can be attributed to Fr´enod and Sonnendr¨ucker [13] (see Theorem 1.5) even if the setting of this paper is a charged particle beam Vlasov-Poisson model not involving any electron density. Nonetheless, the proof of [13] works again in the setting of model (2.16) which involves an electron density.

Theorem 1 (Fr´enod & Sonnendr¨ucker [13]). We assume that, for a fixed p≥2, f˜⁰ is in L¹(R⁴)∩L^p(R⁴), is positive everywhere and such that

Z

R⁴

|˜v|²f˜⁰(˜x,v)˜ d˜xd˜v<+∞. (2.17) We also assume thatn˜e does not depend ont, is inL¹(R²)∩L^3/2(R²)and satisfies

Z

R⁴

f˜⁰(˜x,v)˜ d˜xd˜v= Z

R²

˜

ne(˜x)d˜x. (2.18)

Then, the sequence( ˜f,E˜) >0 is boundedL^∞ 0, T;L^p(R⁴)

× L^∞ 0, T;W^1,3/2(R²)² independently of. Furthermore, by extracting some subsequences,

f˜ −→ F˜= ˜F(˜x,v, τ, t)˜ two-scale inL^∞ 0, T;L^∞_# 0,2π;L^p(R⁴) , E˜ −→ E˜= ˜E(˜x, τ, t) two-scale in

L^∞ 0, T;L^∞_# 0,2π;W^1,3/2(R²)²

. (2.19) Moreover,F˜ is linked withG˜= ˜G(˜y,u, t)˜ ∈L^∞ 0, T;L^p(R⁴)

by the relation

F˜(˜x,v, τ, t) = ˜˜ G x˜+R(−τ) ˜v, R(−τ) ˜v, t

, (2.20)

and( ˜G,E)˜ is the solution of











∂tG(˜˜ y,u, t) +˜

"

Z 2π 0

R(−σ) ˜E y˜+R(σ) ˜u, σ, t dσ

#

· ∇y˜G(˜˜ y,u, t)˜ +

"

Z 2π 0

R(−σ) ˜E y˜+R(σ) ˜u, σ, t dσ

#

· ∇u˜G(˜˜ y,u, t) = 0˜ , G(˜˜ y,u,˜ 0) = 1

2π

f˜⁰(˜y,u)˜ , E(˜˜ x, τ, t) =−∇˜xΦ(˜˜ x, τ, t),

−∆˜xΦ(˜˜ x, τ, t) = Z

R²

G˜ ˜x+R(−τ) ˜v, R(−τ) ˜v, t

d˜v− 1

2π˜ne(˜x),

(2.21)

where

R(τ) =

sinτ 1−cosτ cosτ−1 sinτ

, R(τ) =

cosτ sinτ

−sinτ cosτ

. (2.22)

In this theorem,L^∞_#(0,2π;L^p(R⁴)) stands for the space of functions ˜h= ˜h(˜x,v, τ˜ ) being inL^∞(0,2π;L^p(R⁴)) and 2π-periodic with respect toτ.

As a consequence of this theorem, extracting some subsequences, we have f˜

*∗ f˜ inL^∞ 0, T;L^p(R⁴) , E˜

*∗ E˜ in L^∞ 0, T;W^1,3/2(R²)²

, (2.23)

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where

f˜(˜x,v, t) =˜ Z 2π

0

F˜(˜x,v, τ, t)˜ dτ , and E(˜˜ x, t) = Z 2π

0

E˜(˜x, τ, t)dτ . (2.24) By using the relation between ˜F and ˜G, we can easily remark that ( ˜f ,E) is solution of˜











∂_tf˜(˜x,v, t)˜ +

Z 2π 0

"

Z 2π 0

R(τ−σ) ˜E x˜+R(σ−τ) ˜v, σ, t dσ

#

· ∇x˜F˜(˜x,v, τ, t)˜ dτ +

Z 2π 0

"

Z 2π 0

R(τ−σ) ˜E ˜x+R(σ−τ) ˜v, σ, t dσ

#

· ∇˜vF(˜˜ x,v, τ, t)˜ dτ = 0, f˜(˜x,v,˜ 0) = 1

2π Z 2π

0

f˜⁰ x˜+R(−τ) ˜v, R(−τ) ˜v dτ ,

−∇x˜φ(˜˜ x, t) = ˜E(˜x, t), −∆˜xφ(˜˜x, t) = Z

R²

f˜(˜x,v, t)˜ d˜v−˜ne(˜x).

(2.25)

We notice that these equations still involve ˜F and ˜E. In order to make these dependencies disappear, Bostan has proposed in [4] a reformulation of the Vlasov equation in guiding- center coordinates. Before presenting it, we introduce the sequence ( ˘f) >0 defined by

f˜(˜x,v, t) = ˘˜ f

˜ x+

v˜2

−˜v1

,v, t˜

, (2.26)

and, in the same spirit, we define the initial guiding-center distribution ˘f⁰by f˜⁰(˜x,v, t) = ˘˜ f⁰

˜ x+

˜v2

−˜v1

,v, t˜

. (2.27)

Theorem 2(Bostan [4]). We assume thatn˜e= 1, and thatf˜⁰ is2π-periodic inx˜1andx˜2, is positive everywhere and satisfies

Z

R²

Z 2π 0

f˜⁰(˜x,v)˜ d˜xd˜v= 1, Z

R²

Z 2π 0

|˜v|²f˜⁰(˜x,v)˜ d˜xd˜v<+∞. (2.28) We also assume that there existsF˜⁰∈L^∞(R⁺)∩L¹(R⁺;r dr) such that

∀(˜x,v)˜ ∈[0,2π]²×R², f˜⁰(˜x,v)˜ ≤F˜⁰ |˜v|

. (2.29)

We also assume that( ˜E)_>₀admits a strong limit denoted withE˜ in L² 0, T;L²_# [0,2π]²² . Then, up to a subsequence,f˘ weakly-* converges to a functionf˘= ˘f(x,v, t)inL^∞ [0, T)× R²;L^∞_# [0,2π]²

verifying

f˘(x,v, t) = 1

2πg x,|v|² 2 , t

, (2.30)

whereg=g(x, k, t)is the solution of











∂tg+hE2i∂x₁g− hE1i∂x₂g= 0, g(x, k,0) =

Z 2π 0

f˘⁰(x,√

2kcosα,√

2ksinα)dα , hEi(x, k, t) = 1

2π Z 2π

0

E˜ x₁−√

2ksinα, x₂+√

2kcosα, t dα ,

− ∇˜xφ(˜˜x, t) = ˜E(˜x, t),

−∆˜xφ(˜˜x, t) = 1 2π

Z +∞

0

Z 2π 0

g x˜1+

√

2ksinα,x˜2−√

2k cosα, k, t

dα dk−1.

(2.31)

(7)

In this theorem, L²_# [0,2π]²

stands for the space of functions ˜h = ˜h(˜x) being in L² [0,2π]²

and 2π-periodic with respect to ˜x1and ˜x2.

This last result introduces a mathematical justification of the approximation of the Vlasov-Poisson model (2.16) by the finite Larmor radius model which is exactly (2.30)-(2.31).

However, in order to prove this convergence result, Bostan considered stronger assumptions on ˜f⁰ and ˜n_e than needed to get existence of the weak-* limit ( ˜f ,E) from Theorem 1: the˜ initial distribution ˜f⁰ is supposed to be 2π-periodic in ˜x₁ and ˜x₂ and ( ˜E)_>₀ is supposed to admit a strong limit in some Banach space.

Even if Theorems 1 and 2 induce common results, i.e. the convergence in a weak sense of the solution of the 2D Vlasov-Poisson model with a strong magnetic field towards the solution of a 2D finite Larmor radius model, they are built on quite different assumptions.

Then, it seems pertinent to gather the two-scale and weak-* convergence results within a unique theorem based on common assumptions which are as weak as possible. This is what we do in the next sections.

3 Synthetic convergence result

This section is devoted to the gathering of two-scale convergence and weak-* convergence results under a common assumption set. For this purpose, we firstly reformulate the Vlasov- Poisson system (2.16) in a new set of variables which are so-called canonical gyrokinetic coordinates. Then, we prove a two-scale convergence result from which we are able to deduce a weak-* convergence corollary straightforwardly.

3.1 Reformulation of Vlasov equation

Following the ideas of Littlejohn [20], Lee [18, 19], and Brizard et al. [5, 6], we define the variables (x1, x2, k, α)∈R²×R+×[0,2π] by linking them with (˜x1,x˜2,v˜1,v˜2)∈R⁴by

x˜₁ = x₁−v˜₂,

˜

x2 = x2+ ˜v1,

˜

v₁ = √

2kcosα ,

˜

v₂ = √

2ksinα . (3.1)

This set of variables is so-called canonical gyrokinetic coordinates: indeed, if we define the characteristicsX1, X2, K, A linked withx1, x2, k, αby

X˜1 = X1−V˜2, X˜2 = X2+ ˜V1,

V˜1 = √

2K cosA , V˜2 = √

2K sinA , (3.2)

where ˜X1,X˜2,V˜1,V˜2are the characteristics associated with the Vlasov equation (2.16.a),i.e.

satisfying











∂tX˜1(t) = 1

V˜1(t),

∂tX˜2(t) = 1

V˜2(t),

∂tV˜1(t) = E˜,1 X˜1(t),X˜2(t), t +1

V˜2(t),

∂tV˜2(t) = E˜,2 X˜1(t),X˜2(t), t

−1

V˜1(t),

(3.3)

we have











∂tX1(t) = −∂x₂H X1(t), X2(t), K(t), A(t), t ,

∂tX2(t) = ∂x₁H X1(t), X2(t), K(t), A(t), t ,

∂tK(t) = ∂αH X1(t), X2(t), K(t), A(t), t ,

∂tA(t) = −∂kH X1(t), X2(t), K(t), A(t), t ,

(3.4)

(8)

where the hamiltonian functionHis defined by H(x1, x2, k, α, t) = k

+φ(x1, x2, k, α, t), (3.5) andφis linked with ˜φ by the relation

φ(x1, x2, k, α, t) = ˜φ x1−√

2ksinα, x2+

√

2kcosα, t

. (3.6)

Then it is straightforward to see that, in the gyrokinetic canonical coordinates, the Vlasov- Poisson system (2.16) has the following shape:











∂_tf+E_,2∂_x₁f−E_,1∂_x₂f+√

2k(E_,1 cosα+E_,2sinα)∂_kf +E,2 cosα−E,1sinα

√2k ∂αf−1

∂αf= 0, f(x, k, α,0) = ˜f⁰ x1−√

2ksinα, x2+

√

2k cosα,

√

2k cosα,

√

2ksinα , E(x, k, α, t) = ˜E x₁−√

2ksinα, x₂+√

2kcosα, t , E˜(˜x, t) =−∇˜xφ˜(˜x, t),

−∆x˜φ˜(˜x, t) = Z 2π

0

Z +∞

0

f x˜₁+√

2ksinα,x˜₂−√

2k cosα, k, α, t

dk dα−n˜_e(˜x, t),

(3.7)

where ˜x= (˜x1,x˜2) andx= (x1, x2), and wheref=f(x, k, α, t) andE=E(x, k, α, t) are linked with ˜fand ˜Eby

f(x, k, α, t) = ˜f x1−√

2ksinα, x2+

√

2kcosα,

√

2kcosα,

√

2ksinα, t , E(x, k, α, t) = ˜E x1−√

2ksinα, x2+

√

2kcosα, t .

(3.8)

3.2 Two-scale convergence

We set the following notations

Ω =R²×R+×[0,2π], Γ =R²×R+, S=R+×[0,2π], (3.9) and we consider the following Banach spaces, involving periodicity with respect toα:

L^p_# 0,2π;L^p(Γ)

=n

f ∈L^p(Ω) : f is periodic in αo , W_#^1,p 0,2π;W^1,p(Γ)

=n

f ∈W^1,p(Ω), f(., .,0) =f(., .,2π) : f is periodic inαo , W_#^2,p 0,2π;W^2,p(Γ)

=n

f ∈W^2,p(Ω), f(., .,0) =f(., .,2π), ∂_αf(., .,0) =∂_αf(., .,2π) : f is periodic inαo

, and we can state the following theorem.

Theorem 3. We assume that, for a fixedp≥2,f˜⁰and˜nesatisfy the assumptions of Theo- rem 1. Then sequences(f) >0and( ˜E) >0 of system (3.7) are bounded independently of

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inL^∞ 0, T;L^p_#(0,2π;L^p(Γ))

and L^∞ 0, T;W^1,3/2(R²)²

respectively. As a consequence, there existF=F(x, k, α, τ, t)andE˜= ˜E(˜x, τ, t)such that, extracting some subsequences,

f −→ F two-scale in L^∞ 0, T;L^∞_# 0,2π;L^p_#(0,2π;L^p(Γ)) , E˜ −→ E˜ two-scale in L^∞ 0, T;L^∞_#(0,2π;W^1,3/2(R²))²

. (3.10)

Furthermore, there existG=G(x, k, α, t)∈L^∞ 0, T;L^p_#(0,2π;L^p(Γ))

andE=E(x, k, α, τ, t)∈ L^∞ 0, T;L^∞_#(0,2π;W_#^1,3/2(0,2π;W^1,3/2(Γ)))²

such that

F(x, k, α, τ, t) =G(x, k, α+τ, t), (3.11) E(x, k, α, τ, t) = ˜E x1−√

2ksinα, x2+√

2kcosα, τ, t

, (3.12)

and verifying











∂_tG+hE₂i∂_x₁G− hE₁i∂_x₂G+hF_αi∂_αG= 0, G(x, k, α,0) = 1

2π

f˜⁰ x₁−√

2k sinα, x₂+√

2kcosα,√

2ksinα ,

−∇x˜Φ(˜˜ x, τ, t) = ˜E(˜x, τ, t),

−∆x˜Φ(˜˜ x, τ, t) = Z

S

G x˜1+√

2ksinα,x˜2−√

2kcosα, k, α+τ, t dk dα

− 1 2πn˜_e(˜x), Fα(x, k, α, τ, t) = E2(x, k, α, τ, t) cosα− E1(x, k, α, τ, t) sinα

√

2k ,

(3.13)

where the notationh·istands for

hui(x, k, t) = Z 2π

0

u(x, k,−τ, τ, t)dτ . (3.14)

Proof of Theorem 3. Several parts of this proof are only sketched since they can be redundant with [13]. However, more details can be found in Mouton [22].

Following the same way as in [13], we prove that, under the assumptions of Theorem 1, we have

f(·, t) _Lp

#(0,2π;L^p(Γ))=kf˜⁰k_Lp(R⁴),∀t≥0, (3.15) and, defining ˜ρ as

˜

ρ(˜x, t) = Z

S

f x˜1+√

2k sinα,x˜2−√

2kcosα, k, α, t

dk dα , (3.16) that the sequence ( ˜ρ)_>₀ is bounded inL^∞ 0, T;L^3/2(R²)

independently of . Then, we deduce that ( ˜φ) >0 and ( ˜E) >0 are bounded independently of inL^∞ 0, T;W^2,3/2(R²) and L^∞ 0, T;W^1,3/2(R²)²

respectively. As a consequence, there existF =F(x, k, α, τ, t), Φ = ˜˜ Φ(˜x, τ, t) and ˜E= ˜E(˜x, τ, t) such that

f −→ F two-scale inL^∞ 0, T;L^∞_# 0,2π;L^p_#(0,2π;L^p(Γ)) , φ˜ −→ Φ˜ two-scale inL^∞ 0, T;L^∞_# 0,2π;W^2,3/2(R²)

, E˜ −→ E˜ two-scale in L^∞ 0, T;L^∞_# 0,2π;W^1,3/2(R²)²

.

(3.17)

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Considering a compact setK⊂Γ, we easily remark that the sequence (φ)_>₀defined by (3.6) is bounded inL^∞ 0, T;W_#^1,3/2(0,2π;W^1,3/2(K))

and that all its second order derivatives except ∂_k²φ are bounded independently of in L^∞ 0, T;L^3/2_# 0,2π;L^3/2(K))). The sequence (E)_>₀ defined by (3.8.b) is bounded in L^∞ 0, T;W_#^1,3/2(0,2π;W^1,3/2(K))2

independently of . As a consequence, we claim that there exist Φ = Φ(x, k, α, τ, t) and E=E(x, k, α, τ, t) such that

φ −→ Φ two-scale inL^∞ 0, T;L^∞_# 0,2π;W_#^1,3/2(0,2π;W^1,3/2(K)) , E −→ E two-scale in L^∞ 0, T;L^∞_# 0,2π;W_#^1,3/2(0,2π;W^1,3/2(K))²

. (3.18) Furthermore, we remark that Φ andE are linked with ˜Φ and ˜E by the formula

E(x, k, α, τ, t) = ˜E x1−√

2ksinα, x2+

√

2kcosα, τ, t , Φ(x, k, α, τ, t) = ˜Φ x1−√

2k sinα, x2+

√

2kcosα, τ, t .

(3.19) Then the vector functionA defined by

A=







−∂_x₂φ

∂_x₁φ

∂αφ

−∂kφ







=







E,2

−E,1

√

2k E,1 cosα+E,2sinα E,2cosα−E,1 sinα

√2k







, (3.20)

has its three first components which are bounded inL^∞ 0, T;W_#^1,3/2(0,2π;W^1,3/2(K)) , independently ofand its fourth one inL^∞ 0, T;L^3/2_# (0,2π;L^3/2(K))

and admits a two-scale limit denoted A = A(x, k, α, τ, t) in L^∞ 0, T;L^∞_#(0,2π;W_#^1,3/2(0,2π;W^1,3/2(K)))³

× L^∞ 0, T;L^∞_#(0,2π;L^3/2_# (0,2π;L^3/2(K)). The convergence of the three first components is the consequence of classical embedding of Sobolev spaces inL^p spaces. Concerning the convergence of the fourth one we need to use thatE,1andE,2are bounded inL^∞ 0, T;W_#^1,3/2(0,2π;

W^1,3/2(K)) and consequently inL^∞ 0, T;W_#^1,3/2(0,2π;L⁶(K)) and that (2k)^−1/2is, for any kmax and anyq <2, inL^q(0, kmax).

This vector function is linked with Φ andE as follows:

A=







−∂x₂Φ

∂x₁Φ

∂αΦ

−∂kΦ







=







E2

−E1

√

2k E1cosα+E2sinα E2 cosα− E1 sinα

√2k







. (3.21)

In order to establish the two-scale limit model, we cannot simply apply Theorem 1.3 of [13]:

indeed, the formulation (3.7.a) of Vlasov equation does not fit with the assumptions which are needed for applying this theorem since the differential operatorf 7→ −1

∂αf cannot be written under the form

f 7→ 1

M





 x1

x2

k α





 +N

· ∇f , (3.22)

whereMis a constant square matrix satisfyingT r(M) = 0, andN∈Im(M). However, the approach which is considered in [13] can be adapted to the present case.

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Firstly, we prove that there exists a functionGsuch thatF(x, k, α, τ, t) =G(x, k, α+τ, t).

To reach such a result, we consider a test functionψ=ψ(x, k, α, τ, t) on Ω×[0,2π]×[0, T] which is 2π-periodic in αand τ. If we multiply (3.7.a) byψ(x, k, α,^t, t) and integrate over Ω×[0, T], we obtain

Z T 0

Z

Ω

f(x, k, α, t)h

∂_tψ x, k, α,t , t

+1

∂_τψ x, k, α,t , t +A(x, k, α, t)· ∇f(x, k, α, t)−1

∂αψ x, k, α,t , ti

dxdk dα dt

=− Z

Ω

f˜⁰(x1−√

2ksinα, x2+√

2kcosα,√

2ksinα)

×ψ(x₁, x₂, k, α,0,0)dx₁dx₂dk dα .

(3.23)

Multiplying (3.23) byand letting→0, we obtain the weak formulation of∂_τF−∂αF = 0, which indicates that there exists a functionG∈L^∞ 0, T;L^p_#(0,2π;L^p(Γ))

such that F(x, k, α, τ, t) =G(x, k, α+τ, t). (3.24) Secondly, we introduce the sequence (g) >0defined by

g(x, k, α, t) =f

x, k, α−t , t

. (3.25)

In the spirit of [13], we prove thatgstrongly converges to 2π Gin a given Banach space. For that, we notice that, up to a subsequence,gtwo-scale converges toGinL^∞ 0, T;L^∞_# 0,2π;

L^p_#(0,2π;L^p(Γ))

since we have Z T

0

Z

Ω

g(x, k, α, t)ψ x, k, α,t , t

dxdk dα dt

= Z T

0

Z

Ω

f(x, k, α, t)ψ x, k, α+t ,t

, t

dxdk dα dt

→ Z 2π

0

Z T 0

Z

Ω

F(x, k, α, τ, t)ψ(x, k, α+τ, τ, t)dxdk dα dt dτ

= Z 2π

0

Z T 0

Z

Ω

G(x, k, α, t)ψ(x, k, α, τ, t)dxdk dα dt dτ ,

(3.26)

for any test functionψon Ω×[0,2π]×[0, T] which is 2π-periodic inαandτ. As a consequence, gweakly-* converges to 2π Gin L^∞ 0, T;L^p_#(0,2π;L^p(Γ))

.

Let us prove that this weak-* convergence is a strong convergence in a given Banach space. This is the aim of the following lemma:

Lemma 4. For any compact subset K ofΓ, and up to a subsequence,g strongly converges to2πGinL^∞ 0, T; (W_#^1,3/2(0,2π;W₀^1,3/2(K)))^∗

.

In this Lemma (W_#^1,3/2(0,2π;W₀^1,3/2(K)))^∗stands for the dual of W_#^1,3/2(0,2π;W₀^1,3/2(K)).

Proof of Lemma 4. For any compact subsetK of Γ, (g)_>₀ and (A)_>₀ are respectively bounded inL^∞ 0, T;L^p_#(0,2π;L^p(K))

and L^∞ 0, T;L^∞_#(0,2π;W_#^1,3/2(0,2π;W^1,3/2(K)))3

×L^∞ 0, T;L^∞_#(0,2π;L^3/2_# (0,2π;L^3/2(K)), independently of. Then, remarking thatgis solution of







∂tg(x, k, α, t) +A x, k, α−t , t

· ∇g(x, k, α, t) = 0, g(x, k, α,0) = ˜f⁰(x1−√

2k sinα, x2+√

2kcosα,√

2ksinα),

(3.27)

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we use similar arguments as the ones given after equation (3.20) to deduce 1. (A) >0 is bounded independently of in L^∞ 0, T;L^q_#(0,2π;L^q(K))⁴

for anyq∈ [1,³₂[,

2. g(x, k, α, t)A(x, k, α−^t, t)

>0is bounded in L^∞ 0, T;L^r_#(0,2π;L^r(K))⁴ independently ofwithrdefined by ¹_r = ¹_p+¹_q (r∈]1,³₂[),

3. (∂tg) >0 is bounded inL^∞ 0, T; W_#^1,r^∗(0,2π;W₀^1,r^∗(K))∗

independently of with

1

r^∗ +¹_r = 1.

If r^∗ ≥ 3/2, the embedding W_#^1,r^∗(0,2π;W₀^1,r^∗(K)) ⊂ W_#^1,3/2(0,2π;W₀^1,3/2(K)) is compact with density. Furthermore, Rellich-Kondrakov’s theorem (see [1]) gives the compact embedding L^p_#(0,2π;L^p(K)) ⊂ (W_#^1,3/2(0,2π;W₀^1,3/2(K)))^∗ since p ≥ 2. Then, we apply Aubin-Lions’ lemma (see [21]) and we prove that the functional spaceU defined by

U =n

u∈L^∞ 0, T;L^p_#(0,2π;L^p(K)) :

∂tu∈L^∞ 0, T; W_#^1,r^∗(0,2π;W₀^1,r^∗(K))∗o ,

(3.28)

is compactly embedded inL^∞ 0, T; (W_#^1,3/2(0,2π;W₀^1,3/2(K)))^∗

. Sinceg∈ U for all, we deduce that the weak-* convergence ofg to 2π GinL^∞ 0, T;L^p_#(0,2π;L^p(K))

is a strong convergence inL^∞ 0, T; (W_#^1,3/2(0,2π;W₀^3/2(K)))^∗

.

Ifr^∗<3/2, the compact embedding W_#^1,r^∗(0,2π;W₀^1,r^∗(K))∗

⊂ W_#^1,3/2(0,2π;W₀^1,3/2(K))∗

is gotten directly. If we introduce the functional spaceU⁰ defined by U⁰=n

u∈L^∞ 0, T;L^p_#(0,2π;L^p(K)) :

∂_tu∈L^∞ 0, T; W_#^1,3/2(0,2π;W₀^1,3/2(K))∗o ,

(3.29)

we remark that the sequence (g) >0 is bounded inU⁰ independently of. By using Aubin- Lions’ lemma, we prove thatU⁰ ⊂L^∞ 0, T; (W_#^1,3/2(0,2π;W₀^1,3/2(K)))^∗

is a compact embedding. Then,g strongly converges to 2π Gin L^∞ 0, T; (W_#^1,3/2(0,2π;W₀^1,3/2(K)))^∗

.

Let us finish the proof of Theorem 3 by establishing a transport equation satisfied byG.

Let us consider a test functionψ=ψ(x, k, α, t) on Ω. We denote its compact support in Γ byKand we assume thatψand its first order derivatives are 2π-periodic in theαdirection.

Then we have Z T

0

Z 2π 0

Z

K

g(x, k, α, t)∂tψ(x, k, α, t)dxdk dα dt +

Z T 0

Z 2π 0

Z

K

g(x, k, α, t)A x, k, α−t , t

· ∇ψ(x, k, α, t)i

dxdk dα dt

+ Z 2π

0

Z

K

f˜⁰(x1−√

2ksinα, x2+√

2kcosα,√

2ksinα)

×ψ(x, k, α,0)dxdk dα= 0. (3.30)

Sincegconverges to 2π Gstrongly inL^∞ 0, T; (W_#^1,3/2(0,2π;W₀^1,3/2(K)))^∗

and weakly-* in L^∞ 0, T;L^p_#(0,2π;L^p(K))

, andA(x, k, α−^t, t) two-scale converges toA(x, k, α−τ, τ, t) in

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L^∞ 0, T;L^∞_# 0,2π;W_#^1,3/2(0,2π;W^1,3/2(K))³

×L^∞ 0, T;L^∞_# 0,2π;L^3/2_# (0,2π;L^3/2(K)) , we obtain the weak formulation of











∂tG(x, k, α, t) +

"

Z 2π 0

A(x, k, α−τ, τ, t)dτ

#

· ∇G(x, k, α, t) = 0,

G(x, k, α,0) = 1 2π

f˜⁰(x₁−√

2k sinα, x₂+√

2kcosα,√

2ksinα),

(3.31)

when→0.

In order to obtain Poisson type equations (3.13.c) and (3.13.d), we consider a test function ψ˜= ˜ψ(˜x, τ, t) onR²×[0,2π]×[0, T] which is 2π-periodic inτ, we multiply ˜E(˜x, t),∇˜xφ˜(˜x, t) and ∆_x_˜φ˜(˜x, t) by ˜ψ(˜x,^t, t) and we integrate in ˜x andt. We obtain

Z T 0

Z

R²

E˜(˜x, t)ψ ˜x,t , t

d˜xdt→ Z 2π

0

Z T 0

Z

R²

E˜(˜x, τ, t)ψ(˜x, τ, t)d˜xdt dτ , (3.32)

Z T 0

Z

R²

∇˜xφ˜(˜x, t)ψ ˜x,t , t

d˜xdt→ Z 2π

0

Z T 0

Z

R²

∇˜xΦ(˜˜ x, τ, t)ψ(˜x, τ, t)d˜xdt dτ , (3.33) Z T

0

Z

R²

∆˜xφ˜(˜x, t)ψ ˜x,t , t

d˜xdt→ Z 2π

0

Z T 0

Z

R²

∆˜xΦ(˜˜ x, τ, t)ψ(˜x, τ, t)d˜xdt dτ , (3.34) whenconverges to 0. SinceF is the two-scale limit of (f)_>₀, we also have

Z T 0

Z

R²

Z

S

f x˜1+√

2ksinα,x˜2−√

2ksinα, k, α, t)dk dα

! ψ x,˜ t

, t d˜xdt

→ Z 2π

0

Z T 0

Z

R²

Z

S

F x˜₁+√

2k sinα,x˜₂−√

2ksinα, k, α, τ, t)

×ψ(˜x, τ, t)dk dα d˜xdt dτ ,

(3.35)

when→0. Then, gathering convergence results (3.32)-(3.35), we obtain the weak formulation of (3.13.c) and (3.13.d).

To summarize the work which has already been done, we have proved that, up to a subsequence the solution (f,E˜)_>₀ two-scale converges to a couple (F,E˜) such that

F(x, k, α, τ, t) =G(x, k, α+τ, t), (3.36) whereGis solution of











∂_tG(x, k, α, t) +

"

Z 2π 0

A(x, k, α−τ, τ, t)dτ

#

· ∇G(x, k, α, t) = 0,

G(x, k, α,0) = 1 2π

f˜⁰(x1−√

2k sinα, x2+√

2kcosα,√

2ksinα), E(x, k, α, τ, t) = ˜E(x1−√

2ksinα, x2+

√

2k cosα, τ, t),

−∇x˜Φ(˜˜ x, τ, t) = ˜E(˜x, τ, t),

−∆x˜Φ(˜˜ x, τ, t) = Z

S

G(˜x1+√

2ksinα,˜x2−√

2kcosα, k, α, t)dk dα− 1 2πn˜e(˜x).

(3.37)

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In order to complete the proof of the present theorem, we have to prove that the function (x, k, α, t)7→

Z 2π 0

A(x, k, α−τ, τ, t)dτ does not depends onα, and this can be viewed as a direct consequence of the following lemma:

Lemma 5. For all (x, k, α, t)∈Ω×[0, T], we have

∂α

Z 2π 0

Φ(x, k, α−τ, τ, t)dτ

!

= 0. (3.38)

Indeed, this lemma allows us to claim that Z 2π

0

A(x, k, α−τ, τ, t)dτ = Z 2π

0

A(x, k,−τ, τ, t)dτ =hAi(x, k, t), (3.39) where theh·inotation is defined by (3.14), and that

Z 2π 0

√

2k E1(x, k, α−τ, τ, t) cos(α−τ) +E2(x, k, α−τ, τ, t) sin(α−τ)

dτ = 0, (3.40) for all (x, k, α, t)∈Ω×[0, T]. Then model (3.37) reduces itself to (3.13), which concludes the proof of Theorem 3.

Proof of Lemma 5. We consider a compact subsetK of Γ. As it has been previously men- tioned,φtwo-scale converges to Φ inL^∞ 0, T;L^∞_# 0,2π;W_#^1,3/2(0,2π;W^1,3/2(K))

, so we have

Z T 0

Z

Ω

∂_αφ

x, k, α−t , t

ψ(x, k, α, t)dxdk dα dt

→ Z T

0

Z

Ω

Z 2π 0

∂αΦ(x, k, α−τ, t)ψ(x, k, α, t)dτ dxdk dα dt ,

(3.41)

for any regular test functionψwhich support in Γ is included inK, and which is 2π-periodic inα. It means that we have the following weak-* convergence result

∂αφ

x, k, α−t , t _∗

* Z 2π

0

∂αΦ(x, k, α−τ, t)dτ , (3.42) inL^∞ 0, T;L^3/2_# (0,2π;L^3/2(K))

. Considering such a test fonctionψ, we define ¯ψby ψ(x, k, α, τ, t) =¯ ψ(x, k, α+τ, t). (3.43) Then we have

Z T 0

Z

Ω

∂_αφ

x, k, α−t , t

=−

Z T 0

Z

Ω

φ(x, k, α, t)∂τψ¯

x, k, α−t , t

dxdk dα dt .

(3.44)

Since (φ)_>₀ is bounded independently of in L^∞ 0, T;W_#^1,3/2(0,2π;W^1,3/2(K)) , there exists a constantC >0 which only depends on the initial data ˜f⁰and ˜n_e such that

Z T 0

Z

Ω

∂αφ

x, k, α−t , t

≤C × max

K×[0,2π]×[0,T]|∂αψ|. (3.45)

(15)

We deduce that

∂_αφ

x, k, α− t , t _∗

*0 inL^∞ 0, T;L^3/2_# (0,2π;L^3/2(K))

. (3.46)

When coupled with (3.42), this result allows us to finish the proof of the lemma by using the uniqueness of the weak-* limit of ∂_αφ(x, k, α−^t, t)

>0.

3.3 Weak-* convergence

As it has been announced previously, we have chosen the variables (x, k, α) in order to present the weak-* convergence of (f,E˜) >0 to the solution of the 2D finite Larmor radius model as a direct consequence of the two-scale convergence result we have just proved.

Firstly, we have a direct corollary of Theorem 3:

Corollary 6. Up to some subsequences,

• f weakly-* converges to f ∈L^∞ 0, T;L^p_#(0,2π;L^p(Γ))

withf defined by

f(x, k, α, t) = Z 2π

0

F(x, k, α, τ, t)dτ , (3.47)

• E˜ weakly-* converges toE˜ ∈ L^∞ 0, T;W^1,3/2(R²)²

with E˜ defined by E(x, t) =˜

Z 2π 0

E(x, τ, t)˜ dτ , (3.48)

• φ˜ weakly-* converges toφ˜∈L^∞ 0, T;W^2,3/2(R²)

with φ˜defined by φ(x, t) =˜

Z 2π 0

Φ(x, τ, t)˜ dτ . (3.49)

Then, we have the following theorem:

Theorem 7. There exists a function g defined on Γ×[0, T]such that f(x, k, α, t) = 1

2πg(x, k, t), ∀(x, k, α, t)∈Ω×[0, T], (3.50) and verifying











∂_tg+hE2i∂_x₁g− hE1i∂_x₂g= 0, g(x, k,0) =

Z 2π 0

f˜⁰(x1−√

2ksinα, x2+√

2k cosα,√

2kcosα,√

2ksinα)dα ,

hEi(x, k, t) = 1 2π

Z 2π 0

E˜ x1−√

2ksinα, x2+√

2kcosα, t)dα ,

−∇x˜φ(˜˜ x, t) = ˜E(˜x, t),

−∆x˜φ(˜˜ x, t) = 1 2π

Z

S

g x˜₁+√

2ksinα,x˜₂−√

2k cosα, k, t)dk dα−n˜_e(˜x).

(3.51)