Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field

Mod´elisation Math´ematique et Analyse Num´erique M2AN, Vol. 36, N 6, 2002, pp. 1071–1090 DOI: 10.1051/m2an:2003006

CONVERGENCE OF THE SCHR ¨ ODINGER-POISSON SYSTEM TO THE EULER EQUATIONS UNDER THE INFLUENCE OF A LARGE MAGNETIC FIELD

Marjolaine Puel

Abstract. In this paper, we prove the convergence of the current defined from the Schr¨odinger- Poisson system with the presence of a strong magnetic field toward a dissipative solution of the Euler equations.

Mathematics Subject Classification. 76X05, 76N99, 81Q99, 82D10, 35Q40.

Received: October 26, 2001. Revised: June 18, 2002.

1. Introduction

We consider in this paper the behavior of electrons moving on a positive charged background influenced by a very strong magnetic field. We describe this phenomenon from a quantum point of view. This model describes a magneto-active plasma where the velocities of the particles are small compared to the speed of light (to use the electrostatic approximation) and where the magnetic field is very strong (like in the magneto-sphere). The equations satisfied by the wave functions are a variation from the Schr¨odinger-Poisson system. They depend on three parameters, the Planck constant (h >0), the permittivity of the system (ε >0) and the strength of the magnetic field.

We deal with the asymptotic limit which corresponds to the case when the magnetic field is very strong, the Planck constant and the permittivity are both very small (implying that the electron density is quasi-equal to the ion density). As Brenier in [2], who studied the quasi-neutral limit of the Vlasov–Poisson system (with and without external magnetic field), we obtain at the limit that the curl of the potential involved in the system is a dissipative solution of the Euler equations in the sense of P.-L. Lions.

In [11], we also studied the same asymptotic limit in the case of the Schr¨odinger–Poisson system without magnetic field and obtained that the electron current converges to a dissipative solution to the Euler equations.

Those results are a rigorous justification of the composition of two different asymptotic processes which already have been independently studied in [8, 9] and [5] for the semi-classical limit and, as mentioned above, in [2] for the quasi-neutral limit.

Keywords and phrases. Quasi-neutral plasmas, semi-classical limit, modulated energy.

1 Laboratoire d’analyse num´erique (B 187), Universit´e Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France.

e-mail:mpuel@ann.jussieu.fr

c EDP Sciences, SMAI 2003

1.1. Presentation of the system

We write the Schr¨odinger equation describing the motion of electrons on a charged background submitted to an external magnetic fieldB= curlA. In this case (cf. [4]), the quantum Hamiltonian is given by

H_q= 1

2m[P −qA]²= 1 2m

P²+qAP+qP A+q²A² , whereP is the operator defined byP = ^h_i∇.

This problem is initially stated inR³. Since we neglect the third component of the velocity and the dependence of the two first ones with respect to the vertical variable, our study is set inR². In the following, forZ ∈R², we denote by^⊥Z the vector

⊥Z= Z₂

−Z1

.

We choose a potentialA=−^⊥_2ε^x, which corresponds to a vertical magnetic field. Let us notice thatAcommutes with P because ∇_⊥

2εx

= 0. Assuming q =m = 1, the Schr¨odinger equation ih∂_tψ =H_qψ becomes in our setting

ih∂_tψ^ε+h²

2 ∆ψ^ε+ih

2ε(^⊥x· ∇)ψ^ε−|x|² 8ε²ψ^ε= 0 whereψ is a wave function.

In this paper, we consider a superposition of states (instead of a pure state) where each statek∈Z² has a weightλ_k ≥0 and we add the effect of the electric field. Finally, we deal with an electron density given by

ρ^ε(t, x) =

k∈Z²

λ_kψ^ε_k(t, x)ψ^ε_k(t, x) (1.1)

where the wave functions satisfy the equation ih∂_tψ_k^ε+h²

2 ∆ψ^ε_k+ ih 2ε

_⊥ x· ∇

ψ^ε_k−|x|² 8ε²ψ_k^ε=1

εφ^εψ^ε_k (1.2)

coupled with the Poisson equation

ρ^ε=θ^ε−∆φ^ε. (1.3)

Here, the ion densityθ^ε(x)∈L^∞(R^d) satisfies

(1 +|x|)θis inL¹(R²)∩L²(R²),

0≤θ^ε(x)≤1, (1.4)

θ^ε(x) = 1 ifx∈]−R^ε;R^ε[^d, (1.5)

θ^ε(x) = 0 ifx∈R^d\]−R^ε−γ^ε;R^ε+γ^ε[^d, (1.6)

where (R^ε)⊂R^∗₊, (R^ε)^ε→0−→+∞, and (γ^ε)⊂R^∗₊, (γ^ε)^ε→0−→0. Under this form, (θ^ε) converges inD(R²) to 1 and its derivatives to 0. This is important because we need at the limit a uniform densityθso that∇ ·(θv) = 0 when∇·v= 0. Moreover, in the existence proof, we need some integrability properties forθ^εwhich are certainly true if θ^εis compactly supported.

An adaptation of some results of [1] and [3] enables us to show the existence of a solution of the system (1.2, 1.3) for a fixed ε. We skip the indexεto present the existence result.

Let us define the following Hilbert spaces X=

Γ = (γ_m)_m∈Z2 γ_m∈L²(R²), ∀mand ∞ m=1

λ_m||γm||²_L2(R²)<∞

,

H=

γ (γ,∇γ,∆γ,|x|γ,|x|∇γ,|x|²γ)∈L²(R²)

with their associated norms

||Γ||²_X = ∞ m=1

λ_m||γ_m||²_L2(R²),

||γ||²_H=||γ||²_H1(R²)+||∆γ||²_L2(R²)+|||x|γ||²_L2(R²)+|||x|∇γ||²_L2(R²)+|||x|²γ||²_L2(R²),

Y =

Γ = (γ_m)_m∈Z2 γ_m∈H, ∀mand ∞ m=1

λ_m||γm||²_H <∞,

with the norm

||Γ||²_Y = ∞ m=1

λ_m||γ_m||²_H

and finally, ifθ:R²→R,

Y˜ =

Γ∈Y

R²(θ−n(Γ))dx= 0

wheren(Γ) =_∞

m=1λ_m|γm|².

The following existence result holds.

Proposition 1.1. Under the following assumptions,

– the function θ is inL^∞(R²)and(1 +|x|)θ is in L¹(R²)∩L²(R²), – the initial data are smooth enough, i.e. (ψ_0,k)_k∈Z2 ∈Y˜,

there exists a unique solution to (1.2–1.3) (ψ_k)_k∈Z2 ∈ C([0,∞), Y)∩C¹([0,∞), X) and (ψ_k(t))_k∈Z2 ∈ Y˜ for each time t.

The proof of Proposition 1.1 is very similar to the proof of the existence result in [1] for the pure electrostatic case. The main differences are the following.

First, the definition of the space H is different because of the shape of the operator. Indeed, we need that

|x|∇γ and|x|²γbelong toL²(R²). This implies some additional computations which can be found in [11].

Moreover, we don’t get directly that the conservation of energy gives a time independent bound for||ψ||H¹(R²). However, we obtain using the equation that

d

dt(||h∇ψ^ε_k||_L²+|||x|ψ_k^ε||_L²)≤C||h∇ψ_k^ε+i

⊥x

2εψ_k^ε||_L² ≤C. It implies (see [1]) that

||φ^ε||L^∞+||∇φ^ε||L^∞+||∆φ^ε||L^p≤E₁+E₂t², for 1≤p <∞

with E₁ and E₂ two constants depending only on θ^ε and on the initial energy. That gives us as in [1] the existence result for global solutions.

Remark 1.1. It is probably possible to obtain additional regularity for the solution ψ when the initial data are smooth, but this question is not addressed in this work.

We will use this result for anyε. Let us note that since the ion densitiesθ^εare inL^∞(R²) and are compactly supported, the first assumption is satisfied.

1.2. Formal analysis

In this section, we use the Wigner transform to motivate formally the obtained result. Let us introduce W^ε,h(t, x, ξ) = (2πh)^−d

R^d

k

λ_ke^−ihξ·zψ_k^ε

t, x+z 2

ψ^ε_k

t, x−z 2

dz,

the Wigner transform for the system and let us operate a change of variable W˜^ε(t, x; ˜ξ) =W^ε(t, x, ξ) with ˜ξ=ξ+

⊥x 2ε· The new function ˜W^ε(t, x; ˜ξ) satisfies

∂_tW˜^ε+ ˜ξ· ∇xW˜^ε+1 ε

⊥ξ˜· ∇ξ˜W˜^ε+1

εK˜ ∗W˜^ε= 0 and

∆φ^ε=ρ^ε−θ^ε with

K(t, x,˜ ξ) =˜ i (2π)^d

R^de^−iξ·y˜ (φ^ε(t, x+^hy₂)−φ^ε(t, x−^hy₂))

h dy

and

˜

ρ^ε(t, x) =

W˜^ε

t, x,ξ˜

d ˜ξ.

Denoting ˜J^ε(t, x) the new current defined by J˜_i^ε(t, x) =

ξ˜_iW˜^ε

t, x,ξ˜

d ˜ξ,

we obtain the following equations

∂_tρ˜^ε+∇ ·J˜^ε= 0,

∂_tJ˜_i^ε(t, x) =−

ξ˜_iξ˜_j∂_jW˜^ε

t, x,ξ˜

d ˜ξ+1 ε

⊥J˜_i^ε(t, x)−∂_iφ^ε(t, x) ˜ρ^ε(t, x)

.

Therefore, we obtain formally, when goes to zero that J˜=−˜ρ^⊥∇φ,

∂_tρ˜+∇ ·J˜= 0 and

˜

ρ−1 = ∆φ.

Those equations are nothing but the incompressible Euler system written for w= curlu= ˜ρ−1,u=−^⊥∇φ since we have

∂_tw+∇ ·(uw) = 0,

∇ ·u= 0.

1.3. Main result

Theorem 1.2. Let T >0 andJ⁰=−^⊥∇Φ0 be a divergence free vector inL²(R²). For anyε >0, letθ^ε be the ion density of charge characterized by (1.4–1.6), and let us consider, for any ε,h and each statek, the wave functionsψ^ε_k, solutions to (1.2, 1.3) with initial data in Y˜. Moreover, we assume that

R²

k∈Z²

λ_k|ψ_k^ε(0, x)|²dx=

R²

θ^ε(x)dx, (1.7)

R²|∇φ^ε(0, x)−(^⊥J⁰(x))|²dx^ε,h→0−→ 0, (1.8)

ε

k∈Z²

λ_k

R²

h∇ψ^ε_k(0, x) +i

⊥x

2εψ^ε_k(0, x)

^2ε,h→0−→ 0. (1.9)

Then, up to a subsequence, (−^⊥∇φ^ε) converges in C⁰([0, T];D(R²)) to a dissipative solution of the Euler equations withJ⁰ as initial datum.

Remark 1.3. As in [2], we consider thatJ is a dissipative solution of the Euler equations withJ⁰as the initial datum if, for all smooth divergence free compactly supported vectorv(t, x), and for almost every time,

|J(t, x)−v(t, x)|²dx≤

|J⁰(x)−v(0, x)|²dxexp _t

0 2||d(v(θ))||dθ

+ 2 _t

0 exp _t

s

2||d(v(θ))||dθ A(v)(s, x)·(v−J)(s, x)dx

ds, (1.10)

where d(v) is the symmetric part of Dv = ∂_jv_i, ||d(v(t))|| is the supremum on x of the spectral radius of d(v)(t, x) and

A(v) =∂_tv+v· ∇v. (1.11)

In the second section, we establish the equations which are satisfied by the density, the current and the energy.

Then, in Section 3, we give the proof of Theorem 1.2, which is mainly based on the modulated energy method introduced in [2]. Finally, we build an example of well-prepared initial data.

2. Equations of conservation 2.1. Conservation of mass and total momentum

We define the current ˜J^ε from the densityρ^ε using the equation

∂_tρ^ε+ div ˜J^ε= 0, (2.1)

and we obtain

J˜^ε=J^ε+

⊥x

2ερ^ε (2.2)

withJ^εdefined by

J^ε(t, x) =

k

λ_kIm

h∇ψ_k^ε(t, x)ψ^ε_k(t, x)

. (2.3)

Equation (2.3) is the classical form of the current when there is no magnetic field (see [10] for instance). It is important to note that the classical relation∂_tρ^ε+ divJ^ε= 0 does not hold here.

Remark 2.1. The difference between J^ε and ˜J^ε must be understood like the difference between impulsion and total momentum. Indeed, ˜J^ε corresponds to a real physical quantity (cf. [4]). Assumption (1.9) can be interpreted as an hypothesis on the initial velocity or kinetic energy.

Proposition 2.2. The currentJ˜^ε satisfies the following equation in the distributional sense

∂_tJ˜^ε =−

k

λ_k∇:

h∇ψ_k^ε+i

⊥x 2εψ_k^ε

⊗

h∇ψ^ε_k−i

⊥x 2εψ^ε_k

+1

ε(−ρ^ε∇φ^ε+^⊥J˜^ε)

+

k

λ_k∇ h²

4

∆ψ^ε_kψ^ε_k+ ∆ψ^ε_kψ^ε_k+ 2∇ψ_k^ε· ∇ψ^ε_k

,

(2.4)

which means

v·∂_tJ˜^ε=

k

λ_k

dv:

h∇ψ_k^ε+i

⊥x 2εψ^ε_k

⊗

h∇ψ^ε_k−i

⊥x 2εψ^ε_k

+1

ε

v·(−ρ^ε∇φ^ε+^⊥J˜^ε)

for all divergence free test function v∈ D(]0, T[×R²).

Remark 2.3. We use the compact notations

A⊗B=A_iB_j , (∇:B)i=∂_jB_ij , A⊗B=1 2

A⊗B+A⊗B .

Proof. We compute ∂_tJ˜^εusing (1.2) and obtain

∂_tJ˜^ε =

k

λ_k

∇ h²

4

∆ψ^ε_kψ^ε_k+ ∆ψ^ε_kψ^ε_k+ 2∇ψ_k^ε· ∇ψ^ε_k

−h²∇:

∇ψ^ε_k⊗∇ψ^ε_k

+ 1 2ε

⊥J˜^ε− 1 2ε

_⊥ x· ∇

J^ε−1

ε∇φ^ερ^ε+

⊥x 2ε∂_tρ^ε.

(2.5)

Moreover, we define

I₁=

k

λ_k∇:

h∇ψ^ε_k+i

⊥x 2εψ_k^ε

⊗

h∇ψ^ε_k−i

⊥x 2εψ^ε_k

.

It follows that

I₁=

k

λ_k∇:

h∇ψ^ε_k⊗h∇ψ^ε_k

+1 2∇:

_⊥ x

ε ⊗J^ε+J^ε⊗^⊥x ε

+ 1

4ε²∇:_⊥

x⊗^⊥xρ^ε .

Then we have

I₁ =

k

λ_k∇:

h∇ψ^ε_k⊗h∇ψ^ε_k +

⊥x

2εdivJ^ε+ 1 2ε

⊥J^ε+ 1

2ε(^⊥x· ∇)J^ε− 1 4ε²xρ^ε+

⊥x 2ε

_⊥ x 2ε · ∇ρ^ε

=

k

λ_k∇:

h∇ψ^ε_k⊗h∇ψ^ε_k + 1

2ε

⊥J^ε+ 1

2ε(^⊥x· ∇)J^ε+ 1 2ε

⊥_⊥ x 2ε

ρ^ε−^⊥x 2ε∂_tρ^ε,

because the conservation of mass (2.1) implies

⊥x 2ε

divJ^ε+

⊥x 2ε · ∇ρ^ε

=−^⊥x 2ε∂_tρ^ε. This computation gives us equation (2.4) from (2.5) because

∂_tJ˜^ε+

k

λ_k∇:

h∇ψ^ε_k+i

⊥x 2εψ_k^ε

⊗

h∇ψ^ε_k−i

⊥x 2εψ^ε_k

=∂_tJ˜^ε+

k

λ_k∇:

h∇ψ^ε_k⊗h∇ψ^ε_k + 1

2ε

⊥

J^ε+ 1 2ε

_⊥ x· ∇

J^ε+ 1 2ε

⊥_⊥ x 2ε

ρ^ε−^⊥x 2ε∂_tρ^ε

= 1 ε

−ρ^ε∇φ^ε+^⊥J˜^ε

+

k

λ_k∇ h²

4

∆ψ_k^εψ^ε_k+ ∆ψ^ε_kψ_k^ε+ 2∇ψ^ε_k· ∇ψ^ε_k

.

2.2. Conservation of energy

The energy of the system is the following H˜^ε(t) =

k

λ_k1 2ε

R²

h∇ψ_k^ε(t, x) +i

⊥x

2εψ_k^ε(t, x)

²dx+1 2

R²|∇φ^ε(t, x)|²dx, (2.6) and satisfies:

Proposition 2.4. Energy H^ε remains constant with respect to time.

The proof of this proposition is given in Section 5.

3. Proof of Theorem 1.2 3.1. Definition of the modulated energy

Proposition 3.1. For all divergence free test function v, we define a modulated energy

H˜_v^ε(t) =

k

λ_kε

2 h∇ψ^ε_k+ i 2ε

⊥xψ^ε_k−ivψ_k^ε h∇ψ^ε_k− i 2ε

⊥xψ^ε_k+ivψ^ε_k

+1 2

|∇(φ^ε−Ψ)|² (3.1)

with ^⊥∇Ψ =v. The function H˜_v^ε(t) satisfies d

dtH˜_v^ε(t) =−ε

k

λ_kdv:

h∇ψ_k^ε+ i 2ε

⊥xψ_k^ε−ivψ_k^ε

⊗

h∇ψ^ε_k− i 2ε

⊥xψ^ε_k+ivψ^ε_k

+

dv:∇(φ^ε−Ψ)⊗ ∇(φ^ε−Ψ)

+ε

A(v)·(ρ^εv−J˜^ε) +

θ^εv∇φ^ε+

A(v)·(v+^⊥∇φ^ε).

(3.2)

Proof. Differentiating the modulated energy (3.1) with respect to time, we obtain d

dtH˜_v^ε(t) =

k

λ_k d dt

ε 2h²

∇ψ^ε_k+ i 2ε

⊥xψ^ε_k ²−hε

2iv·

∇ψ^ε_kψ^ε_k−ψ^ε_k∇ψ^ε_k

−v· ^⊥x 2 ρ^ε

+ ε

2|v|²ψ^ε_kψ^ε_k+1

2|∇φ^ε|²− ∇φ^ε· ∇Ψ +1 2|∇Ψ|²

.

It can be expressed in term of the energy ˜H^εin the following way d

dtH˜_v^ε(t) = d

dtH˜^ε+

k

λ_k d

dt −hε

2iv·

∇ψ^ε_kψ^ε_k−ψ^ε_k∇ψ^ε_k

−v·^⊥x 2 ρ^ε+ε

2|v|²ψ_k^εψ^ε_k− ∇φ^ε· ∇Ψ +1 2|∇Ψ|²

.

Thanks to Proposition 2.4, we know that the energy is conserved. Therefore, we obtain the following simplified form for the time derivative of the modulated energy ˜H_v^ε

d

dtH˜_v^ε(t) = d dt

−εJ˜^ε·v+ε

2|v|²ρ^ε− ∇φ^ε· ∇Ψ +1 2|∇Ψ|²

= ε

2|v|²∂_tρ^ε+ε 2

ρ^ε∂_t|v|²−ε

v;∂_tJ˜^ε

D,D−ε

J˜^ε·∂_tv

+1 2

∂_t|v|²−

∇φ^ε·∂_t∇Ψ− ∇Ψ;∂_t∇φ^ε_D,D,

since^⊥∇Ψ =v.

We compute each term of this identity. First of all

−∇Ψ;∂_t∇φ^εD,D =

∂_t∆φ^εΨ =−

∂_tρ^εΨ

=

div ˜J^εΨ =−

J˜^ε· ∇Ψ.

Furthermore, because of equation (2.4) satisfied by∂_tJ˜^ε, we obtain I₃=−ε

v;∂_tJ˜^ε

D,D =−ε

k

λ_kdv:

h∇ψ_k^ε+ i 2ε

⊥xψ_k^ε

⊗

h∇ψ^ε_k− i 2ε

⊥xψ^ε_k

+

ρ^εv· ∇φ^ε−

v·^⊥J˜^ε, that we transform, using ^⊥∇Ψ =vand the Poisson equation (1.3), into

I₃ =−ε

k

λ_kdv:

h∇ψ_k^ε+ i 2ε

⊥xψ_k^ε

⊗

h∇ψ^ε_k− i 2ε

⊥xψ^ε_k

−

∆φ^εv· ∇φ^ε+

θ^εv∇φ^ε+

J˜^ε· ∇Ψ.

Moreover,

−

∆φ^εv· ∇φ^ε =

∂_j∂_jφ^εv_i∂_iφ^ε=

∂_jφ^ε∂_jv_i∂_iφ^ε+∂_jφ^εv_i∂_j∂_iφ^ε

=

dv:∇φ^ε⊗ ∇φ^ε. We have then

I₃ =−ε

k

λ_kdv:

h∇ψ_k^ε+ i 2ε

⊥xψ_k^ε

⊗

h∇ψ^ε_k− i 2ε

⊥xψ^ε_k

+

dv:∇φ^ε⊗ ∇φ^ε+

θ^εv∇φ^ε+

J˜^ε· ∇Ψ.

(3.3)

To obtain equation (3.2), we must study the modulation by v=^⊥ ∇Ψ and by ∇Ψ in the two terms of energy H˜^εgiven by (2.6). Let us study the first term

−ε

k

λ_kdv:

h∇ψ_k^ε+ i 2ε

⊥xψ_k^ε

⊗

h∇ψ^ε_k− i 2ε

⊥xψ^ε_k

=−ε

k

λ_kDv:

h∇ψ^ε_k+ i 2ε

⊥xψ^ε_k

⊗

h∇ψ^ε_k− i 2ε

⊥xψ^ε_k

=−ε

k

λ_kDv:

h∇ψ^ε_k+ i 2ε

⊥xψ^ε_k−ivψ_k^ε

⊗

h∇ψ^ε_k− i 2ε

⊥xψ^ε_k+ivψ^ε_k

−ε

Dv: ˜J^ε⊗v−ε

Dv:v⊗J˜^ε+ε

ρ^εDv:v⊗v.

The conservation of mass ensures that

−ε

Dv:v⊗J˜^ε =−ε

∂_jv_iJ˜_j^εv_i= ε 2

div ˜J^ε|v|²

=−ε 2

∂_tρ^ε|v|². Now, we can use the results shown by Brenier in [2] to obtain

dv:∇φ^ε⊗ ∇φ^ε =

dv:∇(φ^ε−Ψ)⊗ ∇(φ^ε−Ψ) +

Dv:∇Ψ⊗ ∇φ^ε

+

Dv:∇φ^ε⊗ ∇Ψ−

Dv:∇Ψ⊗ ∇Ψ.

In [2], it is also shown that ε

2

ρ^ε∂_t|v|²−ε

J˜^ε·∂_tv−ε

Dv: ˜J^ε⊗v+ε

ρ^εDv:v⊗v=ε

A(v)·

ρ^εv−J˜^ε

,

1 2

∂_t|v|²−

∇φ^ε·∂_tΨ +

Dv:∇Ψ⊗ ∇φ^ε+

Dv:∇φ^ε⊗ ∇Ψ−

Dv:∇Ψ⊗ ∇Ψ =

A(v)·(v+^⊥∇φ^ε) so we obtain (3.2) introducing the modulation in (3.3).

Remark 3.2. Since the functions Ψ andv are test functions with compact support, the integrations by parts do not give any boundary term. And they are well defined because ρ^ε(t,·), ∂tρ^ε(t,·),div ˜J^ε(t,·) are in L¹(R²), J˜^ε(t,·)∈L¹(R²,R²),∇φ^ε(t,·)∈L^∞(R²,R²) and ∆φ^ε(t,·)∈L¹(R²).

3.2. Convergence to the Euler equations

We prove in this section that (−^⊥∇φ^ε) converges in C⁰([0, T],D(R²,R²)) to a dissipative solution of the Euler equations. We first establish a result about the convergence and then we perform some estimates involving the modulated energy to identify the limit.

3.2.1. Convergence

Proposition 3.3. The sequence (ρ^ε)is compact inC⁰([0, T];D(R²)),( ˜J^ε)is bounded inD(]0, T[×R²),(H_v^ε) converges, up to a subsequence, in L^∞([0, T])−w∗ and(∇φ^ε)converges in C⁰([0, T];D(R²;R²)).

Let us first note that the conservation of energy implies that ˜H^ε(t) = ˜H^ε(0) and assumptions (1.9) and (1.8) imply that ( ˜H^ε(0)) converges. Thus, there existsCindependent ontandεsuch that ˜H^ε(t)≤C. This inequality implies that (∇φ^ε) is bounded independently onεin L^∞([0, T];L²(R²,R²)) and then (ρ^ε−θ^ε) is bounded in L^∞([0, T];H⁻¹(R²)) since, for all test functionv, we have

(ρ^ε(t, x)−θ^ε(t, x))v(x)dx =

−

∆φ^ε(t, x)v(x)dx

≤C||∇v||L²(R²).

Remark 3.4. The densityρ^εgiven by (1.1) satisfies

ρ^ε(t, x)dx=

ρ^ε(0, x)dxbut

ρ^ε(0, x)dx−→ ∞when εgoes to zero.

Let us now show that (ε¹²J˜^ε) is bounded inL^∞([0, T]; (W^1,4(R²,R²)∩L²(R²,R²))).

Indeed, since ˜J^ε=

k

λ_kIm

h∇ψ^ε_k+i

⊥x 2εψ_k^ε

ψ^ε_k

, we have

ε¹²J˜^ε·vdx ≤

ε

k

λ_k|h∇ψ_k^ε+i

⊥x 2εψ_k^ε|²

¹₂

k

λ_k|ψ_k^ε|²|v|^{2 1}₂

≤C

||v||²_W1,4+||v||²_L2

¹₂ .

Furthermore, the Poisson equation (1.3), which links ρ^ε and φ^ε, implies that (ρ^ε−θ^ε)∇φ^ε is bounded in L^∞([0, T]; (W^m,p(R²,R²))) withm−1>²_p because for allg∈W^m,p(R²,R²),

g(x)·(ρ^ε(t, x)−θ^ε(t, x))∇φ^ε(t, x)dx =

−

∆φ^ε(t, x)g(x)· ∇φ^ε(t, x)dx

=

(∇φ^ε(t, x)· ∇)g(x)· ∇φ^ε(t, x)dx

− 1

2|∇φ^ε(t, x)|²divg(x)dx

≤C||g||_C¹_(R²₎≤C||g||W^m,p, recalling thatm−1≥2/p. Additionally, we have

∂_t(ρ^ε−ε^⊥∇ ·J˜^ε) +^⊥∇ ·(ρ^ε∇φ^ε) =

k

λ_kε^⊥∇ ·

∇x:

h∇ψ^ε_k+i

⊥x 2εψ_k^ε

⊗

h∇ψ^ε_k−i

⊥x 2εψ^ε_k

. (3.4)

Since ((ρ^ε−θ^ε)∇φ^ε) is bounded inL^∞([0, T]; (W^m,p(R²,R²))) and since the energy is conserved, identity (3.4) implies that

∂_t

ρ^ε−ε^⊥∇.J˜^ε

is bounded in L^∞([0, T]; (W^m+1,p(R²)∩H¹(R²))) and then, using Aubin’s

Lemma locally (cf.appendices of [7] and [6]), (ρ^ε−ε^⊥∇ ·J˜^ε) is compact inC⁰([0, T];D(R²)). Sinceε^⊥∇ ·J˜^ε= O(ε¹²) in C⁰([0, T];D(R²)) we show that (ρ^ε) is compact in C⁰([0, T];D(R²)). Sinceρ^ε=θ^ε−∆φ^ε, (∇φ^ε) is also compact and then, up to a subsequence, (∇φ^ε) converges inC⁰([0, T];D(R²,R²)). The modulated energy ( ˜H_v^ε) is bounded inL^∞([0, T]) and then, up to a subsequence, converges in L^∞([0, T])−w∗.

This concludes the proof of Proposition 3.3.

3.2.2. Identification of the limit

Inequality (3.2) can be written in a weak form in the following way

−

H˜_v^ε(t)z(t)dt−z(0) ˜H_v^ε(0)≤

2||dv||H˜_v^ε(t)z(t)dt+ εA(v)(ρ^εv−J˜^ε)(t, x)

z(t)dtdx

+ A(v)

v+^⊥∇φ^ε

(t, x) +vθ^ε∇φ^ε(t, x)

z(t)dtdx, wherez is a test function belonging toD([0, T[).

As soon as ε is small enough, θ^ε = 1 on the support of v. We can pass to the limit because ( ˜H_v^ε) con- verges in L^∞([0, T])−w∗, (ρ^ε) and ( ˜J^ε) are bounded in the sense of distributions and (∇φ^ε) converges in C⁰([0, T];D(R²,R²)).

Then for all test functionz∈ D([0, T[), we have

−

H˜_v(t)z(t)dt−z(0) ˜H_v,0≤

2||dv||H˜_v(t)z(t)dt+

A(v)

v+^⊥∇φ

(t, x)z(t)dtdx, with ˜H_v,0= lim_ε→0H˜_v^ε(0).

Using Gronwall’s Lemma, we obtain

|^⊥∇(φ)−^⊥∇(Ψ) |²_L2(R²)≤2 ˜H_v(t)≤2 ˜H_0,vexp _t

0 2||dv(θ)||dθ

+ 2 _t

0 exp _t

s

2||dv(θ)||dθ

A(v)(s, x)·(v+^⊥∇φ)(s, x)dx

ds.

Since assumptions (1.8, 1.9) hold, we have ˜H_v,0= lim

ε→0

1

2|∇φ^ε(0, x)− ∇Ψ(0, x)|²dx.

Moreover, we have

||^⊥∇(φ)−^⊥∇(Ψ)||²_L2(R²) =||∇(φ)− ∇(Ψ)||²_L2(R²)≤lim inf||∇(φ^ε)− ∇(Ψ)||

≤lim inf ˜H_v^ε(t) = ˜H_v(t).

And then, if

|∇φ^ε(0, x)− ∇φ0(x)|²dx→0,−^⊥∇φis a dissipative solution of the Euler equations withJ⁰=

−^⊥∇φ0 as an initial condition.

3.3. Construction of an example of well-prepared initial data

To build an initial condition satisfying assumptions (1.8, 1.9), we consider the case whenω₀= curlJ⁰belongs to D(R²; ]−1,+1[) with

ω₀= 0, and we pick upεsmall enough so thatθ^ε= 1 on the support of ω₀. Letφ₀

be a solution inR²of

∆φ₀=ω₀, such thatJ⁰=−^⊥∇φ₀. Assumption (1.8) can be written as

|∇φ^ε(0, x)− ∇φ₀(x)|²dx→0.

We also have to assume the global neutrality at the initial time which can be written

θ^ε(x)dx=

ρ^ε(0, x)dx= 0.

To estimate

|∇φ^ε(0, x)− ∇φ₀(x)|²dx, we use a result from [1], stating that, if ∆φ=f with

f = 0, then

||∇φ||L² ≤C(|||x|f||L¹+||f||L²).

Writef(x) =θ^ε(x)−ρ^ε(0, x)−ω₀(x). If the global neutrality is satisfied, we have

f(x)dx= 0 and then

|∇φ^ε(0, x)− ∇φ0(x)|²dx≤C(|||x|f||L¹+||f||L²)². (3.5)

LetC_k be the elementary box defined byC_k=]y_k−η;y_k+η[² withy_k = (k₁η, k₂η),k∈Z²∩]−_ησ¹ε;_ησ¹_ε[², σ^ε being such that the support ofθ^εis included in I_σ^ε =]−_σ¹ε;_σ¹ε[² andλ_k=η²=|Ck|.

Letθ^ε,η andω₀^η be defined by

∀x∈C_k θ^ε,η(x) = 1

|Ck|

Ck

θ^ε(y)dy,

ω^η₀(x) = 1

|C_k|

Ck

ω₀(y)dy.

We take









ψ^ε,h_k (x) = (2πh)⁻²⁴

!

(θ^ε,η−ω₀^η)(y_k)exp

−(x−y_k)² 2h

exp

i

⊥y_k·x εh

if|x| ≤ 1 σ^ε,

ψ^ε,h_k (x) = 0 elsewhere.

Let us remark that, sinceθ^ε,η(x) = 1 on the support ofω₀, the function"

(θ^ε,η−ω₀^η) is well defined, since we assumed|ω0|<1. We then get











ρ^ε,h₀ (x) =

k

λ_k(2πh)⁻²²(θ^ε,η−ω^η₀)(y_k)exp

−(x−y_k)² h

if|x| ≤ 1 σ^ε ρ^ε,h₀ (x) = 0 elsewhere.