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Zero-Mass-Electrons Limits in Hydrodynamic Models for Plasmas^{1}

Thierry Goudon

Laboratoire J.A.Dieudonné, CNRS UMR 6621, Université de Nice, F-06108 Nice cédex 2, France

                             goudon@math.unice.fr

Ansgar Jüngel

Fachbereich Mathematik, Technische Universität Berlin, Straβe des 17. Juni 136, D-10623 Berlin, Germany

                           jungel@math.tu-berlin.de

Yue-Jun Peng

Laboratoire de Mathématiques Appliquées, CNRS UMR 6620, Université Blaise Pascal, F-63177 Aubière cédex, France

                        peng@ucfma.univ-bpclermont.fr

Abstract : We study the zero-mass-electrons limits in the hydrodynamic models for plasmas consisting of electrons and ions. We prove, under suitable assumptions, that the evolution of the ions is governed by the Euler equations coupled with a nonlinear Poisson equation, as the mass of electrons tends to zero.

Keywords : Plasmas, Euler-Poisson system, Zero-electron-mass limits.

I. Introduction

In previous works, we introduced a hierarchy of hydrodynamic models for plasmas and proved rigorously the zero-mass-electrons limits in the drift-diffusion equations [4] and the zero-relaxation-time limits in the hydrodynamic equations [5]. In this paper, we study the zero-mass-electrons limits in the hydrodynamic equations.

Let us consider an unmagnetized plasma consisting of electrons with mass m_{e} and charge q_{e} = −1, and of a single species of ions with mass m_{i} and charge q_{i} = 1. We denote by n_{e}(t, x), j_{e}(t, x) (respectively, n_{i}(t, x), j_{i}(t, x)) the scaled density and electric current of electrons (respectively, ions), and by E(t, x) the scaled electric field, at time t > 0 and position x ∈ IR. For simplicity, set the scaled relaxation time τ_{α} and the scaled Debye length λ equal to 1, then this model is described by the following Euler-Poisson system (HD-EI) :

m_{α}∂_{t}n_{α} + m_{α}∂_{x}j_{α} = 0, t > 0, x ∈ IR,  (1)

m_{α}∂_{t}j_{α} + ∂_{x} (\frac{m_{α}j_{α}^{2}}{n_{α}} + p_{α}(n_{α})) = q_{α}n_{α}E − m_{α}j_{α}, t > 0, x ∈ IR,  (2)
∂_{x}E = n_{i} − n_{e}, t > 0, x ∈ IR,  (3)

subject to the initial and boundary conditions :

1Applied Mathematics Letters, 12 (1999) 75-79.

n_{α}(0, x) = n_{α0}(x), j_{α}(0, x) = j_{α0}(x), x ∈ IR, lim_{x→−∞} E(t, x) = E^{−}(t), t > 0,  (4)

where the subscripts α = e and α = i stand for electrons and ions respectively, and E^{−} is a given function. We assume the pressure functions are of the form :

p_{α}(n_{α}) = a_{α}^{2}n_{α}^{γ_{α}}, n_{α} ≥ 0,  (5)

where γ_{α} > 1 and a_{α} > 0 are constants.

Usually, the ions are heavy compared to the electrons, i.e. m_{i} ≫ m_{e}. Therefore, the problem of the zero-electron-mass limit of (1)-(3) is to study the asymptotic limit as m_{e} → 0 and m_{i} = 1 fixed. The formal limit of this problem is the model (HD-I) :

∂_{t}n + ∂_{x}j = 0, t > 0, x ∈ IR,  (6)

∂_{t}j + ∂_{x} (\frac{j^{2}}{n} + p(n)) = nE − j, p(n) = a^{2}n^{γ}, t > 0, x ∈ IR,  (7)
∂_{x}p_{e}(n_{e}) = −n_{e}E, ∂_{x}E = n − n_{e}, t > 0, x ∈ IR,  (8)

where we have used a simplified notation n = n_{i}, j = j_{i} and p = p_{i} etc. If we define the electric potential φ by −∂_{x}φ = E, then for n_{e} > 0, we can eliminate the variable n_{e} in (8) and obtain a nonlinear Poisson equation :

−∂_{xx}φ = n − f(φ), t > 0, x ∈ IR,  (9)

where the function f is defined by

q^{−1}(φ) := f(φ) ≥ 0, q′(n_{e}) = p′_{e}(n_{e})/n_{e}, n_{e} > 0,

and q^{−1} is the inverse function of q.

The existence of weak solutions to the system (1)-(3) and (6), (7), (9) has been proved in [10, 11]. The goal of this Note is to give a justification of the asymptotic limit (HD-EI) ⟶ (HD-I) under suitable assumptions. We observe that the system satisfied by (n_{e}, j_{e}) in (1)-(3) is similar to that treated in [5] for which an a priori L^{p} estimate is available. That is why we suppose below a L^{p} type estimate for (n_{e}, j_{e}) uniformly in m_{e}. The proof is based on the kinetic formulation, the monotonicity and compactness arguments.

II. Main results

For simplicity, we denote by ε = m_{e} > 0. Then the system (1)-(3) can be written as :

∂_{t}n^{ε} + ∂_{x}j^{ε} = 0, t > 0, x ∈ IR,  (10)

∂_{t}j^{ε} + ∂_{x} (\frac{(j^{ε})^{2}}{n^{ε}} + p(n^{ε})) = n^{ε}E^{ε} − j^{ε}, t > 0, x ∈ IR,  (11)
ε∂_{t}n_{e}^{ε} + ε∂_{x}j_{e}^{ε} = 0, t > 0, x ∈ IR,  (12)

ε∂_{t}j_{e}^{ε} + ∂_{x} (\frac{ε(j_{e}^{ε})^{2}}{n_{e}^{ε}} + p_{e}(n_{e}^{ε})) = −εj_{e}^{ε} − n_{e}^{ε}E^{ε}, t > 0, x ∈ IR,  (13)
∂_{x}E^{ε} = n^{ε} − n_{e}^{ε}, t > 0, x ∈ IR.  (14)

Let T > 0 and Q_{T} = [0, T [×IR. We assume that

(H1) 0 ≤ n_{α0}, n_{α0}, j_{α0}/n_{α0} ∈ L^{∞}(IR), E^{−} ∈ L_{loc}^{∞}(IR^{+}), supp n_{α0} ⊂IR, α = e, i,

(H2) the sequence (n_{e}^{ε})_{ε>0} is bounded in L_{loc}^{2γ_{e}}(Q_{T}) with γ_{e} > 1,

(H3) the sequence ((j_{e}^{ε})^{2}/n_{e}^{ε})_{ε>0} is bounded in L_{loc}^{2}(Q_{T}).

THEOREM. - Under the assumptions (H1)-(H3), there is a subsequence, still denoted by (n^{ε}, n_{e}^{ε}, j^{ε}, j_{e}^{ε}, E^{ε}), such that as ε → 0, (n^{ε}, n_{e}^{ε}, j^{ε}, j_{e}^{ε}, E^{ε}) converges to (n, n_{e}, j, j_{e}, E) in the following sense :

n^{ε} ⟶ n, j^{ε} ⟶ j, in L_{loc}^{p}(Q_{T}) strongly for any p ∈ [1, +∞[,  (15)
n_{e}^{ε} ⟶ n, in L_{loc}^{p}(Q_{T}) strongly for any p ∈ [1, γ_{e} + 1],  (16)
j_{e}^{ε} −⇀ j_{e}, in L_{loc}^{2}(Q_{T}) weakly,  (17)
E^{ε} −⇀ E, in L^{∞}(Q_{T}) weak − ∗,  (18)

where (n, n_{e}, j, E) is a weak solution of the problem (6)-(8).

REMARK. - A direct consequence of (H2)-(H3) is that (j_{e}^{ε})_{ε>0} is bounded in L_{loc}^{2}(Q_{T}), which implies (17). Hypothesis (H3) can be replaced by a weaker one :

(H3’) the sequences (j_{e}^{ε})_{ε>0} and (\sqrt{ε}(j_{e}^{ε})^{2}/n_{e}^{ε})_{ε>0} are bounded in L_{loc}^{2}(Q_{T}).

III. Proof of the Theorem

Under (15)-(18), the passage to the limit (HD-EI) ⟶ (HD-I) as ε → 0 is evident. According to the previous remark, it suffices to prove (15)-(16) and (18).

We first show that the sequence (E^{ε})_{ε>0} is bounded in L^{∞}(Q_{T}) which yields (18). Indeed, from (4) and (14), we have

E^{ε}(t, x) = E^{−}(t) + ∫_{−∞}^{x}(n^{ε}(t, y) − n_{e}^{ε}(t, y))dy.

Since j^{ε} and j_{e}^{ε} have compact support in IR_{x} and n_{e}^{ε} ≥ 0, n_{e}^{ε} ≥ 0 (see [10, 11]), integrating (10) and (12) over IR, we obtain

||n^{ε}(t, .)||_{L^{1}(IR)} = ||n_{i0} ||_{L^{1}(IR)}, ||n_{e}^{ε}(t, .)||_{L^{1}(IR)} = ||n_{e0} ||_{L^{1}(IR)} .

Hence

||E^{ε} ||_{L^{∞}(Q_{T})} ≤ ||E^{−} ||_{L^{∞}(0,T)} + ||n_{i0} ||_{L^{1}(IR)} + ||n_{e0} ||_{L^{1}(IR)} .

Next, we show the strong compactness of the sequence (n_{e}^{ε})_{ε>0}. By (H2), there is n_{e} ∈ L_{loc}^{2γ_{e}}(Q_{T}) such that for any p ∈]1, 2γ_{e}], up to a subsequence,

n_{e}^{ε} −⇀ n_{e}, in L_{loc}^{p}(Q_{T}) weakly, as ε → 0.

We define now two sequences U^{ε} = (n_{e}^{ε}, j_{e}^{ε})^{t} and V^{ε} = (ε(j_{e}^{ε})^{2}/n_{e}^{ε} + p_{e}(n_{e}^{ε}), −εj_{e}^{ε})^{t}. It is easy to see that (U^{ε})_{ε>0} and (V^{ε})_{ε>0} are all bounded in (L_{loc}^{2}(Q_{T}))^{2}. Since (εj_{e}^{ε} + n_{e}^{ε}E^{ε})_{ε>0} is bounded in L_{loc}^{2}(Q_{T}), by (12)-(13), (div_{t,x}U^{ε})_{ε>0} is bounded in L_{loc}^{2}(Q_{T}) and (curl_{t,x}V^{ε})_{ε>0} is bounded in (L_{loc}^{2}(Q_{T}))^{4}. Thus, the div-curl lemma [12] can be applied to (U^{ε}, V^{ε}) to obtain

lim_{ε→0} n_{e}^{ε}p_{e}(n_{e}^{ε}) = n_{e}\overline{p}, in D′(Q_{T}),  (19)

where

\overline{p} = lim_{ε→0} p_{e}(n_{e}^{ε}) in L_{loc}^{2}(Q_{T}) weakly.

Since s → p_{e}(s) is monotone increasing on IR^{+}, by monotonicity argument, we deduce (see the proof in the appendix)

\overline{p} = p_{e}(n_{e}).  (20)

Hence, (5) and (19)-(20) actually imply that

lim_{ε→0} ||n_{e}^{ε} ||_{L^{γe+1}(Ω)}=||n_{e} ||_{L^{γe+1}(Ω)}, ∀ Ω ⊂ Q_{T} bounded.

This proves (16).

As a consequence of (16) and (18), we have, passing if necessary to subsequences

n_{e}^{ε}E^{ε} −⇀ n_{e}E, in D′(Q_{T}), as ε → 0.

Now we prove (15). Since (E^{ε})_{ε>0} is bounded in L^{∞}(Q_{T}), from [8], the sequences (n^{ε})_{ε>0} and (j^{ε}/n^{ε})_{ε>0} are bounded in L^{∞}(Q_{T}), as limits of the strong convergence of the approximate solutions (n_{h}^{ε})_{h>0} and (j_{h}^{ε}/n_{h}^{ε})_{h>0}, constructed by the fractional step Lax-Friedrichs scheme or Godunov scheme with the space mesh size h > 0. Following [6], it remains to prove that for any pair of weak entropy-flux (η, q) of (10)-(11), the sequence (Λ^{ε})_{ε>0} := (∂_{t}η(n^{ε}, j^{ε})+∂_{x}q(n^{ε}, j^{ε}))_{ε>0} lies in a compact set of H_{loc}^{−1}(Q_{T}).

To this end, we apply the kinetic formulation to system (10)-(11). Let us denote by

χ^{ε}(t, x, ξ) = [(w_{2}^{ε} − ξ)(ξ − w_{1}^{ε})]_{+}^{λ}, ζ^{ε}(t, x, ξ) = (θξ + (1 − θ)u^{ε})χ^{ε}(t, x, ξ),

with u^{ε} = j^{ε}/n^{ε} and

w_{1}^{ε} = u^{ε} − \frac{2a\sqrt{γ}}{γ − 1} (n^{ε})^{θ}, w_{2}^{ε} = u^{ε} + \frac{2a\sqrt{γ}}{γ − 1} (n^{ε})^{θ}, λ = \frac{3 − γ}{2(γ − 1)}, θ = \frac{γ − 1}{2} .

Then (n^{ε}, j^{ε}) is a solution of (10)-(11) if and only if ([6, 7])

∂_{t}χ^{ε} + ∂_{x}ζ^{ε} = (n^{ε}E^{ε} − j^{ε})\frac{∂χ^{ε}}{∂j^{ε}} + \frac{∂^{2}μ^{ε}}{∂ξ^{2}} , in D′(Q_{T} × IR_{ξ}),

where (μ^{ε})_{ε>0} is a bounded sequence of non-positive measures defined on Q_{T} × IR_{ξ}. Due to (H1), (n^{ε}, j^{ε}) has compact support in x uniformly in ε. Therefore, μ^{ε} has compact support in x and ξ uniformly in ε.

Let g : IR ⟶ IR be an arbitrary smooth function. Then any pair of weak entropy-flux (η^{ε}, q^{ε}) = (η(n^{ε}, j^{ε}), q(n^{ε}, j^{ε})) of (10)-(11) can be written as :

η^{ε} = ∫_{IR} g(ξ)χ^{ε}(t, x, ξ)dξ, q^{ε} = ∫_{IR} g(ξ)ζ^{ε}(t, x, ξ)dξ.

By using \frac{∂χ^{ε}}{∂j^{ε}} = −\frac{1}{n^{ε}} \frac{∂χ^{ε}}{∂ξ}, we obtain

∂_{t}η^{ε} + ∂_{x}q^{ε} = (E^{ε} − u^{ε}) ∫_{IR} g′(ξ)χ^{ε}(t, x, ξ)dξ + < g′′, μ^{ε} > .  (21)

The first term in the right hand side of (21) is bounded in L^{∞}(Q_{T}) ⊂ compact set of H_{loc}^{−1}(Q_{T}), the second term is a bounded measure on Q_{T}. Since (η^{ε}, q^{ε})_{ε>0} is bounded in L^{∞}(Q_{T}), we conclude from Murat’s lemma [9] that (Λ^{ε})_{ε>0} is compact in H_{loc}^{−1}(Q_{T}). The proof of the Theorem is finished.

Appendix For the sake of completness, let us prove in this appendix the following

LEMMA. - Let Ω ⊂ IR^{N}. Let (ρ_{ε})_{ε>0} satisfy

ρ_{ε} ⇀ ρ weakly-* in L^{∞}(Ω), p(ρ_{ε}) ⇀ \overline{p} weakly-* in L^{∞}(Ω), p : IR ⟶ IR continuous and non decreasing, ρ_{ε} p(ρ_{ε}) ⇀ ρ \overline{p} weakly-* in D′(Ω)

Then, one has \overline{p} = p(ρ).

Let ψ ∈ L^{∞}(Ω). Since p is non decreasing, one gets

(p(ρ_{ε}) − p(ψ)) (ρ_{ε} − ψ) ≥ 0.

Let φ ∈ C_{0}^{∞}(Ω) be a non negative test function. Then, the integral

I_{ε} = ∫_{Ω}(p(ρ_{ε}) − p(ψ)) (ρ_{ε} − ψ) φ dx

is non negative. Furthermore, as ε goes to 0, one has

I_{ε} → ∫_{Ω}(\overline{p} − p(ψ)) (ρ − ψ) φ dx ≥ 0.

Let α > 0 and ζ ∈ L^{∞}(Ω). Choosing ψ = ρ − αζ yields

∫_{Ω}(\overline{p} − p(ρ − αζ)) ζ φ dx ≥ 0.

By using the continuity of p, and letting α → 0, one is led to

∫_{Ω}(\overline{p} − p(ρ)) ζ φ dx ≥ 0,

for any ζ ∈ L^{∞}(Ω) and φ ∈ C_{0}^{∞}(Ω), φ ≥ 0. One concludes that actually

∫_{Ω}(\overline{p} − p(ρ)) ζ φ dx = 0,

holds. Therefore \overline{p} = p(ρ).

Acknowledgments

The second author acknowledges partial support from the DAAD-PROCOPE Program, from the Deutsche Forschungsgemeinschaft, grant numbers MA 1662/ -1 and -2, and from the TMR Project “Asymptotic Methods in Kinetic Theory”, grant number ERB-4061-PL970-396. The research of the third author was partially supported by the TMR network of the Project “Hyperbolic Systems of Conservation Laws” (HCL), grant number ERB FMRX CT96 0033.

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