Scaling limits of non-isentropic Euler-Maxwell equations for plasmas

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Abstract

In this paper, we will discuss asymptotic limit of non-isentropic compressible Euler-Maxwell system arising from plasma physics. Formally, we give some different limit systems according to the corresponding different scalings. Furthermore, some recent results about the convergence of non-isentropic compressible Euler-Maxwell system to the compressible Euler-Poisson equations will be given via the non-relativistic regime.

Keywords

non-isentropic Euler-Maxwell system asymptotic limit convergence 

1 Introduction and the formal limits

Let n, u, and θ denote the scaled macroscopic density, mean velocity vector, and temperature of the electrons and E and B the scaled electric field and magnetic field, respectively. They are functions of a three-dimensional position vector Open image in new window and of the time t > 0, where Open image in new window is the torus. The fields E and B are coupled to the particles through the Maxwell equations and act on the particles via the Lorentz force E + γu × B. These variables satisfy the scaled non-isentropic Euler-Maxwell system for plasma physics in a uniform background of non-moving ions with fixed density b(x) (see [1, 2, 3]):

In the system, Equations 1.1-1.3 are the mass, momentum, and energy balance laws, respectively, while (1.4)-(1.5) are the Maxwell equations. It is well known that two equations in (1.5) are redundant with two equations in (1.4) as soon as they are satisfied by the initial conditions. The non-dimensionalized parameters γ and ε can be chosen independently on each other, according to the desired scaling. Physically, γ and ε are proportional to Open image in new window and the Debye length, where c is the speed of light. Thus, the limit γ → 0 is called the non-relativistic limit while the limit ε → 0 is called the quasi-neutral limit. Starting from one fluid and non-isentropic Euler-Maxwell system, we can derive some different limit systems according to the corresponding different scalings.

Case 1: Non-relativistic limit, Quasi-neutral limit

In this case, we first perform non-relativistic limit and then quasi-neutral limit.

Step 1: Let ε be fixed and γ → 0. Formally, we get the following system:

This limit is the Euler-Poisson system of compressible electron fluid.

Remark 1.1. Equations 1.10 implies B = 0 when the mean value of B(x, t) vanishes, i.e. m (B) = 0. Here,
denotes the mean value of a given scalar or vector function v(x, t) in Open image in new window with respect to x. Furthermore, equation ∇ × E = 0 in (1.11) with m(E) = 0 implies the existence of a potential function ϕ0such that
Step 2: Set ε = 0 in Euler-Poisson system (1.7)-(1.11), we can obtain n - b(x) = 0, which is so-called quasi-neutrality in plasma physics. Then, (u, θ, ϕ) satisfy the following equations:
Remark 1.2. If the ion density b(x) is a constant, say b(x) = 1 for simplicity; then from (1.12)-(1.14), we see that (u, θ, ϕ) satisfy the non-isentropic incompressible Euler equations of ideal fluid:

Hence, one derives the non-isentropic incompressible Euler equations.

Case 2: Quasineutral limit, Non-relativistic limit

In this case, we take b(x) = 1 for simplicity. Contrarily to Case 1, we first perform quasineutral limit and then non-relativistic limit.

Step 1: Let γ be fixed and ε turns to zero, one can gets n - 1 = 0 (quasineutrality), and then we get from the Euler-Maxwell system (1.1)-(1.5) that

This is so-called the non-isentropic e-MHD equations.

Step 2: We set γ = 0 and get that

and the non-isentropic incompressible Euler equations 1.16-1.18 of ideal fluid from the e-MHD system (1.18)-(1.21).

Case 3: Combined quasineutral and non-relativistic limits

Similarly to Case 2, we still take b(x) = 1 for simplicity. Choose ε = γ and let ε = γ → 0, first it is easy to get from the Maxwell system (1.4)-(1.5) that n = 1 (quasi-neutrality) and

Then one gets the non-isentropic incompressible Euler equations 1.15-1.17 of ideal fluid from the Euler-Maxwell system (1.1)-(1.5).

The above formal limits are obvious, but it is very difficult to rigorously prove them, even in isentropic case, see [4, 5, 6]. Since usually it is required to deal with some complex related problems such as the oscillatory behavior of the electric fields, the initial layer problem, the sheath boundary layer problem, and the classical shock problem. The proofs of these convergence are based on the asymptotic expansion of multiple-scale and the careful energy methods, iteration scheme, the entropy methods, etc. In the following section, we will provide a rigorous convergence result when ε is fixed (especially we take ε = 1) and γ → 0. We state our result in the following section. For detail, see [7]. For the other results, see [4, 5, 6] and references therein.

2 Rigorous convergence

Let (n γ , u γ , θ γ , E γ , B γ ) be the classical solutions to problem (1.1)-(1.6) and assume that the initial conditions have the following asymptotic expansion with respect to γ:
Plugging the following ansatz:
into system (1.1)-(1.6), we obtain:
  1. (1)
    The leading profiles (n 0, u 0, θ 0, E 0, B 0) satisfy the following equations:
     
From (2.6), we may take B0 = 0, and equation ∇ × E0 = 0 in (2.5) implies the existence of a potential function ϕ0 such that E0 = -∇ϕ0. Then Equations 2.2-2.5 become a non-isentropic compressible Euler-Poisson system and determine a unique smooth solution (n0, u0, ϕ0) in the class m(ϕ0) = 0 well defined on Open image in new window with T * > 0. Here, we need the following compatibility conditions on (E0, B0):
where ϕ0 satisfies
  1. (2)
    For any j ≥ 1, the profiles (n j , u j , θ j , E j , B j ) can be obtained by induction. Now, we assume that (n k , u k , θ k , E k , B k )0≤kj-1are smooth and already determined in previous steps. Then (n j , u j , θ j , E j , B j ) satisfy the following linearized equations:
     
where f0 = 0, Open image in new window and fj-1((n k , θ k )kj-1) is defined by

Equations 2.14 are of curl-div type and they determine a unique smooth B j in the class m(B j ) = 0 in Open image in new window . Moreover, from div B j = 0, we deduce the existence of a given vector function ω j such that B j = - ∇ × ω j . Then, the first equation in (2.13) becomes ∇ × (E j -∂ t ωj-1) = 0. It follows that there is a potential function ϕ j such that E j = ∂ t ωj-1- ∇ϕ j with ω0 = 0.

Then, (n j , u j , θ j , ϕ j ) solve a compressible linearized Euler-Poisson system:
where gj-1= fj-1+∂ t ω j-1. Then system (2.16)-(2.20) determines a unique smooth solution (n j , u j , θ j , ϕ j )j≥1in the class m(ϕ j ) = 0, in the time interval [0, T*]. Since E j = ∂ t ωj-1- ∇ϕ j , we need the following compatibility conditions on (E j , B j ):
where ϕ j is determined by

for Open image in new window and m(ϕ j ) = 0.

Proposition 2.1. Assume that the initial data (n j , u j , E j , B j )j<0are sufficiently smooth with n0 > 0 in Open image in new window and satisfy the compatibility conditions (2.8)-(2.9) and (2.21)-(2.22). Then there exists a unique asymptotic expansion up to any order of the form (2.1), i.e. there exist the unique smooth profiles (n j u j , E j , B j )j<0, solutions of the problems (2.2)-(2.7) and (2.10)-(2.15) in the time interval [0, T*]. In particular, the formal non-relativistic limit γ → 0 of the non-isentropic compressible Euler-Maxwell system (1.1)-(1.6) is the non-isentropic compressible Euler-Poisson system.

where (n j , u j , θ j , E j , B j ) are those constructed in the previous Proposition 1.1.

For the convergence of the compressible Euler-Maxwell system (1.1)-(1.6), our main result is stated as follows.

Theorem 2.1. For any fixed integer Open image in new window and m ≥ 1, assume that the mean values of E γ (x, t), B γ (x, t) vanish and the ion density b(x) the initial data (n j , u j , θ j )j≥0, satisfy the following conditions:

with N = m and s = m + s0 + 3 hold. Furthermore, if
satisfy the compatibility condition
and initial condition
then, there exists T* ∈ (0, T*] such that problem (1.1)-(1.6) has a unique solution

where (n j , u j , θ j , E j , B j )0≤jmare solutions to problems and C > 0 is a constant independent of γ.

Notes

Acknowledgements

The authors cordially acknowledge partial support from the Research Initiation Project for High-level Talents (no. 201035) of North China University of Water Resources and Electric Power.

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Copyright information

© Yang et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceNorth China University of Water Resources and Electric PowerZhengzhouPR China
  2. 2.Beijing City UniversityBeijingPR China

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