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Scaling limits of nonisentropic EulerMaxwell equations for plasmas
Advances in Difference Equations volume 2011, Article number: 22 (2011)
Abstract
In this paper, we will discuss asymptotic limit of nonisentropic compressible EulerMaxwell 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 nonisentropic compressible EulerMaxwell system to the compressible EulerPoisson equations will be given via the nonrelativistic regime.
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 threedimensional position vector and of the time t > 0, where 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 nonisentropic EulerMaxwell system for plasma physics in a uniform background of nonmoving ions with fixed density b(x) (see [1–3]):
In the system, Equations 1.11.3 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 nondimensionalized parameters γ and ε can be chosen independently on each other, according to the desired scaling. Physically, γ and ε are proportional to and the Debye length, where c is the speed of light. Thus, the limit γ → 0 is called the nonrelativistic limit while the limit ε → 0 is called the quasineutral limit. Starting from one fluid and nonisentropic EulerMaxwell system, we can derive some different limit systems according to the corresponding different scalings.
Case 1: Nonrelativistic limit, Quasineutral limit
In this case, we first perform nonrelativistic limit and then quasineutral limit.
Step 1: Let ε be fixed and γ → 0. Formally, we get the following system:
This limit is the EulerPoisson 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 with respect to x. Furthermore, equation ∇ × E = 0 in (1.11) with m(E) = 0 implies the existence of a potential function ϕ ^{0} such that
Step 2: Set ε = 0 in EulerPoisson system (1.7)(1.11), we can obtain n  b(x) = 0, which is socalled quasineutrality 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 nonisentropic incompressible Euler equations of ideal fluid:
Hence, one derives the nonisentropic incompressible Euler equations.
Case 2: Quasineutral limit, Nonrelativistic limit
In this case, we take b(x) = 1 for simplicity. Contrarily to Case 1, we first perform quasineutral limit and then nonrelativistic 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 EulerMaxwell system (1.1)(1.5) that
This is socalled the nonisentropic eMHD equations.
Step 2: We set γ = 0 and get that
and the nonisentropic incompressible Euler equations 1.161.18 of ideal fluid from the eMHD system (1.18)(1.21).
Case 3: Combined quasineutral and nonrelativistic 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 (quasineutrality) and
Then one gets the nonisentropic incompressible Euler equations 1.151.17 of ideal fluid from the EulerMaxwell 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 [4–6]. 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 multiplescale 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 [4–6] 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)
The leading profiles (n ^{0}, u ^{0}, θ ^{0}, E ^{0}, B ^{0}) satisfy the following equations:
(2.2)(2.3)(2.4)(2.5)(2.6)(2.7)
From (2.6), we may take B ^{0} = 0, and equation ∇ × E ^{0} = 0 in (2.5) implies the existence of a potential function ϕ ^{0} such that E ^{0} = ∇ϕ ^{0}. Then Equations 2.22.5 become a nonisentropic compressible EulerPoisson system and determine a unique smooth solution (n ^{0}, u ^{0}, ϕ ^{0}) in the class m(ϕ ^{0}) = 0 well defined on with T _{ * } > 0. Here, we need the following compatibility conditions on (E _{0}, B _{0}):
where ϕ _{0} satisfies

(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≤k≤j1}are smooth and already determined in previous steps. Then (n ^{j} , u ^{j} , θ ^{j} , E ^{j} , B ^{j} ) satisfy the following linearized equations:
(2.10)(2.11)(2.12)(2.13)(2.14)(2.15)
where f ^{0} = 0, and f ^{j1}((n ^{k} , θ ^{k})_{ k≤j1}) is defined by
Equations 2.14 are of curldiv type and they determine a unique smooth B ^{j} in the class m(B ^{j} ) = 0 in . 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 }ω^{j1}) = 0. It follows that there is a potential function ϕ ^{j} such that E ^{j} = ∂_{ t }ω^{j1} ∇ϕ ^{j} with ω ^{0} = 0.
Then, (n ^{j} , u ^{j} , θ ^{j} , ϕ ^{j} ) solve a compressible linearized EulerPoisson system:
where g ^{j1}= f ^{j1}+∂_{ t }ω ^{j1}. Then system (2.16)(2.20) determines a unique smooth solution (n ^{j}, u ^{j}, θ ^{j}, ϕ ^{j})_{ j≥1}in the class m(ϕ ^{j}) = 0, in the time interval [0, T _{*}]. Since E ^{j}= ∂_{ t } ω ^{j1} ∇ϕ ^{j}, we need the following compatibility conditions on (E _{ j }, B _{ j }):
where ϕ _{ j } is determined by
for and m(ϕ ^{j} ) = 0.
Proposition 2.1. Assume that the initial data (n _{ j }, u _{ j } , E _{ j } , B _{ j } )_{ j<0} are sufficiently smooth with n _{0} > 0 in 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 nonrelativistic limit γ → 0 of the nonisentropic compressible EulerMaxwell system (1.1)(1.6) is the nonisentropic compressible EulerPoisson system.
Set
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 EulerMaxwell system (1.1)(1.6), our main result is stated as follows.
Theorem 2.1. For any fixed integer 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:

,

n _{0}, θ _{0} ≥ δ > 0 for some constant δ,

m(b(x)  n _{0}) = m(n _{ j }) = 0, j ≥ 1
with N = m and s = m + s _{0} + 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
Furthermore,
where (n ^{j} , u ^{j} , θ ^{j} , E ^{j} , B ^{j} )_{0≤j≤m } are solutions to problems and C > 0 is a constant independent of γ.
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Acknowledgements
The authors cordially acknowledge partial support from the Research Initiation Project for Highlevel Talents (no. 201035) of North China University of Water Resources and Electric Power.
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Yang, J., Gao, Q. & Zhang, Q. Scaling limits of nonisentropic EulerMaxwell equations for plasmas. Adv Differ Equ 2011, 22 (2011). https://doi.org/10.1186/16871847201122
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Keywords
 nonisentropic EulerMaxwell system
 asymptotic limit
 convergence