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 Open Access
Scaling limits of nonisentropic EulerMaxwell equations for plasmas
 Jianwei Yang^{1}Email author,
 Qinghua Gao^{2} and
 Qingnian Zhang^{1}
https://doi.org/10.1186/16871847201122
© Yang et al; licensee Springer. 2011
 Received: 22 January 2011
 Accepted: 18 July 2011
 Published: 18 July 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.
Keywords
 nonisentropic EulerMaxwell system
 asymptotic limit
 convergence
1 Introduction and the formal limits
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.
This limit is the EulerPoisson system of compressible electron 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.
This is socalled the nonisentropic eMHD equations.
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
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
 (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≤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:
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.
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.
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
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.
Declarations
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.
Authors’ Affiliations
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