Scaling limits of non-isentropic Euler-Maxwell equations for plasmas
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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.
Keywordsnon-isentropic Euler-Maxwell system asymptotic limit convergence
1 Introduction and the formal limits
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.
This limit is the Euler-Poisson system of compressible electron 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.
This is so-called the non-isentropic e-MHD equations.
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
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 . For the other results, see [4, 5, 6] and references therein.
2 Rigorous convergence
- (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≤j-1are 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)
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.
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:
n0, θ0 ≥ δ > 0 for some constant δ,
m(b(x) - n0) = m(n j ) = 0, j ≥ 1
where (n j , u j , θ j , E j , B j )0≤j≤mare solutions to problems and C > 0 is a constant independent of γ.
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.
- 3.Dinklage A, Klinger T, Marx G, Schweikhard L: Plasma Physics. Lecture Notes in Physics. Volume 670. Springer, Berlin; 2005.Google Scholar
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