The Non-Relativistic Limit for the e-MHD Equations

We investigate the non-relativistic limit for the e-MHD equations in a three-dimension unit periodic torus. With the prepared initial data, our result shows that the small parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems (incompressible Euler equations) have smooth solutions. Moreover, the formal limit is rigorously justified.

The rigorous derivations of the e-MHD equation (1) from Vlasov-Maxwell system equations by a scaling limit and from Euler-Maxwell system by a quasineutral regime are obtained, respectively, in [2] and in [6]. In [7], the incompressible Euler equation (2) of ideal fluid from the e-MHD system (1) via a non-relativistic limit was gotten only in formal derivation. The aim of this work is to give a rigorous justification of the asymptotic limit using the energy method.
Let us recall that the non-relativistic limit → 0 or → ∞ has been investigated in a few fields. For instance, the limit → 0 has been performed in Vlasov-Poisson system in [8], in isentropic relativistic Euler equations [9], in a model system for multiple space dimensions radiation hydrodynamics in [10], and in Euler-Maxwell equations [11,12].
Now we recall some results on the Moser-type calculus inequalities in Sobolev spaces and the local existence of smooth solutions for symmetrizable hyperbolic equations for later use in this paper.
Lemma 1 (Moser-type calculus inequalities; see [13,14]). Let The Scientific World Journal is continuous and one has Recalling the classical result on the existence of sufficiently regular solutions of the incompressible Euler equations (2), we have the following regularity result about ( , ).
Lemma 2 (see [15,16]). Let 0 satisfy 0 ∈ ∞ and div 0 = 0. Then there exist 0 < * < ∞, the maximal existence time, and a unique smooth solution ( , ) of the incompressible Euler equations (2) on [0, * ) with initial datum 0 satisfying, for any 0 < * , This paper is organized as follows. In Section 2 we give some analytical backgrounds and the main result. The proof of the main result is given in Section 3.

Main Result
First we will state the existence of smooth local solutions for system (1) for the smooth initial data given by Proposition 3 (see [2,17]). Assume that ( 0 , 0 ) belongs to ∞ (Ω) and satisfies Then there exist 0 < < +∞, the maximal existence time, which depends only on the initial data, and a unique smooth For the convergence of the e-MHD system (1), our main result is stated as follows.

Proof of Theorem 4
Now we begin to justify the convergence of e-MHD equations to incompressible Euler equations when → 0. To this end, by the local existence theory and extension method, it suffices to obtain the uniform estimates of the smooth solutions to (1) with respect to the parameter so as to guarantee > ⋆ for any given 0 < ⋆ . Denote by = min{ ⋆ , } and by > 0 a constant which depends on 0 .

Derivation of Error Equations and
which satisfies the following problem: with the initial data Based on 2 -conservation of solutions to the e-MHD system, we obtain 2 estimates of the error function ( , , ). Our basic idea is to cancel the oscillations of the electric field and the magnetic field by using the special structures of the e-MHD system.

Lemma 5. For all 0 < < and sufficiently small , it holds that
Proof. Taking the 2 inner product of the first equation in the error system (12) for , by integration by parts, we get where (⋅, ⋅) stands for the 2 inner product of two scalar or vector functions in Ω. Now we estimate each term on the right-hand side of (15). For 1 , by using the equation = (1/ )∇ × in (12) and vector analysis formula we have For 2 and 3 , using the property of the approximate solution , Cauchy-Schwarz's inequality, and Minkowski's inequality, we have Combining (17) with (18), we have Multiplying the second equation in the error system (12) by , Cauchy-Schwarz's inequality, we get It follows from (19) and (20) that This completes the proof of Lemma 5.

High Order Energy Estimates.
We differentiate (12) with for a multi-index ∈ N 3 satisfying | | ≤ 0 with 0 > (3/ 2) + 1 to get that satisfies the following problem: Before performing the energy estimate, we set Lemma 6. For all 0 < < and sufficiently small , it holds that Proof. Taking the 2 inner product of the first equation in (23) with , one gets, by using the third equation in (23) and integration by parts, that By using Cauchy-Schwarz's inequality, the Moser-type calculus inequalities in Lemma 1, and Sobolev's lemma, we have that 4 The Scientific World Journal Combining (27) with (28) together, one gets Taking the 2 inner product of the second equation in (23) with , one gets, by using Cauchy-Schwarz's inequality, Combining (29) and (30), one gets which yields (26).

The End of Proof of Theorem 4. Now we let
Then it follows from Lemmas 5 and 6 that there exists 0 > 0, depending only on 0 , such that, for any 0 < ≤ 0 and any 0 < < , Since ( = 0) ≤ 2 for some positive constant , now applying Gronwall's inequality to (33), one can conclude that there exists an 0 > 0 sufficiently small such that, for any 0 < ≤ 0 and any 0 < < , Thus, using the a priori estimate (10), by the extension argument, we can conclude that > 0 for any 0 < ⋆ . The proof of Theorem 4 is complete.