# The Non-Relativistic Limit for the e-MHD Equations

Authors: Hongli Wang; Jie Zhao
Journal: The Scientific World Journal
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

Let us consider the following so-called electron-magnetohydrodynamics (e-MHD) system [1, 2]:
(1)∂tu+(u·∇)u+E+γu×B=0,∂tB+1γ∇×E=0,u=-1γ∇×B,divB=0,
for (x,t)∈Ω×[0,T], where Ω=(ℝ/2π)3 is the 3-dimensional torus. The unknowns are u∈ℝ3, E∈ℝ3, and B∈ℝ3. In the above equations, the non-dimensionless parameter σ is proportional to 1/c, where c=(ϵ0ν0)-1/2 is the speed of light, with ϵ0 and ν0 being the vacuum permittivity and permeability. Let us notice that, as γ→0 in (1), we get ∇×E=0(⇒E=∇pI), ∇×B=0, divB=0(⇒B=0), and the incompressible Euler equations of ideal fluid:
(2)∂tuI+(uI·∇)uI+∇pI=0,divuI=0.The goal of this paper is to justify the above formal derivation of the incompressible Euler equations for periodic initial-value problems (IVPs) with an emphasis on three space dimensions. Indeed, it is well known that the phenomenon of non-relativistic is important in many physical situation involving various non-equilibrium processes. For instance, important examples occur in inviscid radiation hydrodynamics [3], quantum mechanics [4], Klein-Gordon-Maxwell system [5], and so on.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 limitγ→0 or c→∞ has been investigated in a few fields. For instance, the limit γ→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]).
Lets≥1 be an integer. Suppose u∈Hs(𝒯3), ∇u∈L∞(𝒯3), and v∈Hs-1(𝒯3)∩L∞(𝒯3). Then for all multi-indexes |α|≤s, one has (∂xα(uv)-u∂xαv)∈L2(𝒯3) and
(3)∥∂xα(uv)-u∂xαv∥≤Cs(∥∇u∥0,∞∥D|α|-1v∥+∥D|α|u∥∥v∥0,∞),
where
(4)∥Dhu∥=∑|α|=h‍∥∂xαu∥,∀h∈ℕ.
Moreover, if s≥3, then the embedding Hs-1(𝒯3)↪L∞(𝒯3) is continuous and one has
(5)∥uv∥s-1≤Cs∥u∥s-1∥v∥s-1,∥∂xα(uv)-u∂xαv∥≤Cs∥u∥s∥v∥s-1.Recalling the classical result on the existence of sufficiently regular solutions of the incompressible Euler equations (2), we have the following regularity result about (uI,pI).Lemma 2 (see [15, 16]).
Letu0I satisfy u0I∈C∞ and divu0I=0. Then there exist 0<T*<∞, the maximal existence time, and a unique smooth solution (uI,pI) of the incompressible Euler equations (2) on [0,T*) with initial datum u0I satisfying, for any T0<T*,
(6)(uI,pI)∈C∞(Ω×[0,T0)).This paper is organized as follows. In Section2 we give some analytical backgrounds and the main result. The proof of the main result is given in Section 3.


## 2. Main Result

First we will state the existence of smooth local solutions for system (1) for the smooth initial data given by
(7)u|t=0=u0,B|t=0=B0.Proposition 3 (see [2, 17]).
Assume that(u0,B0) belongs to C∞(Ω) and satisfies
(8)-∇×B0=γu0,divB0=0.
Then there exist 0<Tγ<+∞, the maximal existence time, which depends only on the initial data, and a unique smooth solution (u,B,E)∈C∞(Ω×[0,Tγ)) of the incompressible e-MHD equations (1) defined on [0,Tγ).For the convergence of the e-MHD system (1), our main result is stated as follows.Theorem 4.
Lets0∈ℕ with s0>(3/2)+1. Let u0I∈C∞(Ω) satisfy divu0I=0 and let (u0,E0,B0)∈C∞(Ω) satisfy (8). Assume that
(9)∥(u0-u0I,B0)∥Hs0(Ω)≤Cγ
for some positive constant C independent of γ. Let T⋆ be the maximal existence time of the smooth solution (uI,pI)∈C∞(Ω×[0,T⋆)) of the incompressible Euler equations (2). Then, for any T0<T⋆, there exist constants γ0(T0)>0 and C(T0)>0, depending only on T0 and the initial data, such that the e-MHD system (1) has a classical smooth solution (u,E,B), defined on [0,T0], satisfying
(10)∥(u-uI,B-γbI)∥Hs0(Ω)≤Cγ
for all 0<γ≤γ0 and 0<t≤T0, where the function bI satisfies -∇×bI=uI and divbI=0.


## 3. Proof of Theorem4

Now we begin to justify the convergence of e-MHD equations to incompressible Euler equations whenγ→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 γ so as to guarantee Tγ>T⋆ for any given T0<T⋆. Denote by T=min{T⋆,Tγ} and by C>0 a constant which depends on T0.


### 3.1. Derivation of Error Equations andL2 Estimate

Let(u,E,B) be the unknown solution to the problem (1) and let (uI,pI) be the solution to the incompressible Euler equations (2) defined on [0,T⋆) given by Proposition 3. Denote this by
(11)(U,F,G)=(u-u0,E-∇pI,B-γbI),
which satisfies the following problem:
(12)∂tU+(U+uI)·∇U=-F-U×(G+γbI)∂tU+(U+uI)·∇U=-uI×(G+γbI)-U·∇uI,∂tG-1γ∇×F=-γ∂tbI,U=1γ∇×G,divG=0
with the initial data
(13)(U,G)|t=0=(u0-u0I,B0-γbI(t=0)).Based onL2-conservation of solutions to the e-MHD system, we obtain L2 estimates of the error function (U,F,G). Our basic idea is to cancel the oscillations of the electric field F and the magnetic field G by using the special structures of the e-MHD system.Lemma 5.
For all0<t<T and sufficiently small γ, it holds that
(14)∫Ω(|U|2+|G|2)(t)dx≤∫Ω(|U|2+|G|2)(t=0)dx∫Ω‍(|U|2+|G|2)(t)dx≤+C∫0t(∥U∥2+∥G∥2)(s)ds∫Ω‍(|U|2+|G|2)(t)dx≤+Cγ2.Proof.
Taking theL2 inner product of the first equation in the error system (12) for U, by integration by parts, we get
(15)12ddt(U,U)=-(F,U)-uI×(G+γbI,U)-(U·∇uI,U)12ddt(U,U)=A1+A2+A3,
where (·,·) stands for the L2 inner product of two scalar or vector functions in Ω. Now we estimate each term on the right-hand side of (15).
ForA1, by using the equation U=(1/γ)∇×G in (12) and vector analysis formula
(16)div(f×g)=∇×f·g-∇×g·f,
we have
(17)A1=-1γ(F,∇×G)=-1γ(∇×F,G).
For A2 and A3, using the property of the approximate solution uI, Cauchy-Schwarz’s inequality, and Minkowski’s inequality, we have
(18)A2≤C(γ+∥G∥)∥U∥≤Cγ2+C(∥G∥2+∥U∥2),A3≤C∥U∥2.
Combining (17) with (18), we have
(19)12ddt(U,U)≤-1γ(∇×F,G)+C(∥U∥2+∥G∥2)+Cγ2.
Multiplying the second equation in the error system (12) by G, Cauchy-Schwarz’s inequality, we get
(20)12ddt(G,G)≤1γ(∇×F,G)+C∥G∥2+Cγ2.
It follows from (19) and (20) that
(21)ddt∫Ω(|U|2+|G|2)(t)dx≤C(∥U∥2+∥G∥2)+Cγ2.
This completes the proof of Lemma 5.


### 3.2.  High Order Energy Estimates

We differentiate (12) with ∂xα for a multi-index α∈ℕ3 satisfying |α|≤s0 with s0>(3/2)+1 to get that
(22)(Uα,Fα,Gα)=(∂xαU,∂xαF,∂xαG)
satisfies the following problem:
(23)∂tUα+(U+uI)·∇Uα=-Fα+∑i=14‍ℋ1i,∂tGα-1γ∇×Fα=-γ∂tbαI,Uα=1γ∇×Gα,divGα=0,
where
(24)ℋ11=(U+uI)·∇Uα-[(U+uI)·∇U]α,ℋ12=-[U×(G+γbI)]α,ℋ13=-[uI×(G+γbI)]α,ℋ14=-[U·∇uI]α.Before performing the energy estimate, we set(25)ℰs0(t)=∥(U,G)∥Hs0(Ω).Lemma 6.
For all0<t<T and sufficiently small γ, it holds that
(26)∫Ω‍(|Uα|2+|Gα|2)(t)dx≤∫Ω‍(|Uα|2+|Gα|2)(t=0)dx∫Ω‍(|Uα|2+|Gα|2)(t)dx≤+C∫0t‍(1+ℰs0(s))ℰs02(s)ds∫Ω‍(|Uα|2+|Gα|2)(t)dx≤+Cγ2.Proof.
Taking theL2 inner product of the first equation in (23) with Uα, one gets, by using the third equation in (23) and integration by parts, that
(27)12ddt(Uα,Uα)=-1γ(Fα,∇×G)+∑i=14‍(ℋ1i,Uα)12ddt(Uα,Uα)=-1γ(∇×Fα,Gα)+∑i=14‍(ℋ1i,Uα).
By using Cauchy-Schwarz’s inequality, the Moser-type calculus inequalities in Lemma 1, and Sobolev’s lemma, we have that
(28)(ℋ11,Uα)≤C(∥∇(U+uI)∥L∞∥D|α|-1∇U∥(ℋ11,Uα)≤+∥D|α|(U+uI)∥∥∇U∥Hs0-1)∥Uα∥(ℋ11,Uα)≤C(∥∇(U+uI)∥Hs0-1∥D|α|-1∇U∥(ℋ11,Uα)≤+∥D|α|(U+uI)∥∥∇U∥L∞)∥Uα∥(ℋ11,Uα)≤C(1+∥U∥Hs0)∥U∥Hs02(ℋ11,Uα)≤C(1+ℰs0(t))ℰs02(t),(ℋ12,Uα)≤C(1+ℰs0(t))ℰs02(t),(ℋ13,Uα)≤Cℰs02(t)+Cγ2,(ℋ14,Uα)≤Cℰs02(t).
Combining (27) with (28) together, one gets
(29)12ddt(Uα,Uα)≤-1γ(Fα,∇×Gα)+C(1+ℰs0(t))ℰs02(t)12ddt(Uα,Uα)≤+Cγ2.
Taking theL2 inner product of the second equation in (23) with Gα, one gets, by using Cauchy-Schwarz’s inequality,
(30)12ddt(Gα,Gα)=1γ(∇×Fα,Gα)-(γ∂tbαI,Gα)12ddt(Gα,Gα)≤1γ(∇×Fα,Gα)+C∥Gα∥2+Cγ212ddt(Gα,Gα)≤1γ(∇×Fα,Gα)+Cℰs02(t)+Cγ2.
Combining (29) and (30), one gets
(31)ddt∫Ω‍(|Uα|2+|Gα|2)(t)dx≤C(1+ℰs0(t))ℰs02(t)ddt∫Ω‍(|Uα|2+|Gα|2)(t)dx≤+Cγ2,
which yields (26).


### 3.3. The End of Proof of Theorem4

Now we let(32)z(t)=ℰs02(t).
Then it follows from Lemmas 5 and 6 that there exists γ0>0, depending only on T0, such that, for any 0<γ≤γ0 and any 0<t<T,
(33)z(t)≤Cz(t=0)+C∫0t((1+z)z)(s)ds+Cγ2.
Since z(t=0)≤Cγ2 for some positive constant C, now applying Gronwall’s inequality to (33), one can conclude that there exists an γ0>0 sufficiently small such that, for any 0<γ≤γ0 and any 0<t<T,
(34)z(t)≤Cγ2.
Thus, using the a priori estimate (10), by the extension argument, we can conclude that Tγ>T0 for any T0<T⋆.The proof of Theorem4 is complete.


## 3.1. Derivation of Error Equations andL2 Estimate

Let(u,E,B) be the unknown solution to the problem (1) and let (uI,pI) be the solution to the incompressible Euler equations (2) defined on [0,T⋆) given by Proposition 3. Denote this by
(11)(U,F,G)=(u-u0,E-∇pI,B-γbI),
which satisfies the following problem:
(12)∂tU+(U+uI)·∇U=-F-U×(G+γbI)∂tU+(U+uI)·∇U=-uI×(G+γbI)-U·∇uI,∂tG-1γ∇×F=-γ∂tbI,U=1γ∇×G,divG=0
with the initial data
(13)(U,G)|t=0=(u0-u0I,B0-γbI(t=0)).Based onL2-conservation of solutions to the e-MHD system, we obtain L2 estimates of the error function (U,F,G). Our basic idea is to cancel the oscillations of the electric field F and the magnetic field G by using the special structures of the e-MHD system.Lemma 5.
For all0<t<T and sufficiently small γ, it holds that
(14)∫Ω(|U|2+|G|2)(t)dx≤∫Ω(|U|2+|G|2)(t=0)dx∫Ω‍(|U|2+|G|2)(t)dx≤+C∫0t(∥U∥2+∥G∥2)(s)ds∫Ω‍(|U|2+|G|2)(t)dx≤+Cγ2.Proof.
Taking theL2 inner product of the first equation in the error system (12) for U, by integration by parts, we get
(15)12ddt(U,U)=-(F,U)-uI×(G+γbI,U)-(U·∇uI,U)12ddt(U,U)=A1+A2+A3,
where (·,·) stands for the L2 inner product of two scalar or vector functions in Ω. Now we estimate each term on the right-hand side of (15).
ForA1, by using the equation U=(1/γ)∇×G in (12) and vector analysis formula
(16)div(f×g)=∇×f·g-∇×g·f,
we have
(17)A1=-1γ(F,∇×G)=-1γ(∇×F,G).
For A2 and A3, using the property of the approximate solution uI, Cauchy-Schwarz’s inequality, and Minkowski’s inequality, we have
(18)A2≤C(γ+∥G∥)∥U∥≤Cγ2+C(∥G∥2+∥U∥2),A3≤C∥U∥2.
Combining (17) with (18), we have
(19)12ddt(U,U)≤-1γ(∇×F,G)+C(∥U∥2+∥G∥2)+Cγ2.
Multiplying the second equation in the error system (12) by G, Cauchy-Schwarz’s inequality, we get
(20)12ddt(G,G)≤1γ(∇×F,G)+C∥G∥2+Cγ2.
It follows from (19) and (20) that
(21)ddt∫Ω(|U|2+|G|2)(t)dx≤C(∥U∥2+∥G∥2)+Cγ2.
This completes the proof of Lemma 5.


## 3.2.  High Order Energy Estimates

We differentiate (12) with ∂xα for a multi-index α∈ℕ3 satisfying |α|≤s0 with s0>(3/2)+1 to get that
(22)(Uα,Fα,Gα)=(∂xαU,∂xαF,∂xαG)
satisfies the following problem:
(23)∂tUα+(U+uI)·∇Uα=-Fα+∑i=14‍ℋ1i,∂tGα-1γ∇×Fα=-γ∂tbαI,Uα=1γ∇×Gα,divGα=0,
where
(24)ℋ11=(U+uI)·∇Uα-[(U+uI)·∇U]α,ℋ12=-[U×(G+γbI)]α,ℋ13=-[uI×(G+γbI)]α,ℋ14=-[U·∇uI]α.Before performing the energy estimate, we set(25)ℰs0(t)=∥(U,G)∥Hs0(Ω).Lemma 6.
For all0<t<T and sufficiently small γ, it holds that
(26)∫Ω‍(|Uα|2+|Gα|2)(t)dx≤∫Ω‍(|Uα|2+|Gα|2)(t=0)dx∫Ω‍(|Uα|2+|Gα|2)(t)dx≤+C∫0t‍(1+ℰs0(s))ℰs02(s)ds∫Ω‍(|Uα|2+|Gα|2)(t)dx≤+Cγ2.Proof.
Taking theL2 inner product of the first equation in (23) with Uα, one gets, by using the third equation in (23) and integration by parts, that
(27)12ddt(Uα,Uα)=-1γ(Fα,∇×G)+∑i=14‍(ℋ1i,Uα)12ddt(Uα,Uα)=-1γ(∇×Fα,Gα)+∑i=14‍(ℋ1i,Uα).
By using Cauchy-Schwarz’s inequality, the Moser-type calculus inequalities in Lemma 1, and Sobolev’s lemma, we have that
(28)(ℋ11,Uα)≤C(∥∇(U+uI)∥L∞∥D|α|-1∇U∥(ℋ11,Uα)≤+∥D|α|(U+uI)∥∥∇U∥Hs0-1)∥Uα∥(ℋ11,Uα)≤C(∥∇(U+uI)∥Hs0-1∥D|α|-1∇U∥(ℋ11,Uα)≤+∥D|α|(U+uI)∥∥∇U∥L∞)∥Uα∥(ℋ11,Uα)≤C(1+∥U∥Hs0)∥U∥Hs02(ℋ11,Uα)≤C(1+ℰs0(t))ℰs02(t),(ℋ12,Uα)≤C(1+ℰs0(t))ℰs02(t),(ℋ13,Uα)≤Cℰs02(t)+Cγ2,(ℋ14,Uα)≤Cℰs02(t).
Combining (27) with (28) together, one gets
(29)12ddt(Uα,Uα)≤-1γ(Fα,∇×Gα)+C(1+ℰs0(t))ℰs02(t)12ddt(Uα,Uα)≤+Cγ2.
Taking theL2 inner product of the second equation in (23) with Gα, one gets, by using Cauchy-Schwarz’s inequality,
(30)12ddt(Gα,Gα)=1γ(∇×Fα,Gα)-(γ∂tbαI,Gα)12ddt(Gα,Gα)≤1γ(∇×Fα,Gα)+C∥Gα∥2+Cγ212ddt(Gα,Gα)≤1γ(∇×Fα,Gα)+Cℰs02(t)+Cγ2.
Combining (29) and (30), one gets
(31)ddt∫Ω‍(|Uα|2+|Gα|2)(t)dx≤C(1+ℰs0(t))ℰs02(t)ddt∫Ω‍(|Uα|2+|Gα|2)(t)dx≤+Cγ2,
which yields (26).


## 3.3. The End of Proof of Theorem4

Now we let(32)z(t)=ℰs02(t).
Then it follows from Lemmas 5 and 6 that there exists γ0>0, depending only on T0, such that, for any 0<γ≤γ0 and any 0<t<T,
(33)z(t)≤Cz(t=0)+C∫0t((1+z)z)(s)ds+Cγ2.
Since z(t=0)≤Cγ2 for some positive constant C, now applying Gronwall’s inequality to (33), one can conclude that there exists an γ0>0 sufficiently small such that, for any 0<γ≤γ0 and any 0<t<T,
(34)z(t)≤Cγ2.
Thus, using the a priori estimate (10), by the extension argument, we can conclude that Tγ>T0 for any T0<T⋆.The proof of Theorem4 is complete.



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