HALL-MHD SYSTEM AS A SIMPLIFIED
ONE-FLUID ION MODEL DERIVED FROM
TWO-FLUID EULER-MAXWELL EQUATIONS

Yue-Jun Peng, Shuai Xi, Liang Zhao

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Yue-Jun Peng, Shuai Xi, Liang Zhao. HALL-MHD SYSTEM AS A SIMPLIFIED ONE-FLUID ION MODEL DERIVED FROM TWO-FLUID EULER-MAXWELL EQUATIONS. 2025. [hal-04889318](https://uca.hal.science/hal-04889318v1)

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Distributed under a Creative Commons Attribution 4.0 International License
HALL-MHD SYSTEM AS A SIMPLIFIED ONE-FLUID ION MODEL
DERIVED FROM TWO-FLUID EULER-MAXWELL EQUATIONS

YUE-JUN PENG¹, SHUAI XI² AND LIANG ZHAO³

¹Université Clermont Auvergne, CNRS, Laboratoire de Mathématiques Blaise Pascal
F-63000 Clermont-Ferrand, France

²College of Mathematics and Systems Science,
Shandong University of Science and Technology, Qingdao 266590, P. R. China

³ Department of Mathematics, National University of Singapore,
Singapore 119076, Singapore

**Abstract.** We derive the one-fluid ion model from the partial-compressible two-fluid Euler-Maxwell system with collision mechanism in which the ions are considered compressible while electrons are not. Deriving such a model based solely on the zero-electron-mass limit or the quasi-neutral limit is quite open because of the singularities of the electric field and the strong coupling effects of the Lorentz forces. In this work, we apply the combined zero-electron-mass and quasi-neutral limit. We prove the global-in-time existence and the combined limit of smooth solutions. Furthermore, for well-prepared initial data, we establish the error estimates of the convergence between smooth solutions to one-fluid and two-fluid models. It is worth mentioning that the velocities of ions and electrons are different in this study.

**Keywords:** Euler-Maxwell system; Hall-MHD equations; one-fluid; two-fluid; zero-electron-mass limit; quasi-neutral limit

**AMS Subject Classification (2020):** 35B40, 35L60, 35Q35, 78A25, 35K45.

1. INTRODUCTION

Plasma is one of the four fundamental states of matter, which can be regarded as a collection of fast-moving charged particles. Especially at high temperatures, ions and electrons in a plasma tend to retain two separate fluids due to their different physical properties. Hence, the evolution of the charged particles can be described by the following two-fluid compressible isentropic Euler-Maxwell system [27, 7, 5]:

$$ 
\left\{
\begin{array}{l}
\partial_t \rho_\nu + \text{div}(\rho_\nu u_\nu) = 0, \\
m_\nu \partial_t (\rho_\nu u_\nu) + m_\nu \text{div}(\rho_\nu u_\nu \otimes u_\nu) + \nabla p_\nu(\rho_\nu) = q_\nu \rho_\nu \mathbf{F}_\nu^{\text{EM}} - \rho_\nu \text{Diss}_\nu, \\
\lambda^2 \partial_t E - \nabla \times B = \rho_e u_e - \rho_i u_i, \quad \lambda^2 \text{div} E = \rho_i - \rho_e, \\
\partial_t B + \nabla \times E = 0, \quad \text{div} B = 0,
\end{array}
\right.
\qquad (1.1) $$

with initial data

$$ t = 0: \quad (\rho_\nu, u_\nu, E, B) = (\rho_{\nu,0}, u_{\nu,0}, E_0, B_0), \qquad (1.2) $$
for $v = i, e$. Here $t > 0$ is the time variable and $x = (x_1, x_2, x_3) \in \mathbb{R}^3$ is the space variable, the subscripts $i$ and $e$ stand for the ions and the electrons, $\rho_v$ and $u_v$ are the scaled densities and the velocity vectors of particles, $E$ and $B$ are the electric field and the magnetic field, respectively. The parameters $m_v > 0$ and $q_v$ stand for the mass and the charge of the particle with $q_i = -q_e = 1$, and $\lambda > 0$ is the Debye length. The term $\mathbf{F}_v^{\text{EM}} := E + u_v \times B$ is the Lorentz force and $\text{Diss}_v$ is one kind of dissipation structure, which will be specified later. The pressure functions $p_v$ are supposed to be smooth and strictly increasing, namely,

$$p'_{v}(\rho) > 0, \quad \forall \rho > 0.$$

For the classical damping mechanisms, we have $\text{Diss}_v = m_v u_v$. System (1.1) is symmetrizable hyperbolic for $\rho_v > 0$. Hence, the local-in-time existence and global-in-time well-posedness of smooth solutions near equilibrium $(\rho_v, u_v, E, B) = (1, 0, 0, \tilde{B})$ are established for any constant vector $\tilde{B} \in \mathbb{R}^3$. We refer readers to [19, 17, 22] for local-in-time well-posedness and to Peng [24], Duan-Liu-Zhu [11], Xu-Xiong-Kawashima [35], Guo-Ionescu-Pausader [14], Li-Zhu [21] and references therein for various global-in-time well-posedness results.

It is well-known that electrons and ions have large differences in mass and velocity. Consequently, the study of (1.1) often requires a separate focus on the dynamics of each species of particles based on certain physical simplifications, resulting in decoupled self-consistent one-fluid models. This problem is both numerically and theoretically significant. However, deriving the one-fluid ion model from (1.1) is very complicated. A natural attempt to derive the one-fluid model is to apply the zero-electron-mass limit ($m_i = \lambda = 1, m_e \to 0$), which has been used to derive the ion Euler-Poisson system from the two-fluid model in Li-Peng-Xi [20], Grenier-Guo-Pausader-Suzuki [13] and Xi-Zhao [33, 34], see also [16, 2, 36]. However, this limit is infeasible for (1.1) because of the strong coupling effects induced by Lorentz forces. In the Euler-Poisson system, $\mathbf{F}_v^{\text{EM}}$ is replaced by the electric force $\mathbf{F}_v^{\text{EP}} := \nabla \varphi$ for some electric potential function $\varphi$. Clearly $\mathbf{F}_v^{\text{EP}}$ is irrotational while $\mathbf{F}_v^{\text{EM}}$ is not. This implies that $\mathbf{F}_v^{\text{EM}}$ exerts an additional rotational effect on the fluid compared with $\mathbf{F}_v^{\text{EP}}$, which, together with the attraction between ions and electrons, makes these two kinds of particles more likely to unite. As a result, rather than separating electrons from ions, it is more reasonable to regard ions and electrons together as a group called quasi-charge-neutral fluid, which can be achieved by applying the quasi-neutral limit ($m_e = m_i = 1, \lambda \to 0$) to (1.1). We refer to Peng-Liu [25] for this limit applied to two-fluid Euler-Poisson or Euler-Maxwell systems, and to [8, 30, 31, 15] for other relevant studies. Nevertheless, since electrons move extremely fast, usually a condition $u_{e,0} - u_{i,0} = O(\lambda)$ should be imposed in these results to ensure that $u_e - u_i = O(\lambda)$ for all time so that the particle group remains stable after the limit is applied. It is uncertain that, if $u_e$ and $u_i$ differ significantly, after a collision between ions and electrons, their attraction is enough to form a stable particle group, or the electric force is strong enough to let electrons escape from ions. In other words, the common velocity of the particle group is hard to define without this condition. We refer to Cordier-Grenier-Guo [9] for a rigorous justification of instability for the two-fluid Euler-Poisson system if two velocities differ a lot. However this condition also eliminates the possibility that $u_{e,0} - u_{i,0}$ can be large, which is a more common physical observation. For these reasons, the rigorous mathematical analysis for either the zero-electron-mass limit or the quasi-neutral limit still remains open for less restrictive initial data.

Based on the analysis above, to derive the one-fluid ion model, apart from taking the combined zero-electron-mass and quasi-neutral limit, more intrinsic physical phenomena and mathematical structures should be identified. In physics, Debye length $\lambda$ is inversely proportional to the density for a fixed temperature [7]. Consequently, when we consider the quasi-neutral limit $\lambda \to 0$, usually the density is higher, meaning particles are closer together on average, resulting in more chaotic and irregular motion of particles and significant mutual collisions between particles. Therefore,
for small $\lambda$, the collision effects should be taken into consideration [1]. Hence, apart from the usual damping term $m_\nu u_\nu$, Diss$_\nu$ are modified as

$$ \widetilde{\text{Diss}}_{\nu} = \text{Coll}_{\nu} + m_{\nu} \rho_{\nu}, \quad \text{with} \quad \text{Coll}_{i} = \rho_{e}(u_{i} - u_{e}), \quad \text{Coll}_{e} = \rho_{i}(u_{e} - u_{i}). $$

The collision effects of electrons and ions act as a damping mechanism which sums up to zero because of the total momentum conservation. Besides, to further weaken the effects of electrons, we assume that electrons are incompressible, which is a direct consequence of the zero-electron-mass limit, see more explanations in [3, 36]. Based on the above considerations, (1.1) becomes

$$ \left\{ \begin{array}{l} m_e \partial_t u_e + m_e (u_e \cdot \nabla) u_e + \nabla p_e = -E - u_e \times B - \rho_i (u_e - u_i) - m_e u_e, \quad \text{div} u_e = 0, \\ \partial_t \rho_i + \text{div}(\rho_i u_i) = 0, \\ m_i \partial_t (\rho_i u_i) + m_i \text{div}(\rho_i u_i \otimes u_i) + \nabla p_i (\rho_i) = \rho_i (E + u_i \times B) - \rho_i (u_i - u_e) - m_i \rho_i u_i, \\ \lambda^2 \partial_t E - \nabla \times B = u_e - \rho_i u_i, \quad \lambda^2 \text{div} E = \rho_i - 1, \\ \partial_t B + \nabla \times E = 0, \quad \text{div} B = 0, \end{array} \right. \quad (1.3) $$

with $t > 0$, $x \in \mathbb{R}^3$ and initial conditions

$$ t = 0: \quad (\rho_i, u_\nu, E, B) = (\rho_{i,0}, u_{\nu,0}, E_0, B_0), \quad x \in \mathbb{R}^3. \quad (1.4) $$

Here the independent unknowns of this system become $(u_e, p_e, \rho_i, u_i, E, B)$. In order to make $p_e$ uniquely determined, we assume

$$ p_e(t, x) \to 0, \quad \text{sufficiently fast as } |x| \to \infty. $$

Formally, if $(\rho_i, u_\nu, p_e, E, B) \to (\bar{\rho}_i, \bar{u}_\nu, \bar{p}_e, \bar{E}, \bar{B})$ as $(m_e, \lambda) \to 0$, then

$$ \bar{\rho}_i = 1 \Rightarrow \text{div} \bar{u}_i = 0, \quad \nabla \times \bar{B} = \bar{u}_i - \bar{u}_e =: \bar{J}, \quad \text{div} \bar{B} = 0, \quad \partial_t \bar{B} + \nabla \times \bar{E} = 0, \quad (1.5) $$

where $\bar{J}$ is called the electric current. Adding the first and third equations in (1.3), we obtain

$$ \begin{aligned} & \partial_t (m_e u_e + m_i \rho_i u_i) + m_e (u_e \cdot \nabla) u_e + m_i \text{div}(\rho_i u_i \otimes u_i) + \nabla (p_e + p_i(\rho_i)) \\ & = (\rho_i - 1)E + (u_i - u_e) \times B + (\rho_i - 1)(u_i \times B) - m_e u_e - m_i \rho_i u_i, \end{aligned} \quad (1.6) $$

which is called center-of-mass equation. Let $m_i = 1$. Applying the combined limit $(m_e, \lambda) \to 0$ to (1.6), we obtain

$$ \partial_t \bar{u}_i + (\bar{u}_i \cdot \nabla) \bar{u}_i + \nabla \bar{p}_e = \bar{J} \times \bar{B} - \bar{u}_i. \quad (1.7) $$

Besides, taking $\nabla \times$ to the first equation in (1.3) and passing to the limit $(m_e, \lambda) \to 0$, we get

$$ \partial_t \bar{B} - \nabla \times ((\bar{u}_i - \bar{J}) \times \bar{B}) = -\nabla \times \bar{J}. \quad (1.8) $$

Combining (1.5), (1.7) and (1.8), we obtain the incompressible Hall-MHD equations with zero viscosity for ion velocity:

$$ \left\{ \begin{array}{l} \partial_t \bar{u}_i + (\bar{u}_i \cdot \nabla) \bar{u}_i + \nabla \bar{p}_e = (\nabla \times \bar{B}) \times \bar{B} - \bar{u}_i, \quad \text{div} \bar{u}_i = 0, \\ \partial_t \bar{B} - \nabla \times (\bar{u}_i \times \bar{B}) = -\nabla \times ((\nabla \times \bar{B}) \times \bar{B}) + \Delta \bar{B}, \quad \text{div} \bar{B} = 0. \end{array} \right. \quad (1.9) $$

It is worth emphasizing that when $m_e \to 0$, ions dominate the center-of-mass equation (1.6). Hence (1.9) can be regarded as the one-fluid ion model when the effects of electrons are neglected.

The Hall-MHD system (1.9) is a very important model in physics and has been studied intensively, see Chae-Degond-Liu [6], Danchin-Tan [10], Weng [32] and references therein. However, this physically accurate limit passage from (1.3) to (1.9) has not been rigorously justified in mathematics. We refer to Acheritogaray-Degond-Frouvelle-Liu [1] for the formal derivation from (1.1) to the compressible Hall-MHD equations and to Peng-Wang-Xu [23] for the derivation from the totally incompressible Navier-Stokes-Maxwell equations to the incompressible Hall-MHD equations, see also Arsénio-Ibrahim-Masmoudi [4] for a relevant study.
The main focus of the present paper is to prove rigorously that smooth solutions near equilibrium to system (1.3) converges in appropriate norms to those of (1.9) when the combined limit $((m_e, \lambda) \to 0, m_i = 1)$ is applied. It is worth mentioning that we do not need the condition $u_{e,0} - u_{i,0} = O(\lambda)$. To achieve this, we first establish the local-in-time existence of smooth solutions to (1.3) and then extend those to be global ones based on the uniform estimates of solutions with respect to time and the small parameters $(m_e, \lambda)$. The global-in-time convergence of the combined limit is attained through careful compactness while the strong convergence is obtained through error estimates between solutions to these two systems. However, there are still some essential obstacles occurring in the proof, toward which our strategies are as follows:

1) the local-in-time existence of smooth solutions to (1.3), which is non-trivial because the system does not fit into the classical framework of first-order quasi-linear symmetrizable hyperbolic system as there is neither evolutionary equation for $p_e$, nor can the pressure term $\nabla p_e$ be regarded as a source term. To overcome this, we first approximate the system by adding an artificial viscosity term for $u_e$. Then the local-in-time existence of smooth solutions to these approximate systems are guaranteed by [18, 28]. Hence, the local-in-time solvability of (1.3) is obtained by taking the vanishing viscosity limit of these approximate systems.

2) rapid oscillations and high singularities of $E$ in the Lorentz forces $F_\nu^{\text{EM}}$ with respect to $\lambda$. To tackle this obstacle, motivated by the different orders of singularities of $\text{div}E$ and $\nabla \times E$ implied by the Maxwell equations, by taking good advantage of the Maxwell equations, the symmetrizable hyperbolic structure as well as the symmetry of the source terms, we establish elaborate uniform estimates of $\text{div}(\sqrt{m_\nu}u_\nu, \lambda E, B)$ and $\nabla \times (\sqrt{m_\nu}u_\nu, \lambda E, B)$ and recover those of $(\sqrt{m_\nu}u_\nu, \lambda E, B)$ by using Hodge theories.

3) non-uniform estimates of $E$ with respect to $\lambda$. As mentioned above, we can only obtain the uniform estimates of $\lambda E$ instead of $E$, which causes great troubles in the convergence analysis because the limit behaviors of $E$ are uncertain. To take the limit, we introduce the generalized vorticities $B + q_\nu \sqrt{m_\nu} \nabla \times u_\nu$. As a bridge, we can enhance the convergence of $\nabla \times u_i$ and $B$, which is vital in our convergence analysis. Alternatively, we construct a class of initial data, which are in accordance with physical facts, such that by studying the time derivative of solutions, we can mitigate the singularities and determine the limit status of $E$. Based on this, the global-in-time error estimates between solutions to (1.3) and (1.9) are established, and thus the convergences of $u_i$ and $B$ are further improved.

It is worth mentioning that the collision terms play a vital role in our analysis. Mathematically speaking, these terms provide a dissipation structure for $u_e - u_i$, which enables us to get rid of the initial restriction $u_{e,0} - u_{i,0} = O(\lambda)$. In the limiting MHD equations, this dissipation structure exhibits as the diffusion of the magnetic field $B$. At the same time, this structure also compensates for the vanishing of dissipation $\text{Diss}_e$ as $m_e \to 0$, mitigating the singularities of $u_e$.

The paper is organized as follows. In §2, we introduce some notations and present our main results. §3 is devoted to the local-in-time existence of smooth solutions to (1.3). The global-in-time convergence and error estimates are respectively established in §4 and §5.

## 2. NOTATIONS AND MAIN RESULTS

In this section, we introduce some notations and present our main results. Set $m_i = 1$. We define the small parameter $\varepsilon = \lambda + \sqrt{m_e}$ and an integer $s \ge 3$. Thus we need to study the limit $\varepsilon \to 0$. For a multi-index $\alpha = (\alpha_1, \alpha_2, \alpha_3) \in \mathbb{N}^3$, we denote

$$ \partial_x^\alpha = \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1} \partial x_2^{\alpha_2} \partial x_3^{\alpha_3}} \quad \text{with} \quad |\alpha| = \alpha_1 + \alpha_2 + \alpha_3. $$
For simplicity, we denote by $\|\cdot\|$, $\|\cdot\|_{\infty}$ and $\|\cdot\|_{l}$ the usual norms of $L^2 \stackrel{def}{=} L^2(\mathbb{R}^3)$, $L^{\infty} \stackrel{def}{=} L^{\infty}(\mathbb{R}^3)$ and $H^l \stackrel{def}{=} H^l(\mathbb{R}^3)$ for all integers $l \ge 1$, respectively. The inner product in $L^2(\mathbb{R}^3)$ is denoted as $\langle \cdot, \cdot \rangle$. Throughout the paper, we denote $\nu = i, e$, where $i$ stands for ions and $e$ stands for electrons. For smooth solutions with $\rho_i > 0$, the momentum equations for ions in (1.3) are equivalent to

$$ \partial_t u_i^\varepsilon + (u_i^\varepsilon \cdot \nabla)u_i^\varepsilon + \nabla h_i(\rho_i^\varepsilon) = E^\varepsilon + u_i^\varepsilon \times B^\varepsilon - u_i^\varepsilon - (u_i^\varepsilon - u_e^\varepsilon), $$

where $h_i$ is the enthalpy function, defined by

$$ h_i'(\rho) = \frac{p_i'(\rho)}{\rho}. $$

Since $p_i$ is strictly increasing, so is $h_i$. As a result, (1.3)-(1.4) become

$$ \left\{ \begin{array}{l} \partial_t \rho_i^\varepsilon + \text{div}(\rho_i^\varepsilon u_i^\varepsilon) = 0, \\ \partial_t u_i^\varepsilon + (u_i^\varepsilon \cdot \nabla)u_i^\varepsilon + \nabla h_i(\rho_i^\varepsilon) = E^\varepsilon + u_i^\varepsilon \times B^\varepsilon - (u_i^\varepsilon - u_e^\varepsilon) - u_i^\varepsilon, \\ m_e \partial_t u_e^\varepsilon + m_e(u_e^\varepsilon \cdot \nabla)u_e^\varepsilon + \nabla p_e^\varepsilon = -E^\varepsilon - u_e^\varepsilon \times B^\varepsilon - \rho_i^\varepsilon(u_e^\varepsilon - u_i^\varepsilon) - m_e u_e^\varepsilon, \quad \text{div} u_e^\varepsilon = 0, \\ \lambda^2 \partial_t E^\varepsilon - \nabla \times B^\varepsilon = u_e^\varepsilon - \rho_i^\varepsilon u_i^\varepsilon, \quad \lambda^2 \text{div} E^\varepsilon = \rho_i^\varepsilon - 1, \\ \partial_t B^\varepsilon + \nabla \times E^\varepsilon = 0, \quad \text{div} B^\varepsilon = 0, \\ t = 0: \quad (\rho_i^\varepsilon, u_\nu^\varepsilon, E^\varepsilon, B^\varepsilon) = (\rho_{i,0}^\varepsilon, u_{\nu,0}^\varepsilon, E_0^\varepsilon, B_0^\varepsilon). \end{array} \right. \quad (2.1) $$

We tend to find smooth solutions to (2.1) near a constant equilibrium state

$$ (\rho_i, u_e, u_i, p_e, E, B) = (1, 0, 0, 0, 0, \tilde{B}), $$

where $\tilde{B} \in \mathbb{R}^3$ is an arbitrary constant vector. The following compatibility conditions at $t = 0$ should be satisfied to avoid the appearance of the initial layers:

$$ \text{div} u_{e,0}^{\varepsilon} = 0, \quad \lambda^2 \text{div} E_0^{\varepsilon} = \rho_{i,0}^{\varepsilon} - 1, \quad \text{div} B_0^{\varepsilon} = 0. \quad (2.2) $$

The main results of the present paper are as follows. Theorem 2.1 concerns the global-in-time existence and uniform estimates of smooth solutions to (2.1) with respect to $\varepsilon$.

**Theorem 2.1.** (Uniform global existence) Let $s \ge 3$ be an integer and (2.2) hold. Then there exist positive constants $C_1$ and $\delta$ such that for all $\varepsilon \in (0, 1]$, if

$$ \|\rho_{i,0}^{\varepsilon} - 1, \sqrt{m_e} u_{e,0}^{\varepsilon}, u_{i,0}^{\varepsilon}, \lambda E_0^{\varepsilon}, B_0^{\varepsilon} - \tilde{B})\|_s + \frac{1}{\lambda} \|\rho_{i,0}^{\varepsilon} - 1\|_{s-1} \le \delta, \quad (2.3) $$

the problem (1.3)-(1.4) admits a unique global solution $(\rho_i^\varepsilon, p_e^\varepsilon, u_\nu^\varepsilon, E^\varepsilon, B^\varepsilon)$ satisfying for all $t \ge 0$,

$$ \begin{aligned} & \|(\rho_i^\varepsilon - 1, \sqrt{m_e}u_e^\varepsilon, u_i^\varepsilon, \lambda E^\varepsilon, B^\varepsilon - \tilde{B})(t)\|_s^2 + \frac{1}{\lambda^2}\|\rho_i^\varepsilon(t) - 1\|_{s-1}^2 \\ & \qquad + \int_0^t (\|(\nabla\rho_i^\varepsilon, \lambda E^\varepsilon, \lambda^{-1}(\rho_i^\varepsilon - 1))(\tau)\|_{s-1}^2 + \|\nabla \times B^\varepsilon(\tau)\|_{s-2}^2 + \|(u_e^\varepsilon, u_i^\varepsilon)(\tau)\|_s^2 d\tau \\ & \le C_1 \|(\rho_{i,0}^\varepsilon - 1, \sqrt{m_e}u_{e,0}^\varepsilon, u_{i,0}^\varepsilon, \lambda E_0^\varepsilon, B_0^\varepsilon - \tilde{B})\|_s^2 + \frac{1}{\lambda^2}\|\rho_{i,0}^\varepsilon - 1\|_{s-1}^2. \end{aligned} \quad (2.4) $$

The next theorem concerns the global-in-time convergence from (1.3) to (1.9).

**Theorem 2.2.** (Uniform global convergence) Let $(\rho_i^\varepsilon, u_\nu^\varepsilon, p_e^\varepsilon, E^\varepsilon, B^\varepsilon)$ be the unique global smooth solution obtained in Theorem 2.1. If as $\varepsilon \to 0$,

$$ (u_{i,0}^{\varepsilon}, B_0^{\varepsilon} - \tilde{B}) \to (\bar{u}_{i,0}, \bar{B}_0 - \tilde{B}), \quad \text{weakly in } H^s, $$

then there exist $(\bar{u}_i, \bar{u}_e, \bar{p}_e, \bar{B})$ with $(\bar{u}_i, \bar{B} - \tilde{B}) \in L^\infty(\mathbb{R}^+; H^s)$, $\bar{u}_e \in L^2(\mathbb{R}^+; H^s)$ and $\nabla \bar{p}_e \in L^2(\mathbb{R}^+; H^{s-1})$ such that as $\varepsilon \to 0$, up to subsequences,

$$ \rho_i^\varepsilon - 1 \to 0, \quad \text{strongly in } L^\infty(\mathbb{R}^+; H^{s-1}), \quad (2.5) $$
$$ (u_i^\epsilon, B^\epsilon - \tilde{B}) \stackrel{*}{\rightharpoonup} (\bar{u}_i, \bar{B} - \tilde{B}), \quad \text{weakly-} * \text{ in } L^\infty(\mathbb{R}^+; H^s), \qquad (2.6) $$

$$ u_e^\epsilon \to \bar{u}_e, \quad \text{weakly in } L^2(\mathbb{R}^+; H^s), \qquad (2.7) $$

where $(\bar{u}_i, \bar{B}, \bar{p}_e)$ satisfies (1.9) with initial conditions

$$ t = 0 : (\bar{u}_i, \bar{B}) = (\bar{u}_{i,0}, \bar{B}_0). $$

The next theorem focuses on the global-in-time convergence rates.

**Theorem 2.3.** (Convergence rates) Let $(\rho_i^\epsilon, u_\nu^\epsilon, p_e^\epsilon, E^\epsilon, B^\epsilon)$ be the unique solution obtained in Theorem 2.1 and $(\bar{u}_\nu, \bar{p}_e, \bar{B})$ be obtained in Theorem 2.2. Assuming that

$$ \|E_0^\epsilon\|_s + \frac{1}{\lambda} \|u_{e,0}^\epsilon - \rho_{i,0}^\epsilon u_{i,0}^\epsilon + \nabla \times B_0^\epsilon\|_{s-1} + \frac{1}{\sqrt{m_e}} \|\nabla \times (E_0^\epsilon + u_{e,0}^\epsilon \times B_0^\epsilon + \rho_{i,0}^\epsilon (u_{e,0}^\epsilon - u_{i,0}^\epsilon))\|_{s-2} \le C, \quad (2.8) $$

then $E^\epsilon$ is uniformly bounded in $L^2(\mathbb{R}^+; H^{s-1})$. Consequently, there exists $\bar{E} \in L^2(\mathbb{R}^+; H^{s-1})$, such that as $\epsilon \to 0$, up to subsequences,

$$ (\nabla p_e^\epsilon, E^\epsilon) \to (\nabla \bar{p}_e, \bar{E}), \quad \text{weakly in } L^2(\mathbb{R}^+; H^{s-1}). $$

Furthermore, if the following initial error estimates are satisfied for $\zeta > 0$:

$$ \|(\rho_{i,0}^\epsilon - 1, u_{i,0}^\epsilon - \bar{u}_{i,0}, \sqrt{m_e}u_{e,0}^\epsilon, \lambda E_0^\epsilon, B_0^\epsilon - \bar{B}_0)\|_{s-1} \le |\epsilon|^\zeta, $$

then there exists a constant $C_2$ independent of $\epsilon$, such that for all $\epsilon \in (0, 1]$, it holds

$$ \sup_{t \in \mathbb{R}^+} \|(\rho^\epsilon - 1, \sqrt{m_e}(u^\epsilon_e - \bar{u}_e), u^\epsilon_i - \bar{u}_i, \lambda(E - \bar{E}), B^\epsilon - \bar{B})(t)\|_{s-1}^2 \\ + \int_0^\infty \left( \frac{1}{\lambda^2} \|\rho^\epsilon(\tau) - 1\|_{s-2}^2 + \|(u^\epsilon_e - \bar{u}_e, u^\epsilon_i - \bar{u}_i)(\tau)\|_{s-1}^2 d\tau \le C_2 |\epsilon|^{2\min\{1, \zeta\}}. $$

**Remark 2.1.** We claim that the initial data $(\rho_{i,0}^\epsilon, u_\nu^\epsilon, E_0^\epsilon, B_0^\epsilon)$ satisfying (2.8) actually exist. Let $(E_0^\epsilon, u_{e,0}^\epsilon)$ be uniformly bounded in $H^{s+1}$, decay sufficiently fast at infinity and $\text{div}u_{e,0}^\epsilon = 0$. Let $\rho_{i,0}^\epsilon = 1 + \lambda^2 \text{div}E_0^\epsilon$ so that $\rho_{i,0}^\epsilon - 1 \in H^s$ and $\lambda^{-1}(\rho_{i,0}^\epsilon - 1) \in H^{s-1}$. Let $B_0^\epsilon - \tilde{B}$ be decaying sufficiently fast and be the solution to the following elliptic equation with respect to $G_0^\epsilon$:

$$ \Delta G_0^\epsilon + \nabla \times (u_{e,0}^\epsilon \times G_0^\epsilon) = -\nabla \times (E_0^\epsilon + \rho_{i,0}^\epsilon u_{e,0}^\epsilon - \sqrt{m_e} \chi^\epsilon + u_{e,0}^\epsilon \times \tilde{B} - u_{e,0}^\epsilon - \lambda \iota^\epsilon), \quad (2.9) $$

for some given functions $\iota^\epsilon$ and $\chi^\epsilon$, uniformly bounded in $H^{s+1}$ and decaying sufficiently fast. Consequently, $\text{div}B_0^\epsilon = 0$ and for small $\|u_{e,0}\|_s$ and $B_0^\epsilon - \tilde{B} \in H^{s+1}$. Define

$$ u_{i,0}^\epsilon = \frac{1}{\rho_{i,0}^\epsilon} (u_{e,0}^\epsilon + \nabla \times B_0^\epsilon + \lambda \iota^\epsilon) \in H^s. $$

It follows from (2.9) that

$$ \nabla \times (E_0^\epsilon + u_{e,0}^\epsilon \times B_0^\epsilon + \rho_{i,0}^\epsilon (u_{e,0}^\epsilon - u_{i,0}^\epsilon)) = \sqrt{m_e} \nabla \times \chi^\epsilon. $$

Hence (2.2), (2.3) and (2.8) are all satisfied.

### 3. LOCAL EXISTENCE OF SMOOTH SOLUTIONS

In this section, we establish local-in-time existence of smooth solutions to (2.1). For simplicity, we assume $m_e = \lambda = 1$ and drop the superscripts $\epsilon$.

The main result of this section is as follows.
**Theorem 3.1.** Let $s \ge 3$ be an integer. Suppose $(\rho_{i,0} - 1, u_{\nu,0}, E_0, B_0 - \tilde{B}) \in H^s$ such that

$$ \rho_{i,0} \ge 2\underline{n}, \quad \mathrm{div}u_{e,0} = 0, \quad \mathrm{div}E_0 = \rho_{i,0} - 1, \quad \mathrm{div}B_0 = 0, $$

for some given constant $\underline{n} > 0$. Then there exists a $T^{**} > 0$ (defined in (3.17)) such that (2.1) admits a unique smooth solution $(\rho_i, u_i, p_e, u_e, E, B)$ satisfying $\rho_i \ge \underline{n}$ in $[0, T^{**}] \times \mathbb{R}^3$ and

$$ (\rho_i - 1, u_\nu, E, B - \tilde{B}) \in C([0, T^{**}]; H^s) \cap C^1([0, T^{**}]; H^{s-1}), \quad \nabla p_e \in L^2([0, T^{**}]; H^{s-1}). $$

The rest of this section is devoted to the proof of Theorem 3.1.

**3.1. Approximate system.** Taking div to the third equation in (2.1) yields

$$ -\Delta p_e = \mathrm{div}((u_e \cdot \nabla)u_e) + \mathrm{div}E + \mathrm{div}(u_e \times B) + \mathrm{div}(\rho_i(u_e - u_i)). $$

To solve (2.1), it is equivalent to solve the following problem

$$ \left\{ \begin{array}{l} \partial_t \rho_i + \mathrm{div}(\rho_i u_i) = 0, \\ \partial_t u_i + (u_i \cdot \nabla)u_i + \nabla h_i(\rho_i) = E + u_i \times B - (u_i - u_e) - u_i, \\ \partial_t u_e + (u_e \cdot \nabla)u_e + \nabla p_e = -E - u_e \times B - \rho_i(u_e - u_i) - u_e, \\ -\Delta p_e = \mathrm{div}((u_e \cdot \nabla)u_e) + \mathrm{div}E + \mathrm{div}(u_e \times B) + \mathrm{div}(\rho_i(u_e - u_i)), \quad (3.1) \\ \partial_t E - \nabla \times B = u_e - \rho_i u_i, \quad \mathrm{div}E = \rho_i - 1, \\ \partial_t B + \nabla \times E = 0, \quad \mathrm{div}B = 0, \\ t = 0: \quad (\rho_i, u_\nu, E, B) = (\rho_{i,0}, u_{\nu,0}, E_0, B_0), \end{array} \right. $$

due to the fact that the third and the fourth equations of the above system lead to the incompressibility of $u_e$ if it is so initially. However, the system does not fill into the classical framework of the first-order quasi-linear symmetrizable hyperbolic system because there is no evolutionary equation for $p_e$. Besides, the elliptic theories show that

$$ \|\nabla p_e\|_s \le C \|(u_e \cdot \nabla)u_e + E + u_e \times B + \rho_i(u_e - u_i)\|_s, $$

in which $u_e \in H^{s+1}$ is needed. Consequently, $\nabla p_e$ cannot be regarded as a source term. To overcome this difficulty, we introduce an artificial viscosity term $\eta\Delta u_e$ in (3.1) with $\eta > 0$ being a constant and first establish the local-in-time existence of smooth solutions to the following approximate problem

$$ \left\{ \begin{array}{l} \partial_t \rho_i^\eta + \mathrm{div}(\rho_i^\eta u_i^\eta) = 0, \\ \partial_t u_i^\eta + (u_i^\eta \cdot \nabla)u_i^\eta + \nabla h_i(\rho_i^\eta) = E^\eta + u_i^\eta \times B^\eta - (u_i^\eta - u_e^\eta) - u_i^\eta, \\ \partial_t u_e^\eta + (u_e^\eta \cdot \nabla)u_e^\eta + \nabla p_e^\eta = \eta \Delta u_e^\eta - E^\eta - u_e^\eta \times B^\eta - \rho_i^\eta(u_e^\eta - u_i^\eta) - u_e^\eta, \\ -\Delta p_e^\eta = \mathrm{div}((u_e^\eta \cdot \nabla)u_e^\eta) + \mathrm{div}E^\eta + \mathrm{div}(u_e^\eta \times B^\eta) + \mathrm{div}(\rho_i^\eta(u_e^\eta - u_i^\eta)), \quad (3.2) \\ \partial_t E^\eta - \nabla \times B^\eta = u_e^\eta - \rho_i^\eta u_i^\eta, \quad \mathrm{div}E^\eta = \rho_i^\eta - 1, \\ \partial_t B^\eta + \nabla \times E^\eta = 0, \quad \mathrm{div}B^\eta = 0, \\ t = 0: \quad (\rho_i^\eta, u_\nu^\eta, E^\eta, B^\eta) = (\rho_{i,0}, u_{\nu,0}, E_0, B_0). \end{array} \right. $$

Then we want to pass to the limit $\eta \to 0$ in (3.2) to obtain a solution to (3.1).

We first study the following linearized problem:

$$ \left\{ \begin{array}{l} \partial_t \rho_i^\eta + v_i \cdot \nabla \rho_i^\eta + \Psi_i \mathrm{div} u_i^\eta = 0, \\ \partial_t u_i^\eta + (v_i \cdot \nabla) u_i^\eta + h_i'(\Psi_i) \nabla \rho_i^\eta = Q + v_i \times H - (v_i - v_e) - v_i, \\ \partial_t u_e^\eta + (v_e \cdot \nabla) u_e^\eta - \eta \Delta u_e^\eta = -Q - v_e \times H - \Psi_i(v_e - v_i) - \nabla p_e^\eta - v_e, \\ -\Delta p_e^\eta = \mathrm{div}((v_e \cdot \nabla) u_e^\eta) + \mathrm{div} Q + \mathrm{div}(v_e \times H) + \mathrm{div}(\Psi_i(v_e - v_i)), \\ \partial_t E^\eta - \nabla \times B^\eta = v_e - \Psi_i v_i, \quad \partial_t B^\eta + \nabla \times E^\eta = 0, \\ t = 0: \quad (\rho_i^\eta, u_\nu^\eta, E^\eta, B^\eta) = (\rho_{i,0}, u_{\nu,0}, E_0, B_0). \end{array} \right. \quad (3.3) $$
Here $(\Psi_i, v_\nu, Q, H)$ is the linearization of $(\rho_i, u_\nu, E, B)$ independent of $\eta$ satisfying

$$ (\Psi_i, v_\nu, Q, H)(0, x) = (\rho_{i,0}, u_{\nu,0}, E_0, B_0). $$

The first two equations in (3.3) can be rewritten as

$$ \partial_t \mathcal{U}_i + \sum_{j=1}^{3} \mathcal{A}_j(\Psi_i, v_i) \partial_{x_j} \mathcal{U}_i = \mathcal{Q}, \quad (3.4) $$

with $v_i = (v_i^1, v_i^2, v_i^3)^\top$,

$$ \mathcal{U}_i = \begin{pmatrix} \rho_i - 1 \\ u_i \end{pmatrix}, \quad \mathcal{A}_j(\Psi_i, v_i) = \begin{pmatrix} v_i^j & \Psi_i \xi_j^\top \\ h_i'(\Psi_i) \xi_j & v_i^j \mathbb{I}_3 \end{pmatrix}, \quad \mathcal{Q} = \begin{pmatrix} 0 \\ Q + v_i \times H - (v_i - v_e) - v_i \end{pmatrix}, $$

where $\{\xi_j\}_{j=1}^3$ is the canonical basis in $\mathbb{R}^3$, $\mathbb{I}_3$ is the 3×3 unit matrix. System (3.4) is symmetrizable hyperbolic with symmetrizer

$$ \mathcal{A}_0(\Psi_i) = \begin{pmatrix} h_i'(\Psi_i) & 0 \\ 0 & \Psi_i \mathbb{I}_3 \end{pmatrix}, $$

which is symmetric and positive definite if $\Psi_i$ has a positive lower bound. It follows that

$$ \tilde{\mathcal{A}}_j(\Psi_i, v_i) := \mathcal{A}_0(\Psi_i) \mathcal{A}_j(\Psi_i, v_i) = \begin{pmatrix} h_i'(\Psi_i) v_i^j & h_i'(\Psi_i) \Psi_i \xi_j^\top \\ h_i'(\Psi_i) \Psi_i \xi_j & \Psi_i v_i^j \mathbb{I}_3 \end{pmatrix}, $$

which is symmetric. As a result, the global well-posedness of classical solutions to the Cauchy problem (3.3) can be stated as follows.

**Lemma 3.1.** Let $\eta > 0$. Suppose the conditions on initial data in Theorem 3.1 hold. Then the Cauchy problem (3.3) admits a unique classical solution $(\rho_i^\eta, u_i^\eta, p_e^\eta, u_e^\eta, E^\eta, B^\eta)$ satisfying

$$ (\rho_i^\eta - 1, u_i^\eta, u_e^\eta, E^\eta, B^\eta) \in C(\mathbb{R}^+; H^s), \quad \mathrm{div} u_e^\eta = 0, \quad \nabla p_e^\eta \in L^2(\mathbb{R}^+; H^{s-1}). $$

*Proof.* The local existence of a smooth solution follows from classical theories [18, 28] and elliptic theories [12]. Since (3.3) is linear, this solution can be extended to a global one. Taking div to the third equation in (3.3) yields

$$ \partial_t \mathrm{div} u_e^\eta - \eta \Delta \mathrm{div} u_e^\eta = 0, \quad \mathrm{div} u_{e,0} = 0, $$

which implies $\mathrm{div} u_e^\eta = 0$. The regularity of $\nabla p_e$ follows directly from elliptic theories. $\square$

Now we give a uniform estimate for the solution of (3.3) with respect to $\eta$. The superscript $\eta$ is omitted for simplicity. We fix positive constants $c_0$ and $\tilde{c}_0$ large enough such that

$$ \|\rho_{i,0}\|_{\infty} + \|(\rho_{i,0} - 1, u_{i,0}, u_{e,0}, E_0, B_0 - \tilde{B})\|_{s} \le c_0, \quad \rho_{i,0} \ge 2\underline{n}, \quad 1 + \|h\|_{C^s[2\underline{n},c_0]} \le \tilde{c}_0, \quad (3.5) $$

and

$$ \sup_{0 \le t \le T^*} (\|\Psi_i(t)\|_{\infty} + \|(\Psi_i - 1, v_\nu, Q, H - \tilde{B})(t)\|_{s}) \le c_1, \quad \sup_{0 \le t \le T^*} \|\partial_t \Psi_i(t)\|_{\infty} \le c_2, \quad \mathrm{div} v_e = 0. \quad (3.6) $$

Hereinafter, $\tilde{c}_0$ and $c_j(j = 0, 1, 2)$ are positive constants independent of $\eta$ satisfying $c_2 > c_1 > \tilde{c}_0 > c_0 > 1, T^* > 0$ will be determined later (see (3.10)). We also denote by $C$ a generic positive constant independent of $\eta$, which may be different from line to line.

In Lemmas 3.2-3.3 below, we assume (3.5) and (3.6).

**Lemma 3.2.** (Lower bound of $\Psi_i$). Let $c_1 > \frac{1}{2\underline{n}}$. Then $\Psi_i(t, x) \ge c_1^{-1}$ for all $t \le \frac{1}{2c_2}(2\underline{n} - c_1^{-1})$.

*Proof.* By the Fundamental theorem of Calculus, it holds

$$ \Psi_i(t, x) = \rho_{i,0}(x) + \int_0^t \partial_\tau \Psi_i(\tau) d\tau \ge 2\underline{n} - c_2 t, $$

which yields the lower bound of $\Psi_i$ if $t < c_2^{-1}(2\underline{n} - c_1^{-1})$. $\square$
Based on this lemma, for some sufficiently large positive constants $c_3$ and $c_1$, we may require

$$1 + \|h\|_{C^s[c_1^{-1}, c_1]} \le c_3, \quad h'(q) \ge c_1^{-1}, \quad \text{for } q \in [c_1^{-1}, c_1]. \qquad (3.7)$$

**Lemma 3.3.** (Uniform a priori estimate) For $t < \min\{\frac{1}{2c_2}(2\underline{n} - c_1^{-1}), c_1^{-s-1}c_2^{-1}c_3^{-2}\}$, it holds

$$\|(\rho_i - 1, u_i, u_e, E, B - \tilde{B})(t)\|_s^2 + \|\rho_i(t)\|_{\infty}^2 \le Cc_1\tilde{c}_0c_0^3, \quad \|\partial_t\rho_i\|_{\infty}^2 \le Cc_1^3\tilde{c}_0c_0^3, \quad \rho_i(t, x) \ge c_1^{-1}. \quad (3.8)$$

*Proof.* For simplicity, we introduce the following functional

$$\mathbb{E}(t) = \|(\rho_i - 1, u_i, u_e, E, B - \tilde{B})(t)\|_s^2.$$

For all multi-index $\alpha \in \mathbb{N}^3$ such that $|\alpha| \le s$, applying $\partial_x^\alpha$ to (3.4), the third and the fifth to the sixth equations in (3.3), and making the inner product of the resulting equations with $2\mathcal{A}_0(\Psi_i)\partial_x^\alpha u_i$, $2\partial_x^\alpha u_e$ and $2((\partial_x^\alpha E)^\top, (\partial_x^\alpha(B - \tilde{B}))^\top)$, respectively, we have

$$\begin{aligned}
& \frac{d}{dt} (\langle \mathcal{A}_0(\Psi_i) \partial_x^\alpha u_i, \partial_x^\alpha u_i \rangle + \|\partial_x^\alpha u_e\|^2 + \|\partial_x^\alpha (E, B - \tilde{B})\|^2) + \eta \|\nabla \partial_x^\alpha u_e\|^2 \\
& \le \left\langle \left( \partial_t \mathcal{A}_0(\Psi_i) + \sum_{j=1}^3 \partial_{x_j} \tilde{\mathcal{A}}_j(\Psi_i, v_i) \right) \partial_x^\alpha u_i, \partial_x^\alpha u_i \right\rangle \\
& \quad + 2 \sum_{j=1}^3 \langle \mathcal{A}_j(\Psi_i, v_i) \partial_{x_j} \partial_x^\alpha u_i - \partial_x^\alpha (\mathcal{A}_j(\Psi_i, v_i) \partial_{x_j} u_i), \mathcal{A}_0(\Psi_i) \partial_x^\alpha u_i \rangle \\
& \quad + 2 \langle \Psi_i \partial_x^\alpha u_i, \partial_x^\alpha (Q + v_i \times H - (v_i - v_e) - v_i) \rangle - 2 \langle \partial_x^\alpha u_e, \partial_x^\alpha (Q + v_e \times H + \Psi_i(v_e - v_i) + v_e) \rangle \\
& \quad + 2 \langle \partial_x^\alpha E, \partial_x^\alpha (v_e - \Psi_i v_i) \rangle + 2 \langle \partial_x^\alpha u_e, (v_e \cdot \nabla) \partial_x^\alpha u_e - \partial_x^\alpha (v_e \cdot \nabla) u_e \rangle := \sum_{j=1}^6 \mathcal{I}_j,
\end{aligned} \qquad (3.9)$$

with the natural correspondence of $\{\mathcal{I}_j\}_{j=1}^6$. By Moser-type calculus inequalities, it holds

$$\begin{aligned}
|\mathcal{I}_1| + |\mathcal{I}_2| + |\mathcal{I}_6| &\le Cc_1^{s+1}c_2c_3^2\mathbb{E}(t), \\
|\mathcal{I}_3| + |\mathcal{I}_4| + |\mathcal{I}_5| &\le Cc_1^3\mathbb{E}(t) + Cc_1^3.
\end{aligned}$$

Substituting the above estimates into (3.9), integrating the resulting equations over $[0, t]$ and noticing that $\mathcal{A}_0(\Psi_i)$ is positive definite, we have

$$c_1^{-1} \|\partial_x^\alpha u_i\|^2 + \|\partial_x^\alpha (u_e, E, B - \tilde{B})\|^2 \le C\tilde{c}_0c_0^3 + Cc_1^3t + \int_0^t Cc_1^{s+1}c_2c_3^2\mathbb{E}(\tau)d\tau.$$

Summing the above for all $|\alpha| \le s$ and using the Gronwall inequality yield

$$c_1^{-1}\mathbb{E}(t) \le (C\tilde{c}_0c_0^3 + Cc_1^3t)e^{Cc_1^{s+1}c_2c_3^2t} \le C\tilde{c}_0c_0^3, \quad \text{for } t < \min\{(2c_2)^{-1}(2\underline{n} - c_1^{-1}), c_1^{-s-1}c_2^{-1}c_3^{-2}\}.$$

Next, by using the first equation in (3.3) and the above estimate, we have

$$\begin{aligned}
\|\partial_t \rho_i\|_{\infty}^2 &\le \|v_i \cdot \nabla \rho_i\|_{\infty}^2 + \|\Psi_i \operatorname{div} u_i\|_{\infty}^2 \le Cc_1^3 \tilde{c}_0 c_0^3, \\
\|\rho_i(t)\|_{\infty}^2 &\le Cc_0^2 + C \left( \int_0^t \|v_i\|_{\infty} \|\nabla \rho_i\|_{\infty} + \|\Psi\|_{\infty} \|\operatorname{div} u_i\|_{\infty} d\tau \right)^2 \\
&\le Cc_0^2 + Ct^2 c_1^3 \tilde{c}_0 c_0^3 \le Cc_1 \tilde{c}_0 c_0^3, \quad \text{for } t < \min\{(2c_2)^{-1}(2\underline{n} - c_1^{-1}), c_1^{-s-1}c_2^{-1}c_3^{-2}\}.
\end{aligned}$$

The lower bound of $\rho_i$ can be obtained similarly as in Lemma 3.2. These estimates imply (3.8). $\square$

Based on (3.8), we define the constants $c_1, c_2$ and time $T^*$ by

$$c_1 = 6C\tilde{c}_0c_0^3, \quad c_2 = 6\sqrt{6C^2c_0^2c_0^6}, \quad T^* = \min\{(2c_2)^{-1}(2\underline{n} - c_1^{-1}), c_1^{-s-1}c_2^{-1}c_3^{-2}\}, \quad (3.10)$$
and $c_3$ by (3.7). We then deduce that for $0 \le t \le T^*$,

$$ \sup_{0 \le t \le T^*} (\|\rho_i(t)\|_{\infty} + \|(\rho_i - 1, u_i, u_e, E, B - \tilde{B})(t)\|_{s}) \le c_1, \quad \sup_{0 \le t \le T^*} \|\partial_t \rho_i\|_{\infty} \le c_2. \quad (3.11) $$

In other words, given fixed $c_0$ and $\tilde{c}_0$, there exist positive constants $c_1, c_2, c_3$ and time $T^*$, depending solely on $c_0, \tilde{c}_0$ and the generic constant $C$ such that if (3.5)-(3.6) hold for $(\Psi_i, v_i, v_e, Q, H)$, then (3.11) holds for the solution to (3.3) in $[0, T^*] \times \mathbb{R}^3$. Here, it should be noted that the definitions of $c_1, c_2, c_3$ and the time $T^*$ are all independent of the parameter $\eta$.

### 3.2. Passing to the limit $\eta \to 0$
For every $\eta > 0$, there exists a unique classical solution $(\rho_i^\eta, u_\nu^\eta, p_e^\eta, E^\eta, B^\eta)$ satisfying (3.11) to the linear system (3.3). Besides, the fourth equation in (3.3) implies that $\{\nabla p_e^\eta\}_{\eta>0}$ is bounded in $L^2([0, T^*]; H^{s-1})$. Hence, there exists a subsequence of solutions, still denoted by $(\rho_i^\eta, u_\nu^\eta, p_e^\eta, E^\eta, B^\eta)$, which converges to a limit $(\bar{\rho}_i, \bar{u}_\nu, \bar{p}_e, \bar{E}, \bar{B})$ as $\eta \to 0$:

$$ \begin{aligned} (\rho_i^\eta, u_\nu^\eta, E^\eta, B^\eta) &\to (\bar{\rho}_i, \bar{u}_\nu, \bar{E}, \bar{B}), \quad \text{weakly-}^* \text{ in } L^\infty([0, T^*]; H^s), \\ \nabla p_e^\eta &\to \nabla \bar{p}_e, \quad \text{weakly-}^* \text{ in } L^2([0, T^*]; H^{s-1}). \end{aligned} \quad (3.12) $$

Since (3.3) is linear, these estimates are sufficient to pass to the limit $\eta \to 0$ in (3.3) such that the limit $\bar{U} := (\bar{\rho}_i, \bar{u}_i, \bar{p}_e, \bar{u}_e, \bar{E}, \bar{B} - \tilde{B})$ satisfies the following problem

$$ \left\{ \begin{aligned} & \partial_t \bar{\rho}_i + v_i \cdot \nabla \bar{\rho}_i + \Psi_i \operatorname{div} \bar{u}_i = 0, \\ & \partial_t \bar{u}_i + (v_i \cdot \nabla) \bar{u}_i + h_i'(\Psi_i) \nabla \bar{\rho}_i = Q + v_i \times H - (v_i - v_e) - v_i, \\ & \partial_t \bar{u}_e + (v_e \cdot \nabla) \bar{u}_e + \nabla \bar{p}_e = -Q - v_e \times H - \Psi_i (v_e - v_i) - v_e, \\ & -\Delta \bar{p}_e = \operatorname{div}((v_e \cdot \nabla) \bar{u}_e) + \operatorname{div} Q + \operatorname{div}(v_e \times H) + \operatorname{div}(\Psi_i (v_e - v_i)), \\ & \partial_t \bar{E} - \nabla \times \bar{B} = v_e - \Psi_i v_i, \quad \partial_t \bar{B} + \nabla \times \bar{E} = 0, \\ & t = 0: \quad (\bar{\rho}_i, \bar{u}_i, \bar{E}, \bar{B}) = (\rho_{i,0}, u_{i,0}, E_0, B_0). \end{aligned} \right. \quad (3.13) $$

The lower semi-continuity of weak convergence implies that $\bar{U}$ still satisfies estimates (3.11).

Besides, notice that (3.3) shows that $\{\partial_t(\rho_i^\eta, u_\nu^\eta, E^\eta, B^\eta)\}_{\eta>0}$ is bounded in $L^2([0, T^*]; H^{s-1})$. Then, by the classical compactness arguments [29], for any $s' \in [0, s)$,

$$ (\bar{\rho}_i - 1, \bar{u}_i, \bar{E}, \bar{B} - \tilde{B}) \in C([0, T^*]; H^{s'} \cap \operatorname{weak-H}^s). $$

Using the same arguments as in the proof of Lemma 3.3, we have

$$ \limsup_{t \to 0} \|(\bar{\rho}_i - 1, \bar{u}_i, \bar{E}, \bar{B} - \tilde{B})(t)\|_s \le \|(\rho_{i,0} - 1, u_{i,0}, E_0, B_0 - \tilde{B})(t)\|_s, $$

which implies that $(\bar{\rho}_i - 1, \bar{u}_i, \bar{E}, \bar{B} - \tilde{B})$ is right continuous at $t = 0$ in $H^s$ space. The time reversibility of equations in (3.13) yields that $(\bar{\rho}_i - 1, \bar{u}_i, \bar{E}, \bar{B} - \tilde{B}) \in C([0, T^*]; H^s)$.

For uniqueness, let $(\bar{\rho}_i^l - 1, \bar{u}_i^l, \bar{p}_e^l, \bar{u}_e^l, \bar{E}^l, \bar{B}^l - \tilde{B})$ with $l = 1, 2$ be two solutions to (3.13). Set

$$ \tilde{\rho}_i = \bar{\rho}_i^1 - \bar{\rho}_i^2, \quad \tilde{u}_i = \bar{u}_i^1 - \bar{u}_i^2, \quad \tilde{E} = \bar{E}^1 - \bar{E}^2, \quad \tilde{G} = \bar{B}^1 - \bar{B}^2, \quad \tilde{p}_e = \bar{p}_e^1 - \bar{p}_e^2. $$

Let $\tilde{U}_i = (\tilde{\rho}_i, \tilde{u}_i^\top)^\top$. Then $(\tilde{U}_i, \tilde{u}_e, \tilde{E}, \tilde{B}, \tilde{p}_e)$ satisfies

$$ \left\{ \begin{aligned} & \partial_t \tilde{U}_i + \sum_{j=1}^3 A_j(\Psi_i, u_i) \partial_{x_j} \tilde{U}_i = 0, \\ & \partial_t \tilde{u}_e + (v_e \cdot \nabla) \tilde{u}_e + \nabla \tilde{p}_e = 0, \\ & \partial_t \tilde{E} - \nabla \times \tilde{G} = 0, \quad \partial_t \tilde{G} + \nabla \times \tilde{E} = 0, \end{aligned} \right. $$

with zero initial data. Hence $(\tilde{U}_i, \tilde{u}_e, \tilde{E}, \tilde{B}) = 0$ and thus $\tilde{p}_e$ satisfies the Laplace equation, which implies $\tilde{p}_e = 0$ because $\tilde{p}_e$ decays sufficiently fast at infinity. Thus the uniqueness is proved. $\square$
### 3.3. Proof of Theorem 3.1.
The proof is based on the classical iteration scheme and the existence results for the linearized problem obtained in the previous subsection.

**Step 1:** Existence. Let the initial guess of our solution sequence be

$$ (\rho_i^0, u_\nu^0, E^0, B^0) = (\rho_{i,0}, u_{\nu,0}, E_0, B_0). $$

Since $c_1 > c_0$, we have

$$ \sup_{0 \le t \le T^*} (\|\rho_i^0(t)\|_{\infty} + \|(\rho_i^0 - 1, u_i^0, u_e^0, E^0, B^0 - \tilde{B})(t)\|_s) \le c_0 \le c_1, \quad \sup_{0 \le t \le T^*} \|\partial_t \rho_i^0\|_{s-1} = 0 \le c_2. $$

For $k \ge 1$, assume that $\mathcal{U}^k := (\rho_i^k, u_i^k, p_i^k, u_e^k, E^k, B^k)$ is defined and satisfies (3.11). Let $\mathcal{U}^{k+1}$ be the unique solution to (3.13) with $(\Psi_i, v_\nu, Q, H)$ being replaced by $(\rho_i^k, u_\nu^k, E^k, B^k)$, namely,

$$ \left\{ \begin{aligned} & \partial_t \rho_i^{k+1} + u_i^k \cdot \nabla \rho_i^{k+1} + \rho_i^k \operatorname{div} u_i^{k+1} = 0, \\ & \partial_t u_i^{k+1} + (u_i^k \cdot \nabla) u_i^{k+1} + h_i'(\rho_i^k) \nabla \rho_i^{k+1} = E^k + u_i^k \times B^k - (u_i^k - u_e^k) - u_i^k, \\ & \partial_t u_e^{k+1} + (u_e^k \cdot \nabla) u_e^{k+1} + \nabla p_e^{k+1} = -E^k - u_e^k \times B^k - \rho_i^k (u_e^k - u_i^k) - u_e^k, \\ & -\Delta p_e^{k+1} = \operatorname{div}((u_e^k \cdot \nabla) u_e^{k+1}) + \operatorname{div} E^k + \operatorname{div}(u_e^k \times B^k) + \operatorname{div}(\rho_i^k (u_e^k - u_i^k)), \\ & \partial_t E^{k+1} - \nabla \times B^{k+1} = u_e^k - \rho_i^k u_i^k, \quad \partial_t B^{k+1} + \nabla \times E^{k+1} = 0, \\ & t = 0: \quad (\rho_i^{k+1}, u_\nu^{k+1}, E^{k+1}, B^{k+1}) = (\rho_{i,0}, u_{\nu,0}, E_0, B_0). \end{aligned} \right. \quad (3.14) $$

Now we prove the strong convergence of the whole solution sequence $\{\mathcal{U}^k\}_{k \ge 1}$. Set

$$ \tilde{\rho}_i^{k+1} = \rho_i^{k+1} - \rho_i^k, \quad \tilde{u}_\nu^{k+1} = u_\nu^{k+1} - u_\nu^k, \quad \tilde{E}^{k+1} = E^{k+1} - E^k, \quad \tilde{G}^{k+1} = B^{k+1} - B^k, \quad \tilde{p}_e^{k+1} = p_e^{k+1} - p_e^k. $$

Then it follows from (3.14) that $(\tilde{\rho}_i^{k+1}, \tilde{u}_\nu^{k+1}, \tilde{E}^{k+1}, \tilde{G}^{k+1}, \tilde{p}_e^{k+1})$ satisfy

$$ \left\{ \begin{aligned} & \partial_t \tilde{\rho}_i^{k+1} + u_i^k \cdot \nabla \tilde{\rho}_i^{k+1} + \rho_i^k \operatorname{div} \tilde{u}_i^{k+1} = - \tilde{u}_i^k \cdot \nabla \rho_i^k - \rho_i^k \operatorname{div} u_i^k, \\ & \partial_t \tilde{u}_i^{k+1} + (u_i^k \cdot \nabla) \tilde{u}_i^{k+1} + h_i'(\rho_i^k) \nabla \tilde{\rho}_i^{k+1} = -(\tilde{u}_i^k \cdot \nabla) u_i^k - (h_i'(\rho_i^k) - h_i'(\rho_i^{k-1})) \nabla \rho_i^k + \tilde{E}^k \\ & \qquad + u_i^k \times B^k - u_i^{k-1} \times B^{k-1} - (\tilde{u}_i^k - \tilde{u}_e^k) - \tilde{u}_i^k, \\ & \partial_t \tilde{u}_e^{k+1} + (u_e^k \cdot \nabla) \tilde{u}_e^{k+1} + \nabla \tilde{p}_e^{k+1} = -(\tilde{u}_e^k \cdot \nabla) u_e^k - \tilde{E}^k - u_e^k \times B^k + u_e^{k-1} \times B^{k-1} \\ & \qquad - (\rho_i^k (u_e^k - u_i^k) - \rho_i^{k-1} (u_e^{k-1} - u_i^{k-1})) - \tilde{u}_e^k, \\ & \partial_t \tilde{E}^{k+1} - \nabla \times \tilde{G}^{k+1} = \tilde{u}_e^k - (\rho_i^k u_i^k - \rho_i^{k-1} u_i^{k-1}), \quad \partial_t \tilde{G}^{k+1} + \nabla \times \tilde{E}^{k+1} = 0, \\ & t = 0: \quad (\tilde{\rho}_i^{k+1}, \tilde{u}_\nu^{k+1}, \tilde{E}^{k+1}, \tilde{G}^{k+1}) = (0, 0, 0, 0). \end{aligned} \right. \quad (3.15) $$

Making the inner product of (3.15) with $h_i'(\rho_i^k)\tilde{\rho}_i^{k+1}$, $\rho_i^k\tilde{u}_i^{k+1}$, $\tilde{u}_e^{k+1}$, $\tilde{E}^{k+1}$ and $\tilde{G}^{k+1}$ respectively yields

$$ \begin{aligned} & \frac{d}{dt} \left( \langle h_i'(\rho_i^k) \tilde{\rho}_i^{k+1}, \tilde{\rho}_i^{k+1} \rangle + \langle \rho_i^k \tilde{u}_i^{k+1}, \tilde{u}_i^{k+1} \rangle + \|(\tilde{u}_e^{k+1}, \tilde{E}^{k+1}, \tilde{G}^{k+1})\|^2 \right) \\ & \le C \|(\tilde{\rho}_i^{k+1}, \tilde{u}_\nu^{k+1}, \tilde{E}^{k+1}, \tilde{G}^{k+1})\|^2 + C \|(\tilde{\rho}_i^k, \tilde{u}_\nu^k, \tilde{E}^k, \tilde{G}^k)\|^2. \end{aligned} $$

Integrating the above over $[0, t]$ for $t < T^*$, we have

$$ \| (\tilde{\rho}_i^{k+1}, \tilde{u}_\nu^{k+1}, \tilde{E}^{k+1}, \tilde{G}^{k+1})(t) \|^2 \le C \int_0^t \left( \| (\tilde{\rho}_i^{k+1}, \tilde{u}_\nu^{k+1}, \tilde{E}^{k+1}, \tilde{G}^{k+1})(\tau) \|^2 + \| (\tilde{\rho}_i^k, \tilde{u}_\nu^k, \tilde{E}^k, \tilde{G}^k)(\tau) \|^2 \right) d\tau, $$
which, by the Gronwall inequality, implies

$$ \| (\tilde{\rho}_i^{k+1}, \tilde{u}_\nu^{k+1}, \tilde{E}^{k+1}, \tilde{G}^{k+1})(t) \|^2 \le Cte^{Ct} \sup_{\tau \in [0,t]} \| (\tilde{\rho}_i^k, \tilde{u}_\nu^k, \tilde{E}^k, \tilde{G}^k)(\tau) \|^2. \quad (3.16) $$

Since $Cte^{Ct} \to 0$ as $t \to 0$, we choose $T_0 > 0$ small enough such that $Cte^{Ct} < \frac{1}{2}$ when $t \le T_0$. Let

$$ T^{**} = \frac{1}{2} \min\{T^*, T_0\}. \quad (3.17) $$
It follows from (3.16) for all $k \ge 1$ that

$$ \sum_{k=1}^{\infty} \sup_{\tau \in [0, T^{**}]} \|(\tilde{\rho}_i^{k+1}, \tilde{u}_\nu^{k+1}, \tilde{E}^{k+1}, \tilde{G}^{k+1})(\tau)\|^2 \le C \sum_{k=1}^{\infty} \frac{1}{2^k} < \infty. $$

Thus, the sequence $(\rho_i^k, u_\nu^k, E^k, B^k)$ converges to a limit $(\hat{\rho}_i, \hat{u}_\nu, \hat{E}, \hat{B})$ in the following strong sense:

$$ (\rho_i^k, u_\nu^k, E^k, B^k) \to (\hat{\rho}_i, \hat{u}_\nu, \hat{E}, \hat{B}), \quad \text{strongly in } L^\infty([0, T^{**}]; L^2). \quad (3.18) $$

Due to (3.11) and the lower-continuity of norm for weak convergence, $(\hat{\rho}_i, \hat{u}_\nu, \hat{E}, \hat{B})$ still satisfies (3.11). Combining the strong convergence (3.18) and the weak convergence (3.12), we obtain that $(\hat{\rho}_i, \hat{u}_i, \hat{p}_e, \hat{u}_e, \hat{E}, \hat{B})$ satisfies problem (3.1) in the sense of distributions. Notice that at this point the following two relations automatically hold if they are satisfied initially, since

$$ \partial_t \mathrm{div} \hat{E} = -\mathrm{div}(\hat{\rho}_i \hat{u}_i) = \partial_t \hat{\rho}_i, \quad \partial_t \mathrm{div} \hat{B} = 0. $$

Hence the existence of a classical solution is proved.

**Step 2:** Uniqueness. Let $(\hat{\rho}_i^l, \hat{u}_i^l, \hat{p}_e^l, \hat{u}_e^l, \hat{E}^l, \hat{B}^l)(l = 1, 2)$ be two solutions to (3.1) satisfying the uniform a priori estimate (3.11). Set

$$ \underline{\rho}_i = \hat{\rho}_i^1 - \hat{\rho}_i^2, \quad \underline{u}_\nu = \hat{u}_\nu^1 - \hat{u}_\nu^2, \quad \underline{E} = \hat{E}^1 - \hat{E}^2, \quad \underline{G} = \hat{B}^1 - \hat{B}^2, \quad \underline{p}_e = \hat{p}_e^1 - \hat{p}_e^2. $$

Then $(\underline{\rho}_i, \underline{u}_\nu, \underline{E}, \underline{G}, \underline{p}_e)$ satisfies the following system

$$ \left\{ \begin{aligned} \partial_t \underline{\rho}_i + \hat{u}_i^1 \cdot \nabla \underline{\rho}_i + \hat{\rho}_i^1 \mathrm{div} \underline{u}_i &= -\underline{u}_i \cdot \nabla \hat{\rho}_i^2 - \underline{\rho}_i \mathrm{div} \hat{u}_i^2, \\ \partial_t \underline{u}_i + (\hat{u}_i^1 \cdot \nabla) \underline{u}_i + \nabla h_i(\hat{\rho}_i^1) - \nabla h_i(\hat{\rho}_i^2) &= -(\underline{u}_i \cdot \nabla) \hat{u}_i^2 + \underline{E} + \hat{u}_i^1 \times \hat{B}^1 - \hat{u}_i^2 \times \hat{B}^2 - (\underline{u}_i - \underline{u}_e) - \underline{u}_i, \\ \partial_t \underline{u}_e + (\hat{u}_e^1 \cdot \nabla) \underline{u}_e + \nabla \underline{p}_e &= -(\underline{u}_e \cdot \nabla) \hat{u}_e^2 - \underline{E} - (\hat{u}_e^1 \times \hat{B}^1 - \hat{u}_e^2 \times \hat{B}^2) - (\hat{\rho}_i^1(\hat{u}_e^1 - \hat{u}_e^1) - \hat{\rho}_i^2(\hat{u}_e^2 - \hat{u}_e^2)) - \underline{u}_e, \\ \partial_t \underline{E} - \nabla \times \underline{G} &= \underline{u}_e - (\hat{\rho}_i^1 \hat{u}_i^1 - \hat{\rho}_i^2 \hat{u}_i^2), \quad \mathrm{div} \underline{E} = \underline{\rho}_i, \\ \partial_t \underline{G} + \nabla \times \underline{E} &= 0, \quad \mathrm{div} \underline{G} = 0, \\ t = 0: \quad (\underline{\rho}_i, \underline{u}_\nu, \underline{E}, \underline{G}) &= (0, 0, 0, 0). \end{aligned} \right. $$

By using classical energy estimates, we obtain that

$$ \frac{d}{dt} \|(\underline{\rho}_i, \underline{u}_\nu, \underline{E}, \underline{G})(t)\|^2 \le C \|(\underline{\rho}_i, \underline{u}_\nu, \underline{E}, \underline{G})(t)\|^2, $$

which, by the Gronwall inequality, implies that $(\underline{\rho}_i, \underline{u}_\nu, \underline{E}, \underline{G}) = (0, 0, 0, 0)$. As a result, $\underline{p}_e = 0$ since $\underline{p}_e$ satisfies the Laplace equation. Hence the uniqueness is obtained.

**Step 3:** Time continuity can be obtained via the same arguments as in the end of Subsection 3.2. Hence the proof of Theorem 3.1 is complete. □

**Remark 3.1.** Our method is also applicable for the existence of smooth solutions to other partial-compressible models, see for instance system (4.5) in [26].

**Remark 3.2.** The collision and the dissipation structures are not necessary in proving Theorem 3.1.
## 4. UNIFORM GLOBAL ESTIMATES

In this section, we denote by $s \ge 3$ an integer and $C$ a generic positive constant independent of the small parameter $\lambda$, $m_e$ and any time. We first prove Theorem 2.1. In what follows, we drop the superscript $\varepsilon$ for simplicity and assume that the conditions in Theorem 2.1 hold. Let

$$n_i = \rho_i - 1, \quad G = B - \tilde{B}, \quad U_i = \begin{pmatrix} n_i \\ u_i \end{pmatrix}, \quad U = (U_i, p_e, u_e^\top, E^\top, G^\top)^\top.$$

Without loss of generality, let $T > 0$ and $U$ be the local smooth solution of (2.1) defined on the interval $[0, T]$. For $k = 0, 1$, we introduce

$$\begin{aligned} \mathcal{E}_k(t) &= \|\partial_t^k (n_i, \sqrt{m_e}u_e, u_i, \lambda E, G)(t)\|_{s-k}^2 + \frac{1}{\lambda^2} \|\partial_t^k n_i\|_{s-1-k}^2, \\ \mathcal{D}_k(t) &= \|\partial_t^k (\nabla n_i(t), \lambda^{-1}n_i, \lambda E)(t)\|_{s-1-k}^2 + \|\partial_t^k (u_e, u_i)(t)\|_{s-k}^2 + \|\partial_t^k \nabla \times B(t)\|_{s-2-k}^2. \end{aligned}$$

For simplicity, we denote

$$\mathcal{H}_k(t) := \mathcal{E}_k^{1/2}(t) \mathcal{D}_0^{1/2}(t) \mathcal{D}_k^{1/2}(t) + \mathcal{E}_0^{1/2}(t) \mathcal{D}_k(t), \quad \mathcal{E}_T = \sup_{0 \le t \le T} \mathcal{E}_0(t).$$

In this section, we assume that $\mathcal{E}_T$ is uniformly sufficiently small with respect to $T$ and $\varepsilon$ and it is worth mentioning that we do not require $\mathcal{E}_1(0)$ is uniformly bounded in this section. Besides, because of the smallness of $\mathcal{E}_T$, it is reasonable to assume

$$\frac{1}{2} \le \rho_i \le \frac{3}{2} \quad \text{and} \quad h'_i(\rho_i) \ge h_1, \qquad (4.1)$$

where $h_1$ is a positive constant independent of the small parameter $\varepsilon$ and any time.

### 4.1. $L^2$ estimate. We first have

**Lemma 4.1.** (L$^2$-estimate) It holds

$$\| (n_i, \sqrt{m_e}u_e, u_i, \lambda E, G)(t) \|^2 + \int_0^t \|(u_e, u_i)(\tau)\|^2 d\tau \le C \mathcal{E}_0(0). \qquad (4.2)$$

*Proof.* Set

$$\begin{aligned} \eta(U) &= \frac{1}{2}\rho_i|u_i|^2 + H_i(\rho_i) + \frac{m_e}{2}|u_e|^2 + \frac{1}{2}|\lambda E|^2 + \frac{1}{2}|G|^2, \\ \psi(U) &= \frac{1}{2}\rho_i|u_i|^2u_i + \rho_ih_i(\rho_i)u_i + \frac{1}{2}m_eu_e|u_e|^2 + p_eu_e + E \times G, \end{aligned}$$

where $H'_i(\rho_i) = h_i(\rho_i)$. Then from the system (2.1), we obtain

$$\partial_t \eta(U) + \rho_i |u_e - u_i|^2 + \rho_i |u_i|^2 + m_e |u_e|^2 + \operatorname{div} \psi(U) = 0. \qquad (4.3)$$

By the Taylor formula, it holds

$$H_i(\rho_i) = H_i(1) + h_i(1)n_i + \frac{1}{2}h'_i(\rho_i^*)n_i^2,$$

where $\rho_i^*$ is between 1 and $\rho_i$. Since $\partial_t n_i = -\operatorname{div}(\rho_i u_i)$, we have

$$\partial_t H_i(\rho_i) = -h_i(1)\operatorname{div}(\rho_i u_i) + \frac{1}{2}\partial_t (h'_i(\rho_i^*)n_i^2).$$

Substituting the above into (4.3), integrating the resulting equation over $[0, T] \times \mathbb{R}^3$ and noticing the lower bounds of $\rho_i$ and $h'_i(\rho_i)$ stated in (4.1) yield (4.2). $\square$
4.2. Div-curl estimate. Taking $\nabla$ and div operators to the first two equations in (2.1), respectively, we obtain

$$ egin{cases} 
\partial_t(\nabla n_i) + (u_i \cdot \nabla)(\nabla n_i) + \rho_i \nabla \mathrm{div} u_i = -R_n, \\
\partial_t \mathrm{div} u_i + (u_i \cdot \nabla)\mathrm{div} u_i + h'_i(\rho_i)\mathrm{div}(\nabla n_i) = \mathrm{div} E + \mathrm{div}(u_i \times B) - 2\mathrm{div} u_i - R_u,
\end{cases} $$

in which

$$ R_n = (\nabla n_i \cdot \nabla)u_i + \nabla n_i \times (\nabla \times u_i) + \mathrm{div} u_i \nabla n_i, $$

$$ R_u = \mathrm{div}((u_i \cdot \nabla)u_i) - (u_i \cdot \nabla)\mathrm{div}u_i + \nabla h'_i(\rho_i) \cdot \nabla n_i. $$

Let $V = ((\nabla n_i)^\top, \mathrm{div} u_i)^\top$, then $V$ satisfies

$$ \partial_t V + \sum_{j=1}^{3} D_j(U_i) \partial_{x_j} V = f - R, \qquad (4.4) $$

where

$$ D_j(U_i) = \begin{pmatrix} u_i^j \mathbb{I}_3, & \rho_i \xi_j \\ h'_i(\rho_i) \xi_j^\top & u_i^j \end{pmatrix}, \quad f = \begin{pmatrix} 0 \\ \mathrm{div} E + \mathrm{div}(u_i \times B) - 2\mathrm{div} u_i \end{pmatrix}, \quad R = \begin{pmatrix} R_n \\ R_u \end{pmatrix}. $$

Here $u_i = (u_i^1, u_i^2, u_i^3)^\top$. Define a symmetrizer $D_0(\rho_i)$ such that

$$ D_0(\rho_i) = \mathrm{diag} \left( \frac{h'_i(\rho_i)}{\rho_i} \mathbb{I}_3, 1 \right), \quad \tilde{D}_j(U_i) := D_0(\rho_i) D_j(U_i) = \begin{pmatrix} \frac{h'_i(\rho_i)}{\rho_i} u_i^j \mathbb{I}_3 & h'_i(\rho_i) \xi_j \\ h'_i(\rho_i) \xi_j^\top & u_i^j \end{pmatrix}. $$

Then $D_0(\rho_i)$ is symmetric and positive definite when $\rho_i > 0$, and $\tilde{D}_j(U_i)$ are symmetric. In the rest of the section, let $\alpha, \gamma \in \mathbb{N}^3$ be multi-indexes and $k$ be an integer satisfying

$$ k + |\alpha| \le s - 1, \quad k + |\gamma| \le s - 2, \quad k = 0, 1. $$

We then have the estimates for $\nabla n_i$, $\mathrm{div} u_i$ and $\mathrm{div} E$.

**Lemma 4.2.** It holds

$$ \frac{d}{dt} \left( \langle D_0(\rho_i) \partial_x^\alpha \partial_t^k V, \partial_x^\alpha \partial_t^k V \rangle + \|\lambda \partial_x^\alpha \partial_t^k \mathrm{div} E \|^2 \right) + 4 \|\partial_x^\alpha \partial_t^k \mathrm{div} u_i \|^2 \le C H_k(t) + r_{\mathrm{div}}^{\alpha,k}, \quad (4.5) $$

where

$$ r_{\mathrm{div}}^{\alpha,k} = 2 \left\langle \partial_x^\alpha \partial_t^k \mathrm{div} u_i, \partial_x^\alpha \partial_t^k \mathrm{div}(u_i \times \tilde{B}) \right\rangle. \quad (4.6) $$

*Proof.* Applying $\partial_x^\alpha \partial_t^k$ to the equation (4.4), and making the inner product with $2D_0(\rho_i) \partial_x^\alpha \partial_t^k V$ in $L^2$ yield the energy equality:

$$ \frac{d}{dt} \langle D_0(\rho_i) \partial_x^\alpha \partial_t^k V, \partial_x^\alpha \partial_t^k V \rangle = \langle \mathrm{div} D \partial_x^\alpha \partial_t^k V, \partial_x^\alpha \partial_t^k V \rangle + 2 \langle D_0(\rho_i) \partial_x^\alpha \partial_t^k f, \partial_x^\alpha \partial_t^k V \rangle \\ -2 \langle D_0(\rho_i) \partial_x^\alpha \partial_t^k R, \partial_x^\alpha \partial_t^k V \rangle + 2 \langle D_0(\rho_i) g_{\alpha,k}^{\mathrm{div}}, \partial_x^\alpha \partial_t^k V \rangle := \sum_{j=1}^4 I_j, $$

with the natural correspondence of $\{I_j\}_{j=1}^4$ and

$$ \mathrm{div} D = \partial_t D_0(\rho_i) + \sum_{j=1}^3 \partial_{x_j} \tilde{D}_j(U_i), $$

$$ g_{\alpha,k}^{\mathrm{div}} = -\partial_x^\alpha \partial_t^k \left( \sum_{j=1}^3 D_j(U_i) \partial_{x_j} V \right) + \sum_{j=1}^3 D_j(U_i) \partial_{x_j} \partial_x^\alpha \partial_t^k V. $$
Notice that

$$|I_1| \le C \|\mathrm{div} D\|_{\infty} \|\partial_x^\alpha \partial_t^k V\|^2 \le C (\|\partial_t n_i\|_{\infty} + C \|\nabla U_i\|_{\infty}) \|\partial_x^\alpha \partial_t^k V\|^2 \le C\mathcal{H}_k(t). \quad (4.7)$$

For $I_2$, since

$$I_2 = 2 \langle \partial_x^\alpha \partial_t^k \mathrm{div} E, \partial_x^\alpha \partial_t^k \mathrm{div} u_i \rangle - 4 \| \partial_x^\alpha \partial_t^k \mathrm{div} u_i \|^2 + 2 \langle \partial_x^\alpha \partial_t^k \mathrm{div}(u_i \times G), \partial_x^\alpha \partial_t^k \mathrm{div} u_i \rangle + r_{\mathrm{div}}^{\alpha,k},$$

where $r_{\mathrm{div}}^{\alpha,k}$ is defined in (4.6). Since

$$|2 \langle \partial_x^\alpha \partial_t^k \mathrm{div}(u_i \times G), \partial_x^\alpha \partial_t^k \mathrm{div} u_i \rangle| \le C\mathcal{H}_k(t),$$

we obtain that

$$I_2 \le 2 \langle \partial_x^\alpha \partial_t^k \mathrm{div} E, \partial_x^\alpha \partial_t^k \mathrm{div} u_i \rangle - 4 \| \partial_x^\alpha \partial_t^k \mathrm{div} u_i \|^2 + C\mathcal{H}_k(t) + r_{\mathrm{div}}^{\alpha,k}. \quad (4.8)$$

By Moser-type inequalities, we have directly that

$$|I_3| \le C \|\partial_x^\alpha \partial_t^k R\| \|\partial_x^\alpha \partial_t^k V\| \le C\mathcal{H}_k(t), \quad (4.9)$$

$$|I_4| \le C\mathcal{H}_k(t). \quad (4.10)$$

Combining the estimates (4.7)-(4.10) yields

$$\frac{d}{dt} \langle D_0(\rho_i) \partial_x^\alpha \partial_t^k V, \partial_x^\alpha \partial_t^k V \rangle \le 2 \langle \partial_x^\alpha \partial_t^k \mathrm{div} E, \partial_x^\alpha \partial_t^k \mathrm{div} u_i \rangle - 4 \| \partial_x^\alpha \partial_t^k \mathrm{div} u_i \|^2 + C\mathcal{H}_k(t) + r_{\mathrm{div}}^{\alpha,k}.$$

To estimate the first term on the right hand side of the above estimate, we need to combine the Maxwell equations. Taking $\partial_x^\alpha \partial_t^k \mathrm{div}$ to the fourth evolutionary equation in (2.1), and making the inner product of the resulting equation with $\partial_x^\alpha \partial_t^k \mathrm{div} E$ yield

$$\frac{d}{dt} \| \lambda \mathrm{div} \partial_x^\alpha \partial_t^k E \|^2 = -2 \langle \partial_x^\alpha \partial_t^k \mathrm{div} E, \partial_x^\alpha \partial_t^k \mathrm{div}(\rho_i u_i) \rangle.$$

Combining the above two estimates, we obtain

$$\begin{align*}
\frac{d}{dt} \left( \langle D_0(\rho_i) \partial_x^\alpha \partial_t^k V, \partial_x^\alpha \partial_t^k V \rangle + \| \lambda \partial_x^\alpha \partial_t^k \mathrm{div} E \|^2 \right) + 4 \| \partial_x^\alpha \partial_t^k \mathrm{div} u_i \|^2 \\
&\le -2 \langle \partial_x^\alpha \partial_t^k \mathrm{div} E, \partial_x^\alpha \partial_t^k \mathrm{div}(n_i u_i) \rangle + r_{\mathrm{div}}^{\alpha,k} + C\mathcal{H}_k(t) \\
&\le -\frac{2}{\lambda^2} \langle \partial_x^\alpha \partial_t^k n_i, (\partial_x^\alpha \partial_t^k \nabla n_i) \cdot u_i \rangle - \frac{2}{\lambda^2} \langle \partial_x^\alpha \partial_t^k n_i, \partial_x^\alpha \partial_t^k (\nabla n_i \cdot u_i) - (\partial_x^\alpha \partial_t^k \nabla n_i) \cdot u_i \rangle \\
&\quad -\frac{2}{\lambda^2} \langle \partial_x^\alpha \partial_t^k n_i, \partial_x^\alpha \partial_t^k (n_i \mathrm{div} u_i) \rangle + r_{\mathrm{div}}^{\alpha,k} + C\mathcal{H}_k(t) \\
&\le \frac{C}{\lambda^2} \| \mathrm{div} u_i \|_{\infty} \| \partial_x^\alpha \partial_t^k n_i \|^2 + C\mathcal{H}_k(t) + r_{\mathrm{div}}^{\alpha,k} \le C\mathcal{H}_k(t) + r_{\mathrm{div}}^{\alpha,k},
\end{align*}$$

where the Gaussian theorem is used. Hence the proof is complete. □

The next lemma concerns the estimates for $\nabla \times u_\nu$, $\nabla \times E$ and $\nabla \times G$.

**Lemma 4.3.** It holds

$$\begin{align}
\frac{d}{dt} ( \| \partial_x^\alpha \partial_t^k (\sqrt{m_e} \omega_e, \omega_i, \lambda \nabla \times \partial_x^\alpha \partial_t^k E, \nabla \times \partial_x^\alpha \partial_t^k G) \|^2 - 2 \langle \partial_x^\alpha \partial_t^k G, \partial_x^\alpha \partial_t^k \nabla \times (n_i u_i) \rangle ) \nonumber \\
+ 2 \| \partial_x^\alpha \partial_t^k (\omega_e - \omega_i, \omega_i) \|^2 \le C\mathcal{H}_k(t) + r_{\mathrm{rot},i}^{\alpha,k}, \tag{4.11}
\end{align}$$

where $\omega_\nu = \nabla \times u_\nu$ and

$$r_{\mathrm{rot},i}^{\alpha,k} = 2 \langle \partial_x^\alpha \partial_t^k \nabla \times (u_i \times \tilde{B}), \partial_x^\alpha \partial_t^k \omega_i \rangle. \quad (4.12)$$
*Proof.* For simplicity, we introduce the following auxiliary parameters:

$$ \beta_e = 1, \quad \beta_i = 0. $$

Applying $\nabla \times$ to the second and third equation in (2.1), with $\nu = e, i$, yields

$$ m_e^{\beta\nu} \partial_t \omega_\nu + m_e^{\beta\nu} \left( (u_\nu \cdot \nabla)\omega_\nu - (\omega_\nu \cdot \nabla)u_\nu + \omega_\nu \operatorname{div} u_\nu \right) = q_\nu \nabla \times (E + u_\nu \times B) - \operatorname{diss}_\nu, \quad (4.13) $$

in which

$$ \operatorname{diss}_e = \nabla \times (\rho_i(u_e - u_i)) + m_e \omega_e, \quad \operatorname{diss}_i = \nabla \times (u_i - u_e) + \omega_i. $$

Applying $\partial_x^\alpha \partial_t^k$ to (4.13), and making the inner product with $2\partial_x^\alpha \partial_t^k \omega_\nu$ in $L^2$ yield:

$$ \begin{aligned} m_e^{\beta\nu} \frac{d}{dt} \|\partial_x^\alpha \partial_t^k \omega_\nu\|^2 = & m_e^{\beta\nu} \langle \operatorname{div} u_\nu \partial_x^\alpha \partial_t^k \omega_\nu, \partial_x^\alpha \partial_t^k \omega_\nu \rangle + 2 \langle q_\nu \nabla \times \partial_x^\alpha \partial_t^k E, \partial_x^\alpha \partial_t^k \omega_\nu \rangle \\ & - 2m_e^{\beta\nu} \langle \partial_x^\alpha \partial_t^k ((u_\nu \cdot \nabla)\omega_\nu) - (u_\nu \cdot \nabla)\partial_x^\alpha \partial_t^k \omega_\nu, \partial_x^\alpha \partial_t^k \omega_\nu \rangle \\ & - 2m_e^{\beta\nu} \langle \partial_x^\alpha \partial_t^k ((\omega_\nu \cdot \nabla)u_\nu - \omega_\nu \operatorname{div} u_\nu), \partial_x^\alpha \partial_t^k \omega_\nu \rangle \\ & + 2 \langle q_\nu \nabla \times \partial_x^\alpha \partial_t^k (u_\nu \times B), \partial_x^\alpha \partial_t^k \omega_\nu \rangle - 2 \langle \partial_x^\alpha \partial_t^k \operatorname{diss}_\nu, \partial_x^\alpha \partial_t^k \omega_\nu \rangle := \sum_{j=1}^6 J_j^\nu, \end{aligned} $$

with the natural correspondence of $\{J_j^\nu\}_{j=1}^6$. The estimates of $J_1^\nu, J_3^\nu$ and $J_4^\nu$ follow directly from the Moser-type calculus inequalities:

$$ |J_1^\nu| + |J_3^\nu| + |J_4^\nu| \le C\mathcal{H}_k(t). $$

We keep $J_2^\nu$ for the moment, since these two estimates should be treated together with the Maxwell equations and the corresponding estimates for the other fluid. For $J_5^\nu$, we obtain

$$ J_5^\nu = 2 \langle q_\nu \nabla \times \partial_x^\alpha \partial_t^k (u_\nu \times G), \partial_x^\alpha \partial_t^k \omega_\nu \rangle + r_{\text{rot},\nu}^{\alpha,k} \le C\mathcal{H}_k(t) + r_{\text{rot},\nu}^{\alpha,k}, $$

where $r_{\text{rot},i}^{\alpha,k}$ is defined in (4.12) and

$$ r_{\text{rot},e}^{\alpha,k} := -2 \langle \partial_x^\alpha \partial_t^k \nabla \times (u_e \times \tilde{B}), \partial_x^\alpha \partial_t^k \omega_e \rangle = 2 \langle \partial_x^\alpha \partial_t^k u_e \times \tilde{B}, \Delta \partial_x^\alpha \partial_t^k u_e \rangle = 0. $$

Notice that $J_6^e + J_6^i$ admits the form

$$ \begin{aligned} J_6^e + J_6^i = & -2 \|\partial_x^\alpha \partial_t^k \omega_i\|^2 - 2 \|\partial_x^\alpha \partial_t^k (\omega_e - \omega_i)\|^2 - \langle \partial_x^\alpha \partial_t^k \omega_e, \partial_x^\alpha \partial_t^k \nabla \times (n_i(u_e - u_i)) \rangle \\ & \le -2 \|\partial_x^\alpha \partial_t^k \omega_i\|^2 - 2 \|\partial_x^\alpha \partial_t^k (\omega_e - \omega_i)\|^2 + C\mathcal{H}_k(t). \end{aligned} $$

Combining these estimates, we conclude that

$$ \frac{d}{dt} \| \partial_x^\alpha \partial_t^k (\sqrt{m_e \omega_e}, \omega_i) \|^2 + 2 \| \partial_x^\alpha \partial_t^k \omega_i \|^2 + 2 \| \partial_x^\alpha \partial_t^k (\omega_e - \omega_i) \|^2 \le J_2^i + J_2^e + C\mathcal{H}_k(t) + r_{\text{rot},i}^{\alpha,k}. \quad (4.14) $$

Next, applying $\partial_x^\alpha \partial_t^k \nabla \times$ to the fourth and fifth evolutionary equations in (2.1) respectively, and making the inner product of the resulting equation with $(\nabla \times \partial_x^\alpha \partial_t^k E, \nabla \times \partial_x^\alpha \partial_t^k G)$ in $L^2$ yield

$$ \frac{d}{dt} (\|\lambda \nabla \times \partial_x^\alpha \partial_t^k E\|^2 + \|\nabla \times \partial_x^\alpha \partial_t^k G\|^2) = 2 \langle \nabla \times \partial_x^\alpha \partial_t^k E, \partial_x^\alpha \partial_t^k \omega_e - \partial_x^\alpha \partial_t^k \nabla \times (\rho_i u_i) \rangle =: J_7. \quad (4.15) $$

Direct calculations show that

$$ \begin{aligned} \frac{1}{2}(J_2^i + J_2^e + J_7) &= - \langle \nabla \times \partial_x^\alpha \partial_t^k E, \partial_x^\alpha \partial_t^k \nabla \times (n_i u_i) \rangle \\ &= \langle \partial_t \partial_x^\alpha \partial_t^k G, \partial_x^\alpha \partial_t^k \nabla \times (n_i u_i) \rangle \\ &= \frac{d}{dt} \langle \partial_x^\alpha \partial_t^k G, \partial_x^\alpha \partial_t^k \nabla \times (n_i u_i) \rangle - \langle \nabla \times \partial_x^\alpha \partial_t^k G, \partial_x^\alpha \partial_t^k (\partial_t n_i u_i) \rangle \end{aligned} $$
$$ - \langle \nabla \times \partial_x^\alpha \partial_t^k G, \partial_x^\alpha \partial_t^k (n_i \partial_t u_i) \rangle := \sum_{k=8}^{10} J_k, $$

with the natural correspondence of {$J_k$}$_{k=8}^{10}$. We keep $J_8$ for the moment. We obtain through direct calculations and the mass conservation for $\rho_i$ that $J_9$ is non-singular having the estimate:

$$ |J_9| \le C\mathcal{H}_k(t). $$

To treat $J_{10}$, by using the momentum equation for $u_i$ in (2.1), we obtain directly that

$$ |J_{10}| \le C\mathcal{H}_k(t) + C\|\nabla \times \partial_x^\alpha \partial_t^k G\|\|\lambda^{-1} \partial_t^k n_i\|_{s-1-k} \|\lambda \partial_t^k E\|_{s-1-k} \le C\mathcal{H}_k(t). $$

These estimates show that

$$ J_2^i + J_2^e + J_7 \le 2 \frac{d}{dt} \langle \partial_x^\alpha \partial_t^k G, \partial_x^\alpha \partial_t^k \nabla \times (n_i u_i) \rangle + C\mathcal{H}_k(t). $$

Adding (4.14) and (4.15) and combining the above estimates end the proof. □

### 4.3. Dissipation estimates.

**Lemma 4.4. (Dissipation of $\nabla n_i$) It holds**

$$ \frac{d}{dt} \langle \partial_x^\alpha \partial_t^k u_i, \partial_x^\alpha \partial_t^k \nabla n_i \rangle + \frac{h_1}{2} \| \partial_x^\alpha \partial_t^k \nabla n_i \|^2 + \frac{1}{\lambda^2} \| \partial_x^\alpha \partial_t^k n_i \|^2 \le C \| \partial_t^k (u_i, u_e) \|_{s-k}^2 + C\mathcal{H}_k(t). \quad (4.16) $$

*Proof.* Applying $\partial_x^\alpha \partial_t^k$ to the momentum equation of ions in (2.1) yields

$$ h'_i(\rho_i) \partial_x^\alpha \partial_t^k \nabla n_i - \partial_x^\alpha \partial_t^k E = - \partial_t \partial_x^\alpha \partial_t^k u_i - \partial_x^\alpha \partial_t^k ((u_i \cdot \nabla) u_i) - \partial_x^\alpha \partial_t^k (u_i - u_e) - \partial_x^\alpha \partial_t^k u_i \\ + \partial_x^\alpha \partial_t^k (u_i \times B) + (h'_i(\rho_i) \partial_x^\alpha \partial_t^k \nabla n_i - \partial_x^\alpha \partial_t^k (h'_i(\rho_i) \nabla n_i)). $$

Taking the inner product of the above equation with $\partial_x^\alpha \partial_t^k \nabla n_i$, we have

$$ \begin{aligned} & \langle \partial_x^\alpha \partial_t^k \nabla n_i, h'_i(\rho_i) \partial_x^\alpha \partial_t^k \nabla n_i \rangle - \langle \partial_x^\alpha \partial_t^k \nabla n_i, \partial_x^\alpha \partial_t^k E \rangle \\ &= - \langle \partial_x^\alpha \partial_t^k \nabla n_i, \partial_t \partial_x^\alpha \partial_t^k u_i \rangle + \langle \partial_x^\alpha \partial_t^k \nabla n_i, (h'_i(\rho_i) \partial_x^\alpha \partial_t^k \nabla n_i - \partial_x^\alpha \partial_t^k (h'_i(\rho_i) \nabla n_i)) \rangle \\ & \quad - \langle \partial_x^\alpha \partial_t^k \nabla n_i, \partial_x^\alpha \partial_t^k (((u_i \cdot \nabla)u_i) + u_i - u_e + u_i - u_i \times B) \rangle \\ & \le - \frac{d}{dt} \langle \partial_x^\alpha \partial_t^k u_i, \partial_x^\alpha \partial_t^k \nabla n_i \rangle + \frac{h_1}{2} \| \partial_x^\alpha \partial_t^k \nabla n_i \|^2 + C \| \partial_t^k (u_i, u_e - u_i) \|_{s-k}^2 + C\mathcal{H}_k(t). \end{aligned} $$

It is clear that by (4.1),

$$ \langle \partial_x^\alpha \partial_t^k \nabla n_i, h'_i(\rho_i) \partial_x^\alpha \partial_t^k \nabla n_i \rangle \ge h_1 \| \partial_x^\alpha \partial_t^k \nabla n_i \|^2, $$

and an integration by parts gives

$$ - \langle \partial_x^\alpha \partial_t^k \nabla n_i, \partial_x^\alpha \partial_t^k E \rangle = \langle \partial_x^\alpha \partial_t^k n_i, \partial_x^\alpha \partial_t^k \mathrm{div} E \rangle = \frac{1}{\lambda^2} \| \partial_x^\alpha \partial_t^k n_i \|^2. $$

Combining all these estimates, we obtain (4.16). □

**Lemma 4.5. (Dissipation estimate for $E$ and $G$)** For $\theta \in (0, 1)$ to be determined, it holds

$$ \begin{aligned} -\frac{d}{dt} \langle \lambda^2 \partial_x^\alpha \partial_t^k E, \partial_x^\alpha \partial_t^k u_i \rangle + \frac{\lambda^2}{2} \| \partial_x^\alpha \partial_t^k E \|^2 \le C \theta^{-1} \| \partial_t^k (u_i, u_e) \|_{s-k}^2 + C \| \partial_t^k \nabla n_i \|_{s-1-k}^2 & \quad (4.17) \\ + \theta \| \partial_t^k \nabla G \|_{s-2-k}^2, & \end{aligned} $$

and

$$ \| \nabla \times \partial_t^k \partial_x^\gamma G \|^2 \le 2 \frac{d}{dt} \langle \lambda^2 \partial_t^k \partial_x^\gamma E, \nabla \times \partial_t^k \partial_x^\gamma G \rangle + 2\lambda^2 \| \nabla \times \partial_t^k \partial_x^\gamma E \|^2 + C \| \partial_t^k (u_i, u_e) \|_{s-k}^2. \quad (4.18) $$
Proof. Notice

$$E = \partial_t u_i + (u_i \cdot \nabla)u_i + \nabla h_i(\rho_i) - u_i \times B + (u_i - u_e) + u_i =: \partial_t u_i + f_E, \quad (4.19)$$

with the natural correspondence of $f_E$. Applying $\partial_x^\alpha \partial_t^k$ to (4.19), and making the inner product of the resulting equation with $\lambda^2 \partial_x^\alpha \partial_t^k E$ in $L^2$, we have

$$\lambda^2 \|\partial_x^\alpha \partial_t^k E\|^2 = \langle \lambda^2 \partial_x^\alpha \partial_t^k E, \partial_t \partial_x^\alpha \partial_t^k u_i \rangle + \langle \partial_x^\alpha \partial_t^k f_E, \lambda^2 \partial_x^\alpha \partial_t^k E \rangle. \quad (4.20)$$

The first term on the right hand side of the above equation is treated as follows:

$$\begin{aligned} \langle \lambda^2 \partial_x^\alpha \partial_t^k E, \partial_t \partial_x^\alpha \partial_t^k u_i \rangle &= \frac{d}{dt} \langle \lambda^2 \partial_x^\alpha \partial_t^k E, \partial_x^\alpha \partial_t^k u_i \rangle - \langle \lambda^2 \partial_t \partial_x^\alpha \partial_t^k E, \partial_x^\alpha \partial_t^k u_i \rangle \\ &= \frac{d}{dt} \langle \lambda^2 \partial_x^\alpha \partial_t^k E, \partial_x^\alpha \partial_t^k u_i \rangle - \langle \nabla \times \partial_x^\alpha \partial_t^k G, \partial_x^\alpha \partial_t^k u_i \rangle \\ &\quad - \langle \partial_x^\alpha \partial_t^k (u_e - \rho_i u_i), \partial_x^\alpha \partial_t^k u_i \rangle, \end{aligned}$$

where

$$\begin{aligned} |\langle \nabla \times \partial_t^k G, \partial_t^k u_i \rangle| &\le \theta \|\nabla \partial_t^k G\|_{s-k-2}^2 + C \theta^{-1} \|\partial_t^k u_i\|_{s-k}^2, \\ |\langle \nabla \times \partial_x^\alpha \partial_t^k G, \partial_x^\alpha \partial_t^k u_i \rangle| &= |\langle \partial_x^\alpha \partial_t^k G, \nabla \times \partial_x^\alpha \partial_t^k u_i \rangle| \\ &\le \theta \|\nabla \partial_t^k G\|_{s-k-2}^2 + C \theta^{-1} \|\partial_t^k u_i\|_{s-k}^2, \quad \text{when } |\alpha| \ge 1, \\ |\langle \partial_x^\alpha \partial_t^k (u_e - \rho_i u_i), \partial_x^\alpha \partial_t^k u_i \rangle| &\le C \|\partial_t^k (u_i, u_e)\|_{s-k}^2, \end{aligned}$$

for $\theta > 0$ sufficiently small to be determined. For the term containing $f_E$ in (4.20), we have

$$| \langle \partial_x^\alpha \partial_t^k f_E, \lambda^2 \partial_x^\alpha \partial_t^k E \rangle | \le \frac{1}{2} \lambda^2 \| \partial_x^\alpha \partial_t^k E \|^2 + C \| \partial_t^k (u_i, u_e) \|_{s-k}^2 + C \| \partial_t^k \nabla n_i \|_{s-1-k}^2,$$

Combining all these estimates yields (4.17).

Applying $\partial_t^k \partial_x^\gamma$ to the equation for $\partial_t E$ in (2.1), we obtain

$$\nabla \times \partial_t^k \partial_x^\gamma G = \lambda^2 \partial_t \partial_t^k \partial_x^\gamma E - \partial_t^k \partial_x^\gamma (u_e - \rho_i u_i).$$

Taking the inner product of the above with $\nabla \times \partial_t^k \partial_x^\gamma G$ in $L^2$, and using the Maxwell equations for $E$ and $B$ in (2.1), the Cauchy-Schwarz inequality and the Moser-type inequalities, we have

$$\begin{aligned} \|\nabla \times \partial_t^k \partial_x^\gamma G\|^2 &= \frac{d}{dt} \langle \lambda^2 \partial_t^k \partial_x^\gamma E, \nabla \times \partial_t^k \partial_x^\gamma G \rangle - \langle \lambda^2 \partial_t^k \partial_x^\gamma E, \nabla \times \partial_t^k \partial_x^\gamma \partial_t G \rangle \\ &\quad - \langle \partial_t^k \partial_x^\gamma (u_e - \rho_i u_i), \nabla \times \partial_t^k \partial_x^\gamma G \rangle \\ &\le \frac{d}{dt} \langle \lambda^2 \partial_t^k \partial_x^\gamma E, \nabla \times \partial_t^k \partial_x^\gamma G \rangle + \lambda^2 \|\nabla \times \partial_t^k \partial_x^\gamma E\|^2 \\ &\quad + \frac{1}{2} \|\nabla \times \partial_t^k \partial_x^\gamma G\|^2 + C \|\partial_t^k (u_i, u_e)\|_{s-k}^2, \end{aligned}$$

which implies (4.18). $\square$

4.4. **Time derivative estimate.** It remains to estimate the first-order time derivative of solutions. The first two equations in (2.1) can be written into the following

$$\partial_t U_i + \sum_{j=1}^{3} A_j(U_i) \partial_{x_j} U_i = Q(U_i, u_e, E, B), \quad (4.21)$$

in which

$$A_j(U_i) = \begin{pmatrix} u_i^j & \rho_i \xi_j^\top \\ h_i'(\rho_i) \xi_j & u_i^j \mathbb{I}_3 \end{pmatrix}, \quad Q(U_i, u_e, E, B) = \begin{pmatrix} 0 \\ E + (u_i \times B) - (u_i - u_e) - u_i \end{pmatrix}.$$
**Lemma 4.6.** (First-order time derivative) It holds

$$ \frac{d}{dt} \left( \langle A_0(\rho_i) \partial_t U_i, \partial_t U_i \rangle + \| (\lambda \partial_t E, \partial_t G, \sqrt{m_e} \partial_t u_e) \|^2 \right) + 2 \| (\partial_t (u_e - u_i), u_i) \|^2 \le C \mathcal{H}_1(t), \quad (4.22) $$

where the symmetrizer $A_0(\rho_i) = \text{diag}(h'_i(\rho_i)/\rho_i, \mathbb{I}_3)$.

*Proof.* Applying time derivative to (4.21) and making the inner product of the resulting equation with $2A_0(\rho_i)\partial_t U_i$ yield

$$ \begin{aligned} \frac{d}{dt} \langle A_0(\rho_i) \partial_t U_i, \partial_t U_i \rangle = & \left\langle \left( \partial_t A_0(\rho_i) + \sum_{j=1}^3 \partial_{x_j} A_j(U_i) \right) \partial_t U_i, \partial_t U_i \right\rangle \\ & - \left\langle \sum_{j=1}^3 \partial_t (A_j(U_i)) \partial_{x_j} U_i, 2A_0(\rho_i) \partial_t U_i \right\rangle \\ & + 2 \langle A_0(\rho_i) \partial_t U_i, \partial_t Q(U_i, u_e, E, B) \rangle := K_1 + K_2 + K_3, \end{aligned} $$

with the natural correspondence of $K_1, K_2$ and $K_3$. We obtain

$$ |K_1 + K_2| \le C \mathcal{H}_1(t). $$

For $K_3$, it holds

$$ \begin{aligned} K_3 &= 2 \langle \partial_t u_i, \partial_t E \rangle + 2 \langle \partial_t u_i, (u_i \times \partial_t B) \rangle - 2 \langle \partial_t (u_i - u_e), \partial_t u_i \rangle - 2 \| \partial_t u_i \|^2 \\ &\le 2 \langle \partial_t u_i, \partial_t E \rangle - 2 \langle \partial_t (u_i - u_e), \partial_t u_i \rangle - 2 \| \partial_t u_i \|^2 + C \mathcal{H}_1(t). \end{aligned} $$

Therefore,

$$ \frac{d}{dt} \langle A_0(\rho_i) \partial_t U_i, \partial_t U_i \rangle \le 2 \langle \partial_t u_i, \partial_t E \rangle - 2 \langle \partial_t (u_i - u_e), \partial_t u_i \rangle - 2 \| \partial_t u_i \|^2 + C \mathcal{H}_1(t). $$

Then, taking time derivative to Maxwell equations and the third equation in (2.1) and making the inner product of the resulting equations with $(\partial_t E, \partial_t G)$ and $\partial_t u_e$ respectively, we obtain

$$ \frac{d}{dt} \| (\lambda \partial_t E, \partial_t G) \|^2 \le -2 \langle \partial_t E, \partial_t (\rho_i u_i) \rangle + 2 \langle \partial_t E, \partial_t u_e \rangle, $$

and

$$ \begin{aligned} \frac{d}{dt} \| \sqrt{m_e} \partial_t u_e \|^2 &= m_e \langle \operatorname{div} u_e, |\partial_t u_e|^2 \rangle - 2m_e \langle (\partial_t u_e \cdot \nabla) u_e, \partial_t u_e \rangle - 2 \langle \partial_t E, \partial_t u_e \rangle \\ &\quad - 2 \langle \partial_t u_e, u_e \times \partial_t B \rangle - 2 \langle \partial_t (\rho_i(u_e - u_i)), \partial_t u_e \rangle - 2m_e \| \partial_t u_e \|^2 \\ &\le -2 \langle \partial_t E, \partial_t u_e \rangle - 2 \langle \partial_t (u_e - u_i), \partial_t u_e \rangle + C \mathcal{H}_1(t). \end{aligned} $$

Combining the above three estimates yields

$$ \begin{aligned} \frac{d}{dt} \left( \langle A_0(\rho_i) \partial_t U_i, \partial_t U_i \rangle + \| (\lambda \partial_t E, \partial_t G, \sqrt{m_e} \partial_t u_e) \|^2 \right) &\le -2 \| (\partial_t (u_e - u_i), \partial_t u_i) \|^2 - 2 \langle \partial_t E, \lambda^2 \partial_t (\operatorname{div} E u_i) \rangle + C \mathcal{H}_1(t) \\ &\le -2 \| (\partial_t (u_e - u_i), \partial_t u_i) \|^2 + C \mathcal{H}_1(t). \end{aligned} $$

This ends the proof.

<figure><img src="image_2.png" /></figure>

**Lemma 4.7.** Let the conditions in Theorem 2.1 hold such that $\mathcal{E}_0(0)$ is uniformly bounded with respect to $\varepsilon$. Then for $\forall t \ge 0$, it holds

$$ \mathcal{E}_k(t) + \int_0^t \mathcal{D}_k(\tau) d\tau \le C \mathcal{E}_k(0) + C \int_0^t \mathcal{H}_k(\tau) d\tau + C \mathcal{E}_0^{1/2}(t) \mathcal{E}_k(t), \quad \text{for } k = 0, 1. \quad (4.23) $$
In particular, for the case $k = 0$, we have

$$ \mathcal{E}_0(t) + \int_0^t \mathcal{D}_0(\tau) d\tau \le C \mathcal{E}_0(0). \quad (4.24) $$

**Remark 4.1.** It is worth emphasizing that we do not require $\mathcal{E}_1(0)$ is uniformly bounded to get (4.23). Thus (4.23) does not necessarily lead to the uniform boundedness of $\mathcal{E}_1(t)$ at present.

*Proof.* We first notice that

$$ r_{\text{rot},i}^{\alpha,k} + r_{\text{div}}^{\alpha,k} = \langle \partial_x^\alpha \partial_t^k u_i \times \tilde{B}, -\Delta \partial_x^\alpha \partial_t^k u_i \rangle = 0. \quad (4.25) $$

Now combining (4.5) and (4.11), we obtain

$$ \frac{d}{dt} \mathbb{E}^{\alpha,k}(t) + 4 \| \partial_x^\alpha \partial_t^k \mathrm{div} u_i \|^2 + 2 \| \partial_x^\alpha \partial_t^k (\omega_e - \omega_i, \omega_i) \|^2 \le C \mathcal{H}_k(t), \quad (4.26) $$

in which

$$ \begin{aligned} \mathbb{E}^{\alpha,k}(t) = & \langle D_0(\rho_i) \partial_x^\alpha \partial_t^k V, \partial_x^\alpha \partial_t^k V \rangle + \| \partial_x^\alpha \partial_t^k (\sqrt{m_e} \omega_e, \omega_i, \lambda \mathrm{div} E, \lambda \nabla \times E, \nabla \times G) \|^2 \\ & - 2 \langle \partial_x^\alpha \partial_t^k G, \partial_x^\alpha \partial_t^k \nabla \times (n_i u_i) \rangle. \end{aligned} $$

Integrating (4.26) over $[0, t]$ for any $t \in [0, T]$ yields

$$ \mathbb{E}^{\alpha,k}(t) + \int_0^t (4 \|\partial_x^\alpha \partial_t^k \mathrm{div} u_i(\tau)\|^2 + 2 \|\partial_x^\alpha \partial_t^k (\omega_e, \omega_i)(\tau)\|^2) d\tau \le C \mathcal{E}_k(0) + C \int_0^t \mathcal{H}_k(\tau) d\tau. \quad (4.27) $$

Summing for $|\alpha| \le s - 1 - k$ for given $k = 0$ or $k = 1$ yields

$$ \sum_{|\alpha| \le s-1-k} \mathbb{E}^{\alpha,k}(t) + \int_0^t \| \partial_t^k \nabla (u_i, u_e)(\tau) \|_{s-k-1}^2 d\tau \le C \mathcal{E}_k(0) + C \int_0^t \mathcal{H}_k(\tau) d\tau. $$

Notice that

$$ \left| \langle \partial_x^\alpha \partial_t^k G, \partial_x^\alpha \partial_t^k \nabla \times (n_i u_i) \rangle \right| \le C \mathcal{E}_0^{1/2}(t) \mathcal{E}_k(t). $$

Integrate (4.22) over $[0, t]$ yields

$$ \| \partial_t U_i(t) \|^2 + \| \partial_t (\lambda E, G, \sqrt{m_e} u_e)(t) \|^2 + \int_0^t \| \partial_t (u_i, u_e)(\tau) \|^2 d\tau \le C \mathcal{E}_1(0) + C \int_0^t \mathcal{H}_1(\tau) d\tau. $$

It follows from $\lambda^{-1}n_i = \lambda \mathrm{div} E$, (4.2) and the above three estimates that

$$ \mathcal{E}_k(t) + \int_0^t \| \partial_t^k (u_i, u_e)(\tau) \|_{s-k-1}^2 d\tau \le C \mathcal{L}_k(t). \quad (4.28) $$

with

$$ \mathcal{L}_k(t) := \mathcal{E}_k(0) + \int_0^t \mathcal{H}_k(\tau) d\tau + \mathcal{E}_0^{1/2}(t) \mathcal{E}_k(t). $$

Next, integrating (4.16), (4.17) and (4.18) over $[0, t]$ respectively and summing the resulting equations for $|\alpha| \le s - 1 - k$ or $|\gamma| \le s - 2 - k$ yield

$$ \begin{aligned} & \sum_{|\alpha| \le s-1-k} \langle \partial_x^\alpha \partial_t^k u_i, \partial_x^\alpha \partial_t^k \nabla n_i \rangle + \int_0^t \left( \frac{h_1}{2} \| \partial_t^k \nabla n_i(\tau) \|_{s-k-1}^2 + \lambda^{-2} \| n_i(\tau) \|_{s-k-1}^2 \right) d\tau \\ & \le C \mathcal{E}_k(0) + C \int_0^t \| \partial_t^k (u_i, u_e)(\tau) \|_{s-1-k}^2 d\tau + C \int_0^t \mathcal{H}_k(\tau) d\tau, \quad (4.29) \\ & - \sum_{|\alpha| \le s-1-k} \langle \lambda^2 \partial_x^\alpha \partial_t^k E, \partial_x^\alpha \partial_t^k u_i \rangle + \frac{1}{2} \int_0^t \| \partial_t^k (\lambda E)(\tau) \|_{s-1-k}^2 d\tau - \theta \int_0^t \| \partial_t^k \nabla G(\tau) \|_{s-2-k}^2 d\tau \end{aligned} $$
$$ \leq C \mathcal{E}_k(0) + C \int_0^t \left( \frac{1}{\theta} \| \partial_t^k (u_i, u_e)(\tau) \|_{s-k}^2 + \| \partial_t^k \nabla n_i(\tau) \|_{s-1-k}^2 \right) d\tau, \quad (4.30) $$

$$ -2 \sum_{|\gamma| \le s-2-k} \langle \lambda^2 \partial_t^k \partial_x^\gamma E, \nabla \times \partial_t^k \partial_x^\gamma G \rangle + \int_0^t \| \partial_t^k \nabla \times G(\tau) \|_{s-2-k}^2 d\tau $$

$$ \leq C \mathcal{E}_k(0) + C \int_0^t (\| \partial_t^k (u_i, u_e)(\tau) \|_{s-1-k}^2 + \| \lambda \partial_t^k E(\tau) \|_{s-1-k}^2) d\tau. \quad (4.31) $$

Substituting (4.28) into (4.29) and (4.30), we obtain,

$$ \int_0^t \| \partial_t^k (\nabla n_i, \lambda^{-1} n_i)(\tau) \|_{s-1-k}^2 d\tau \le C \mathcal{L}_k(t), \quad (4.32) $$

$$ \int_0^t \| \partial_t^k (\lambda E)(\tau) \|_{s-1-k}^2 d\tau \le \frac{C}{\theta} \mathcal{L}_k(t) + C \int_0^t \| \partial_t^k \nabla n_i(\tau) \|_{s-1-k}^2 d\tau + \theta \| \partial_t^k \nabla G(\tau) \|_{s-2-k}^2 d\tau $$

$$ \le \frac{C}{\theta} \mathcal{L}(t) + \theta \int_0^t \| \partial_t^k \nabla G(\tau) \|_{s-2-k}^2 d\tau, \quad (4.33) $$

where (4.32) is used in the last inequality. By using (4.28) and (4.33), we obtain that there exists a constant $c_4 > 1$ independent of $\varepsilon$ such that (4.31) takes the form

$$ \int_0^t \| \partial_t^k \nabla \times G(\tau) \|_{s-2-k}^2 d\tau \le \frac{c_4}{\theta} \mathcal{L}_k(t) + c_4 \theta \int_0^t \| \partial_t^k \nabla G(\tau) \|_{s-2-k}^2 d\tau. $$

Choosing $\theta = \frac{1}{2c_4}$, we obtain

$$ \int_0^t \| \partial_t^k \nabla \times G(\tau) \|_{s-2-k}^2 d\tau \le 2c_4^2 \mathcal{L}_k(t), $$

substituting which into (4.33) and combining (4.32) we have

$$ \int_0^t \| \partial_t^k (\nabla n_i, \lambda^{-1} n_i, \lambda E)(\tau) \|_{s-1-k}^2 + \| \partial_t^k \nabla \times G(\tau) \|_{s-2-k}^2 d\tau \le C \mathcal{L}_k(t). $$

Combining the above estimate and (4.28), we obtain (4.23). Consequently, for $k = 0$, (4.23) yields

$$ \mathcal{E}_0(t) + \int_0^t \mathcal{D}_0(\tau) d\tau \le C \mathcal{E}_0(0) + C \int_0^t \mathcal{E}_0^{1/2}(\tau) \mathcal{D}_0(\tau) d\tau + C \mathcal{E}_0^{1/2}(t) \mathcal{E}_0(t), $$

$$ \le C \mathcal{E}_0(0) + C \mathcal{E}_T^{1/2} \int_0^t \mathcal{D}_0(\tau) d\tau + C \mathcal{E}_T^{1/2} \mathcal{E}_0(t), $$

which implies (4.24) since $\mathcal{E}_T$ is sufficiently small. $\square$

**Proof of Theorem 2.1.** The estimate (4.24) shows that the smooth solution is uniformly bounded in $L^\infty([0, T]; H^s)$ with respect to $T$ and small parameters $m_e$ and $\lambda$. By the bootstrap principle, it yields uniformly global solution. In particular, this estimate gives (2.4). $\square$

**Proof of Theorem 2.2.** From (2.4), the solution sequences $\{\frac{1}{\lambda}(\rho_i^\varepsilon - 1)\}_{\varepsilon>0}$ is bounded in $L^\infty(\mathbb{R}^+; H^{s-1})$, $\{u_i^\varepsilon\}_{\varepsilon>0}$ and $\{B^\varepsilon - \bar{B}\}_{\varepsilon>0}$ are bounded in $L^\infty(\mathbb{R}^+; H^s)$ and $\{u_e^\varepsilon\}_{\varepsilon>0}$ is bounded in $L^2(\mathbb{R}^+; H^s)$. These imply that there exist functions $(\bar{u}_\nu, \bar{B})$, such that (2.5)-(2.7) hold.

Now we pass to the limit $\varepsilon \to 0$ in (2.1). Based on (2.4), in the sense of distributions, we have

$$ \lambda^2 \partial_t E^\varepsilon \to 0, \quad u_e^\varepsilon - \rho_i^\varepsilon u_i^\varepsilon \to \bar{u}_e - \bar{u}_i =: -\bar{J}, $$

with $\bar{J}$ naturally defined and

$$ \nabla \times B^\varepsilon \to \nabla \times \bar{B}, \quad \operatorname{div} B^\varepsilon \to \operatorname{div} \bar{B}, $$
which results in

$$ \bar{J} = \nabla \times \bar{B}. $$

Taking the rotation operator to the momentum equation for electrons and combining the equation for $\partial_t B^\epsilon$ in (2.1), we have

$$ m_e \nabla \times (\partial_t u_e^\epsilon + (u_e^\epsilon \cdot \nabla)u_e^\epsilon + u_e^\epsilon) = \partial_t B^\epsilon - \nabla \times (u_e^\epsilon \times B^\epsilon) - \nabla \times (\rho_i^\epsilon (u_e^\epsilon - u_i^\epsilon)). \quad (4.34) $$

From (2.4), we have $\{\partial_t(B^\epsilon - m_e \nabla \times u_e^\epsilon)\}_{\epsilon>0}$ is bounded $L^2(\mathbb{R}^+; H^{s-2})$. Hence, by a classical compactness theorem [29], we obtain that

$$ B^\epsilon - m_e \nabla \times u_e^\epsilon \to \bar{B}, \quad \text{strongly in } C([0, T]; H_{\text{loc}}^{s'}) \text{, for } s' \in [0, s-1). $$

Since $m_e \nabla \times u_e^\epsilon \to 0$ strongly in $L^\infty([0, T]; H^{s-1})$, we conclude

$$ B^\epsilon \to \bar{B}, \quad \text{strongly in } C([0, T]; H_{\text{loc}}^{s'}) \text{, for } s' \in [0, s-1). $$

Next, taking the rotation operator to the second equation in (2.1), we have

$$ \partial_t(\nabla \times u_i^\epsilon + B_i^\epsilon) + \nabla \times ((u_i^\epsilon \cdot \nabla)u_i^\epsilon) = \nabla \times (u_i^\epsilon \times B^\epsilon - 2u_i^\epsilon + u_e^\epsilon). $$

Similarly, from (2.4), we have $\{\partial_t B^\epsilon + \nabla \times u_i^\epsilon\}_{\epsilon>0}$ is bounded $L^2(\mathbb{R}^+; H^{s-2})$. This implies that $\{\partial_t B^\epsilon + \nabla \times u_i^\epsilon\}_{\epsilon>0}$ is relatively compact in $C([0, T]; H_{\text{loc}}^{s'})$ for $s' \in [0, s-1)$. By noticing the strong convergence of $B^\epsilon$, we obtain

$$ \nabla \times u_i \to \nabla \times \bar{u}_i, \quad \text{strongly in } C([0, T]; H_{\text{loc}}^{s'}) \text{, for } s' \in [0, s-1). $$

Adding the momentum equations for ions and electrons in (2.1), and taking the rotation operator to the resulting equation, we have

$$ \begin{aligned} & \nabla \times \partial_t u_i^\epsilon + (u_i^\epsilon \cdot \nabla)(\nabla \times u_i^\epsilon) - ((\nabla \times u_i^\epsilon) \cdot \nabla)u_i + (\nabla \times u_i^\epsilon)\text{div}u_i^\epsilon \\ & = -\nabla \times ((u_e^\epsilon - u_i^\epsilon) \times B^\epsilon + (\rho_i^\epsilon - 1)(u_e^\epsilon - u_i^\epsilon) + u_i^\epsilon) - \nabla \times (m_e \partial_t u_e^\epsilon + m_e(u_e^\epsilon \cdot \nabla)u_e^\epsilon + m_e u_e^\epsilon). \end{aligned} $$

By noticing the strong convergences of $\{B^\epsilon, \nabla \times u_i^\epsilon, \rho_i^\epsilon\}_{\epsilon>0}$, we obtain that $\bar{u}_i$ satisfies

$$ \nabla \times (\partial_t \bar{u}_i + (\bar{u}_i \cdot \nabla) \bar{u}_i) = \nabla \times (\bar{J} \times \bar{B}) - \nabla \times \bar{u}_i, $$

which implies that there exists a function $\bar{p}_e$, such that

$$ \partial_t \bar{u}_i + (\bar{u}_i \cdot \nabla) \bar{u}_i + \nabla \bar{p}_e = \bar{J} \times \bar{B} - \bar{u}_i, \quad \text{div} \bar{u}_i = 0, \quad (4.35) $$

where $\text{div} \bar{u}_i = 0$ follows from the strong convergence of $\rho_i^\epsilon$ and the first equation in (2.1).

Finally, the strong convergences of $\{B^\epsilon\}_{\epsilon>0}$ and $\{\rho_i^\epsilon\}_{\epsilon>0}$ enable us to pass the limit to the equation (4.34) to get

$$ \partial_t \bar{B} - \nabla \times (\bar{u}_e \times \bar{B}) + \nabla \times \bar{J} = 0. $$

Noticing that $\bar{J} = \nabla \times \bar{B} = -(\bar{u}_e - \bar{u}_i)$, we obtain

$$ \partial_t \bar{B} + \nabla \times ((\bar{J} - \bar{u}_i) \times \bar{B}) + \nabla \times (\nabla \times \bar{B}) = 0, $$

which is equivalent to the following

$$ \partial_t \bar{B} - \nabla \times (\bar{u}_i \times \bar{B}) = -\nabla \times ((\nabla \times \bar{B}) \times \bar{B}) + \Delta \bar{B}, \quad \text{div} \bar{B} = 0. \quad (4.36) $$

Combining (4.35) and (4.36) yields the hyperbolic MHD equation. $\square$

**Remark 4.2.** It has to be pointed out that the limit status of E is uncertain by this moment. We only have the information of $\nabla \times \bar{E}$, which equals to $-\partial_t \bar{B}$.
## 5. GLOBAL CONVERGENCE RATE

In this section, we study the global error estimates between the smooth solutions to (1.3) and (1.9) under the extra condition that $\mathcal{E}_1(0)$ is uniformly bounded with respect to $\varepsilon$. We first show that under this condition, $E^\varepsilon$ is uniformly bounded in $L^2(\mathbb{R}^+; H^{s-1})$, based on which the strong convergence of $\{\rho_i^\varepsilon, u_\nu^\varepsilon, p_e^\varepsilon, E^\varepsilon, B^\varepsilon\}_{\varepsilon>0}$ is established.

### 5.1. Improved uniform estimates.
We first show the uniform boundedness of $E^\varepsilon$ and $p_e^\varepsilon$.

**Lemma 5.1.** Let the conditions in Theorem 2.3 hold. Then $\mathcal{E}_1(0)$ is uniformly bounded with respect to $\varepsilon$. Furthermore, we have

$$ \mathcal{E}_0(t) + \mathcal{E}_1(t) + \int_0^t (\mathcal{D}_0(\tau) + \mathcal{D}_1(\tau) + \|(E^\varepsilon, \nabla p_e^\varepsilon)(\tau)\|_{s-1}^2) d\tau \le C(\mathcal{E}_0(0) + \mathcal{E}_1(0)), \quad \forall t \ge 0. \quad (5.1) $$

*Proof.* The initial data of time derivatives of solutions are defined by (2.1), namely

$$ \left\{ \begin{aligned} \partial_t \rho_i^\varepsilon \Big|_{t=0} &= -\operatorname{div}(\rho_{i,0}^\varepsilon u_{i,0}^\varepsilon), \\ \partial_t u_i^\varepsilon \Big|_{t=0} &= -(u_{i,0}^\varepsilon \cdot \nabla)u_{i,0}^\varepsilon - \nabla h_i(\rho_{i,0}^\varepsilon) + E_0^\varepsilon + u_{i,0}^\varepsilon \times B_0^\varepsilon - (u_{i,0}^\varepsilon - u_{e,0}^\varepsilon) - u_{i,0}^\varepsilon, \\ \lambda \partial_t E^\varepsilon \Big|_{t=0} &= \lambda^{-1}(\nabla \times B_0^\varepsilon + u_{e,0}^\varepsilon - \rho_{i,0}^\varepsilon u_{i,0}^\varepsilon), \\ \partial_t B^\varepsilon \Big|_{t=0} &= -\nabla \times E_0^\varepsilon, \\ \sqrt{m_e} \partial_t u_e^\varepsilon \Big|_{t=0} &= -\sqrt{m_e}((u_{e,0}^\varepsilon \cdot \nabla)u_e^\varepsilon + u_{e,0}^\varepsilon) - \frac{1}{\sqrt{m_e}} Y_0^\varepsilon, \end{aligned} \right. \quad (5.2) $$

with

$$ Y_0^\varepsilon := (E_0^\varepsilon + u_{e,0}^\varepsilon \times B_0^\varepsilon + \rho_{i,0}^\varepsilon (u_{e,0}^\varepsilon - u_{i,0}^\varepsilon) - \nabla p_{e,0}^\varepsilon), $$

and $p_{e,0}^\varepsilon$ being the solution to the following Poisson equation

$$ -\Delta p_{e,0}^\varepsilon = m_e \operatorname{div}((u_{e,0}^\varepsilon \cdot \nabla)u_{e,0}^\varepsilon) + \operatorname{div} E_0^\varepsilon + \operatorname{div}(u_{e,0}^\varepsilon \times B_0^\varepsilon) + \operatorname{div}(\rho_{i,0}^\varepsilon (u_{e,0}^\varepsilon - u_{i,0}^\varepsilon)). \quad (5.3) $$

We require that $p_e^\varepsilon(0)$ decays sufficiently fast at infinity. Under (2.8), we obtain that

$$ \begin{aligned} \|u_{e,0}^\varepsilon\|_{s-1} &\le C\lambda + \|\rho_{i,0}^\varepsilon u_{i,0}^\varepsilon - \nabla \times B_0^\varepsilon\|_{s-1} \le C, \\ \|\nabla p_{e,0}^\varepsilon\|_{s-1} &\le C\|m_e((u_{e,0}^\varepsilon \cdot \nabla)u_{e,0}^\varepsilon) + E_0^\varepsilon + (u_{e,0}^\varepsilon \times B_0^\varepsilon) + (\rho_{i,0}^\varepsilon(u_{e,0}^\varepsilon - u_{i,0}^\varepsilon))\|_{s-1} \le C. \end{aligned} $$

Combining (2.3) and (2.8), we get the uniform boundedness of $\|\partial_t(\rho_i, u_i, E, B)\|_{s-1}(0)$. Applying $\operatorname{div} \times \nabla$ to the last equation in (5.2) yields

$$ \left\{ \begin{aligned} \left| \sqrt{m_e} \operatorname{div} \partial_t u_e^\varepsilon \right|_{t=0} &= 0, \\ \left| \sqrt{m_e} \nabla \times \partial_t u_e^\varepsilon \right|_{t=0} &= -\sqrt{m_e} \nabla \times \left( (u_{e,0}^\varepsilon \cdot \nabla) u_{e,0}^\varepsilon + u_{e,0}^\varepsilon \right) - \frac{1}{\sqrt{m_e}} \nabla \times Y_0^\varepsilon, \end{aligned} \right. $$

which implies the boundedness of $\|\sqrt{m_e} \partial_t u_e^\varepsilon\|_{s-1}|_{t=0}$. Hence $\mathcal{E}_1(0)$ is uniformly bounded. Let $T > 0$ be arbitrarily given. For $k = 1$, (4.23) together with (4.24) imply

$$ \begin{aligned} \mathcal{E}_1(t) + \int_0^t \mathcal{D}_1(\tau) d\tau &\le C \mathcal{E}_1(0) + C \mathcal{E}_0^{1/2}(t) \mathcal{E}_1(t) \\ &\quad + C \int_0^t (\mathcal{E}_1^{1/2}(\tau) \mathcal{D}_0^{1/2}(\tau) \mathcal{D}_1^{1/2}(\tau) + \mathcal{E}_0^{1/2}(\tau) \mathcal{D}_1(\tau)) d\tau \\ &\le C \mathcal{E}_1(0) + (C \mathcal{E}_0^{1/2}(0) + \frac{1}{2}) \int_0^t \mathcal{D}_1(\tau) d\tau + C \mathcal{E}_0^{1/2}(0) \sup_{t \in [0,T]} \mathcal{E}_1(t), \end{aligned} $$

in which (4.24) is used. This implies

$$ \sup_{t \in [0,T]} \mathcal{E}_1(t) + \int_0^T \mathcal{D}_1(\tau) d\tau \le C \mathcal{E}_1(0), $$
provided that $\mathcal{E}_0(0)$ is sufficiently small. Since $T$ is arbitrary, we conclude

$$ \mathcal{E}_1(t) + \int_0^t \mathcal{D}_1(\tau) \mathrm{d}\tau \le C\mathcal{E}_1(0), \quad \forall t \ge 0, \qquad (5.4) $$

which implies the uniform boundedness of $\mathcal{E}_1(t)$. By the definition of $\mathcal{D}_1(t)$, we obtain that $\{\partial_t u_i^\epsilon\}_{\epsilon>0}$ is bounded in $L^2(\mathbb{R}^+; H^{s-1})$. Hence, the second equation in (2.1) implies the uniform boundedness of $\{E^\epsilon\}_{\epsilon>0}$ in $L^2(\mathbb{R}^+; H^{s-1})$. Taking div to the third equation in (2.1) yields

$$ -\Delta p_e = m_e \mathrm{div}((u_e \cdot \nabla)u_e) + \mathrm{div}E + \mathrm{div}(u_e \times B) + \mathrm{div}(\rho_i(u_e - u_i)), $$

which, along with (4.24), (5.4) and classical elliptic theories, implies (5.1). □

Based on the above lemma, we obtain the estimates for the limit equations:

**Lemma 5.2.** The solution $(\bar{u}_i, \bar{B})$ to (1.9) satisfies

$$ \|(\bar{u}_i, \bar{B} - \tilde{B})(t)\|_s^2 + \int_0^t (\|(\bar{u}_i, \bar{u}_e, \nabla \bar{B})(\tau)\|_s^2 + \|\partial_t \bar{u}_e(\tau)\|_{s-1}^2) \mathrm{d}\tau \le C\delta^2. \qquad (5.5) $$

Furthermore, there exists a function $\bar{E} \in L^2(\mathbb{R}^+; H^{s-1})$ such that as $\epsilon \to 0$, up to subsequences,

$$ (E^\epsilon, \nabla p_e^\epsilon) \to (\bar{E}, \nabla \bar{p}_e), \quad \text{weakly in } L^2(\mathbb{R}^+; H^{s-1}), \qquad (5.6) $$

where $\bar{p}_e$ is introduced in (4.35) and $(\bar{E}, \nabla \bar{p}_e)$ satisfies

$$ \int_0^t (\|(\bar{E}, \nabla \bar{p}_e)(\tau)\|_{s-1}^2 + \|\partial_t \bar{E}(\tau)\|_{s-2}^2) \mathrm{d}\tau \le C\delta^2. \qquad (5.7) $$

**Proof.** By the semi-lower continuity of the norm, it is clear that the limit solution still satisfies

$$ \|(\bar{u}_i, \bar{B} - \tilde{B})(t)\|_s^2 + \int_0^t (\|(\bar{u}_i, \bar{u}_e)(\tau)\|_s^2 + \|\nabla \bar{B}(\tau)\|_{s-2}^2 + \|\partial_t \bar{u}_e(\tau)\|_{s-1}^2) \mathrm{d}\tau \le C\delta^2. $$

In addition, we have

$$ J^\epsilon := \rho_i^\epsilon u_i^\epsilon - u_e^\epsilon \to \bar{J} = \nabla \times \bar{B}, \quad \text{weakly in } L^2(\mathbb{R}^+; H^s). $$

Since $\mathrm{div}\bar{B} = 0$, (5.5) is obtained.

Adding the momentum equations for ions and electrons in (2.1), we have

$$ \begin{aligned} & \partial_t u_i^\epsilon + (u_i^\epsilon \cdot \nabla)u_i^\epsilon + \nabla(h_i(\rho_i) + p_e^\epsilon) \\ & = (u_i^\epsilon - u_e^\epsilon) \times B^\epsilon + (\rho_i^\epsilon - 1)(u_i^\epsilon - u_e^\epsilon) - u_i^\epsilon - m_e(\partial_t u_e^\epsilon + (u_e^\epsilon \cdot \nabla)u_e^\epsilon + u_e^\epsilon). \end{aligned} \qquad (5.8) $$

In order to pass to the limit $\epsilon \to 0$ in the above equation, we need to study the convergence of $p_e^\epsilon$ and $u_i^\epsilon$ more carefully. Due to (5.1), we obtain that there exists a function $(\bar{E}, \nabla \tilde{P}) \in L^2(\mathbb{R}^+; H^{s-1})$ such that as $\epsilon \to 0$, up to subsequences,

$$ (E^\epsilon, \nabla p_e^\epsilon) \to (\bar{E}, \nabla \tilde{P}), \quad \text{weakly in } L^2(\mathbb{R}^+; H^{s-1}). \qquad (5.9) $$

Besides, since $\{\partial_t u_i^\epsilon\}_{\epsilon>0}$ is bounded in $L^2(\mathbb{R}^+; H^{s-1})$, by classical compactness argument [29], $\{u_i^\epsilon\}_{\epsilon>0}$ is relatively compact in $C([0, T]; H_{\mathrm{loc}}^{s-1})$. Consequently, combining (2.5)–(2.7), (5.9) and the strong convergence of $\{(u_i^\epsilon, \rho_i^\epsilon, B^\epsilon)\}_{\epsilon>0}$, we can pass to the limit $\epsilon \to 0$ in (5.8) to obtain that

$$ \partial_t \bar{u}_i + (\bar{u}_i \cdot \nabla) \bar{u}_i + \nabla \tilde{P} = \bar{J} \times \bar{B} - \bar{u}_i. \qquad (5.10) $$

Comparing the above with (4.35) yields that $\nabla \tilde{P} = \nabla \bar{p}_e$. It follows from (5.9) that (5.6) holds.

The estimate of $\partial_t \bar{E}$ depends on those of $\partial_{tt} \bar{u}_i$. For a multi-index $\beta \in \mathbb{N}^3$ with $|\beta| \le s - 2$, taking $\partial_t \partial_x^\beta$ to (5.10) and making the inner product of the resulting equation with $\partial_{tt} \partial_x^\beta \bar{u}_i$ yield

$$ \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \|\partial_t \partial_x^\beta \bar{u}_i\|^2 + \|\partial_{tt} \partial_x^\beta \bar{u}_i\|^2 \le \frac{1}{2} \|\partial_{tt} \partial_x^\beta \bar{u}_i\|^2 + C \|\partial_t \partial_x^\beta ((\bar{u}_i \cdot \nabla) \bar{u}_i - \bar{J} \times \bar{B})\|^2, $$
where the Young's inequality is used. This implies that $\partial_{tt}\bar{u}_i \in L^2(\mathbb{R}^+; H^{s-2})$. Because of the strong convergence of $\{u_i^\varepsilon, \rho_i^\varepsilon, B_i^\varepsilon\}_{\varepsilon>0}$ and the weak convergence (5.6), applying $\varepsilon \to 0$ to the second equation in (2.1) yields

$$ \bar{E} = \partial_t \bar{u}_i + (\bar{u}_i \cdot \nabla) \bar{u}_i - \bar{u}_i \times \bar{B} + \bar{u}_i + (\bar{u}_i - \bar{u}_e). $$

It follows from the regularity of $\partial_{tt}\bar{u}_i$ and (5.1) that $\bar{E} \in L^2(\mathbb{R}^+; H^{s-1})$ and $\partial_t \bar{E} \in L^2(\mathbb{R}^+; H^{s-2})$. The estimates for $\nabla \bar{p}_e$ follows from (5.1) and the semi-lower continuity of the norms. $\square$

The aim of this section is to establish global-in-time error estimates between $(\rho_i^\varepsilon, u_i^\varepsilon, p_e^\varepsilon, E^\varepsilon, B^\varepsilon)$ and its corresponding limit $(1, \bar{u}_\nu, \bar{p}_e, \bar{E}, \bar{B})$ in some energy spaces. For simplicity, the superscripts of $\varepsilon$ are omitted in the rest of this section. Besides, we denote $\gamma \in \mathbb{N}^3$ be a multi-index such that $|\gamma| \le s-3$ and $C > 0$ a generic constant independent of $\varepsilon$ and any time. Let $T > 0$. For convenience, we denote $\zeta_1 = \min\{1, \zeta\}$ and

$$ (\mathcal{N}_i, w_\nu, \mathcal{F}, \mathcal{G}, \mathcal{P}) = (\rho_i - 1, u_\nu - \bar{u}_\nu, E - \bar{E}, B - \bar{B}, p_e - \bar{p}_e). $$

We also introduce the following functionals:

$$ \mathcal{E}(t) = \|(w_e, w_i, \lambda \mathcal{F}, \mathcal{G})(t)\|_{s-2}^2, \quad \mathcal{D}(t) = \|(w_e, w_i, \lambda \mathcal{F})(t)\|_{s-2}^2 + \|(\lambda^{-1}\mathcal{N}_i, \nabla \times \mathcal{G})(t)\|_{s-3}^2. $$

Moreover, we continuously use the notations $\mathcal{E}_k(t)$ and $\mathcal{D}_k(t)$ and set

$$ \mathcal{E}(t) := \mathcal{E}_0(t) + \mathcal{E}_1(t) + \|(\bar{u}_i, \bar{B} - \tilde{B})(t)\|_s^2, $$

$$ \mathcal{D}(t) := \mathcal{D}_0(t) + \mathcal{D}_1(t) + \|(\bar{u}_i, \bar{u}_e, \nabla \bar{B})(\tau)\|_s^2 + \|\bar{E}(\tau)\|_{s-1}^2 + \|\partial_t \bar{E}\|_{s-2}^2, $$

$$ \mathcal{K}(t) := (\mathcal{E}^{1/2}(t) + \kappa)\mathcal{D}(t) + \kappa^{-1}\mathcal{E}(t)\mathcal{D}(t) + \kappa^{-1}\varepsilon^{2\zeta_1}\mathcal{D}(t), $$

where $\kappa > 0$ is a sufficiently small constant, of which the value is determined later. It follows from (4.24), (5.1), (5.5) and (5.7) that

$$ \mathcal{E}(t) + \int_0^t \mathcal{D}(\tau) d\tau \le C\mathcal{E}(0), \quad \forall t \ge 0. $$

## 5.2. Energy estimates for error functions. We first have the $L^2$ estimates.

**Lemma 5.3.** (L$^2$-estimates) It holds

$$ \frac{\mathrm{d}}{\mathrm{d}t} \|(w_i, \sqrt{m_e}w_e, \lambda \mathcal{F}, \mathcal{G})\|^2 + \|(w_i - w_e, w_i)\|^2 \le C \mathcal{K}(t), \quad (5.11) $$

and

$$ -\frac{\mathrm{d}}{\mathrm{d}t} \langle \lambda^2 \mathcal{F}, w_i \rangle + \frac{1}{2} \|\lambda \mathcal{F}\|^2 \le C \mathcal{K}(t) + C \kappa^{-1} \|(w_i, w_e)\|^2. \quad (5.12) $$

**Proof.** The error system for $(w_e, w_i, \mathcal{F}, \mathcal{G})$ takes the form

$$ \left\{ \begin{array}{l} \partial_t w_i + (u_i \cdot \nabla) w_i + h_i'(\rho_i) \nabla \mathcal{N}_i = \mathcal{F} + w_i \times G + w_i \times \tilde{B} + \bar{u}_i \times \mathcal{G} \\ \qquad - (w_i \cdot \nabla) \bar{u}_i - (w_i - w_e) - w_i, \\ m_e \partial_t w_e + \nabla \mathcal{P} = -(\mathcal{F} + w_e \times G + w_e \times \tilde{B} + \bar{u}_e \times \mathcal{G} + \mathcal{N}_i(u_e - u_i) + (w_e - w_i)) \\ \qquad - m_e (\partial_t \bar{u}_e + (u_e \cdot \nabla) u_e + u_e), \\ \lambda^2 \partial_t \mathcal{F} - \nabla \times \mathcal{G} = w_e - w_i - \mathcal{N}_i u_i - \lambda^2 \partial_t \bar{E}, \quad \lambda^2 \mathrm{div} \mathcal{F} = \mathcal{N}_i - \lambda^2 \mathrm{div} \bar{E}, \\ \partial_t \mathcal{G} + \nabla \times \mathcal{F} = 0, \quad \mathrm{div} \mathcal{G} = 0. \end{array} \right. \quad (5.13) $$

Making the inner product of the above system with $(w_i^\top, w_e^\top, \mathcal{F}^\top, \mathcal{G}^\top)^\top$ yields

$$ \begin{aligned} & \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \|(w_i, \sqrt{m_e}w_e, \lambda \mathcal{F}, \mathcal{G})\|^2 + \|(w_i - w_e, w_i)\|^2 \\ & = \langle \bar{u}_i \times \mathcal{G} - (w_i \cdot \nabla) \bar{u}_i - (u_i \cdot \nabla) w_i - h_i'(\rho_i) \nabla \mathcal{N}_i, w_i \rangle - \langle \mathcal{F}, \mathcal{N}_i u_i + \lambda^2 \partial_t \bar{E} \rangle \end{aligned} $$
$$-\langle \bar{u}_e \times \mathcal{G} + \mathcal{N}_i(u_e - u_i) + m_e(\partial_t \bar{u}_e + (u_e \cdot \nabla)u_e + u_e), w_e \rangle \leq CK(t),$$

which implies (5.11).

Next, making the inner product of the second equation of (5.13) with $\lambda^2 \mathcal{F}$ yields

$$\begin{aligned}
\|\lambda \mathcal{F}\|^2 & \leq \langle \lambda^2 \mathcal{F}, \partial_t w_i \rangle \\
& \quad + \langle \lambda^2 \mathcal{F}, (u_i \cdot \nabla)w_i + \nabla h_i(\rho_i) - w_i \times B - \bar{u}_i \times \mathcal{G} + (w_i \cdot \nabla)\bar{u}_i + 2w_i - w_e \rangle \\
& \leq \frac{d}{dt} \langle \lambda^2 \mathcal{F}, w_i \rangle - \langle \nabla \times \mathcal{G} + w_e - w_i - \mathcal{N}_i u_i - \lambda^2 \partial_t \bar{E}, w_i \rangle + \frac{1}{2} \|\lambda \mathcal{F}\|^2 + CK(t) \\
& \leq \frac{d}{dt} \langle \lambda^2 \mathcal{F}, w_i \rangle + \frac{1}{2} \|\lambda \mathcal{F}\|^2 + CK(t) + C\kappa^{-1} \|(w_i, w_e)\|^2,
\end{aligned}$$

which implies (5.12). □

We still need a div-curl decomposition to get desired estimates.

**Lemma 5.4.** It holds

$$\frac{d}{dt}(\|\partial_x^\gamma \mathrm{div}(w_i, \lambda \mathcal{F})\|^2) + 4\|\partial_x^\gamma \mathrm{div} w_i\|^2 \leq CK(t) + \tilde{r}_{\mathrm{div}}^\gamma, \quad (5.14)$$

where

$$\tilde{r}_{\mathrm{div}}^{\gamma} = 2 \langle \partial_x^{\gamma} \mathrm{div} w_i, \partial_x^{\gamma} \mathrm{div}(w_i \times \tilde{B}) \rangle. \quad (5.15)$$

*Proof.* Applying $\partial_x^\gamma \mathrm{div}$ to the first equation in (5.13) and making the inner product of the resulting equation with $2\partial_x^\gamma \mathrm{div} w_i$ yields

$$\begin{aligned}
& \frac{d}{dt} \|\partial_x^\gamma \mathrm{div} w_i\|^2 + 4 \|\partial_x^\gamma \mathrm{div} w_i\|^2 \\
& = \langle \mathrm{div} u_i, |\partial_x^\gamma \mathrm{div} w_i|^2 \rangle - 2 \langle \partial_x^\gamma \mathrm{div} w_i, \partial_x^\gamma ((u_i \cdot \nabla) \mathrm{div} w_i) - (u_i \cdot \nabla) \partial_x^\gamma \mathrm{div} w_i \rangle \\
& \quad - 2 \langle \partial_x^\gamma \mathrm{div} w_i, \partial_x^\gamma (\mathrm{div}((u_i \cdot \nabla)w_i) - (u_i \cdot \nabla)\mathrm{div} w_i) \rangle \\
& \quad - 2 \langle \partial_x^\gamma \mathrm{div} w_i, \partial_x^\gamma \mathrm{div}(h_i'(\rho_i)\nabla \mathcal{N}_i) \rangle + 2 \langle \partial_x^\gamma \mathrm{div} w_i, \partial_x^\gamma \mathrm{div} \mathcal{F} \rangle \\
& \quad + 2 \langle \partial_x^\gamma \mathrm{div} w_i, \partial_x^\gamma \mathrm{div}(w_i \times G + \bar{u}_i \times \mathcal{G} - (w_i \cdot \nabla)\bar{u}_i) \rangle + \tilde{r}_{\mathrm{div}}^\gamma \\
& \leq 2 \langle \partial_x^\gamma \mathrm{div} w_i, \partial_x^\gamma \mathrm{div} \mathcal{F} \rangle + CK(t) + \tilde{r}_{\mathrm{div}}^\gamma.
\end{aligned} \quad (5.16)$$

To estimate the first term on the right hand side of the above estimate, we need to combine the Maxwell equations. Applying $\partial_x^\gamma \mathrm{div}$ to the fourth equation in (5.13), and making the inner product of the resulting equation with $\partial_x^\gamma \mathrm{div} \mathcal{F}$ yield

$$\frac{d}{dt} \|\lambda \mathrm{div} \partial_x^\gamma \mathcal{F}\|^2 = -2 \langle \partial_x^\gamma \mathrm{div} \mathcal{F}, \partial_x^\gamma \mathrm{div} w_i \rangle - 2 \langle \partial_x^\gamma \mathrm{div} \mathcal{F}, \partial_x^\gamma \mathrm{div}(\mathcal{N}_i u_i) \rangle - 2 \langle \partial_x^\gamma \mathrm{div} \mathcal{F}, \partial_x^\gamma \lambda^2 \partial_t \mathrm{div} \bar{E} \rangle.$$

By using the error Gaussian theorem, it holds

$$\begin{aligned}
\langle \partial_x^\gamma \mathrm{div} \mathcal{F}, \partial_x^\gamma \mathrm{div}(\mathcal{N}_i u_i) \rangle &= -\frac{1}{2\lambda^2} \langle |\partial_x^\gamma \mathcal{N}_i|^2, \mathrm{div} u_i \rangle + \frac{1}{\lambda^2} \langle \partial_x^\gamma \mathcal{N}_i, \partial_x^\gamma (\nabla \mathcal{N}_i \cdot u_i) - (\partial_x^\gamma \nabla \mathcal{N}_i) \cdot u_i \rangle \\
& \quad + \frac{1}{\lambda^2} \langle \partial_x^\gamma \mathcal{N}_i, \partial_x^\gamma (\mathcal{N}_i \mathrm{div} u_i) \rangle - \langle \partial_x^\gamma \mathrm{div} \bar{E}, \partial_x^\gamma \mathrm{div}(\mathcal{N}_i u_i) \rangle \leq CK(t), \\
|\langle \partial_x^\gamma \mathrm{div} \mathcal{F}, \lambda^2 \partial_x^\gamma \partial_t \mathrm{div} \bar{E} \rangle| & \leq CK(t).
\end{aligned}$$

Combining the above estimates, we obtain

$$\frac{d}{dt} \|\lambda \mathrm{div} \partial_x^\gamma \mathcal{F}\|^2 \leq -2 \langle \partial_x^\gamma \mathrm{div} \mathcal{F}, \partial_x^\gamma \mathrm{div} w_i \rangle + CK(t).$$

Substituting the above into (5.16) ends the proof. □

The next lemma is the convergence rate for $\nabla \times (w_\nu, \mathcal{F}, \mathcal{G})$.
**Lemma 5.5.** It holds

$$ \frac{d}{dt}(\|\nabla \times \partial_x^\gamma(\sqrt{m_e}w_e, w_i, \lambda\mathcal{F}, \mathcal{G})\|^2 + \langle \partial_x^\gamma \mathcal{G}, \partial_x^\gamma \nabla \times (\mathcal{N}_i u_i) \rangle) \\ +2\|\nabla \times \partial_x^\gamma(w_e - w_i, w_i)\|^2 \leq CK(t) + \tilde{r}_{\text{rot}}^\gamma, \qquad (5.17) $$

where

$$ \tilde{r}_{\text{rot}}^{\gamma} = 2 \langle \nabla \times \partial_x^{\gamma}(w_i \times \tilde{B}), \nabla \times \partial_x^{\gamma} w_i \rangle. \qquad (5.18) $$

*Proof.* Applying $\partial_x^\gamma \nabla \times$ to the second to fourth equations in (5.13) and making the inner product of the resulting equation with $2\partial_x^\gamma \nabla \times (w_i^\top, w_e^\top, \mathcal{F}^\top, \mathcal{G}^\top)^\top$, we have

$$ \begin{aligned} & \frac{d}{dt} \|\partial_x^\gamma \nabla \times (w_i, \sqrt{m_e}w_e, \lambda\mathcal{F}, \mathcal{G})\|^2 + 2\|\partial_x^\gamma \nabla \times (w_i - w_e, w_i)\|^2 \\ & \leq \langle \operatorname{div} u_i \partial_x^\gamma \nabla \times w_i, \partial_x^\gamma \nabla \times w_i \rangle - 2 \langle \partial_x^\gamma \nabla \times w_i, \nabla \times \partial_x^\gamma ((u_i \cdot \nabla)w_i) - (u_i \cdot \nabla)(\nabla \times \partial_x^\gamma w_i) \rangle \\ & \quad + 2 \langle \partial_x^\gamma \nabla \times w_i, \nabla \times \partial_x^\gamma (w_i \times (G + \tilde{B}) + \bar{u}_i \times \mathcal{G} - (w_i \cdot \nabla)\bar{u}_i) \rangle \\ & \quad - 2 \langle \partial_x^\gamma \nabla \times w_e, \nabla \times \partial_x^\gamma (w_e \times (G + \tilde{B}) + \bar{u}_e \times \mathcal{G} + \mathcal{N}_i(u_e - u_i)) \rangle \\ & \quad - 2 \langle \nabla \times \partial_x^\gamma \mathcal{F}, \nabla \times \partial_x^\gamma (\mathcal{N}_i u_i + \lambda^2 \partial_t \bar{E}) \rangle - 2 \langle \partial_x^\gamma \nabla \times w_e, m_e \nabla \times \partial_x^\gamma (\partial_t \bar{u}_e + (u_e \cdot \nabla)u_e) + u_e \rangle \\ & \leq CK(t) + \tilde{r}_{\text{rot}}^\gamma - 2 \langle \nabla \times \partial_x^\gamma \mathcal{F}, \nabla \times \partial_x^\gamma (\mathcal{N}_i u_i + \lambda^2 \partial_t \bar{E}) \rangle + 2 \langle \partial_x^\gamma w_e \times \tilde{B}, \Delta \partial_x^\gamma w_e \rangle, \end{aligned} $$

where the Moser-type inequalities are used and $\tilde{r}_{\text{rot}}^\gamma$ is defined in (5.18). Notice that

$$ \begin{aligned} & \langle \nabla \times \partial_x^\gamma \mathcal{F}, \nabla \times \partial_x^\gamma (\mathcal{N}_i u_i) \rangle \\ &= - \langle \partial_t \partial_x^\gamma \mathcal{G}, \partial_x^\gamma \nabla \times (\mathcal{N}_i u_i) \rangle \\ &= - \frac{d}{dt} \langle \partial_x^\gamma \mathcal{G}, \partial_x^\gamma \nabla \times (\mathcal{N}_i u_i) \rangle + \langle \nabla \times \partial_x^\gamma \mathcal{G}, \partial_x^\gamma (\partial_t \mathcal{N}_i u_i) \rangle + \langle \nabla \times \partial_x^\gamma \mathcal{G}, \partial_x^\gamma (\mathcal{N}_i \partial_t u_i) \rangle \\ & \leq - \frac{d}{dt} \langle \partial_x^\gamma \mathcal{G}, \partial_x^\gamma \nabla \times (\mathcal{N}_i u_i) \rangle + CK(t), \end{aligned} $$

and

$$ \begin{aligned} & |\lambda^2 \langle \nabla \times \partial_x^\gamma \mathcal{F}, \nabla \times \partial_x^\gamma (\partial_t \bar{E}) \rangle| \leq CK(t), \\ & \langle \partial_x^\gamma w_e \times \tilde{B}, \Delta \partial_x^\gamma w_e \rangle = 0. \end{aligned} $$

Combining these estimates ends the proof.

<figure><img src="image_3.png" /></figure>

The next lemma concerns the dissipation estimates for $\mathcal{N}_i$, $\mathcal{F}$ and $\mathcal{G}$.

**Lemma 5.6.** (Dissipation estimates) It holds

$$ \begin{aligned} & -\frac{d}{dt} (\langle \partial_x^\gamma \operatorname{div} w_i, \partial_x^\gamma \mathcal{N}_i \rangle + \langle \lambda^2 \partial_x^\gamma \mathcal{F}, \nabla \times \partial_x^\gamma \mathcal{G} \rangle) + \frac{1}{\lambda^2} \|\partial_x^\gamma \mathcal{N}_i\|^2 + \frac{1}{2} \|\nabla \times \partial_x^\gamma \mathcal{G}\| + \|\lambda \nabla \times \partial_x^\gamma \mathcal{F}\|^2 \\ & \leq CK(t) + C \|(w_e, w_i)\|_{s-2}^2. \end{aligned} \qquad (5.19) $$

*Proof.* Applying $\partial_x^\gamma$ to the second equation in (5.13) yields

$$ \partial_x^\gamma \nabla h_i(\rho_i) - \partial_x^\gamma \mathcal{F} = -\partial_t \partial_x^\gamma w_i + \partial_x^\gamma (w_i \times B + \bar{u}_i \times \mathcal{G} - (w_i \cdot \nabla)\bar{u}_i - 2w_i - w_e - (u_i \cdot \nabla)w_i). \qquad (5.20) $$

Making the inner product of (5.20) with $\partial_x^\gamma \nabla \mathcal{N}_i$ yields

$$ \begin{aligned} & h_1 \|\nabla \partial_x^\gamma \mathcal{N}_i\|^2 - \langle \partial_x^\gamma \nabla \mathcal{N}_i, \partial_x^\gamma \mathcal{F} \rangle \\ &= - \langle \partial_t \partial_x^\gamma w_i, \partial_x^\gamma \nabla \mathcal{N}_i \rangle + \langle \partial_x^\gamma \nabla \mathcal{N}_i, \partial_x^\gamma (w_i \times B + \bar{u}_i \times \mathcal{G} - (w_i \cdot \nabla)\bar{u}_i - 2w_i - w_e - (u_i \cdot \nabla)w_i) \rangle \\ & \quad - \langle \partial_x^\gamma \nabla \mathcal{N}_i, \partial_x^\gamma (h'_i(\rho_i) \nabla \mathcal{N}_i) - h'_i(\rho_i) \nabla \partial_x^\gamma \mathcal{N}_i \rangle \end{aligned} $$
$$ \leq \frac{d}{dt} \langle \partial_x^\gamma \mathrm{div} w_i, \partial_x^\gamma \mathcal{N}_i \rangle - \langle \partial_x^\gamma \mathrm{div} w_i, \partial_x^\gamma \partial_t \mathcal{N}_i \rangle + \frac{h_1}{2} \| \nabla \partial_x^\gamma \mathcal{N}_i \|^2 + C\mathcal{K}(t) + C \|(w_e, w_i)\|_{s-2}^2 $$

$$ \leq \frac{d}{dt} \langle \partial_x^\gamma \mathrm{div} w_i, \partial_x^\gamma \mathcal{N}_i \rangle + C\mathcal{K}(t) + C \|(w_e, w_i)\|_{s-2}^2. $$

Notice that

$$ - \langle \partial_x^\gamma \nabla \mathcal{N}_i, \partial_x^\gamma \mathcal{F} \rangle = \langle \partial_x^\gamma \mathcal{N}_i, \mathrm{div} \partial_x^\gamma \mathcal{F} \rangle = \frac{1}{\lambda^2} \| \partial_x^\gamma \mathcal{N}_i \|^2 - \langle \partial_x^\gamma \mathcal{N}_i, \mathrm{div} \partial_x^\gamma \bar{E} \rangle \geq \frac{1}{\lambda^2} \| \partial_x^\gamma \mathcal{N}_i \|^2 - C\mathcal{K}(t). $$

Combining these estimates yields

$$ -\frac{d}{dt} \langle \partial_x^\gamma \mathrm{div} w_i, \partial_x^\gamma \mathcal{N}_i \rangle + \frac{1}{\lambda^2} \| \partial_x^\gamma \mathcal{N}_i \|^2 \leq C\mathcal{K}(t) + C \|(w_e, w_i)\|_{s-2}^2. \quad (5.21) $$

Next, applying $\nabla \times \partial_x^\gamma$ to the second equation in (5.13) and making the inner product of the resulting equation with $\lambda^2 \nabla \times \partial_x^\gamma \mathcal{F}$ yield

$$ \begin{aligned} \| \lambda \nabla \times \partial_x^\gamma \mathcal{F} \|^2 & \leq - \langle m_e \nabla \times \partial_x^\gamma (\partial_t u_e + (u_e \cdot \nabla) u_e + u_e), \lambda^2 \nabla \times \partial_x^\gamma \mathcal{F} \rangle \\ & \quad - \langle \lambda^2 \nabla \times \partial_x^\gamma \mathcal{F}, \nabla \times \partial_x^\gamma (w_e \times B + \bar{u}_e \times \mathcal{G} + \mathcal{N}_i(u_e - u_i) + (w_e - w_i)) \rangle \\ & \leq \frac{1}{2} \| \lambda \nabla \times \partial_x^\gamma \mathcal{F} \|^2 + C\mathcal{K}(t), \end{aligned} $$

which implies

$$ \| \lambda \nabla \times \partial_x^\gamma \mathcal{F} \|^2 \leq C\mathcal{K}(t). \quad (5.22) $$

It remains to estimate the dissipation of $\nabla \times \mathcal{G}$. Applying $\partial_x^\gamma$ to the equation for $\partial_t \mathcal{F}$ in (5.13) and making the inner product of the resulting equation with $\nabla \times \partial_x^\gamma \mathcal{G}$ yields

$$ \begin{aligned} \| \nabla \times \partial_x^\gamma \mathcal{G} \|^2 &= \frac{d}{dt} \langle \lambda^2 \partial_x^\gamma \mathcal{F}, \nabla \times \partial_x^\gamma \mathcal{G} \rangle - \langle \lambda^2 \partial_x^\gamma \mathcal{F}, \nabla \times \partial_x^\gamma \partial_t \mathcal{G} \rangle \\ & \quad - \langle \partial_x^\gamma (w_e - w_i - \mathcal{N}_i u_i - \lambda^2 \partial_t \bar{E}), \nabla \times \partial_x^\gamma \mathcal{G} \rangle \\ & \leq \frac{d}{dt} \langle \lambda^2 \partial_x^\gamma \mathcal{F}, \nabla \times \partial_x^\gamma \mathcal{G} \rangle + \lambda^2 \| \nabla \times \partial_x^\gamma \mathcal{F} \|^2 \\ & \quad + \frac{1}{2} \| \nabla \times \partial_x^\gamma \mathcal{G} \|^2 + C \|(w_i, w_e)\|_{s-2}^2 + C\mathcal{K}(t), \end{aligned} $$

which, along with (5.22), yields

$$ -\frac{d}{dt} \langle \lambda^2 \partial_x^\gamma \mathcal{F}, \nabla \times \partial_x^\gamma \mathcal{G} \rangle + \frac{1}{2} \| \nabla \times \partial_x^\gamma \mathcal{G} \|^2 \leq C\mathcal{K}(t) + C \|(w_i, w_e)\|_{s-2}^2. \quad (5.23) $$

Combining (5.21)–(5.23) yields (5.19). $\square$

**Proof of Theorem 2.3.** Integrating (5.11), (5.14) and (5.17) over $[0, t]$ for $t \in [0, T]$ and summing the resulting equations for all $|\gamma| \leq s - 3$. We conclude

$$ \begin{aligned} & \|(w_i, \sqrt{m_e} w_e, \lambda \mathcal{F}, \mathcal{G})(t)\|_{s-2}^2 + \int_0^t \|(w_e, w_i)(\tau)\|_{s-2}^2 d\tau \\ & \leq C \varepsilon^{2\zeta_1} + C \int_0^t \mathcal{K}(\tau) d\tau + C \mathcal{E}(t) \mathcal{E}(t), \end{aligned} \quad (5.24) $$

where we have used the fact that $\tilde{r}_{\mathrm{div}}^\gamma + \tilde{r}_{\mathrm{rot}}^\gamma = 0$, a similar proof of which is stated in (4.25), and

$$ |\langle \partial_x^\gamma \mathcal{G}, \partial_x^\gamma \nabla \times (\mathcal{N}_i u_i) \rangle| \leq C \|u_i\|_s^2 \|\mathcal{G}\|_{s-3}^2 + C \|\mathcal{N}_i\|_{s-1}^2 \leq C \varepsilon^{2\zeta_1} + C \mathcal{E}(t) \mathcal{E}(t). $$
Besides, integrating (5.19) over $[0, t]$ for $t \in [0, T]$, summing the resulting equations for all $|\gamma| \le s - 3$ and using (5.24), we obtain

$$ 
\begin{aligned}
& \int_0^t \|\lambda^{-1}\mathcal{N}_i, \lambda\nabla \times \mathcal{F}, \nabla \times \mathcal{G}(\tau)\|_{s-3}^2 d\tau \\
& \le C\varepsilon^{2\zeta_1} + C\|(\mathcal{N}_i, w_i, \lambda\mathcal{F}, \mathcal{G})(t)\|_{s-2}^2 + C\int_0^t \mathcal{K}(\tau)d\tau + C\int_0^t \|(w_e, w_i)(\tau)\|_{s-2}^2 d\tau, \\
& \le C\varepsilon^{2\zeta_1} + C\int_0^t \mathcal{K}(\tau)d\tau + C\mathcal{E}(t)\mathcal{E}(t).
\end{aligned}
$$

Combining the above and (5.24), we have

$$ \mathcal{E}(t) + \int_0^t \mathcal{D}(\tau)d\tau \le C\varepsilon^{2\zeta_1} + C(\delta + \kappa)\int_0^t \mathcal{D}(\tau)d\tau + C\kappa^{-1}\delta^2 \sup_{t \in [0,T]} \mathcal{E}(t) + C\delta^2\kappa^{-1}\varepsilon^{2\zeta_1}. $$

Choosing $\kappa = \delta$ and let $\delta > 0$ sufficiently small, we obtain

$$ \sup_{t \in [0,T]} \mathcal{E}(t) + \int_0^T \mathcal{D}(\tau)d\tau \le C\varepsilon^{2\zeta_1}, $$

which ends the proof.

<figure><img src="image_4.png" /></figure>

**Data availability.** Data sharing is not applicable to this article as no data sets were generated or analysed during the current study.

**Conflict of interest statement.** There is no conflict of interest.

**Acknowledgments.** Shuai Xi would like to thank the National Natural Science Foundation of China 12201360, Natural Science Foundation of Shandong Province ZR2020QA016.

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