# Linear stability analysis of the Couette flow for the 2D Euler-Poisson
system 00footnotetext: ∗Corresponding author. 00footnotetext:  E-mail
addresses: puxueke@gmail.com(X. Pu), wywlzhou@163.com(W. Zhou),
biandongfen@bit.edu.cn(D. Bian)

Xueke Pu1, Wenli Zhou1 and Dongfen Bian2,∗
1School of Mathematics and Information Science, Guangzhou University,
Guangzhou 510006, PR China
2School of Mathematics and Statistics, Beijing Institute of Technology,
Beijing 100081, PR China

> Abstract This paper is concerned with the linear stability analysis for the
> Couette flow of the Euler-Poisson system for both ionic fluid and electronic
> fluid in the domain $\mathbb{T}\times\mathbb{R}$. We establish the upper and
> lower bounds of the linearized solutions of the Euler-Poisson system near
> Couette flow. In particular, the inviscid damping for the solenoidal
> component of the velocity is obtained.
>
> Keywords: Euler-Poisson equations; Couette flow; inviscid damping
>
> Mathematics Subject Classification: 35M30; 76E05

## 1\. Introduction

Consider the following two fluid Euler-Poisson system in a two dimensional
domain $\mathbb{T}\times\mathbb{R}$,

$\begin{split}&\partial_{t}{\widetilde{\eta}_{\pm}}+\nabla\cdot(\widetilde{\eta}_{\pm}\widetilde{v}_{\pm})=0,\\\
&\widetilde{\eta}_{\pm}m_{\pm}[\partial_{t}\widetilde{v}_{\pm}+\widetilde{v}_{\pm}\cdot\nabla\widetilde{v}_{\pm}]+T_{\pm}\nabla\widetilde{p}_{\pm}=\mp
e\widetilde{\eta}_{\pm}\nabla\widetilde{\phi},\\\
&-\Delta\widetilde{\phi}=4\pi
e[\widetilde{\eta}_{+}-\widetilde{\eta}_{-}],\end{split}$ (1.1)

where ${\widetilde{\eta}_{\pm}}(x,y,t)$ and ${\widetilde{v}_{\pm}}(x,y,t)$
denote the density and the velocity of the ion fluid $(+)$ and the electron
fluid $(-)$ in a plasma, $\nabla\widetilde{\phi}$ represents the self-
consistent electric field, $\nabla{\widetilde{p}}_{\pm}$ denote the pressure
of ions and electrons, $m_{\pm}$ denote the mass density of the ions and the
electrons, $e$ is the charge for an electron and $T_{\pm}$ denote the
effective temperatures of the ions and the electrons. Such a system is the
simplified model from the two fluid Euler-Maxwell system as the light speed
tends to infinity and is very fundamental in plasma physics. This Euler-
Poisson system is also an important origin of many famous dispersive equations
such as KdV, KP, NLS equations and is widely studied in the literature in
recent years. See a series of papers of the first author and his collaborators
[44, 40, 21, 41, 39] for the rigorous mathematical justifications of these
dispersive equations from the Euler-Poisson system. For more physics
background and the mathematical development of such a system, one may see the
concise review paper [17] and the references cited therein.

The two-fluid Euler-Poisson system has rich and complex dynamics with distinct
physical parameters. In particular, since the mass ratio of electrons and ions
is of $10^{-3}\ll 1$, the system (1.1) can be simplified to the following one-
fluid Euler-Poisson system for the electron fluid dynamics of
$({\widetilde{\eta}}_{-},{\widetilde{v}}_{-},\widetilde{\phi})$ in an ion
background (Langmuir waves, [18, 19])

$\displaystyle\partial_{t}\widetilde{\eta}_{-}+\nabla\cdot(\widetilde{\eta}_{-}\widetilde{v}_{-})=0,$
(1.2)
$\displaystyle\widetilde{\eta}_{-}\left[\partial_{t}\widetilde{v}_{-}+(\widetilde{v}_{-}\cdot\nabla)\widetilde{v}_{-}\right]+\frac{1}{m_{-}}\nabla{\widetilde{p}}_{-}(\widetilde{\eta}_{-})=\frac{e}{m_{-}}\widetilde{\eta}_{-}\nabla\widetilde{\phi},$
(1.3) $\displaystyle\Delta\widetilde{\phi}=4\pi e\widetilde{\eta}_{-}-4\pi e.$
(1.4)

On the other hand, assume
$\widetilde{p}_{-}(\widetilde{\eta}_{-})=T_{-}\widetilde{\eta}_{-}$ in (1.1)
with a constant temperature $T_{-}$ and take formally $m_{-}/m_{+}\to 0$ (such
a limit is rigorously justified by Grenier et al. recently in [16]), we deduce
$\nabla\widetilde{p}_{-}=T_{-}\nabla\widetilde{\eta}_{-}=e\widetilde{\eta}_{-}\nabla\tilde{\phi}$,
from which the celebrated Boltzmann relation for the electron density can be
derived:

$\widetilde{\eta}_{-}=\eta_{0}\exp{\\{e\widetilde{\phi}/T_{-}\\}},$

where $\eta_{0}$ is a constant, set to be one for simplicity. Then we obtain
the reduced Euler-Poisson system for ion dynamics for
$(\widetilde{\eta}_{+},\widetilde{v}_{+},\widetilde{\phi})$

$\displaystyle\partial_{t}\widetilde{\eta}_{+}+\nabla\cdot(\widetilde{\eta}_{+}\widetilde{v}_{+})=0,$
(1.5)
$\displaystyle\widetilde{\eta}_{+}\left[\partial_{t}\widetilde{v}_{+}+(\widetilde{v}_{+}\cdot\nabla)\widetilde{v}_{+}\right]+\frac{T_{+}}{m_{+}}\nabla\widetilde{\eta}_{+}=-\frac{e}{m_{+}}\widetilde{\eta}_{+}\nabla\widetilde{\phi},$
(1.6) $\displaystyle\Delta\widetilde{\phi}=4\pi
e\left[\exp\left(\frac{e}{T_{-}}\widetilde{\phi}\right)-\widetilde{\eta}_{+}\right].$
(1.7)

For global well-posedness of such ion dynamics system in $\mathbb{R}^{3+1}$,
one may refer to [20].

In this paper, we are interested in another topic of this system, i.e.,
stability of the Couette flow of the above Euler-Poisson system. To be
precise, we will consider the linear stability of the Couette flow for the two
Euler-Poisson systems (1.2)-(1.4) and (1.5)-(1.7) in a periodic strip
$\mathbb{T}\times\mathbb{R}$ with $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. It is
clear that there is a stationary solution

$(\widetilde{\eta}_{s},\widetilde{v}_{s},\widetilde{\phi}_{s})=({1,(y,0)^{\top},0})$

for both system (1.5)-(1.7) and (1.2)-(1.4). We also call this solution
Couette flow, following the name in Navier-Stokes equation in fluid dynamics.
Formal stability analysis of plasma equilibria to the related Poisson-Vlasov
and Maxwell-Vlasov systems in one, two and three dimensions is referred to the
paper of Holm et al [22]. To the best of our knowledge, no rigorous
mathematical stability analysis of the Couette flow for the Euler-Poisson
system either for the one-fluid Euler-Poisson system or the two-fluid Euler-
Poisson system can be found in the literature and we believe that this topic
is new and interesting for the Euler-Poisson system. The purpose of the
present paper is to study the linear stability of the Couette flow for both
the ion and electron one-fluid Euler-Poisson dynamics.

Let us deviate slightly to briefly introduce some related results on the
stability analysis of the Couette flow in fluid dynamics, in particular for
the Navier-Stokes or the Euler system. Indeed, from the mathematical point of
view, the stability of the Couette flow and related topics have been
extensively studied and many fruitful results were obtained in the past
decades. For an incompressible fluid, the classical results on the linear
stability analysis of the Couette flow were obtained by Rayleigh[45] and
Kelvin[34]. Inviscid damping for the Euler equations on the domain
$\mathbb{T}\times\mathbb{R}$ was studied by Bedrossian and Masmoudi[7], which
is a significant result on nonlinear stability of the planar Couette flow. For
nonlinear inviscid damping for a class of monotone shear flows for the 2D
Euler equation in finite channel for initial perturbation in Gevrey class with
compact support, one may refer to [42]. See also [24] for the inviscid damping
near the Couette flow in a channel for the 2D Euler equation, where the
authors showed that the velocity field converges to a nearby shear flow when
the initial perturbation is small in a suitable Gevrey sapce. One may also
refer to other nice work [11, 50, 25, 29, 30, 38, 43, 47] for the stability
analysis of the Couette flow or general shear flows for the Euler equation in
various situations. In the presence of viscosity, some interesting results
have been obtained in the study of transition threshold problem related to
enhanced dissipation (see, e.g., [9, 3, 5, 6, 8, 46, 10, 43] for various
situations). More results on the stability of the two-dimensional and three-
dimensional Couette flow in the Navier-Stokes equations at high Reynolds
number can be found in the review paper[4].

In the case of compressible fluid, the linear stability of plane Couette flow
has been studied in the past decades in physics literature in the past nearly
a century starting from the ’40s [35, 22, 12]. Glatzel [14] considered the
inviscid stability of this problem based on a normal mode analysis, and the
effects of viscosity were then taken into account by the same author [15].
Duck et al.[13] investigated the compressible stability of the plane Couette
flow for realistic compressible flow models, where they show that the details
of the mean flow profile have a profound effect on the stability of the flow.
Hu and Zhong[23] studied the linear stability of viscous supersonic plane
Couette flow for a perfect gas governed by Sutherland viscosity law by using a
fourth-order finite-difference method and a spectral collocation method. For
rigorous mathematical analysis, when the initial value is sufficiently close
to the plane Couette flow, Kagei[31] proved that the plane Couette flow is
asymptotically stable provided that the Reynolds and Mach numbers are
sufficiently small, see also Kagei[32, 33] for more general parallel flows.
Subsequently, the constraint on the Reynolds number in [31], which is used to
ensure the stability of the plane Couette flow, was relaxed by Li and
Zhang[37]. The inviscid damping and enhanced dissipation phenomena were
detected by Antonelli et al.[1, 2] for the homogeneous Couette flow in a 2D
isentropic compressible fluid. In addition, we refer to [49] for a inviscid
damping result on the non-isentropic compressible Euler equations on the
domain $\mathbb{T}\times\mathbb{R}$. For the 3D case, Zeng et al.[48]
established the linear stability result of the Couette flow in the isentropic
compressible Navier-Stokes equations on the domain
$\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. However, the rigorous
mathematical study is less developed compared to the incompressible case, and
a lot of work is to be established in both the linear and the nonlinear level
in the future. In particular, there is no stability analysis for the Couette
flow for the Euler-Poisson system studied in this paper.

In the following, we will discuss the linear stability of the Couette flow for
the ion Euler-Poisson system and the electron Euler-Poisson system separately.
Compared to the classical Euler system and the Navier-Stokes system, the
analysis includes some new insights, since the system is nonlocal due to the
presence of the electric field.

### 1.1. Ion dynamics case

To study the linear stability for the Couette flow in system (1.5)-(1.7), we
consider a perturbation around the above stationary solution, i.e., let

$\widetilde{\eta}_{+}=\eta_{+}+\widetilde{\eta}_{s},\quad\widetilde{v}_{+}=u_{+}+\widetilde{v}_{s},\quad\widetilde{\phi}=\phi+\widetilde{\phi}_{s}.$

We first consider the ion dynamics system (1.5)-(1.7). Then the linearized
system around the stationary solution
$(\widetilde{\eta}_{s},\widetilde{v}_{s},\widetilde{\phi}_{s})$ for system
(1.5)-(1.7) is written as

$\displaystyle\partial_{t}\eta_{+}+y\partial_{x}\eta_{+}+(\nabla\cdot
u_{+})=0,$ (1.8)
$\displaystyle\partial_{t}u_{+}+y\partial_{x}u_{+}+\begin{pmatrix}u^{y}_{+}\\\
0\end{pmatrix}+\frac{T_{+}}{m_{+}}\nabla\eta_{+}=-\frac{e}{m_{+}}\nabla\phi,$
(1.9) $\displaystyle\Delta\phi=4\pi
e\left(\frac{e}{T_{-}}\phi-\eta_{+}\right).$ (1.10)

Let $\psi_{+}=\nabla\cdot u_{+}$ and $\omega_{+}=\nabla^{\perp}\cdot u_{+}$
with $\nabla^{\perp}=(-\partial_{y},\partial_{x})^{\top}$. Then the Helmholtz
projection operators are defined as

$u_{+}=(u^{x}_{+},u^{y}_{+})^{\top}=\nabla\Delta^{-1}\psi_{+}+\nabla^{\perp}\Delta^{-1}\omega_{+}:=\mathbb{Q}[u_{+}]+\mathbb{P}[u_{+}].$
(1.11)

Then the linearized system (1.8)-(1.10) is transformed into the form with
respect to $(\eta_{+},\psi_{+},\omega_{+})$

$\displaystyle\partial_{t}\eta_{+}+y\partial_{x}\eta_{+}+\psi_{+}=0,$ (1.12)
$\displaystyle\partial_{t}\psi_{+}+y\partial_{x}\psi_{+}+2\partial_{x}u^{y}_{+}+\frac{T_{+}}{m_{+}}\Delta\eta_{+}=-\frac{4\pi
e^{2}}{m_{+}}\Delta\left(-\Delta+\frac{4\pi
e^{2}}{T_{-}}\right)^{-1}\eta_{+},$ (1.13)
$\displaystyle\partial_{t}\omega_{+}+y\partial_{x}\omega_{+}-\psi_{+}=0.$
(1.14)

Denote the average of a function in the $x$ direction by

$f_{a}(y)=\frac{1}{2\pi}\int_{\mathbb{T}}f(x,y)dx.$

We take the $x$-average for equations (1.12), (1.13) and (1.14) to obtain the
following system

$\displaystyle\partial_{t}\eta_{+,a}+\psi_{+,a}=0,$ (1.15)
$\displaystyle\partial_{t}\psi_{+,a}+\frac{T_{+}}{m_{+}}\partial_{yy}\eta_{+,a}=-\frac{4\pi
e^{2}}{m_{+}}\Delta\left(-\Delta+\frac{4\pi
e^{2}}{T_{-}}\right)^{-1}\eta_{+,a},$ (1.16)
$\displaystyle\partial_{t}\omega_{+,a}-\psi_{+,a}=0.$ (1.17)

Combining (1.15) with (1.17) yields
$\partial_{t}\left(\eta_{+,a}+\omega_{+,a}\right)=0$. It is easy to see that

$\eta_{+,a}+\omega_{+,a}=\eta^{in}_{+,a}+\omega^{in}_{+,a},$ (1.18)

where $(\eta^{in}_{+,a},\omega^{in}_{+,a})$ represents the initial data. In
the rest of the paper, we still use similar notation to represent the initial
data.

According to (1.15) and (1.16), we get

$\partial_{tt}\eta_{+,a}-\frac{T_{+}}{m_{+}}\partial_{yy}\eta_{+,a}-\frac{4\pi
e^{2}}{m_{+}}\Delta\left(-\Delta+\frac{4\pi
e^{2}}{T_{-}}\right)^{-1}\eta_{+,a}=0~{}in~{}\mathbb{R}.$ (1.19)

Next we supply system (1.15)-(1.17) with the initial condition

$\left(\eta_{+,a},\psi_{+,a},\omega_{+,a}\right)|_{t=0}=\left(\eta^{in}_{+,a},\psi^{in}_{+,a},\omega^{in}_{+,a}\right)=(0,0,0).$
(1.20)

Then $\left(\eta_{+,a},\psi_{+,a},\omega_{+,a}\right)=(0,0,0)$ is the unique
solution to system (1.15)-(1.17) with initial condition (1.20). In fact,
taking the $L^{2}(\mathbb{R})$ inner product of (1.19) with
$\partial_{t}\eta_{+,a}$, then it follows from integration by parts that

$\frac{1}{2}\frac{d}{dt}\left(\left\lVert\partial_{t}\eta_{+,a}\right\rVert^{2}_{L^{2}}+\frac{T_{+}}{m_{+}}\left\lVert\partial_{y}\eta_{+,a}\right\rVert^{2}_{L^{2}}+\frac{4\pi
e^{2}}{m_{+}}\left\lVert\nabla\left(-\Delta+\frac{4\pi
e^{2}}{T_{-}}\right)^{-\frac{1}{2}}\eta_{+,a}\right\rVert^{2}_{L^{2}}\right)=0,$

which implies that $\eta_{+,a}=0$ is the unique solution to the cauchy
problem, equation (1.19) with $\eta^{in}_{+,a}=0$. By virtue of (1.18) and
(1.15), we deduce that $\left(\psi_{+,a},\omega_{+,a}\right)=(0,0)$. Based on
this fact, for simplicity of notation, we study the dynamics of
$(\eta_{+},\psi_{+},\omega_{+})$ with
$\left(\eta^{in}_{+,a},\psi^{in}_{+,a},\omega^{in}_{+,a}\right)=(0,0,0)$,
instead of studying the dynamics for
$(\eta_{+}-\eta_{+,a},\psi_{+}-\psi_{+,a},\omega_{+}-\omega_{+,a})$.

The main results for the ion dynamics are as follows.

######  Theorem 1.1.

Suppose that $(\eta^{in}_{+},\omega^{in}_{+})\in H^{1}_{x}H^{2}_{y}$ and that
$\psi^{in}_{+}\in H^{-\frac{1}{2}}_{x}L^{2}_{y}$ with
$\left(\eta^{in}_{+,a},\psi^{in}_{+,a},\omega^{in}_{+,a}\right)=(0,0,0)$. Let
$(\eta_{+},u_{+},\phi)$ be a smooth solution for the system (1.8)-(1.10). Then
the following estimates hold

$\displaystyle\left\lVert\mathbb{P}[u_{+}]^{x}(t)\right\rVert_{L^{2}}+\left\lVert\phi(t)\right\rVert_{L^{2}}$
$\displaystyle\lesssim\sqrt{\frac{m_{+}}{T_{+}}}\frac{1}{\left\langle
t\right\rangle^{\frac{1}{2}}}\left(\left\lVert\sqrt{\frac{T_{+}}{m_{+}}}\eta^{in}_{+}\right\rVert_{H^{-\frac{1}{2}}_{x}L^{2}_{y}}+\left\lVert\eta^{in}_{+}+\omega^{in}_{+}\right\rVert_{H^{-\frac{1}{2}}_{x}H^{\frac{1}{2}}_{y}}\right)$
$\displaystyle\quad+\frac{1}{\left\langle
t\right\rangle}\left\lVert\eta^{in}_{+}+\omega^{in}_{+}\right\rVert_{H^{-1}_{x}H^{1}_{y}}+\sqrt{\frac{m_{+}}{T_{+}}}\frac{1}{\left\langle
t\right\rangle^{\frac{1}{2}}}\left\lVert\psi^{in}_{+}\right\rVert_{H^{-\frac{1}{2}}_{x}H^{-1}_{y}},$
(1.21) $\displaystyle\left\lVert\mathbb{P}[u_{+}]^{y}(t)\right\rVert_{L^{2}}$
$\displaystyle\lesssim\sqrt{\frac{m_{+}}{T_{+}}}\frac{1}{\left\langle
t\right\rangle^{\frac{3}{2}}}\left(\left\lVert\sqrt{\frac{T_{+}}{m_{+}}}\eta^{in}_{+}\right\rVert_{H^{-\frac{1}{2}}_{x}H^{1}_{y}}+\left\lVert\eta^{in}_{+}+\omega^{in}_{+}\right\rVert_{H^{-\frac{1}{2}}_{x}H^{\frac{3}{2}}_{y}}\right)$
$\displaystyle\quad+\frac{1}{\left\langle
t\right\rangle^{2}}\left\lVert\eta^{in}_{+}+\omega^{in}_{+}\right\rVert_{H^{-1}_{x}H^{2}_{y}}+\sqrt{\frac{m_{+}}{T_{+}}}\frac{1}{\left\langle
t\right\rangle^{\frac{3}{2}}}\left\lVert\psi^{in}_{+}\right\rVert_{H^{-\frac{1}{2}}_{x}L^{2}_{y}},$
(1.22)

and

$\displaystyle\left\lVert\mathbb{Q}[u_{+}](t)\right\rVert_{L^{2}}+\sqrt{\frac{T_{+}}{m_{+}}}\left\lVert\eta_{+}(t)\right\rVert_{L^{2}}$
$\displaystyle\qquad\lesssim\left\langle
t\right\rangle^{\frac{1}{2}}\left(\left\lVert\sqrt{\frac{T_{+}}{m_{+}}}\eta^{in}_{+}\right\rVert_{L^{2}}+\left\lVert\psi^{in}_{+}\right\rVert_{H^{-1}}+\left\lVert\eta^{in}_{+}+\omega^{in}_{+}\right\rVert_{H^{1}}\right).$
(1.23)

######  Theorem 1.2.

Assume that $\left(\eta^{in}_{+},\omega^{in}_{+}\right)\in
L^{2}_{x}H^{-\frac{1}{2}}_{y}$ and that $\psi^{in}_{+}\in
H^{-\frac{3}{2}}_{x}H^{-2}_{y}$ with
$\left(\eta^{in}_{+,a},\psi^{in}_{+,a},\omega^{in}_{+,a}\right)=(0,0,0)$. Let
$(\eta_{+},u_{+},\phi)$ be a smooth solution for the system (1.8)-(1.10). Then
it holds that

$\left\lVert\mathbb{Q}[u_{+}](t)\right\rVert_{L^{2}}+\sqrt{\frac{T_{+}}{m_{+}}}\left\lVert\eta_{+}(t)\right\rVert_{L^{2}}\gtrsim\mathcal{C}^{in}_{\delta}\left(\eta^{in}_{+},\psi^{in}_{+},\omega^{in}_{+}\right)\left\langle
t\right\rangle^{\frac{1}{2}},$

where the constant
$\mathcal{C}^{in}_{\delta}\left(\eta^{in}_{+},\psi^{in}_{+},\omega^{in}_{+}\right)$
is a suitable combination of
$\left\lVert\eta^{in}_{+}\right\rVert_{L^{2}_{x}H^{-\frac{1}{2}}_{y}}$,
$\left\lVert\psi^{in}_{+}\right\rVert_{H^{-\frac{3}{2}}_{x}H^{-2}_{y}}$ and
$\left\lVert\omega^{in}_{+}\right\rVert_{L^{2}_{x}H^{-\frac{1}{2}}_{y}}$.

The above two theorems give upper and lower bounds of the linearized solutions
of the Euler-Poisson system near Couette flow. In particular, the inviscid
damping for the solenoidal component of the velocity is established, while the
$L^{2}$-norm of the velocity grows as $t^{1/2}$. We also remark that it may
happen that the constant $\mathcal{C}^{in}_{\delta}$ vanishes at some positive
time $t>0$, however we can prove that the set of initial data for which the
RHS of the lower bound in Theorem 1.2 vanishes at some time has empty interior
in any Sobolev space in which the initial data are taken. The details are
similar to that for the Euler system in [1] and are omitted here.

### 1.2. Electron dynamics case

To study the linear stability for the Couette flow in system (1.2)-(1.4), we
also carry out a perturbation around the above stationary solution, i.e.,

$\widetilde{\eta}_{-}=\eta_{-}+\widetilde{\eta}_{s},\quad\widetilde{v}_{-}=u_{-}+\widetilde{v}_{s},\quad\widetilde{\phi}=\phi+\widetilde{\phi}_{s}.$

Then the linearized system around the stationary solution
$(\widetilde{\eta}_{s},\widetilde{v}_{s},\widetilde{\phi}_{s})$ for system
(1.2)-(1.4) can be rewritten as

$\displaystyle\partial_{t}\eta_{-}+y\partial_{x}\eta_{-}+(\nabla\cdot
u_{-})=0,$ (1.24)
$\displaystyle\partial_{t}u_{-}+y\partial_{x}u_{-}+\begin{pmatrix}u^{y}_{-}\\\
0\end{pmatrix}+\frac{1}{m_{-}}\nabla\eta_{-}=\frac{e}{m_{-}}\nabla\phi,$
(1.25) $\displaystyle\Delta\phi=4\pi e\eta_{-}.$ (1.26)

Let $\psi_{-}=\nabla\cdot u_{-}$ and $\omega_{-}=\nabla^{\perp}\cdot u_{-}$
with $\nabla^{\perp}=(-\partial_{y},\partial_{x})^{\top}$. Then the Helmholtz
projection operators are defined as

$u_{-}=(u^{x}_{-},u^{y}_{-})^{\top}=\nabla\Delta^{-1}\psi_{-}+\nabla^{\perp}\Delta^{-1}\omega_{-}:=\mathbb{Q}[u_{-}]+\mathbb{P}[u_{-}].$
(1.27)

The linearized system (1.24)-(1.26) is rewritten as (in the variables
$(\eta_{-},\psi_{-},\omega_{-})$)

$\displaystyle\partial_{t}\eta_{-}+y\partial_{x}\eta_{-}+\psi_{-}=0,$ (1.28)
$\displaystyle\partial_{t}\psi_{-}+y\partial_{x}\psi_{-}+2\partial_{x}u^{y}_{-}+\frac{1}{m_{-}}\Delta\eta_{-}=\frac{4\pi
e^{2}}{m_{-}}\eta_{-},$ (1.29)
$\displaystyle\partial_{t}\omega_{-}+y\partial_{x}\omega_{-}-\psi_{-}=0.$
(1.30)

Just as in the ion dynamics case, we take the $x$-average for equations
(1.28), (1.29) and (1.30) to obtain the following system

$\displaystyle\partial_{t}\eta_{-,a}+\psi_{-,a}=0,$ (1.31)
$\displaystyle\partial_{t}\psi_{-,a}+\frac{1}{m_{-}}\partial_{yy}\eta_{-,a}=\frac{4\pi
e^{2}}{m_{-}}\eta_{-,a},$ (1.32)
$\displaystyle\partial_{t}\omega_{-,a}-\psi_{-,a}=0.$ (1.33)

Again, combining (1.31) with (1.33) yields
$\partial_{t}\left(\eta_{-,a}+\omega_{-,a}\right)=0$ and hence

$\eta_{-,a}+\omega_{-,a}=\eta^{in}_{-,a}+\omega^{in}_{-,a},$ (1.34)

where $(\eta^{in}_{-,a},\omega^{in}_{-,a})$ represents the initial data.
According to (1.31) and (1.32), we get

$\partial_{tt}\eta_{-,a}-\frac{1}{m_{-}}\partial_{yy}\eta_{-,a}+\frac{4\pi
e^{2}}{m_{-}}\eta_{-,a}=0~{}in~{}\mathbb{R},$ (1.35)

which is the famous Klein-Gordon equation. The fact that at the linearized
level (near the constant solution), the electron Euler-Poisson system
satisfies the Klein-Gordon equation was found by Guo [18] and the global
irrotational flows in the large to the electron Euler-Poisson system was hence
obtained in $\mathbb{R}^{3+1}$. For global well-posedness of the lower
dimensional Euler-Poisson system, one may refer to [28, 27, 26, 36].

Similar to the ion dynamics case, we supply system (1.31)-(1.33) with the
initial condition

$\left(\eta_{-,a},\psi_{-,a},\omega_{-,a}\right)|_{t=0}=\left(\eta^{in}_{-,a},\psi^{in}_{-,a},\omega^{in}_{-,a}\right)=(0,0,0).$
(1.36)

From (1.34) and the explicit representation formula for the Klein-Gordon
equation, we deduce that
$\left(\eta_{-,a},\psi_{-,a},\omega_{-,a}\right)=(0,0,0)$ is the unique
solution to system (1.31)-(1.33) with initial condition (1.36). Based on this
fact, for simplicity of notation, we study the dynamics of
$(\eta_{-},\psi_{-},\omega_{-})$ with
$\left(\eta^{in}_{-,a},\psi^{in}_{-,a},\omega^{in}_{-,a}\right)=(0,0,0)$,
instead of studying the dynamics for
$(\eta_{-}-\eta_{-,a},\psi_{-}-\psi_{-,a},\omega_{-}-\omega_{-,a})$.

The main results for the linear stability of the electron Euler-Poisson system
are stated as follows.

######  Theorem 1.3.

Suppose that $(\eta^{in}_{-},\omega^{in}_{-})\in H^{1}_{x}H^{2}_{y}$ and that
$\psi^{in}_{-}\in H^{-\frac{1}{2}}_{x}L^{2}_{y}$ with
$\left(\eta^{in}_{-,a},\psi^{in}_{-,a},\omega^{in}_{-,a}\right)=(0,0,0)$. Let
$(\eta_{-},u_{-},\phi)$ be a smooth solution for the system (1.24)-(1.26).
Then the following estimates hold

$\displaystyle\left\lVert\mathbb{P}[u_{-}]^{x}(t)\right\rVert_{L^{2}}+\left\lVert\phi(t)\right\rVert_{L^{2}}$
$\displaystyle\lesssim\frac{\sqrt{m_{-}}}{\left\langle
t\right\rangle^{\frac{1}{2}}}\left(\left\lVert\sqrt{\frac{1}{m_{-}}}\eta^{in}_{-}\right\rVert_{H^{-\frac{1}{2}}_{x}L^{2}_{y}}+\left\lVert\eta^{in}_{-}+\omega^{in}_{-}\right\rVert_{H^{-\frac{1}{2}}_{x}H^{\frac{1}{2}}_{y}}\right)$
$\displaystyle\quad+\frac{1}{\left\langle
t\right\rangle}\left\lVert\eta^{in}_{-}+\omega^{in}_{-}\right\rVert_{H^{-1}_{x}H^{1}_{y}}+\frac{\sqrt{m_{-}}}{\left\langle
t\right\rangle^{\frac{1}{2}}}\left\lVert\psi^{in}_{-}\right\rVert_{H^{-\frac{1}{2}}_{x}H^{-1}_{y}},$
(1.37) $\displaystyle\left\lVert\mathbb{P}[u_{-}]^{y}(t)\right\rVert_{L^{2}}$
$\displaystyle\lesssim\frac{\sqrt{m_{-}}}{\left\langle
t\right\rangle^{\frac{3}{2}}}\left(\left\lVert\sqrt{\frac{1}{m_{-}}}\eta^{in}_{-}\right\rVert_{H^{-\frac{1}{2}}_{x}H^{1}_{y}}+\left\lVert\eta^{in}_{-}+\omega^{in}_{-}\right\rVert_{H^{-\frac{1}{2}}_{x}H^{\frac{3}{2}}_{y}}\right)$
$\displaystyle\quad+\frac{1}{\left\langle
t\right\rangle^{2}}\left\lVert\eta^{in}_{-}+\omega^{in}_{-}\right\rVert_{H^{-1}_{x}H^{2}_{y}}+\frac{\sqrt{m_{-}}}{\left\langle
t\right\rangle^{\frac{3}{2}}}\left\lVert\psi^{in}_{-}\right\rVert_{H^{-\frac{1}{2}}_{x}L^{2}_{y}},$
(1.38)

and

$\displaystyle\left\lVert\mathbb{Q}[u_{-}](t)\right\rVert_{L^{2}}+\sqrt{\frac{1}{m_{-}}}\left\lVert\eta_{-}(t)\right\rVert_{L^{2}}$
$\displaystyle\qquad\lesssim\left\langle
t\right\rangle^{\frac{1}{2}}\left(\left\lVert\sqrt{\frac{1}{m_{-}}}\eta^{in}_{-}\right\rVert_{L^{2}}+\left\lVert\psi^{in}_{-}\right\rVert_{H^{-1}}+\left\lVert\eta^{in}_{-}+\omega^{in}_{-}\right\rVert_{H^{1}}\right).$
(1.39)

######  Theorem 1.4.

Assume that $\left(\eta^{in}_{-},\omega^{in}_{-}\right)\in
L^{2}_{x}H^{-\frac{1}{2}}_{y}$ and that $\psi^{in}_{-}\in
H^{-\frac{3}{2}}_{x}H^{-2}_{y}$ with
$\left(\eta^{in}_{-,a},\psi^{in}_{-,a},\omega^{in}_{-,a}\right)=(0,0,0)$. Let
$(\eta_{-},u_{-},\phi)$ be a smooth solution for the system (1.24)-(1.26).
Then we have

$\left\lVert\mathbb{Q}[u_{-}](t)\right\rVert_{L^{2}}+\sqrt{\frac{1}{m_{-}}}\left\lVert\eta_{-}(t)\right\rVert_{L^{2}}\gtrsim\mathcal{C}^{in}_{\sigma}\left(\eta^{in}_{-},\psi^{in}_{-},\omega^{in}_{-}\right)\left\langle
t\right\rangle^{\frac{1}{2}},$

where the constant
$\mathcal{C}^{in}_{\sigma}\left(\eta^{in}_{-},\psi^{in}_{-},\omega^{in}_{-}\right)$
depends on Sobolev norms of the initial data.

######  Remark 1.1.

It should be noted that at the linearized level, the dynamics of the ion
Euler-Poisson system and the electron Euler-Poisson system are depicted by
different linear PDE systems, (1.19) for the ion dynamics and the (1.35) for
the electron dynamics. The linear dispersion relation behaves much closer to
the wave dispersion for the ions and to the Klein-Gordon dispersion for the
electrons. We investigate the linear stability the Couette flow for ion and
electron dynamics separately below.

######  Remark 1.2.

Since the analysis in frequency space can be given by Lemma 3.1 or 4.1,
estimates (1.23) and (1.39) are also valid in the general Sobolev space
$H^{-s}(s>\frac{1}{2})$.

This paper is organized as follows. In the next section, we will give some
notations and the definitions of Sobolev spaces. We will also write out the
coordinate transform explicitly. The linear stability of the ion dynamics near
the Couette flow will be analyzed in Section 3 and the linear stability of the
electron dynamics near the Couette flow will be analyzed in Section 4.

## 2\. Notations

The Fourier transform for a function $f(x,y)$ is denoted by

$\widehat{f}(k,\xi)=\frac{1}{2\pi}\iint_{\mathbb{T}\times\mathbb{R}}e^{-i\left(kx+\xi
y\right)}f(x,y)dxdy.$

Then

$f(x,y)=\frac{1}{2\pi}\sum_{k\in\mathbb{Z}}\int_{\mathbb{R}}e^{i\left(kx+\xi
y\right)}\widehat{f}(k,\xi)d\xi.$

The anisotropic Sobolev space is defined by

$H^{r}_{x}H^{s}_{y}({\mathbb{T}\times\mathbb{R}})=\left\\{f:\left\lVert
f\right\rVert^{2}_{H^{r}_{x}H^{s}_{y}({\mathbb{T}\times\mathbb{R}})}=\sum_{k}\int\left\langle
k\right\rangle^{2r}\left\langle\xi\right\rangle^{2s}\left|\widehat{f}\right|^{2}(k,\xi)d\xi<+\infty\right\\},$

where $\left\langle a\right\rangle=(1+a^{2})^{1/2}$ for $a\in\mathbb{R}$. In
the same way, the usual Sobolev space is defined as

$H^{s}({\mathbb{T}\times\mathbb{R}})=\left\\{f:\left\lVert
f\right\rVert^{2}_{H^{s}({\mathbb{T}\times\mathbb{R}})}=\sum_{k}\int\left\langle
k,\xi\right\rangle^{2s}\left|\widehat{f}\right|^{2}(k,\xi)d\xi<+\infty\right\\}.$

Here $\left\langle a,b\right\rangle=(1+a^{2}+b^{2})^{1/2}$ for
$a,b\in\mathbb{R}$.

In order to deal with the transport term in system (1.12)-(1.14) or
(1.28)-(1.30), we introduce a set of coordinate transformations

$X=x-yt,~{}Y=y.$ (2.1)

So the differential operators can be represented as

$\partial_{x}=\partial_{X},~{}\partial_{y}=\partial_{Y}-t\partial_{X},~{}\Delta=:\Delta_{L}=\partial_{XX}+\left(\partial_{Y}-t\partial_{X}\right)^{2}.$
(2.2)

We denote the symbol for $-\Delta_{L}$ by

$\alpha(t,k,\xi)=k^{2}+(\xi-kt)^{2}.$ (2.3)

Then the symbol of the operator $2\partial_{X}(\partial_{Y}-t\partial_{X})$ is

$\partial_{t}\alpha(t,k,\xi)=-2k(\xi-kt).$ (2.4)

## 3\. Analysis for the ion dynamics

In this section, we consider the dynamics of the system (1.12)-(1.14). Using
the coordinate transformations (2.1), we define the functions

$\displaystyle\Pi_{+}(t,X,Y)=\eta_{+}(t,X+tY,Y),$ (3.1)
$\displaystyle\Psi_{+}(t,X,Y)=\psi_{+}(t,X+tY,Y),$ (3.2)
$\displaystyle\Gamma_{+}(t,X,Y)=\omega_{+}(t,X+tY,Y).$ (3.3)

Set $F_{+}(t,X,Y):=\Pi_{+}(t,X,Y)+\Gamma_{+}(t,X,Y)$. Combining (1.12) with
(1.14), we have $\partial_{t}F_{+}=0$, which yields

$\Gamma_{+}(t,X,Y)=F^{in}_{+}(X,Y)-\Pi_{+}(t,X,Y).$

Here $F^{in}_{+}=\eta^{in}_{+}+\omega^{in}_{+}$. According to (1.11), we
obtain

$\displaystyle U^{y}_{+}$
$\displaystyle=\left(\partial_{Y}-t\partial_{X}\right)\Delta^{-1}_{L}\Psi_{+}+\partial_{X}\Delta^{-1}_{L}\Gamma_{+}$
$\displaystyle=\left(\partial_{Y}-t\partial_{X}\right)\Delta^{-1}_{L}\Psi_{+}+\partial_{X}\Delta^{-1}_{L}F^{in}_{+}-\partial_{X}\Delta^{-1}_{L}\Pi_{+}.$

In consequence, we can rewrite system (1.12)-(1.14) as

$\displaystyle\partial_{t}\Pi_{+}+\Psi_{+}=0,$ (3.4)
$\displaystyle\partial_{t}\Psi_{+}+2\partial_{XX}\Delta^{-1}_{L}F^{in}_{+}+2\partial_{X}\left(\partial_{Y}-t\partial_{X}\right)\Delta^{-1}_{L}\Psi_{+}$
$\displaystyle-\left(2\partial_{XX}\Delta^{-1}_{L}-\frac{T_{+}}{m_{+}}\Delta_{L}\right)\Pi_{+}+\frac{4\pi
e^{2}}{m_{+}}\Delta_{L}\left(-\Delta_{L}+\frac{4\pi
e^{2}}{T_{-}}\right)^{-1}\Pi_{+}=0.$ (3.5)

Taking the Fourier transform on system (3.4)-(3.5), we get the following
system

$\displaystyle\partial_{t}\widehat{\Pi_{+}}+\widehat{\Psi_{+}}=0,$ (3.6)
$\displaystyle\partial_{t}\widehat{\Psi_{+}}+\frac{2k^{2}}{\alpha}\widehat{F^{in}_{+}}-\frac{\partial_{t}\alpha}{\alpha}\widehat{\Psi_{+}}-\left(\frac{2k^{2}}{\alpha}+\frac{T_{+}\alpha}{m_{+}}+\frac{4\pi
e^{2}}{m_{+}}\frac{\alpha}{\alpha+\frac{4\pi
e^{2}}{T_{-}}}\right)\widehat{\Pi_{+}}=0.$ (3.7)

In order to study system (3.6)-(3.7), we need to find a suitable
symmetrization for this system. Define

$A(t)=\left(A_{1}(t),A_{2}(t)\right)^{\top}=\left(\sqrt{\frac{T_{+}}{m_{+}}}\frac{\widehat{\Pi_{+}}(t)}{\alpha^{1/4}},\frac{\widehat{\Psi_{+}}(t)}{\alpha^{3/4}}\right)^{\top}.$
(3.8)

By a delicate calculation, $A(t)$ satisfy a non-autonomous 2D dynamical system

$\begin{cases}\frac{d}{dt}A(t)=L_{+}(t)A(t)+M_{+}(t)\widehat{F^{in}_{+}},\\\
A(0)=A^{in},\end{cases}$ (3.9)

where

$\displaystyle
L_{+}(t)=\begin{bmatrix}-\frac{1}{4}\alpha^{-1}\partial_{t}\alpha&-\sqrt{\frac{T_{+}}{m_{+}}}\alpha^{1/2}\\\
\frac{4\pi e^{2}}{\sqrt{m_{+}T_{+}}}\frac{\alpha^{1/2}}{\alpha+\frac{4\pi
e^{2}}{T_{-}}}+\sqrt{\frac{m_{+}}{T_{+}}}\frac{2k^{2}}{\alpha^{3/2}}+\sqrt{\frac{T_{+}}{m_{+}}}\alpha^{1/2}&\frac{1}{4}\alpha^{-1}\partial_{t}\alpha\end{bmatrix},$
(3.10) $\displaystyle
M_{+}(t)=\left(0,-\frac{2k^{2}}{\alpha^{7/4}}\right)^{\top},~{}A^{in}=\left(\sqrt{\frac{T_{+}}{m_{+}}}\frac{\widehat{\Pi^{in}_{+}}}{\left(k^{2}+\xi^{2}\right)^{1/4}},\frac{\widehat{\Psi^{in}_{+}}}{\left(k^{2}+\xi^{2}\right)^{3/4}}\right)^{\top}.$
(3.11)

By virtue of Duhamel’s formula, the solution $A(t)$ to system (3.9) is given
by

$A(t)=S_{L}(t,0)A^{in}+\int^{t}_{0}S_{L}(t,s)M_{+}(s)\widehat{F^{in}_{+}}ds.$
(3.12)

Here, $S_{L}$ denotes the solution operator, satisfying the group property

$S_{L}(t,0)S_{L}(0,s)=S_{L}(t,s)$

for any $t,s>0$. In consequence, it suffices to study the homogeneous problem
of system (3.9). For this purpose, we denote by $A(t)$ a solution to system
(3.9) with $\widehat{F^{in}_{+}}=0$. Let

$\mathcal{E}_{+}(t)=\left(\sqrt{\frac{p_{1}}{m_{1}}}|A_{1}|^{2}\right)(t)+2\left(\frac{h_{1}}{\sqrt{m_{1}p_{1}}}\operatorname{Re}\left(A_{1}\bar{A_{2}}\right)\right)(t)+\left(\sqrt{\frac{m_{1}}{p_{1}}}|A_{2}|^{2}\right)(t),$
(3.13)

where

$\displaystyle
h_{1}(t)=\frac{1}{4}\alpha^{-1}\partial_{t}\alpha,~{}m_{1}(t)=\sqrt{\frac{T_{+}}{m_{+}}}\alpha^{1/2},$
(3.14) $\displaystyle p_{1}(t)=\frac{4\pi
e^{2}}{\sqrt{m_{+}T_{+}}}\frac{\alpha^{1/2}}{\alpha+\frac{4\pi
e^{2}}{T_{-}}}+\sqrt{\frac{m_{+}}{T_{+}}}\frac{2k^{2}}{\alpha^{3/2}}+\sqrt{\frac{T_{+}}{m_{+}}}\alpha^{1/2}.$
(3.15)

Then we can get the upper and lower bounds for $\mathcal{E}_{+}(t)$ and the
solution $A(t)$.

######  Lemma 3.1.

There exists positive constants $C_{1}$, $C^{\delta}_{1}$, $C_{2}$,
$C^{\delta}_{2}$ that do not depend on $k$ and $\xi$ such that

$C_{1}\mathcal{E}_{+}(0)\leq\mathcal{E}_{+}(t)\leq
C^{\delta}_{1}\mathcal{E}_{+}(0),$ (3.16)

and

$C_{2}\left|A^{in}\right|\leq\left|A(t)\right|\leq
C^{\delta}_{2}\left|A^{in}\right|.$ (3.17)

###### Proof.

Define

$\displaystyle\lambda_{+}=\sqrt{\frac{p_{1}}{m_{1}}}=\left(1+\frac{2m_{+}k^{2}}{T_{+}\alpha^{2}}+\frac{4\pi
e^{2}}{T_{+}}\frac{1}{\alpha+\frac{4\pi e^{2}}{T_{-}}}\right)^{1/2},$ (3.18)
$\displaystyle\gamma_{+}=\sqrt{m_{1}p_{1}}=\left(\frac{4\pi
e^{2}}{m_{+}}\frac{\alpha}{\alpha+\frac{4\pi
e^{2}}{T_{-}}}+\frac{2k^{2}}{\alpha}+\frac{T_{+}\alpha}{m_{+}}\right)^{1/2}.$
(3.19)

From the definition of $\lambda_{+}$ we get that

$1\leq(\lambda_{+})^{2}\leq 1+\frac{4\pi e^{2}}{T_{+}}+\frac{2m_{+}}{T_{+}}.$
(3.20)

Moreover, we have

$\displaystyle\frac{|h_{1}|}{\gamma_{+}}$
$\displaystyle=\frac{|\partial_{t}\alpha|}{4\alpha}\left(\frac{4\pi
e^{2}}{m_{+}}\frac{\alpha}{\alpha+\frac{4\pi
e^{2}}{T_{-}}}+\frac{2k^{2}}{\alpha}+\frac{T_{+}\alpha}{m_{+}}\right)^{-1/2}$
$\displaystyle=\frac{|\partial_{t}\alpha|}{4\alpha}\left(\frac{m_{+}}{T_{+}\alpha}\right)^{1/2}\left(1+\frac{2m_{+}k^{2}}{T_{+}\alpha^{2}}+\frac{4\pi
e^{2}}{T_{+}}\frac{1}{\alpha+\frac{4\pi e^{2}}{T_{-}}}\right)^{-1/2}$
$\displaystyle\leq\frac{|k|}{2\sqrt{\alpha}}\left(\frac{m_{+}}{T_{+}\alpha}\right)^{1/2}\left(\frac{T_{+}\alpha^{2}}{2m_{+}k^{2}}\right)^{1/2}\leq\frac{\sqrt{2}}{2},$
(3.21)

where we have used the fact that $|\partial_{t}\alpha|\leq 2|k|\sqrt{\alpha}$.

Setting

$\widetilde{\mathcal{E}}_{+}(t)=\left(\lambda_{+}|A_{1}|^{2}\right)(t)+\left(\frac{1}{\lambda_{+}}|A_{2}|^{2}\right)(t).$

Then from (3.21) it can be deduced that

$\left(1-\frac{\sqrt{2}}{2}\right)\widetilde{\mathcal{E}}_{+}(t)\leq\mathcal{E}_{+}(t)\leq\left(1+\frac{\sqrt{2}}{2}\right)\widetilde{\mathcal{E}}_{+}(t).$
(3.22)

Obviously, the coerciveness of $\mathcal{E}_{+}(t)$ is ensured by the above
inequality and (3.20).

According to (3.14), (3.15), (3.18) and (3.19), system (3.9) becomes

$\displaystyle\lambda_{+}\frac{d}{dt}A_{1}=-h_{1}\lambda_{+}A_{1}-\gamma_{+}A_{2},$
(3.23)
$\displaystyle\frac{1}{\lambda_{+}}\frac{d}{dt}A_{2}=\gamma_{+}A_{1}+\frac{h_{1}}{\lambda_{+}}A_{2}.$
(3.24)

Multiplying equations (3.23) and (3.24) by $\bar{A}_{1}$ and $\bar{A}_{2}$,
respectively, and similarly for their conjugate, we get

$\frac{\lambda_{+}}{2}\frac{d}{dt}|A_{1}|^{2}+\frac{1}{2\lambda_{+}}\frac{d}{dt}|A_{2}|^{2}=-h_{1}\lambda_{+}|A_{1}|^{2}+\frac{h_{1}}{\lambda_{+}}|A_{2}|^{2}.$
(3.25)

Note the fact that

$\frac{h_{1}}{\gamma_{+}}\frac{d}{dt}\operatorname{Re}(A_{1}\bar{A}_{2})=h_{1}\lambda_{+}|A_{1}|^{2}-\frac{h_{1}}{\lambda_{+}}|A_{2}|^{2}.$
(3.26)

Adding (3.25) to (3.26), we obtain

$\lambda_{+}\frac{d}{dt}|A_{1}|^{2}+\frac{2h_{1}}{\gamma_{+}}\frac{d}{dt}\operatorname{Re}(A_{1}\bar{A}_{2})+\frac{1}{\lambda_{+}}\frac{d}{dt}|A_{2}|^{2}=0,$

which yields

$\frac{d\mathcal{E}_{+}}{dt}=\lambda_{+}|A_{1}|^{2}\frac{d}{dt}\left(\log\lambda_{+}\right)+2\operatorname{Re}(A_{1}\bar{A_{2}})\frac{d}{dt}\left(\frac{h_{1}}{\gamma_{+}}\right)-\frac{1}{\lambda_{+}}|A_{2}|^{2}\frac{d}{dt}\left(\log\lambda_{+}\right).$

Using Young’s inequality, from (3.22) we conclude that

$\displaystyle\frac{d\mathcal{E}_{+}}{dt}$
$\displaystyle\leq\left(\left|\frac{d}{dt}\left(\frac{h_{1}}{\gamma_{+}}\right)\right|+\left|\frac{d}{dt}\left(\log\lambda_{+}\right)\right|\right)\widetilde{\mathcal{E}}_{+}$
$\displaystyle\leq\left(2+\sqrt{2}\right)\left(\left|\frac{d}{dt}\left(\frac{h_{1}}{\gamma_{+}}\right)\right|+\left|\frac{d}{dt}\left(\log\lambda_{+}\right)\right|\right)\mathcal{E}_{+}.$
(3.27)

An argument similar to that of (3.27) gives

$\frac{d\mathcal{E}_{+}}{dt}\geq-\left(2+\sqrt{2}\right)\left(\left|\frac{d}{dt}\left(\frac{h_{1}}{\gamma_{+}}\right)\right|+\left|\frac{d}{dt}\left(\log\lambda_{+}\right)\right|\right)\mathcal{E}_{+}.$
(3.28)

For the purpose of applying Grönwall’s inequality, we need to estimate the
integrals

$\int^{+\infty}_{0}\left|\frac{d}{dt}\left(\frac{h_{1}}{\gamma_{+}}\right)\right|d\vartheta\
\ \ and\ \ \
\int^{+\infty}_{0}\left|\frac{d}{dt}\left(\log\lambda_{+}\right)\right|d\vartheta.$

First of all, we deal with the first one. By a direct calculation, we have

$\frac{d}{dt}\left(\frac{h_{1}}{\gamma_{+}}\right)=\frac{\mathcal{G}}{8m_{+}(T_{-})^{2}(\gamma_{+})^{3}\alpha^{3}\left(\alpha+\frac{4\pi
e^{2}}{T_{-}}\right)^{3}},$

where

$\displaystyle\mathcal{G}=$ $\displaystyle 16\pi
e^{2}(T_{-})^{2}k^{2}\alpha^{5}+\left(8m_{+}T_{-}k^{4}\alpha^{3}+4T_{+}T_{-}k^{2}\alpha^{5}\right)\left(T_{-}\alpha+4\pi
e^{2}\right)$ $\displaystyle+128\pi^{2}e^{4}T_{-}k^{2}\alpha^{4}+\left(64\pi
e^{2}m_{+}k^{4}\alpha^{2}+32\pi
e^{2}T_{+}k^{2}\alpha^{4}\right)\left(T_{-}\alpha+4\pi e^{2}\right)$
$\displaystyle+256\pi^{3}e^{6}k^{2}\alpha^{3}+\left(128\pi^{2}e^{4}m_{+}k^{4}\alpha+64\pi^{2}e^{4}T_{+}k^{2}\alpha^{3}\right)\left(\alpha+\frac{4\pi
e^{2}}{T_{-}}\right)$ $\displaystyle-8\pi
e^{2}(T_{-})^{2}\alpha^{4}(\partial_{t}\alpha)^{2}-\left[4m_{+}T_{-}k^{2}\alpha^{2}(\partial_{t}\alpha)^{2}+2T_{+}T_{-}\alpha^{4}(\partial_{t}\alpha)^{2}\right]\left(T_{-}\alpha+4\pi
e^{2}\right)$
$\displaystyle-64\pi^{2}e^{4}T_{-}\alpha^{3}(\partial_{t}\alpha)^{2}-\left[32\pi
e^{2}m_{+}k^{2}\alpha(\partial_{t}\alpha)^{2}+16\pi
e^{2}T_{+}\alpha^{3}(\partial_{t}\alpha)^{2}\right]\left(T_{-}\alpha+4\pi
e^{2}\right)$
$\displaystyle-128\pi^{3}e^{6}\alpha^{2}(\partial_{t}\alpha)^{2}-\left[64\pi^{2}e^{4}m_{+}k^{2}(\partial_{t}\alpha)^{2}+32\pi^{2}e^{4}T_{+}\alpha^{2}(\partial_{t}\alpha)^{2}\right]\left(\alpha+\frac{4\pi
e^{2}}{T_{-}}\right)$ $\displaystyle+\left[16\pi
e^{2}m_{+}k^{2}\alpha(\partial_{t}\alpha)^{2}-8\pi
e^{2}T_{+}\alpha^{3}(\partial_{t}\alpha)^{2}+\frac{32\pi^{2}e^{4}m_{+}k^{2}}{T_{-}}(\partial_{t}\alpha)^{2}\right]\left(T_{-}\alpha+4\pi
e^{2}\right)$
$\displaystyle+\left[\left(2m_{+}T_{-}k^{2}-16\pi^{2}e^{4}-\frac{16\pi^{2}e^{4}T_{+}}{T_{-}}\right)\alpha^{2}(\partial_{t}\alpha)^{2}-T_{+}T_{-}\alpha^{4}(\partial_{t}\alpha)^{2}\right]\left(T_{-}\alpha+4\pi
e^{2}\right)$

is a polynomial in time of order 12. Denote by $t_{i}(i=1,2,...,12)$ the
possible positive roots for $\mathcal{G}$. We set $i_{0}=0$ provided that
$\mathcal{G}(0)\leq 0$ and take $i_{0}=1$ for the other cases. Assume that
$t_{0}=0$ and that $t_{13}=+\infty$. By virtue of (3.21), we infer

$\displaystyle\int^{+\infty}_{0}\left|\frac{d}{dt}\left(\frac{h_{1}}{\gamma_{+}}\right)\right|d\vartheta$
$\displaystyle=\sum^{13}_{i=1}(-1)^{i+i_{0}}\left(\frac{h_{1}}{\gamma_{+}}(t_{i})-\frac{h_{1}}{\gamma_{+}}(t_{i-1})\right)$
$\displaystyle\leq 2\sum^{13}_{i=0}\frac{|h_{1}|}{\gamma_{+}}(t_{i})\leq
14\sqrt{2}.$ (3.29)

For the second integral term, we deduce from (3.20) that

$\displaystyle\int^{+\infty}_{0}\left|\frac{d}{dt}\left(\log\lambda_{+}\right)\right|d\vartheta$
$\displaystyle=\frac{1}{2}\int^{\frac{\xi}{k}}_{0}\left|\frac{d}{dt}\left(\log(\lambda_{+})^{2}\right)\right|d\vartheta+\frac{1}{2}\int^{+\infty}_{\frac{\xi}{k}}\left|\frac{d}{dt}\left(\log(\lambda_{+})^{2}\right)\right|d\vartheta$
$\displaystyle\leq\frac{1}{2}\left[\left(\log(\lambda_{+})^{2}\right)\left(\frac{\xi}{k}\right)-\left(\log(\lambda_{+})^{2}\right)\left(0\right)\right]$
$\displaystyle\quad+\frac{1}{2}\left[\left(\log(\lambda_{+})^{2}\right)\left(\frac{\xi}{k}\right)-\left(\log(\lambda_{+})^{2}\right)\left(+\infty\right)\right]$
$\displaystyle\leq\log\left(1+\frac{4\pi
e^{2}}{T_{+}}+\frac{2m_{+}}{T_{+}}\right),$ (3.30)

where we have used the fact that

$\displaystyle\frac{1}{(\lambda_{+})^{2}}\frac{d(\lambda_{+})^{2}}{dt}$
$\displaystyle=(T_{+})^{-1}(\lambda_{+})^{-2}\alpha^{-3}\left(\alpha+\frac{4\pi
e^{2}}{T_{-}}\right)^{-2}$
$\displaystyle\quad\times\left[-4m_{+}k^{2}\left(\alpha+\frac{4\pi
e^{2}}{T_{-}}\right)^{2}-4\pi e^{2}\alpha^{3}\right]\partial_{t}\alpha=0$

has a unique root $t=\xi/k$. Applying Grönwall’s inequality to (3.27) and
(3.28), respectively, from (3.29) and (3.30) we obtain (3.16).

Next, it remains to prove (3.17). Owing to (3.16) and (3.22), it can be seen
that

$C^{\prime}_{1}\widetilde{\mathcal{E}}_{+}(0)\leq\widetilde{\mathcal{E}}_{+}(t)\leq
C^{\delta^{\prime}}_{1}\widetilde{\mathcal{E}}_{+}(0).$ (3.31)

By virtue of (3.20), we get

$\left(1+\frac{4\pi
e^{2}}{T_{+}}+\frac{2m_{+}}{T_{+}}\right)^{-1}\widetilde{\mathcal{E}}_{+}(t)\leq|A(t)|^{2}\leq\left(1+\frac{4\pi
e^{2}}{T_{+}}+\frac{2m_{+}}{T_{+}}\right)\widetilde{\mathcal{E}}_{+}(t).$
(3.32)

Combining (3.31) with (3.32) yields (3.17). This completes the proof. ∎

Based on Lemma 3.1, we give the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

Due to Lemma 3.1, the solution $A(t)$ of system (3.9) can be bounded as

$\displaystyle|A(t)|$
$\displaystyle=\left|S_{L}(t,0)A^{in}+\int^{t}_{0}S_{L}(t,s)M_{+}(s)\widehat{F^{in}_{+}}ds\right|$
$\displaystyle\lesssim|A^{in}|+\left|\widehat{F^{in}_{+}}\right|\int^{t}_{0}\frac{2k^{2}}{\alpha^{7/4}}ds$
$\displaystyle\lesssim|A^{in}|+\left|\widehat{F^{in}_{+}}\right|\int^{\infty}_{0}\frac{1}{|k|^{3/2}(1+(\xi/k-s)^{2})^{7/4}}ds$
$\displaystyle\lesssim|A^{in}|+\left|\widehat{F^{in}_{+}}\right|$ (3.33)

for any $t\geq 0$. By the Helmholtz decomposition and the Plancherel’s
theorem, we get from (3.33) that

$\displaystyle\left\lVert\mathbb{P}[u_{+}]^{x}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle+\left\lVert\phi(t)\right\rVert^{2}_{L^{2}}=\left\lVert\left(\partial_{y}\Delta^{-1}\omega_{+}\right)(t)\right\rVert^{2}_{L^{2}}+\left\lVert
4\pi e\left(-\Delta+\frac{4\pi
e^{2}}{T_{-}}\right)^{-1}\eta_{+}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle=\sum_{k}\int\frac{(\xi-
kt)^{2}}{\alpha^{2}}\left|\widehat{\Gamma_{+}}(t)\right|^{2}d\xi+\sum_{k}\int
16\pi^{2}e^{2}\left(\alpha+\frac{4\pi
e^{2}}{T_{-}}\right)^{-2}\left|\widehat{\Pi_{+}}(t)\right|^{2}d\xi$
$\displaystyle\lesssim\sum_{k}\int\left(\frac{(\xi-
kt)^{2}}{\alpha^{2}}\left|\widehat{F^{in}_{+}}\right|^{2}+\frac{m_{+}(\xi-
kt)^{2}}{T_{+}\alpha^{3/2}}\left|A_{1}(t)\right|^{2}+\frac{m_{+}}{T_{+}\alpha^{3/2}}\left|A_{1}(t)\right|^{2}\right)d\xi$
$\displaystyle\lesssim\sum_{k}\int\left(\alpha^{-1}\left|\widehat{F^{in}_{+}}\right|^{2}+\frac{m_{+}}{T_{+}\alpha^{1/2}}\left|A^{in}\right|^{2}+\frac{m_{+}}{T_{+}\alpha^{1/2}}\left|\widehat{F^{in}_{+}}\right|^{2}\right)d\xi$
$\displaystyle\lesssim\sum_{k}\int\left(\frac{\left\langle\xi\right\rangle^{2}}{\left\langle
t\right\rangle^{2}\left\langle
k\right\rangle^{2}}\left|\widehat{F^{in}_{+}}\right|^{2}+\frac{m_{+}\left\langle\xi\right\rangle}{T_{+}\left\langle
t\right\rangle\left\langle
k\right\rangle}\left|A^{in}\right|^{2}+\frac{m_{+}\left\langle\xi\right\rangle}{T_{+}\left\langle
t\right\rangle\left\langle
k\right\rangle}\left|\widehat{F^{in}_{+}}\right|^{2}\right)d\xi$
$\displaystyle\lesssim\sum_{k}\int\left(\frac{\left\langle\xi\right\rangle^{2}}{\left\langle
t\right\rangle^{2}\left\langle
k\right\rangle^{2}}\left|\widehat{\eta^{in}_{+}}+\widehat{\omega^{in}_{+}}\right|^{2}+\frac{m_{+}\left\langle\xi\right\rangle}{T_{+}\left\langle
t\right\rangle\left\langle
k\right\rangle}\left|\widehat{\eta^{in}_{+}}+\widehat{\omega^{in}_{+}}\right|^{2}\right)d\xi$
$\displaystyle\quad+\sum_{k}\int\frac{m_{+}\left\langle\xi\right\rangle}{T_{+}\left\langle
t\right\rangle\left\langle
k\right\rangle}\left(\frac{T_{+}}{m_{+}}\frac{\left|\widehat{\eta^{in}_{+}}\right|^{2}}{\left(1+\xi^{2}\right)^{1/2}}+\frac{\left|\widehat{\psi^{in}_{+}}\right|^{2}}{\left(1+\xi^{2}\right)^{3/2}}\right)d\xi,$
(3.34)

where we have used the fact that $\left\langle
t\right\rangle\lesssim\left\langle\xi/k-t\right\rangle\left\langle\xi/k\right\rangle$.
Thanks to the definition of the anisotropic Sobolev space, from the above
inequality we deduce (1.21). Moreover, a similar argument as that of (1.21)
yields (1.22).

Using the Helmholtz decomposition and the Plancherel’s theorem, we obtain

$\displaystyle\left\lVert\mathbb{Q}[u_{+}](t)\right\rVert^{2}_{L^{2}}+\frac{T_{+}}{m_{+}}\left\lVert\eta_{+}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle=\left\lVert\left(\nabla\Delta^{-1}\psi_{+}\right)(t)\right\rVert^{2}_{L^{2}}+\frac{T_{+}}{m_{+}}\left\lVert\eta_{+}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle=\left\lVert\left(\partial_{x}\Delta^{-1}\psi_{+}\right)(t)\right\rVert^{2}_{L^{2}}+\left\lVert\left(\partial_{y}\Delta^{-1}\psi_{+}\right)(t)\right\rVert^{2}_{L^{2}}+\frac{T_{+}}{m_{+}}\left\lVert\eta_{+}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle=\sum_{k}\int\left(\frac{1}{k^{2}+(\xi-
kt)^{2}}\left|\widehat{\Psi_{+}}(t)\right|^{2}+\frac{T_{+}}{m_{+}}\left|\widehat{\Pi_{+}}(t)\right|^{2}\right)d\xi.$
(3.35)

Note that the fact that $k^{2}+(\xi-kt)^{2}\lesssim\left\langle
t\right\rangle^{2}\left\langle k,\xi\right\rangle^{2}$. From (3.33) and
(3.35), it follows that

$\displaystyle\left\lVert\mathbb{Q}[u_{+}](t)\right\rVert^{2}_{L^{2}}+\frac{T_{+}}{m_{+}}\left\lVert\eta_{+}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle=\sum_{k}\int\left(\frac{1}{\alpha}\left|\widehat{\Psi_{+}}(t)\right|^{2}+\frac{T_{+}}{m_{+}}\left|\widehat{\Pi_{+}}(t)\right|^{2}\right)d\xi$
$\displaystyle=\sum_{k}\int\alpha^{\frac{1}{2}}\left(\left|\alpha^{-\frac{3}{4}}\widehat{\Psi_{+}}(t)\right|^{2}+\left|\sqrt{\frac{T_{+}}{m_{+}}}\alpha^{-\frac{1}{4}}\widehat{\Pi_{+}}(t)\right|^{2}\right)d\xi$
$\displaystyle=\sum_{k}\int\alpha^{\frac{1}{2}}\left|A(t)\right|^{2}d\xi$
$\displaystyle\lesssim\left\langle
t\right\rangle\left(\left\lVert\sqrt{\frac{T_{+}}{m_{+}}}\eta^{in}_{+}\right\rVert^{2}_{L^{2}}+\left\lVert\psi^{in}_{+}\right\rVert^{2}_{H^{-1}}+\left\lVert\eta^{in}_{+}+\omega^{in}_{+}\right\rVert^{2}_{H^{1}}\right),$
(3.36)

which gives the estimate (1.23). ∎

Now we prove Theorem 1.2.

###### Proof of Theorem 1.2.

Set

$\mathcal{R}\left(t,A^{in},F^{in}_{+}\right)=A^{in}+\int^{t}_{0}S_{L}(0,s)M_{+}(s)\widehat{F^{in}_{+}}ds.$

According to (3.12), we have

$A(t)=S_{L}(t,0)A^{in}+\int^{t}_{0}S_{L}(t,s)M_{+}(s)\widehat{F^{in}_{+}}ds=S_{L}(t,0)\mathcal{R}\left(t,A^{in},F^{in}_{+}\right).$

By virtue of Lemma 3.1, we obtain

$|A(t)|\geq C|\mathcal{R}\left(t,A^{in},F^{in}_{+}\right)|.$

Using the identity in (3.36), we get from the above inequality that

$\displaystyle\left\lVert\mathbb{Q}[u_{+}](t)\right\rVert^{2}_{L^{2}}+\frac{T_{+}}{m_{+}}\left\lVert\eta_{+}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle=\sum_{k}\int\alpha^{\frac{1}{2}}\left|A(t)\right|^{2}d\xi$
$\displaystyle\gtrsim\sum_{k}\int\alpha^{\frac{1}{2}}|\mathcal{R}\left(t,A^{in},F^{in}_{+}\right)|^{2}d\xi$
$\displaystyle\gtrsim\sum_{k}\int\left\langle\xi-
kt\right\rangle|\mathcal{R}\left(t,A^{in},F^{in}_{+}\right)|^{2}d\xi$
$\displaystyle\gtrsim\sum_{k}\int\frac{\left\langle
kt\right\rangle}{\left\langle\xi\right\rangle}|\mathcal{R}\left(t,A^{in},F^{in}_{+}\right)|^{2}d\xi,$

where we have used the facts that $\alpha^{\frac{1}{2}}\geq\left\langle\xi-
kt\right\rangle$ and $\left\langle\xi-
kt\right\rangle\left\langle\xi\right\rangle\gtrsim\left\langle
kt\right\rangle$. The definition of the anisotropic Sobolev space gives

$\left\lVert\mathbb{Q}[u_{+}](t)\right\rVert^{2}_{L^{2}}+\frac{T_{+}}{m_{+}}\left\lVert\eta_{+}(t)\right\rVert^{2}_{L^{2}}\gtrsim\left\langle
t\right\rangle\left\lVert\mathcal{R}\left(t,A^{in},F^{in}_{+}\right)\right\rVert^{2}_{L^{2}_{x}H^{-\frac{1}{2}}_{y}}.$

This completes the proof. ∎

## 4\. Analysis for the electron dynamics

In this section, we consider the dynamics of the electron dynamics system
(1.28)-(1.30). Also using coordinate transformations (2.1), we define the
functions

$\displaystyle\Pi_{-}(t,X,Y)=\eta_{-}(t,X+tY,Y),$ (4.1)
$\displaystyle\Psi_{-}(t,X,Y)=\psi_{-}(t,X+tY,Y),$ (4.2)
$\displaystyle\Gamma_{-}(t,X,Y)=\omega_{-}(t,X+tY,Y).$ (4.3)

Set $F_{-}(t,X,Y):=\Pi_{-}(t,X,Y)+\Gamma_{-}(t,X,Y)$. Combining (1.28) with
(1.30), we have $\partial_{t}F_{-}=0$, which yields

$\Gamma_{-}(t,X,Y)=F^{in}_{-}(X,Y)-\Pi_{-}(t,X,Y),$

where $F^{in}_{-}=\eta^{in}_{-}+\omega^{in}_{-}$. According to (1.27), we
obtain

$\displaystyle U^{y}_{-}$
$\displaystyle=\left(\partial_{Y}-t\partial_{X}\right)\Delta^{-1}_{L}\Psi_{-}+\partial_{X}\Delta^{-1}_{L}\Gamma_{-}$
$\displaystyle=\left(\partial_{Y}-t\partial_{X}\right)\Delta^{-1}_{L}\Psi_{-}+\partial_{X}\Delta^{-1}_{L}F^{in}_{-}-\partial_{X}\Delta^{-1}_{L}\Pi_{-}.$

In consequence, we can rewrite system (1.28)-(1.30) as

$\displaystyle\partial_{t}\Pi_{-}+\Psi_{-}=0,$ (4.4)
$\displaystyle\partial_{t}\Psi_{-}+2\partial_{XX}\Delta^{-1}_{L}F^{in}_{-}+2\partial_{X}\left(\partial_{Y}-t\partial_{X}\right)\Delta^{-1}_{L}\Psi_{-}$
$\displaystyle-\left(2\partial_{XX}\Delta^{-1}_{L}-\frac{1}{m_{-}}\Delta_{L}+\frac{4\pi
e^{2}}{m_{-}}\right)\Pi_{-}=0.$ (4.5)

Taking the Fourier transform on system (4.4)-(4.5), we get the following
system

$\displaystyle\partial_{t}\widehat{\Pi_{-}}+\widehat{\Psi_{-}}=0,$ (4.6)
$\displaystyle\partial_{t}\widehat{\Psi_{-}}+\frac{2k^{2}}{\alpha}\widehat{F^{in}_{-}}-\frac{\partial_{t}\alpha}{\alpha}\widehat{\Psi_{-}}-\left(\frac{2k^{2}}{\alpha}+\frac{\alpha}{m_{-}}+\frac{4\pi
e^{2}}{m_{-}}\right)\widehat{\Pi_{-}}=0.$ (4.7)

In order to study system (4.6)-(4.7), we need to find a suitable
symmetrization for this system. Define

$B(t)=\left(B_{1}(t),B_{2}(t)\right)^{\top}=\left(\frac{\widehat{\Pi_{-}}(t)}{\sqrt{m_{-}}\alpha^{1/4}},\frac{\widehat{\Psi_{-}}(t)}{\alpha^{3/4}}\right)^{\top}.$
(4.8)

By a delicate calculation, $B(t)$ satisfy a non-autonomous 2D dynamical system

$\begin{cases}\frac{d}{dt}B(t)=L_{-}(t)B(t)+M_{-}(t)\widehat{F^{in}_{-}},\\\
B(0)=B^{in},\end{cases}$ (4.9)

where

$\displaystyle
L_{-}(t)=\begin{bmatrix}-\frac{1}{4}\alpha^{-1}\partial_{t}\alpha&-\sqrt{\frac{1}{m_{-}}}\alpha^{1/2}\\\
\frac{4\pi
e^{2}}{\sqrt{m_{-}}\alpha^{1/2}}+\frac{2k^{2}\sqrt{m_{-}}}{\alpha^{3/2}}+\sqrt{\frac{1}{m_{-}}}\alpha^{1/2}&\frac{1}{4}\alpha^{-1}\partial_{t}\alpha\end{bmatrix},$
(4.10) $\displaystyle
M_{-}(t)=\left(0,-\frac{2k^{2}}{\alpha^{7/4}}\right)^{\top},~{}B^{in}=\left(\frac{\widehat{\Pi^{in}_{-}}}{\sqrt{m_{-}}\left(k^{2}+\xi^{2}\right)^{1/4}},\frac{\widehat{\Psi^{in}_{-}}}{\left(k^{2}+\xi^{2}\right)^{3/4}}\right)^{\top}.$
(4.11)

By virtue of Duhamel’s formula, the solution $B(t)$ to system (4.9) is given
by

$B(t)=S_{L}(t,0)B^{in}+\int^{t}_{0}S_{L}(t,s)M_{-}(s)\widehat{F^{in}_{-}}ds.$
(4.12)

Here, $S_{L}$ denotes the solution operator, satisfying the group property

$S_{L}(t,0)S_{L}(0,s)=S_{L}(t,s)$

for any $t,s>0$. In consequence, it suffices to study the homogeneous problem
of system (4.9).

######  Lemma 4.1.

Denote by $B(t)$ a solution to system (4.9) with $\widehat{F^{in}_{-}}=0$. Let

$\mathcal{E}_{-}(t)=\left(\sqrt{\frac{p_{2}}{m_{2}}}|B_{1}|^{2}\right)(t)+2\left(\frac{h_{2}}{\sqrt{m_{2}p_{2}}}\operatorname{Re}\left(B_{1}\bar{B_{2}}\right)\right)(t)+\left(\sqrt{\frac{m_{2}}{p_{2}}}|B_{2}|^{2}\right)(t),$
(4.13)

where

$\displaystyle
h_{2}(t)=\frac{1}{4}\alpha^{-1}\partial_{t}\alpha,~{}m_{2}(t)=\sqrt{\frac{1}{m_{-}}}\alpha^{1/2},$
(4.14) $\displaystyle p_{2}(t)=\frac{4\pi
e^{2}}{\sqrt{m_{-}}\alpha^{1/2}}+\frac{2k^{2}\sqrt{m_{-}}}{\alpha^{3/2}}+\sqrt{\frac{1}{m_{-}}}\alpha^{1/2}.$
(4.15)

Then there exists positive constants $C_{1}$, $C^{\sigma}_{1}$, $C_{2}$,
$C^{\sigma}_{2}$ that do not depend on $k$ and $\xi$ such that

$C_{1}\mathcal{E}_{-}(0)\leq\mathcal{E}_{-}(t)\leq
C^{\sigma}_{1}\mathcal{E}_{-}(0),$ (4.16)

and

$C_{2}\left|B^{in}\right|\leq\left|B(t)\right|\leq
C^{\sigma}_{2}\left|B^{in}\right|.$ (4.17)

###### Proof.

Define

$\displaystyle\lambda_{-}=\sqrt{\frac{p_{2}}{m_{2}}}=\left(1+\frac{4\pi
e^{2}}{\alpha}+\frac{2m_{-}k^{2}}{\alpha^{2}}\right)^{1/2},$ (4.18)
$\displaystyle\gamma_{-}=\sqrt{m_{2}p_{2}}=\left(\frac{4\pi
e^{2}}{m_{-}}+\frac{2k^{2}}{\alpha}+\frac{\alpha}{m_{-}}\right)^{1/2}.$ (4.19)

By the definition of $\lambda_{-}$, it holds that

$1\leq(\lambda_{-})^{2}\leq 1+4\pi e^{2}+2m_{-}.$ (4.20)

Moreover, we have

$\displaystyle\frac{|h_{2}|}{\gamma_{-}}$
$\displaystyle=\frac{|\partial_{t}\alpha|}{4\alpha}\left(\frac{4\pi
e^{2}}{m_{-}}+\frac{2k^{2}}{\alpha}+\frac{\alpha}{m_{-}}\right)^{-1/2}$
$\displaystyle=\frac{|\partial_{t}\alpha|}{4\alpha}\left(\frac{m_{-}}{\alpha}\right)^{1/2}\left(1+\frac{4\pi
e^{2}}{\alpha}+\frac{2m_{-}k^{2}}{\alpha^{2}}\right)^{-1/2}$
$\displaystyle\leq\frac{|k|}{2\sqrt{\alpha}}\left(\frac{m_{-}}{\alpha}\right)^{1/2}\left(\frac{\alpha^{2}}{2m_{-}k^{2}}\right)^{1/2}=\frac{\sqrt{2}}{4},$
(4.21)

where we have used the fact that $|\partial_{t}\alpha|\leq 2|k|\sqrt{\alpha}$.

Setting

$\widetilde{\mathcal{E}}_{-}(t)=\left(\lambda_{-}|B_{1}|^{2}\right)(t)+\left(\frac{1}{\lambda_{-}}|B_{2}|^{2}\right)(t).$

Then from (4.21) it follows that

$\left(1-\frac{\sqrt{2}}{4}\right)\widetilde{\mathcal{E}}_{-}(t)\leq\mathcal{E}_{-}(t)\leq\left(1+\frac{\sqrt{2}}{4}\right)\widetilde{\mathcal{E}}_{-}(t).$
(4.22)

Obviously, the coerciveness of $\mathcal{E}_{-}(t)$ is ensured by the above
inequality and (4.20).

According to (4.14), (4.15), (4.18) and (4.19), system (4.9) becomes

$\displaystyle\lambda_{-}\frac{d}{dt}B_{1}=-h_{2}\lambda_{-}B_{1}-\gamma_{-}B_{2},$
(4.23)
$\displaystyle\frac{1}{\lambda_{-}}\frac{d}{dt}B_{2}=\gamma_{-}B_{1}+\frac{h_{2}}{\lambda_{-}}B_{2}.$
(4.24)

Multiplying equations (4.23) and (4.24) by $\bar{B_{1}}$ and $\bar{B_{2}}$,
respectively, we obtain

$\frac{\lambda_{-}}{2}\frac{d}{dt}|B_{1}|^{2}+\frac{1}{2\lambda_{-}}\frac{d}{dt}|B_{2}|^{2}=-h_{2}\lambda_{-}|B_{1}|^{2}+\frac{h_{2}}{\lambda_{-}}|B_{2}|^{2}.$
(4.25)

Note that

$\frac{h_{2}}{\gamma_{-}}\frac{d}{dt}\operatorname{Re}(B_{1}\bar{B_{2}})=h_{2}\lambda_{-}|B_{1}|^{2}-\frac{h_{2}}{\lambda_{-}}|B_{2}|^{2}.$
(4.26)

Adding (4.25) to (4.26), we get

$\lambda_{-}\frac{d}{dt}|B_{1}|^{2}+\frac{2h_{2}}{\gamma_{-}}\frac{d}{dt}\operatorname{Re}(B_{1}\bar{B_{2}})+\frac{1}{\lambda_{-}}\frac{d}{dt}|B_{2}|^{2}=0,$

which yields

$\frac{d\mathcal{E}_{-}}{dt}=\lambda_{-}|B_{1}|^{2}\frac{d}{dt}\left(\log\lambda_{-}\right)+2\operatorname{Re}(B_{1}\bar{B_{2}})\frac{d}{dt}\left(\frac{h_{2}}{\gamma_{-}}\right)-\frac{1}{\lambda_{-}}|B_{2}|^{2}\frac{d}{dt}\left(\log\lambda_{-}\right).$

Using Young’s inequality, from (4.22) we deduce that

$\displaystyle\frac{d\mathcal{E}_{-}}{dt}$
$\displaystyle\leq\left(\left|\frac{d}{dt}\left(\frac{h_{2}}{\gamma_{-}}\right)\right|+\left|\frac{d}{dt}\left(\log\lambda_{-}\right)\right|\right)\widetilde{\mathcal{E}}_{-}$
$\displaystyle\leq\left(2+\frac{\sqrt{2}}{2}\right)\left(\left|\frac{d}{dt}\left(\frac{h_{2}}{\gamma_{-}}\right)\right|+\left|\frac{d}{dt}\left(\log\lambda_{-}\right)\right|\right)\mathcal{E}_{-}.$
(4.27)

An argument similar to that of (4.27) gives

$\frac{d\mathcal{E}_{-}}{dt}\geq-\left(2+\frac{\sqrt{2}}{2}\right)\left(\left|\frac{d}{dt}\left(\frac{h_{2}}{\gamma_{-}}\right)\right|+\left|\frac{d}{dt}\left(\log\lambda_{-}\right)\right|\right)\mathcal{E}_{-}.$
(4.28)

For the purpose of applying Grönwall’s inequality, we need to estimate the
integral terms
$\int^{+\infty}_{0}\left|\frac{d}{dt}\left(\frac{h_{2}}{\gamma_{-}}\right)\right|d\vartheta$
and
$\int^{+\infty}_{0}\left|\frac{d}{dt}\left(\log\lambda_{-}\right)\right|d\vartheta$.
First of all, we deal with the first one. By a direct calculation, we have

$\frac{d}{dt}\left(\frac{h_{2}}{\gamma_{-}}\right)=\frac{\mathcal{M}}{8m_{-}(\gamma_{-})^{3}\alpha^{3}},$

where

$\mathcal{M}=4m_{-}k^{2}(\gamma_{-})^{2}\alpha^{2}-2m_{-}(\gamma_{-})^{2}\alpha(\partial_{t}\alpha)^{2}-\alpha^{2}(\partial_{t}\alpha)^{2}+2m_{-}k^{2}(\partial_{t}\alpha)^{2}$

is a polynomial in time of order 6. Denote by $t_{i}(i=1,2,...,6)$ the
positive root for $\mathcal{M}$. We set $i_{0}=0$ provided that
$\mathcal{M}(0)\leq 0$ and take $i_{0}=1$ for the other cases. Assume that
$t_{0}=0$ and that $t_{7}=+\infty$. By virtue of (4.21), we conclude that

$\int^{+\infty}_{0}\left|\frac{d}{dt}\left(\frac{h_{2}}{\gamma_{-}}\right)\right|d\vartheta=\sum^{7}_{i=1}(-1)^{i+i_{0}}\left(\frac{h_{2}}{\gamma_{-}}(t_{i})-\frac{h_{2}}{\gamma_{-}}(t_{i-1})\right)\leq
2\sum^{7}_{i=0}\frac{|h_{2}|}{\gamma_{-}}(t_{i})\leq 4\sqrt{2}.$ (4.29)

For the second integral term, we get from (4.20) that

$\displaystyle\int^{+\infty}_{0}\left|\frac{d}{dt}\left(\log\lambda_{-}\right)\right|d\vartheta=\frac{1}{2}\int^{\frac{\xi}{k}}_{0}\left|\frac{d}{dt}\left(\log(\lambda_{-})^{2}\right)\right|d\vartheta+\frac{1}{2}\int^{+\infty}_{\frac{\xi}{k}}\left|\frac{d}{dt}\left(\log(\lambda_{-})^{2}\right)\right|d\vartheta$
$\displaystyle\qquad\leq\frac{1}{2}\left[\left(\log(\lambda_{-})^{2}\right)\left(\frac{\xi}{k}\right)-\left(\log(\lambda_{-})^{2}\right)\left(0\right)\right]+\frac{1}{2}\left[\left(\log(\lambda_{-})^{2}\right)\left(\frac{\xi}{k}\right)-\left(\log(\lambda_{-})^{2}\right)\left(+\infty\right)\right]$
$\displaystyle\qquad\leq\log\left(1+4\pi e^{2}+2m_{-}\right).$ (4.30)

Here, we have used the fact that
$\frac{1}{(\lambda_{-})^{2}}\frac{d(\lambda_{-})^{2}}{dt}=(\lambda_{-})^{-2}\alpha^{-3}\left(-4m_{-}k^{2}-4\pi
e^{2}\alpha\right)\partial_{t}\alpha$ changes sign only in $t=\xi/k$. Applying
Grönwall’s inequality to (4.27) and (4.28), respectively, from (4.29) and
(4.30) we obtain (4.16).

Next, it remains to prove (4.17). Owing to (4.16) and (4.22), it can be
deduced that

$C^{\prime}_{1}\widetilde{\mathcal{E}}_{-}(0)\leq\widetilde{\mathcal{E}}_{-}(t)\leq
C^{\sigma^{\prime}}_{1}\widetilde{\mathcal{E}}_{-}(0).$ (4.31)

By virtue of (4.20), we have

$\left(1+4\pi
e^{2}+2m_{-}\right)^{-1}\widetilde{\mathcal{E}}_{-}(t)\leq|B(t)|^{2}\leq\left(1+4\pi
e^{2}+2m_{-}\right)\widetilde{\mathcal{E}}_{-}(t).$ (4.32)

Combining (4.31) with (4.32) yields (4.17). This completes the proof. ∎

Based on Lemma 4.1, we give the proof of Theorem 1.3 as follows.

###### Proof of Theorem 1.3.

Thanks to Lemma 4.1, an argument similar to that of (3.33) yields

$\displaystyle|B(t)|$
$\displaystyle=\left|S_{L}(t,0)B^{in}+\int^{t}_{0}S_{L}(t,s)M_{-}(s)\widehat{F^{in}_{-}}ds\right|$
$\displaystyle\lesssim|B^{in}|+\left|\widehat{F^{in}_{-}}\right|$ (4.33)

for any $t\geq 0$. By virtue of the Helmholtz decomposition and the
Plancherel’s theorem, we deduce from (4.33) that

$\displaystyle\left\lVert\mathbb{P}[u_{-}]^{x}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle+\left\lVert\phi(t)\right\rVert^{2}_{L^{2}}=\left\lVert\left(\partial_{y}\Delta^{-1}\omega_{-}\right)(t)\right\rVert^{2}_{L^{2}}+\left\lVert
4\pi e\left(-\Delta\right)^{-1}\eta_{-}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle=\sum_{k}\int\frac{(\xi-
kt)^{2}}{\alpha^{2}}\left|\widehat{\Gamma_{-}}(t)\right|^{2}d\xi+\sum_{k}\int\frac{16\pi^{2}e^{2}}{\alpha^{2}}\left|\widehat{\Pi_{-}}(t)\right|^{2}d\xi$
$\displaystyle\lesssim\sum_{k}\int\left(\frac{(\xi-
kt)^{2}}{\alpha^{2}}\left|\widehat{F^{in}_{-}}\right|^{2}+\frac{m_{-}(\xi-
kt)^{2}}{\alpha^{3/2}}\left|B_{1}(t)\right|^{2}+\frac{m_{-}}{\alpha^{3/2}}\left|B_{1}(t)\right|^{2}\right)d\xi$
$\displaystyle\lesssim\sum_{k}\int\left(\alpha^{-1}\left|\widehat{F^{in}_{-}}\right|^{2}+\frac{m_{-}}{\alpha^{1/2}}\left|B^{in}\right|^{2}+\frac{m_{-}}{\alpha^{1/2}}\left|\widehat{F^{in}_{-}}\right|^{2}\right)d\xi$
$\displaystyle\lesssim\sum_{k}\int\left(\frac{\left\langle\xi\right\rangle^{2}}{\left\langle
t\right\rangle^{2}\left\langle
k\right\rangle^{2}}\left|\widehat{F^{in}_{-}}\right|^{2}+\frac{m_{-}\left\langle\xi\right\rangle}{\left\langle
t\right\rangle\left\langle
k\right\rangle}\left|B^{in}\right|^{2}+\frac{m_{-}\left\langle\xi\right\rangle}{\left\langle
t\right\rangle\left\langle
k\right\rangle}\left|\widehat{F^{in}_{-}}\right|^{2}\right)d\xi,$ (4.34)

in which we have used the fact that $\left\langle
t\right\rangle\lesssim\left\langle\xi/k-t\right\rangle\left\langle\xi/k\right\rangle$.
By the definition of the anisotropic Sobolev space, we obtain (1.37) from the
above inequality. Moreover, a similar argument as that of (1.37) gives (1.38).

Owing to the Helmholtz decomposition and the Plancherel’s theorem, we conclude
that

$\displaystyle\left\lVert\mathbb{Q}[u_{-}](t)\right\rVert^{2}_{L^{2}}+\frac{1}{m_{-}}\left\lVert\eta_{-}(t)\right\rVert^{2}_{L^{2}}=\left\lVert\left(\nabla\Delta^{-1}\psi_{-}\right)(t)\right\rVert^{2}_{L^{2}}+\frac{1}{m_{-}}\left\lVert\eta_{-}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle\qquad=\left\lVert\left(\partial_{x}\Delta^{-1}\psi_{-}\right)(t)\right\rVert^{2}_{L^{2}}+\left\lVert\left(\partial_{y}\Delta^{-1}\psi_{-}\right)(t)\right\rVert^{2}_{L^{2}}+\frac{1}{m_{-}}\left\lVert\eta_{-}(t)\right\rVert^{2}_{L^{2}}$
$\displaystyle\qquad=\sum_{k}\int\left(\frac{1}{k^{2}+(\xi-
kt)^{2}}\left|\widehat{\Psi_{-}}(t)\right|^{2}+\frac{1}{m_{-}}\left|\widehat{\Pi_{-}}(t)\right|^{2}\right)d\xi.$
(4.35)

Note that the fact that $k^{2}+(\xi-kt)^{2}\lesssim\left\langle
t\right\rangle^{2}\left\langle k,\xi\right\rangle^{2}$. From (4.33) and
(4.35), it holds that

$\displaystyle\left\lVert\mathbb{Q}[u_{-}](t)\right\rVert^{2}_{L^{2}}+\frac{1}{m_{-}}\left\lVert\eta_{-}(t)\right\rVert^{2}_{L^{2}}=\sum_{k}\int\left(\frac{1}{\alpha}\left|\widehat{\Psi_{-}}(t)\right|^{2}+\frac{1}{m_{-}}\left|\widehat{\Pi_{-}}(t)\right|^{2}\right)d\xi$
$\displaystyle\qquad=\sum_{k}\int\alpha^{\frac{1}{2}}\left(\left|\alpha^{-\frac{3}{4}}\widehat{\Psi_{-}}(t)\right|^{2}+\left|\sqrt{\frac{1}{m_{-}}}\alpha^{-\frac{1}{4}}\widehat{\Pi_{-}}(t)\right|^{2}\right)d\xi=\sum_{k}\int\alpha^{\frac{1}{2}}\left|B(t)\right|^{2}d\xi$
$\displaystyle\qquad\lesssim\left\langle
t\right\rangle\left(\left\lVert\sqrt{\frac{1}{m_{-}}}\eta^{in}_{-}\right\rVert^{2}_{L^{2}}+\left\lVert\psi^{in}_{-}\right\rVert^{2}_{H^{-1}}+\left\lVert\eta^{in}_{-}+\omega^{in}_{-}\right\rVert^{2}_{H^{1}}\right),$
(4.36)

which yields estimate (1.39). ∎

The proof of Theorem 1.4 follows from a similar argument as that of Theorem
1.2. Hence we omit its proof here.

#### Acknowledgements.

The work of X. Pu was supported in part by the National Natural Science
Foundation of China (11871172) and the Science and Technology Projects in
Guangzhou (202201020132). The work of D. Bian was supported in part by the
National Natural Science Foundation of China (12271032).

####

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