© Wuhan University 2025

0 Introduction

Cauchy problem of the two-dimensional bipolar Navier-Stokes-Poisson (BNSP) system has the following formulation:

$\{\begin{array}{l}{\rho }_t+\mathrm{d}\mathrm{i}\mathrm{v}m=\mathrm{0},\\ m_t+\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{m\otimes m}{\rho }\right)+\nabla P_{\mathrm{1}}\left(\rho \right)={\mu }_{\mathrm{1}}\mathrm{\Delta }\left(\frac{m}{\rho }\right)+{\mu }_{\mathrm{2}}\nabla \mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{m}{\rho }\right)+Z\rho e\nabla \varphi ,\\ n_t+\mathrm{d}\mathrm{i}\mathrm{v}\omega =\mathrm{0},\\ {\omega }_t+\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\omega \otimes \omega }{n}\right)+\nabla P_{\mathrm{2}}\left(n\right)={\overline{\mu }}_{\mathrm{1}}\mathrm{\Delta }\left(\frac{\omega }{n}\right)+{\overline{\mu }}_{\mathrm{2}}\nabla \mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\omega }{n}\right)-ne\nabla \varphi ,\\ \mathrm{\Delta }\varphi =\mathrm{4}\mathrm{\pi }e\left(Z\rho -n\right),\hspace{1em}\underset{\left|x\right|\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\varphi =\mathrm{0},\\ \left(\rho ,m,n,\omega \right)\left(x,\mathrm{0}\right)=\left({\rho }_{\mathrm{0}},m_{\mathrm{0}},n_{\mathrm{0}},{\omega }_{\mathrm{0}}\right)\left(x\right),\left(x,t\right)\in \left({\mathbb{R}}^{\mathrm{2}}\times {\mathbb{R}}^{+}\right).\end{array}$(1)

Here the unknown functions $\rho$, $n$ represent the density of ions and electrons, $m\left(x,t\right)$, $\omega \left(x,t\right)$ are the momentum of ions and electrons, $\varphi$ is the electrostatic potential. $\nabla$ and div are the usual gradient and the divergence operator. ${\mu }_{\mathrm{1}}$, ${\mu }_{\mathrm{2}}$, ${\overline{\mu }}_{\mathrm{1}}$, ${\overline{\mu }}_{\mathrm{2}}$ are constant positive viscosity coefficients. Pressure functions $P_{\mathrm{1}}\left(\rho \right)$, $P_{\mathrm{2}}\left(n\right)$ have positive derivatives. The electrons have charge $-e$ and the ions have charge $Ze$ where $Z$, $e$ are positive constants. For simplicity, we set $e=\mathrm{1}$.

The BNSP system is used to describe the dynamics of two separate compressible fluids of ions and electrons with their self-consistent electromagnetic field. It is a hyperbolic parabolic coupling system. Because of its physical importance and mathematical challenges, there were extensive studies on the asymptotic and global existence of the BNSP system. For example, Refs. [1-5] dealt with different forms of BNSP, and got the global existence of classical solution and its decay. But most results are about space dimension $n\ge \mathrm{3}$. There are few results about space $n=\mathrm{2}$. This paper studies $L^{\mathrm{2}}$ decay estimate of a linearized two-dimensional BNSP system.

Throughout this paper, $L^p(U)$ denotes the Lebesgue integrable space function, $H^p(U)$ means the Sobolev space function. $C\mathrm{a}\mathrm{n}\mathrm{d}{C}_i$ denote some general positive constants.

1 Reformulation and Linearization

Suppose the initial value $\left({\rho }_{\mathrm{0}},m_{\mathrm{0}},n_{\mathrm{0}},{\omega }_{\mathrm{0}}\right)\left(x\right)$ of (1) tends to equilibrium state $\left(\frac{\overline{\rho }}{Z},\mathrm{0},\overline{\rho },\mathrm{0}\right)$ as $|x|\to \mathrm{\infty }.$ Set $\tilde{\rho }=Z\rho -\overline{\rho }$,$\tilde{m}=m-\mathrm{0}$,$\tilde{n}=n-\overline{\rho }$,$\tilde{\omega }=\omega$. Then (1) can be rewritten as

$\{\begin{array}{l}\frac{{\tilde{\rho }}_t}{Z}+\mathrm{d}\mathrm{i}\mathrm{v}\tilde{m}=\mathrm{0},\\ {\tilde{m}}_t+\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\tilde{m}\otimes \tilde{m}}{\frac{\tilde{\rho }+\overline{\rho }}{Z}}\right)+\nabla P_{\mathrm{1}}\left(\frac{\tilde{\rho }+\overline{\rho }}{Z}\right)={\mu }_{\mathrm{1}}\mathrm{\Delta }\left(\frac{\tilde{m}}{\frac{\tilde{\rho }+\overline{\rho }}{Z}}\right)+{\mu }_{\mathrm{2}}\nabla \mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\tilde{m}}{\frac{\tilde{\rho }+\overline{\rho }}{Z}}\right)+Z\cdot \frac{\tilde{\rho }+\overline{\rho }}{Z}\nabla \varphi ,\\ {\tilde{n}}_t+\mathrm{d}\mathrm{i}\mathrm{v}\tilde{\omega }=\mathrm{0},\\ {\tilde{\omega }}_t+\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\tilde{\omega }\otimes \tilde{\omega }}{\tilde{n}+\overline{\rho }}\right)+\nabla P_{\mathrm{2}}\left(\tilde{n}+\overline{\rho }\right)={\overline{\mu }}_{\mathrm{1}}\mathrm{\Delta }\left(\frac{\tilde{\omega }}{\tilde{n}+\overline{\rho }}\right)+{\overline{\mu }}_{\mathrm{2}}\nabla \mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\tilde{\omega }}{\tilde{n}+\overline{\rho }}\right)-\left(\tilde{n}+\overline{\rho }\right)\nabla \varphi ,\\ \mathrm{\Delta }\varphi =\mathrm{4}\mathrm{\pi }\left(\tilde{\rho }-\tilde{n}\right).\end{array}$(2)

System (2) can be reformulated as a linear part plus a nonlinear part.

$\{\begin{array}{l}{\tilde{\rho }}_t+Z\mathrm{d}\mathrm{i}\mathrm{v}\tilde{m}=\mathrm{0},\\ {\tilde{m}}_t+P_{\mathrm{1}}^{\text{'}}\left(\frac{\overline{\rho }}{Z}\right)\nabla \left(\frac{\tilde{\rho }}{Z}\right)-{\mu }_{\mathrm{1}}Z\cdot \frac{\mathrm{1}}{\overline{\rho }}\mathrm{\Delta }\tilde{m}-{\mu }_{\mathrm{2}}Z\cdot \frac{\mathrm{1}}{\overline{\rho }}\nabla \mathrm{d}\mathrm{i}\mathrm{v}\tilde{m}-\overline{\rho }\nabla \varphi =F_{\mathrm{1}},\\ {\tilde{n}}_t+\mathrm{d}\mathrm{i}\mathrm{v}\tilde{\omega }=\mathrm{0},\\ {\tilde{\omega }}_t+{P_{\mathrm{2}}}^{\mathrm{\text{'}}}\left(\overline{\rho }\right)\nabla \tilde{n}-\frac{{\overline{\mu }}_{\mathrm{1}}}{\overline{\rho }}\mathrm{\Delta }\tilde{\omega }-\frac{{\overline{\mu }}_{\mathrm{2}}}{\overline{\rho }}\nabla \mathrm{d}\mathrm{i}\mathrm{v}\tilde{\omega }+\overline{\rho }\nabla \varphi =F_{\mathrm{2}}.\end{array}$(3)

The left side of (3) is the linearized part of (2) near $\left(\frac{\overline{\rho }}{Z},\mathrm{0},\overline{\rho },\mathrm{0}\right)$, and the right side of (3) is the nonlinearized part. For simplicity, we denote the perturbation $\tilde{\rho }$, $\tilde{m}$, $\tilde{n}$, $\tilde{\omega }$ as $\frac{\rho }{Z}$,$m$,$n$,$\omega$, so the linearized system of (1) near the state $\left(\frac{\overline{\rho }}{Z},\mathrm{0},\overline{\rho },\mathrm{0}\right)$ is

$\{\begin{array}{l}{\rho }_t+\mathrm{d}\mathrm{i}\mathrm{v}m=\mathrm{0},\\ m_t+c_{\mathrm{1}}^{\mathrm{2}}\nabla \rho -\frac{{\mu }_{\mathrm{1}}Z}{\overline{\rho }}\mathrm{\Delta }m-\frac{{\mu }_{\mathrm{2}}Z}{\overline{\rho }}\nabla \mathrm{d}\mathrm{i}\mathrm{v}m-\overline{\rho }\nabla \varphi =\mathrm{0},\\ n_t+\mathrm{d}\mathrm{i}\mathrm{v}\omega =\mathrm{0},\\ {\omega }_t+c_{\mathrm{2}}^{\mathrm{2}}\nabla n-\frac{{\overline{\mu }}_{\mathrm{1}}}{\overline{\rho }}\mathrm{\Delta }\omega -\frac{{\overline{\mu }}_{\mathrm{2}}}{\overline{\rho }}\nabla \mathrm{d}\mathrm{i}\mathrm{v}\omega +\overline{\rho }\nabla \varphi =\mathrm{0},\\ \mathrm{\Delta }\varphi =\mathrm{4}\mathrm{\pi }\left(\rho -n\right),\\ \left(\rho ,m,n,\omega \right)\left(x,\mathrm{0}\right)=\left({\rho }_{\mathrm{0}},m_{\mathrm{0}},n_{\mathrm{0}},{\omega }_{\mathrm{0}}\right)\left(x\right),\left(x,t\right)\in \left({\mathbb{R}}^{\mathrm{2}}\times {\mathbb{R}}^{+}\right),\end{array}$(4)

where $c_{\mathrm{1}}^{\mathrm{2}}=P_{\mathrm{1}}^{\text{'}}\left(\frac{\overline{\rho }}{Z}\right)$, $c_{\mathrm{2}}^{\mathrm{2}}=P_{\mathrm{2}}^{\text{'}}\left(\overline{\rho }\right)$.

Our final result in this paper is the following theorem.

Theorem 1   If the initial data ${\rho }_{\mathrm{0}}-\frac{\overline{\rho }}{Z},n_{\mathrm{0}}-\overline{\rho }$, $m_{\mathrm{0}}$, $w_{\mathrm{0}}\in L^{\mathrm{1}}\bigcap H^{\mathrm{2}},$we have the following estimate:

${\Vert \rho +n-\frac{\overline{\rho }}{Z}-\overline{\rho }\Vert }_{L^{\mathrm{2}}}+{\Vert m+\omega \Vert }_{L^{\mathrm{2}}}\le C{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}\left({\Vert {\rho }_{\mathrm{0}}+n_{\mathrm{0}}-\frac{\overline{\rho }}{Z}-\overline{\rho }\Vert }_{L^{\mathrm{1}}}+{\Vert {\rho }_{\mathrm{0}}+n_{\mathrm{0}}-\frac{\overline{\rho }}{Z}-\overline{\rho }\Vert }_{L^{\mathrm{2}}}\right)+C{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}\left({\Vert m_{\mathrm{20}}\Vert }_{L^{\mathrm{1}}}+{\Vert m_{\mathrm{20}}\Vert }_{L^{\mathrm{2}}}\right)$

$\mathrm{m}\mathrm{i}\mathrm{n}({\Vert \rho +n-\frac{\overline{\rho }}{Z}-\overline{\rho }\Vert }_{L^{\mathrm{2}}},{\Vert m+\omega \Vert }_{L^{\mathrm{2}}})\ge C{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}$

If ${\left|{\widehat{\rho }}_{\mathrm{0}}-{\widehat{n}}_{\mathrm{0}}\right|}_{\left|\xi \right|\le \epsilon }\le C_{\mathrm{1}}{\left|\xi \right|}^{{\epsilon }_{\mathrm{1}}}$, for any positive ${\epsilon }_{\mathrm{1}}$, we have ${\Vert m-\omega \Vert }_{L^{\mathrm{2}}}\le C{t}^{-{\epsilon }_{\mathrm{1}}}+C{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}\left({\Vert m_{\mathrm{0}}-{\omega }_{\mathrm{0}}\Vert }_{L^{\mathrm{1}}}+{\Vert m_{\mathrm{0}}-{\omega }_{\mathrm{0}}\Vert }_{H^{\mathrm{2}}}\right)$，

${\Vert \rho -\frac{\overline{\rho }}{Z}-n+\overline{\rho }\Vert }_{L^{\mathrm{2}}}\le C{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}\left({\Vert {\rho }_{\mathrm{0}}-\frac{\overline{\rho }}{Z}-n_{\mathrm{0}}+\overline{\rho }\Vert }_{L^{\mathrm{1}}}+{\Vert {\rho }_{\mathrm{0}}-\frac{\overline{\rho }}{Z}-n_{\mathrm{0}}+\overline{\rho }\Vert }_{H^{\mathrm{2}}}\right)$

Further on, if ${\left|{\widehat{\rho }}_{\mathrm{0}}-{\widehat{n}}_{\mathrm{0}}\right|}_{\left|\xi \right|\le \epsilon }\ge C_{\mathrm{2}}{\left|\xi \right|}^{{\epsilon }_{\mathrm{1}}}$, for any positive ${\epsilon }_{\mathrm{1}}$, we have ${\Vert m-\omega \Vert }_{L^{\mathrm{2}}}\ge C_{\mathrm{3}}{t}^{-{\epsilon }_{\mathrm{1}}}$, ${\Vert \rho -\frac{\overline{\rho }}{Z}-n+\overline{\rho }\Vert }_{L^{\mathrm{2}}}\ge C_{\mathrm{3}}{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}$.

Remark 1   From Theorem 1, the perturbation of the sum of density and momentum decay at the rate ${\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}$, the perturbation of the difference of density decay at the rate ${\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}$, but the difference of momentum hardly decay at $t$ because of the influence of the electronic field. Due to the low decay rate, it is difficult to go on the global existence of the system.

We want to separate system (4) into several small sets of equations. Considering the fifth equation of (4), we denote ${\rho }_{\mathrm{1}}=\rho +n$, ${\rho }_{\mathrm{2}}=\rho -n$, $m_{\mathrm{1}}=m+\omega$, $m_{\mathrm{2}}=m-\omega$, (4) is equal to the following system:

$\{\begin{array}{l}{\rho }_{\mathrm{1}t}+\mathrm{d}\mathrm{i}\mathrm{v}{m}_{\mathrm{1}}=\mathrm{0},\\ m_{\mathrm{1}t}+\frac{c_{\mathrm{1}}^{\mathrm{2}}+c_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}}\nabla {\rho }_{\mathrm{1}}+\frac{c_{\mathrm{1}}^{\mathrm{2}}-c_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}}\nabla {\rho }_{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{\mu }_{\mathrm{1}}Z+{\overline{\mu }}_{\mathrm{1}}}{\overline{\rho }}\right)\mathrm{\Delta }{m}_{\mathrm{1}}-\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{\mu }_{\mathrm{1}}Z-{\overline{\mu }}_{\mathrm{1}}}{\overline{\rho }}\right)\mathrm{\Delta }{m}_{\mathrm{2}}\\ -\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{\mu }_{\mathrm{2}}Z+{\overline{\mu }}_{\mathrm{2}}}{\overline{\rho }}\right)\nabla \mathrm{d}\mathrm{i}\mathrm{v}{m}_{\mathrm{1}}-\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{\mu }_{\mathrm{2}}Z-{\overline{\mu }}_{\mathrm{2}}}{\overline{\rho }}\right)\nabla \mathrm{d}\mathrm{i}\mathrm{v}{m}_{\mathrm{2}}=\mathrm{0},\\ {\rho }_{\mathrm{2}t}+\mathrm{d}\mathrm{i}\mathrm{v}{m}_{\mathrm{2}}=\mathrm{0},\\ m_{\mathrm{2}t}+\frac{c_{\mathrm{1}}^{\mathrm{2}}-c_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}}\nabla {\rho }_{\mathrm{1}}+\frac{c_{\mathrm{1}}^{\mathrm{2}}+c_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}}\nabla {\rho }_{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{\mu }_{\mathrm{1}}Z-{\overline{\mu }}_{\mathrm{1}}}{\overline{\rho }}\right)\mathrm{\Delta }{m}_{\mathrm{1}}-\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{\mu }_{\mathrm{1}}Z+{\overline{\mu }}_{\mathrm{1}}}{\overline{\rho }}\right)\mathrm{\Delta }{m}_{\mathrm{2}}\\ -\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{\mu }_{\mathrm{2}}Z-{\overline{\mu }}_{\mathrm{2}}}{\overline{\rho }}\right)\nabla \mathrm{d}\mathrm{i}\mathrm{v}{m}_{\mathrm{1}}-\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{\mu }_{\mathrm{2}}Z+{\overline{\mu }}_{\mathrm{2}}}{\overline{\rho }}\right)\nabla \mathrm{d}\mathrm{i}\mathrm{v}{m}_{\mathrm{2}}-\mathrm{2}\overline{\rho }\mathrm{\Delta }\varphi =\mathrm{0},\\ \mathrm{\Delta }\varphi =\mathrm{4}\mathrm{\pi }{\rho }_{\mathrm{2}}.\end{array}$(5)

For simplicity, we suppose $c_{\mathrm{1}}^{\mathrm{2}}=c_{\mathrm{2}}^{\mathrm{2}}$, ${\mu }_{\mathrm{1}}Z={\overline{\mu }}_{\mathrm{1}}$, ${\mu }_{\mathrm{2}}Z={\overline{\mu }}_{\mathrm{2}}$, denote $c^{\mathrm{2}}=\frac{c_{\mathrm{1}}^{\mathrm{2}}+c_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}}$, ${\gamma }_{\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{\mu }_{\mathrm{1}}Z+{\overline{\mu }}_{\mathrm{1}}}{\overline{\rho }}\right)$, ${\gamma }_{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{\mu }_{\mathrm{2}}Z+{\overline{\mu }}_{\mathrm{2}}}{\overline{\rho }}\right)$. System (5) can be separated into the following two systems

$\{\begin{array}{l}{\rho }_{\mathrm{1}t}+\mathrm{d}\mathrm{i}\mathrm{v}{m}_{\mathrm{1}}=\mathrm{0},\\ m_{\mathrm{1}t}+c^{\mathrm{2}}\nabla {\rho }_{\mathrm{1}}-{\gamma }_{\mathrm{1}}\mathrm{\Delta }{m}_{\mathrm{1}}-{\gamma }_{\mathrm{2}}\nabla \mathrm{d}\mathrm{i}\mathrm{v}{m}_{\mathrm{1}}=\mathrm{0}.\end{array}$(6)

$\{\begin{array}{l}{\rho }_{\mathrm{2}t}+\mathrm{d}\mathrm{i}\mathrm{v}{m}_{\mathrm{2}}=\mathrm{0},\\ m_{\mathrm{2}t}+c^{\mathrm{2}}\nabla {\rho }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\mathrm{\Delta }{m}_{\mathrm{2}}-{\gamma }_{\mathrm{2}}\nabla \mathrm{d}\mathrm{i}\mathrm{v}{m}_{\mathrm{2}}-\mathrm{2}\overline{\rho }\nabla \varphi =\mathrm{0},\\ \mathrm{\Delta }\varphi =\mathrm{4}\mathrm{\pi }{\rho }_{\mathrm{2}}.\end{array}$(7)

We find that (6) is a linearized isentropic Navier-Stokes (NS) system while (7) is a linearized unipolar Navier-Stokes-Poisson (NSP) system. We study them respectively in the next step.

2 ${\mathit{L}}^{\mathrm{2}}$ Decay of Linearized NS System

If the initial data of (6) is $\left({\rho }_{\mathrm{1}},m_{\mathrm{1}}\right)\left(x,\mathrm{0}\right)=\left({\rho }_{\mathrm{10}},m_{\mathrm{10}}\right)\left(x\right)$, according to (1.3) in Ref. [6], we have

$\left(\begin{array}{l}{\widehat{\rho }}_{\mathrm{1}}\\ {\widehat{m}}_{\mathrm{1}}\end{array}\right)=\left(\begin{array}{cc}\frac{{\lambda }_{+}{e}^{{\lambda }_{-}t}-{\lambda }_{-}{e}^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}& \frac{e^{{\lambda }_{-}t}-e^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}i{\xi }^{\tau }\\ -i{c}^{\mathrm{2}}\xi \cdot \frac{e^{{\lambda }_{-}t}-e^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}& \frac{{\lambda }_{+}{e}^{{\lambda }_{-}t}-{\lambda }_{-}{e}^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}\frac{\xi {\xi }^{\tau }}{{\left|\xi \right|}^{\mathrm{2}}}+e^{-{\gamma }_{\mathrm{1}}{\left|\xi \right|}^{\mathrm{2}}t}\left(\mathit{I}-\frac{\xi {\xi }^{\tau }}{{\left|\xi \right|}^{\mathrm{2}}}\right)\end{array}\right)\left(\begin{array}{l}{\widehat{\rho }}_{\mathrm{10}}\\ {\widehat{m}}_{\mathrm{10}}\end{array}\right)$(8)

where $\mathit{I}$ is a $\mathrm{2}\times \mathrm{2}$ unit matrix,

${\lambda }_{+}=\frac{-\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right){\left|\xi \right|}^{\mathrm{2}}+\sqrt[]{{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{4}}-\mathrm{4}{c}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{2}}}}{\mathrm{2}}$

${\lambda }_{-}=\frac{-\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right){\left|\xi \right|}^{\mathrm{2}}-\sqrt[]{{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{4}}-\mathrm{4}{c}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{2}}}}{\mathrm{2}}$(9)

From (9), when $\left|\xi \right|$ is small enough, there is no problem with the decay estimate for the Green function; when $\left|\xi \right|$ is bounded and away from zero, ${\lambda }_{+}-{\lambda }_{-}$ may tend to zero, so we need to consider the integrability of the Green function; when $\left|\xi \right|$ is large enough, for example, $\frac{\xi }{{\lambda }_{+}-{\lambda }_{-}}=O(\frac{\mathrm{1}}{\left|\xi \right|})$ has no $L_{\mathrm{2}}$ bounds. Next, we will divide frequency into three different parts and will use different methods to consider the decay rate respectively.

When ${\left|\xi \right|}^{\mathrm{2}}\le \epsilon \le \frac{\mathrm{2}{c}^{\mathrm{2}}}{{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}}$, we have

$\frac{{\lambda }_{+}{e}^{{\lambda }_{-}t}}{{\lambda }_{+}-{\lambda }_{-}}=\left(-\frac{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right){\left|\xi \right|}^{\mathrm{2}}}{\mathrm{2}}+ic\left|\xi \right|+o\left({\left|\xi \right|}^{\mathrm{2}}\right)\right)\left(\frac{\mathrm{1}}{\mathrm{2}ic\left|\xi \right|}+o\left(\left|\xi \right|\right)\right)e^{\left(-\frac{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right){\left|\xi \right|}^{\mathrm{2}}}{\mathrm{2}}-ic\left|\xi \right|+o\left({\left|\xi \right|}^{\mathrm{2}}\right)\right)t}$

We find the construction of ${\lambda }_{\mathrm{1}}$ is like that of ${\lambda }_{\mathrm{2}}$ in Ref. [7]. Initiated by the estimation method in Ref. [7], we get

${\Vert \frac{{\lambda }_{+}{e}^{{\lambda }_{-}t}}{{\lambda }_{+}-{\lambda }_{-}}\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\le \epsilon \right)}\le C{\Vert e^{-\frac{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right){\left|\xi \right|}^{\mathrm{2}}t}{\mathrm{4}}}\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\le \epsilon \right)}\le C{\left({\int }_{\mathrm{0}}^{\mathrm{1}}{e}^{-\frac{{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}{\rho }^{\mathrm{2}}t}{\mathrm{2}}}\rho \mathrm{d}\rho \right)}^{\frac{\mathrm{1}}{\mathrm{2}}}\le C{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}.$

Similarly

${\Vert \frac{e^{{\lambda }_{-}t}-e^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}i{\xi }^{\tau }\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\le \epsilon \right)}\le C{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}.$

Thus

${\Vert {\widehat{\rho }}_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\le \epsilon \right)}={\Vert \frac{{\lambda }_{+}{e}^{{\lambda }_{-}t}-{\lambda }_{-}{e}^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}\Vert }_{L^{\mathrm{2}}}{\Vert {\rho }_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}}+{\Vert \frac{e^{{\lambda }_{-}t}-e^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}i{\xi }^{\tau }\Vert }_{L^{\mathrm{2}}}{\Vert m_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}}\le C{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}\mathrm{m}\mathrm{a}\mathrm{x}\left({\Vert {\rho }_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}},{\Vert m_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}}\right)$(10)

Using the same method, we can get

${\Vert {\widehat{m}}_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\le \epsilon \right)}\le C{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}\mathrm{m}\mathrm{a}\mathrm{x}\left({\Vert {\rho }_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}},{\Vert m_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}}\right).$(11)

When $\epsilon \le {\left|\xi \right|}^{\mathrm{2}}\le \frac{\mathrm{8}{c}^{\mathrm{2}}}{{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}}\le R$, there exist positive constant $b$ such that ${\lambda }_{\pm }\le -b<\mathrm{0}$.

Because $\frac{e^{{\lambda }_{+}t}-e^{{\lambda }_{-}t}}{{\lambda }_{+}-{\lambda }_{-}}=e^{{\lambda }_{-}t}\frac{e^{\left({\lambda }_{+}-{\lambda }_{-}\right)t}-\mathrm{1}}{{\lambda }_{+}-{\lambda }_{-}}$, $\frac{{\lambda }_{+}{e}^{{\lambda }_{-}t}-{\lambda }_{-}{e}^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}={\lambda }_{+}{e}^{{\lambda }_{-}t}\frac{e^{\left({\lambda }_{+}-{\lambda }_{-}\right)t}-\mathrm{1}}{{\lambda }_{+}-{\lambda }_{-}}+e^{{\lambda }_{-}t}$ are smooth functions, we can easily get

${\Vert {\widehat{\rho }}_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}\left(\epsilon \le {\left|\xi \right|}^{\mathrm{2}}\le R\right)},{\Vert {\widehat{m}}_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}\left(\epsilon \le {\left|\xi \right|}^{\mathrm{2}}\le R\right)}\le C{e}^{-bt}\left({\Vert {\rho }_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}}+{\Vert m_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}}\right)$(12)

When ${\left|\xi \right|}^{\mathrm{2}}\ge R$, notice ${\lambda }_{\pm }\le -b$ for some positive $b$, then

$\frac{\mathrm{1}}{{\lambda }_{+}-{\lambda }_{-}}=\frac{\mathrm{1}}{{\left|\xi \right|}^{\mathrm{2}}\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}\left(\mathrm{1}+\frac{\mathrm{2}{c}^{\mathrm{2}}}{{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}}\frac{\mathrm{1}}{{\left|\xi \right|}^{\mathrm{2}}}+o\left(\frac{\mathrm{1}}{{\left|\xi \right|}^{\mathrm{2}}}\right)\right)\le C\frac{\mathrm{1}}{{\left|\xi \right|}^{\mathrm{2}}}$

we have

${\Vert \widehat{G}\left(\begin{array}{l}{\widehat{\rho }}_{\mathrm{10}}\\ {\widehat{m}}_{\mathrm{10}}\end{array}\right)\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\ge R\right)}\le C{e}^{-bt}\left({\Vert {\widehat{\rho }}_{\mathrm{10}}\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\ge R\right)}+{\Vert {\widehat{m}}_{\mathrm{10}}\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\ge R\right)}\right)$(13)

Next, we consider ${\Vert \frac{e^{{\lambda }_{-}t}-e^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}i{\xi }^{\tau }\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\le \epsilon \right)}.$

Since $\sqrt[]{\mathrm{4}{c}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{2}}-{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{4}}}=\mathrm{2}c\left|\xi \right|\cdot \sqrt[]{\mathrm{1}-\frac{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}{\mathrm{4}{c}^{\mathrm{2}}}{\left|\xi \right|}^{\mathrm{2}}}=\mathrm{2}c\left|\xi \right|+O\left({\left|\xi \right|}^{\mathrm{3}}\right),$ we have

$\begin{array}{l}\left|\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{2}c\left|\xi \right|t\right|\le \left|\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{2}c\left|\xi \right|t-\mathrm{s}\mathrm{i}\mathrm{n}\sqrt[]{\mathrm{4}{c}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{2}}-{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{4}}}t\right|+\left|\mathrm{s}\mathrm{i}\mathrm{n}\sqrt[]{\mathrm{4}{c}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{2}}-{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{4}}}t\right|\\ \le \left|\left(O\left({\left|\xi \right|}^{\mathrm{3}}\right)t\right)\right|+\left|\mathrm{s}\mathrm{i}\mathrm{n}\sqrt[]{\mathrm{4}{c}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{2}}-{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{4}}}t\right|,\end{array}$

then

$\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\sqrt[]{\mathrm{4}{c}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{2}}-{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{4}}}t\ge \frac{\mathrm{1}}{\mathrm{2}}\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\mathrm{2}c\left|\xi \right|t+{\left(O\left({\left|\xi \right|}^{\mathrm{3}}\right)t\right)}^{\mathrm{2}},$(14)

From (9) and (14), we have

$\begin{array}{l}{\Vert \frac{e^{{\lambda }_{-}t}-e^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}i{\xi }^{\tau }\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\le \epsilon \right)}\ge C{\Vert e^{-\frac{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right){\left|\xi \right|}^{\mathrm{2}}t}{\mathrm{2}}}\mathrm{s}\mathrm{i}\mathrm{n}\sqrt[]{\mathrm{4}{c}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{2}}-{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)}^{\mathrm{2}}{\left|\xi \right|}^{\mathrm{4}}}t\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\le \epsilon \right)}\\ \ge C{\left({\int }_{{\left|\xi \right|}^{\mathrm{2}}\le \epsilon }{e}^{-\frac{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right){\left|\xi \right|}^{\mathrm{2}}t}{\mathrm{2}}}\cdot \frac{\mathrm{1}}{\mathrm{2}}\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\mathrm{2}c\left|\xi \right|t\mathrm{d}\xi \right)}^{\frac{\mathrm{1}}{\mathrm{2}}}-C{\left({\int }_{{\left|\xi \right|}^{\mathrm{2}}\le \epsilon }{e}^{-\frac{\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right){\left|\xi \right|}^{\mathrm{2}}t}{\mathrm{2}}}{\left(O\left({\left|\xi \right|}^{\mathrm{3}}\right)t\right)}^{\mathrm{2}}\mathrm{d}\xi \right)}^{\frac{\mathrm{1}}{\mathrm{2}}}={\mathit{I}}_{\mathrm{1}}+{\mathit{I}}_{\mathrm{2}}.\end{array}$(15)

If we fix $t\ge \frac{\mathrm{5}\mathrm{\pi }}{\mathrm{8}c\sqrt[]{\epsilon }},$ we have

${\mathit{I}}_{\mathrm{1}}\ge C{\left({\int }_{\left|\xi \right|\le \sqrt[]{\epsilon t}}\int e^{-\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right){\left|\xi \right|}^{\mathrm{2}}}\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}(\mathrm{2}c\left|\xi \right|\sqrt[]{t})\cdot \left|\xi \right|t^{-\mathrm{1}}\mathrm{d}\left|\xi \right|\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}\ge C\cdot {\displaystyle \sum _{k=\mathrm{0}}^{\left[\frac{\mathrm{2}c\sqrt[]{\epsilon }t}{\mathrm{\pi }}-\frac{\mathrm{1}}{\mathrm{4}}\right]}}\frac{\mathrm{1}}{\mathrm{2}}\cdot t^{-\frac{\mathrm{1}}{\mathrm{2}}}{\left({\int }_{\frac{k\mathrm{\pi }+\frac{\mathrm{\pi }}{\mathrm{4}}}{\mathrm{2}c\sqrt[]{t}}}^{\frac{k\mathrm{\pi }+\frac{\mathrm{3}\mathrm{\pi }}{\mathrm{4}}}{\mathrm{2}c\sqrt[]{t}}}{e}^{-\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right)r^{\mathrm{2}}}\cdot r\mathrm{d}r\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}=C{t}^{-\frac{\mathrm{1}}{\mathrm{2}}},$(16)

${\mathit{I}}_{\mathrm{2}}\le C{\left(\int e^{-\left({\gamma }_{\mathrm{1}}+{\gamma }_{\mathrm{2}}\right){\left|\xi \right|}^{\mathrm{2}}}\cdot O\left({\left|\xi \right|}^{\mathrm{3}}\right)\cdot t^{-\frac{\mathrm{3}}{\mathrm{2}}}\left|\xi \right|\cdot t^{-\mathrm{1}}\mathrm{d}\left|\xi \right|\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}\le C{t}^{-\frac{\mathrm{5}}{\mathrm{4}}}.$(17)

From (15), (16), and (17) , when $t$ is large enough, we have

${\Vert \frac{e^{{\lambda }_{-}t}-e^{{\lambda }_{+}t}}{{\lambda }_{+}-{\lambda }_{-}}i{\xi }^{\tau }\Vert }_{L^{\mathrm{2}}\left({\left|\xi \right|}^{\mathrm{2}}\le \epsilon \right)}\ge C{t}^{-\frac{\mathrm{1}}{\mathrm{2}}}.$(18)

From (8) and (18), we have

${\Vert {\widehat{\rho }}_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}}\ge {\Vert {\widehat{\rho }}_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}\left(\left|\xi \right|\le \epsilon \right)}\ge C_{\mathrm{1}}{t}^{-\frac{\mathrm{1}}{\mathrm{2}}}{\Vert m_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}},$(19)

${\Vert {\widehat{m}}_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}}\ge {\Vert {\widehat{m}}_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}\left(\left|\xi \right|\le \epsilon \right)}\ge C_{\mathrm{1}}{t}^{-\frac{\mathrm{1}}{\mathrm{2}}}{\Vert {\rho }_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}},$(20)

Together with (8), (10), (11), (12), (13), (19), and (20), we get our results.

Theorem 2   Suppose $E=\mathrm{m}\mathrm{a}\mathrm{x}\left({\Vert {\rho }_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}},{\Vert m_{\mathrm{10}}\Vert }_{L^{\mathrm{1}}},{\Vert {\rho }_{\mathrm{10}}\Vert }_{L^{\mathrm{2}}},{\Vert m_{\mathrm{10}}\Vert }_{L^{\mathrm{2}}}\right)$, there exists a positive constant $C$ such that max$\left({\Vert {\rho }_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}},{\Vert m_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}}\right)\le C{\left(\mathrm{1}+t\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}E$. Further on, when $t$ is large enough, we have $\mathrm{m}\mathrm{i}\mathrm{n}({\Vert {\rho }_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}},{\Vert m_{\mathrm{1}}\Vert }_{L^{\mathrm{2}}})\ge CE{t}^{-\frac{\mathrm{1}}{\mathrm{2}}}.$