Global solutions to the Cauchy problem of BNSP equations in some classes of large data

1.   Introduction

The bipolar Navier-Stokes-Poisson system has been used to simulate the transport of charged particles under the influence of electrostatic force governed by the self-consistent Poisson. In this paper, we are concerned with the Cauchy problem of the bipolar Navier-Stokes-Poisson system in $3$ dimensions:

with initial data

Here $\rho_{i}(x, t), u_{i}(x, t), \Phi(x, t),$ and $P_{i}(\rho)((x, t)$ represent the fluid density, velocity, self-consistent electric potential and pressure. The viscosity coefficients satisfy the usual physical conditions $\mu_{1} > 0, \ 3\mu_{1}+2\mu_{2} > 0.$ We assume that $P_{i}(\rho)$ satisfies $P_{i}^{\prime}(\rho) > 0$ for all $\rho > 0$ and $P_{i}^{\prime}(\bar{\rho}) = 1$, where $\overline{\rho} > 0$ denotes the prescribed density of positive charged background ions, and in this paper is taken as a positive constant. Without loss of generality, we take $\overline{\rho}$ to be 1. For the initial data $(\rho_{1, 0}, u_{1, 0}, \rho_{2, 0}, u_{2, 0})$, we consider small perturbations of $(\overline{\rho}, 0, \overline{\rho}, 0)$, in which $\overline{\rho}$ is defined as before and taken to be 1, and we assume that $\rho_{1, 0}, \rho_{2, 0}$ has positive lower bound and upper bound.

Now, we review some previous works on the Cauchy problem for some related models. There has been a lot of studies for the compressible Navier-Stokes system (CNS) for either isentropic or non-isentropic cases on the existence, stability, and $L^p$-decay rates with $p\geq2$. For the results of small solutions, see ^[1,2] and ^[3,4,5,6,7], where the authors use the (weighted) energy method together with spectrum analysis, and for the results of partial small solutions (under the setting that the initial data is of small energy but possibly large oscillations), see ^[8,9] and the references therein. On the other hand, many scholars also use the method of Green's function to analyze the asymptotic behavior of a specific system, for example, by using the method of Green's function, Liu and Zeng ^[10] first studied the point-wise estimates of solutions to the general hyperbolic-parabolic equations in one dimension. Later, Liu and Wang ^[11] give the point-wise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimension and explain the generalized Huygens' principle for the Navier-Stokes systems.

For the unipolar Navier-Stokes-Poisson system (NSP), there is also a mass of results for the Cauchy problem when the initial data $(\rho_{0}, u_{0})$ is a small perturbation around the constant state $(\bar{\rho}, 0).$ For instance, the global existence of weak solutions was obtained by ^[12,13]; the framework of Matsumura and Nishda ^[14,15] shows the global existence of small strong solutions in $H^N$ Sobolev spaces. In ^[16,17], the authors obtain the global existence of small solutions in some Besov spaces. For the solutions which are of small energy but possibly large oscillations, see ^[18,19] and the references therein. In fact, the NSP system is a hyperbolic-parabolic system with a nonlocal term arising from the electric field $\nabla\Phi$. From the analysis of Green's function, the symbol of this nonlocal term is singular in the long wave of the Green's function and it destroys the time-decay rate for the velocity. As we know, for the CNS system, when the initial perturbation $\rho_{0}-1, u_{0} \in L^p\cap H^N$, with $p$ near 1, and $N\geq 3$ is a large enough integer for the nonlinear system, then the solutions have $L^2$ optimal decay rate

In ^[15,20,21], the authors survey the decay rate of solutions for the NSP system and they observed that the electric field destroys the decay rate of the solutions, i.e., when the initial perturbation $\rho_{0}-1, u_{0} \in L^p\cap H^N$, with $p \in [1, 2]$, then the solutions have $L^2$ optimal decay rate

In ^[22], the author gives another comprehension toward the effect of the electric field on the decay rate of the solutions for the NSP system. The author believes that it is natural to assume that $\nabla\Phi_{0} \in L^2$, and with this condition in hand, we can obtain the $L^2$ optimal decay rate for the linear NSP system as follows:

In this sense, we see that the electric force enhances the decay rate of the density with the factor $\frac{1}{2}$ compared with the CNS system.

For the bipolar Navier-Stokes-Poisson system (BNSP), Wang and Xu ^[23] obtain the global existence of small solutions and the decay rate of the solutions. ^[24,25,26] observed the large time behavior of the BNSP system. The global existence of the solutions for the BNSP system under the partial smallness of the initial value is still an open problem, and the aim of this paper is to obtain the global existence of the solution to the system (1.1) and (1.2) provided that the initial data is partially small, which means that we require the initial value itself to be small and its derivative only to be bounded. In other words, its initial value is large in some classes, that is, its derivative can be large except for the initial value itself. In this paper, we first establish a regularity criterion to obtain the uniform boundedness of the solutions, and then combine abstract bootstrap argument to extend the local solutions. In order to prove the global existence of solution under the condition of initial data is partially small, the general regularity hypothesis requires that the solution is bounded with respect to time in the sense of some Sobolev norm, and another important condition is that the solution is integrable with respect to the time variable, which is necessary to obtain a consistent estimate of the solution through Gronwall's inequality.

However, in this paper, it is difficult to obtain the regularity condition of integrability due to the more complex nonlinear structure in BNSP equations. The classical regularity criterion is established based on the energy method and the decay property in time variable is usually captured by integrability. It is always difficult and not applicable for the bootstrap argument. In fact, in ^[27] for the global existence of solutions for shallow water equations with partially large initial data, we applied Green's function method and replaced the integrability condition by detailed decaying structure in time variable, which makes the bootstrap argument more clear and concise.

The linear and nonlinear structures of BNSP equations discussed in this paper are far more complicated, and we can imagine that it could be quite difficult and complex for the energy method. Even in the construction of Green's function, compared with the previous case for shallow water equations, BNSP equations are hyperbolic-parabolic-elliptic coupled ones and the structure is very complicated. The elliptic structure provides a nonlocal operator and also causes the lack of symmetry. These are all troubles we need to overcome in this paper.

Before we list the main result, we introduce some notations. Throughout this paper, $\partial_t$ stands for the derivative with respect to time variable and $f_t = \partial_{t}f.$ The symbol $\partial_{i}f(i = 1, 2, 3)$ means partial derivative with respect to $x_i,$

We use the notation $\nabla^kf$ to mean the partial derivative of order k. That is, if k is a nonnegative integer, then

is a set of all partial derivatives of order $k$, endowed with the norm

where

Let $\Lambda$ be a quasi-differential operator defined as follows:

The main result of this paper is the following:

Theorem 1.1. Let $(\rho_{1, 0}-1, u_{1, 0}, \rho_{2, 0}-1, u_{2, 0}, \nabla\Phi_{0}) \in H^{s+1}(s\geq4),$ and

where $E_0$ is sufficiently small, then the Cauchy problem (1.1) and (1.2) has a global solution in time that satisfies

For $2\leq p\leq +\infty$ and $\alpha = (\alpha_{1}, \alpha_{2}, \alpha_{3}),$ $|\alpha|\leq s-1,$ it holds

Remark 1.1. In Theorem 1.1, we only assume the norm of the initial datum $\rho_{1, 0}, \rho_{2, 0}, u_{1, 0}, u_{2, 0}$ and $\nabla\Phi_{0}$ are small enough, but for the derivatives of $\rho_{1, 0}, \rho_{2, 0}, u_{1, 0}, u_{2, 0},$ and $\nabla\Phi_{0}$, we assume that they are bounded.

The rest of this paper is organized as follows. In Section 2, we establish the uniform time estimate of solutions. In Section 3, we analyze the Green's function of the linear BNSP system, and the different properties of the Green function at high and low frequencies are obtained. In Section 4, we complete the partial proof of Theorem 1.1, i.e., we mainly obtain the existence of the solutions. In this paper, we also obtain the decay rate of the solutions, and the reader can see Sections 4 and 5 for details.

Throughout this paper, we denote by $C$ a positive constant that varies from line to line.

2.   Uniform estimation of solutions

We reformulate the Cauchy problem $(1.1)$ and $(1.2)$ about constant state $(1, 0, 1, 0)$ as follows:

where we also note $(\rho_{1}-1, u_{1}, \rho_{2}-1, u_{2})$ as $(\rho_{1}, u_{1}, \rho_{2}, u_{2})$ without causing confusion. In this section, we will get the estimates of the solutions for the system (2.1) under the assumption that for any fixed $0 < T < +\infty, \ t\in[0, T],$

and

where $E_0$ is a positive constant which is sufficiently small. Similar to reference ^[28,29], we call Eq (2.2) the regularity criterion. In this section, based on this regularity criterion, we get a consistent estimate of the solutions of the equations and its derivatives.

We first define

To do this end, from (2.2), (2.3), and Gagliardo-Nirenberg's inequality, we know that there exists a sufficiently small $\varepsilon_{1} > 0$ that satisfies

First of all, from (2.4), we obtain

Hence, by the definition of $F_{1}(\rho), F_{2}(\rho)$, and $H(\rho),$ we immediately have

Let us start with a lemma that will be frequently used later.

Lemma 2.1. ^[22] Assume that $\|\rho\|_{L^{\infty}(\mathbb{R}^3)}\leq 1,$ and $f(\rho)$ is a smooth function of $\rho$ with bounded derivatives of any order, then for any integer $k\geq 1$, we have

We obtain the estimates of the low order derivatives of the system (2.1) first.

Lemma 2. Under the assumption (2.2) and (2.4), we have

where $C$ is a constant depending on $\|(\rho_{1, 0}, \rho_{2, 0}, u_{1, 0}, u_{2, 0}, \nabla\Phi_{0})\|_{L^2}.$

Proof. Multiplying the Eqs $(2.1)_{1}$, $(2.1)_{2}$, $(2.1)_{3}$, and $(2.1)_{4}$ by $\rho_{1}, u_{1}, \rho_{2}, u_{2}$, respectively, and integrating the equations over $\mathbb{R}^3$, we obtain

Now we estimate $A_{i}$ one by one. Because the self-consistent potential $\Phi(x, t)$ is coupled with the density through the Poisson equation, using Hölder's inequality and Cauchy's inequality, for $A_{1}$, it holds

For $A_{4}$ and $A_{5}$, by Hölder's inequality, we easily check that

Integrating by parts, and using Hölder's inequality and Cauchy's inequality, it holds

By the definition of $F_{1}(\rho)$ and $F_{2}(\rho)$, using Hölder's inequality and Cauchy's inequality, and from (2.5), we have

Also integrating by parts, using Hölder's inequality and Cauchy's inequality, and by (2.5) and Lemma (2.1), we have

where we take $\epsilon$ small enough such that $\epsilon\ll1.$ Plugging the estimates for $A_{1}$–$A_{9}$, i.e., $(2.7)$–$(2.11)$ into $(2.6),$ we get

Using Gronwall's inequality, we complete the proof of the lemma. □

In the following, we would like to give the high regularity estimates of the solutions.

Lemma 2.3. Under the assumption (2.2) and (2.4), we have

where $C$ is a constant depending on $\|(\nabla\rho_{1, 0}, \nabla\rho_{2, 0}, \nabla u_{1, 0}, \nabla u_{2, 0}, \nabla^2\Phi_{0})\|_{L^2}.$

Proof. We operate each equation of (2.1) with operator $\nabla$ to derive

and multiplying the above equations by $\nabla\rho_{1}, \nabla u_{1}, \nabla\rho_{2}, and \nabla u_{2}$, respectively, and integrating over $\mathbb{R}^3$ yields

For the last term on the left-hand side of (2.14), since $\Phi(x, t)$ satisfies the Poisson equation, we have

For the term $\int_{\mathbb{R}^3}\nabla^2\Phi div(\rho_{1}u_{1}-\rho_{2}u_{2})dx,$ using Hölder's inequality and Young's inequality, we have

Now, we estimate each term on the righthand side of (2.14). Hölder's inequality and Young's inequality gives

where $\epsilon$ is a positive number that is small enough to be determined, as $\epsilon$ appears in the following inequalities. Similar to the estimate of $B_{1},$ we obtain

Simple computation gives

By the definition of $F_{1}$ and $F_{2},$ integrating by parts, and using Hölder's inequality and Young's inequality, from (2.5) we have

Integrating by parts, by (2.5) and Lemma 2.1, we obtain

Consequently, by $(2.14)$–$(2.20)$ and taking $\epsilon\ll1$, we deduce

With the help of Gronwall's inequality, we complete the proof of the lemma. □

Lemma 2.4. Under the assumption (2.2) and (2.4), we have

where $C$ is a constant depending on $\|(\nabla^2\rho_{1, 0}, \nabla^2\rho_{2, 0}, \nabla^2 u_{1, 0}, \nabla^2 u_{2, 0}, \nabla^3\Phi_{0})\|_{L^2}.$

Proof. Operating $\nabla^2$ on each equation of (2.1) gives

and multiplying the above equations by $\nabla^2\rho_{1}, \nabla^2 u_{1}, \nabla^2\rho_{2}, and\ \nabla^2 u_{2}$, respectively, and integrating over $\mathbb{R}^3$ gives

Similar to the proof of Lemma 2.3, for the last term on the left-hand side of (2.23), we get

For the term $\int_{\mathbb{R}^3}\nabla^3\Phi\nabla^2(\rho_{1}u_{1}-\rho_{2}u_{2})dx$, we can easily check that

Now we estimate $C_{i}.$ We hereby declare that $\epsilon$ occurring in the following inequalities is a sufficiently small positive number to be determined. To begin, it is easy to check that

Using Hölder's inequality and Young's inequality, we have

Similarly, we also have

Also, integrating by parts, using Hölder's inequality and Cauchy's inequality, and by (2.5) and Lemma 2.1, we have

Similarly to the estimate of $C_5,$ we obtain that

For the rest estimates of $C_i,$ it is easy to check that

and

Combining (2.23)–(2.32), we have

By Gronwall's inequality, we complete the proof of the lemma. □

Lemma 2.5. Under the assumption (2.2) and (2.4), for $3\leq l\leq s+1$, we have

where $C$ is a constant that depends only on $\|(\nabla^l\rho_{1, 0}, \nabla^l\rho_{2, 0}, \nabla^l u_{1, 0}, \nabla^l u_{2, 0}, \nabla^{l+1}\Phi_{0})\|_{L^2}.$

Proof. Similar to the proof of Lemma 2.4, we can obtain the conclusion of the lemma. So we omit it. □

3.   Green's function of the linearized system

In order to see the Green's function of the linear part of the system better, we reformulate the system (2.1) slightly. Let

which equivalently gives

Then, the Cauchy problem (2.1) can be reformulated into the following form:

where $(n_{0}, v_{0}, m_{0}, w_{0}, \nabla\Phi_{0})(x) = (\rho_{1, 0}+\rho_{2, 0}, u_{1, 0}+u_{2, 0}, \rho_{2, 0}-\rho_{1, 0}, u_{2, 0}-u_{1, 0}, \nabla\Phi_{0})(x)$ and

The linearized system of (3.3) is

We can also rewrite (3.8) as

where

Consider the Green's function $\mathbb{G}$ of (3.9), i.e., the solution to the following Cauchy problem

where $\delta(x)$ denotes the Dirac function and $I_{8\times8}$ denotes the unit matrix. By direct calculation, we obtain the Fourier transform of the Green's function $\mathbb{G}$ as

where

and

For the convenience of writing, we also give the following definition,

where $\widehat{\mathbf{G}}$ is the Fourier transform of the Green's function $\mathbf{G}$. In this paper, we divide Green's function into a high frequency part and a low frequency part since Green's function has different properties in high and low frequency. Let $\chi(\xi)$ be a smooth cutoff function

We denote $\mathbb{G} = \mathbb{G}_L+\mathbb{G}_{RH}+S,$ where $\mathbb{G}_L$ stands for the lower frequency part, $\mathbb{G}_{RH}$ stands for the regular part of the high frequency, and the $S$ stands for the sigular part. $\mathbb{G}_L, \mathbb{G}_{RH}$, and $S$ have the following forms:

Here, $\mu = \mu_{1}+\mu_{2}$. For the convenience of the description in the fifth part of this paper, we redefine smooth cutoff functions $\chi_1(\xi)$ and $\chi_2(\xi)$

Let us redefine the low frequency of $\mathbb{G}$ and the high frequency of $\mathbf{G}$,

where

Here, $\mu = \mu_{1}+\mu_{2}$. From the definition of $\chi(\xi)$, $\chi_1(\xi)$, and $\chi_2(\xi)$ in (3.14) and (3.16), we can obtain

In this paper, we use $\mathbb{G}^{i, j}$ to represent the element in row $i$ and column $j$ of $\mathbb{G}.$

Below, we list some properties of Green's function, and the readers can refer to ^{[11,21,30,31]} for details.

Lemma 3.1. If $\epsilon_{1} > 0$ is small enough, then for $|\xi| < \epsilon_{1},$ we have

and

Proof. The readers can refer to ^{[11,21,30,31]} for details, so we omit the proof. □

Lemma 3.2. If $1\leq p\leq +\infty$ and $\alpha = (\alpha_{1}, \alpha_{2}, \alpha_{3}),$ $\alpha_i\geq 0,$ $x\in \mathbb{R}^3,$ we have

Proof. By the representation of $\mathbb{G}$ and Lemma 3.1, we can obtain the proof of the lemma, so we omit it. □

Lemma 3.3. If $K > 0$ is large enough, then for $|\xi| > K,$ we have

Here, $\mu = \mu_{1}+\mu_{2}$, and all $e_{j}, l_{j}$ are real constants.

Proof. See ^{[11,21,30,31]} for details, and we omit the proof. □

Remark: This lemma states that in $\mathbb{G},$ only the terms related to

will occur in a singular part, and the other terms will bear at least a first derivative.

Lemma 3.4. If $1\leq p\leq +\infty$ and $\alpha = (\alpha_{1}, \alpha_{2}, \alpha_{3}),$ $\alpha_i\geq 0,$ $|\alpha|\leq 1,$ we have

where $C_0 > 0$ is the fixed normal number associated with $\mu$.

Proof. The readers can refer to ^{[11,21,30,31]} for details, so we omit the proof. □

Below we derive an estimation method that combines the advantages of the Green's function and energy estimate. We consider the system:

where $B(D)$ is an operator and $R(U)$ is nonlinear terms. The Green's function $\mathcal{G}(x, t)$ corresponding to the system (3.18) is the fundamental solution of the Cauchy problem of the linear equations of its corresponding system, i.e., $\mathcal{G}(x, t)$ is the solution of the following Cauchy problem:

where $\delta(x)$ denotes the Dirac function and $I_{n\times n}$ denotes the unit matrix.

The solution of (3.18) is usually discussed in terms of energy estimate or the Green's function. The following lemma combines the advantages of Green's function and energy estimate, which we may call the G-E estimate. The advantage of this estimate is that on the one hand, the fine decaying estimate of the solution can be obtained with the Green's function; on the other hand, the derivative in the singular part of the high frequency can be shared through integration by parts similar to the energy estimate.

Lemma 3.5. ^[27] If $B(\xi)$ is a complex normal matrix (i.e., $B^*B = BB^*, \ B^* = \overline{B}^T$), then it holds

where $\mathcal{G}$ is Green's function about (3.19).

Remark 3.1. If $\mathcal{G}^T = \mathcal{G}$, then we have

The Eq (3.20) is in the form of row vector times column vector, while the Eq (3.21) is in the form of the inner product of vectors.

Remark 3.2. As can be seen from (3.10), $\mathbf{G}$, which we defined in (3.13), satisfies $\mathbf{G}^T = \mathbf{G}$.

4.   The global existence of solutions

In this section, we first give the local existence theory.

4.1.   The local existence of solutions

According to (2.1), we construct the approximate solution sequence $\{\widetilde{V}^{n}(t)\}$ by the following linearized iteration scheme:

where $\{\widetilde{V}^{n}(t)\}$ is defined as $\widetilde{V}^{n}(t) = (\rho_{1}^{n}(t), u_{1}^{n}(t), \rho_{2}^{n}(t), u_{2}^{n}(t), \nabla\Phi^{n}(t)), n\geq0,$ and $\widetilde{V}^{0}(t) = (0, 0, 0, 0, 0).$ For any given integer $s\geq[\frac{3}{2}]+3,$ we define

as the suitable space for the solutions, where

It is easy to show that $X_{T, E}^{s}$, equipped with the norm $\|\cdot\|_{X^{s}}$, is a nonempty Banach space. To obtain the local soluiton, we need the following two lemmas. To do this end, we first give a prior assumption. For sufficiently small $\varepsilon_{1} > 0$, we have

Lemma 4.1. Under the assumption (4.2), when T is small enough, there exists a constant $E > 0$ such that $\{\widetilde{V}^{n}(x, t)\}\subseteq X_{T, E}^{s+1}.$

Proof. We imply the inductive method to accomplish the proof. To start, when $n = 0,$ we have

By the energy estimate, we have

We take $E = 2\|(\|\rho_{1}(x, 0), u_{1}(x, 0), \rho_{2}(x, 0), u_{2}(x, 0))\|_{H^{s}}$, then we get $\widetilde{V}^{1}(x, t)\in X_{T, E}^{s}.$ Now, assuming that $\{\widetilde{V}^{j}(x, t)\}\in X_{T, E}^{s}$ for all $j\leq n$, we need to prove it holds for $j = n+1.$

Applying the energy method to (4.1), we have

Similar to the proof of Lemma 2.2, we have the following estimates for each $G_i, \ \ 1\leq i\leq9.$

where we take $\epsilon$ small enough such that $\epsilon\ll1.$ Plugging the estimates for $G_{1}-G_{9}$ into (4.4), and integrating with respect to t, we obtain

Taking $T$ small enough, we can get the following estimate from (4.5),

To derive higher-order estimates, similar to the proof of Lemmas 2.3–2.5, we can obtain

Combining (4.6) and (4.7) yields

which means $\tilde{V}^{n+1}(x, t)\in X_{T, E}^{s+1}$. We complete the proof of the lemma. □

Lemma 4.2. Under the assumption (4.2), when T is small enough, $\{\widetilde{V}^{n}(x, t)\}$ is a Cauchy sequence in $X_{T, E}^{s}$.

Proof. We set

and define

then we only need to verify that

where $0 < \kappa < 1.$ From (4.1), we get $Y^{n+1}$ satisfies

Applying the energy method to (4.8), we have

Similar to the proof of Lemma 2.2, we have the following estimates for each $J_i, \ \ 1\leq i\leq9.$

For $J_{1}$, we have

For $J_{2}$, we have

Similar to the proof of $J_{2}$, for $J_{3}$, we can obtain

For $J_{4}$, we have

Similar to the estimate of $J_{4}$, for $J_{5}$, we can obtain

For $J_{6}$, we have

Similar to the estimate of $J_{6}$, for $J_{9}$, we can obtain

For $J_{7}$, we have

Similar to the estimate of $J_{7}$, for $J_{8}$, we have

Plugging the estimates for $J_{1}$–$J_{9}$ into (4.9), and integrating with respect to t over [0, T], we obtain

where T is small enough. To derive higher-order estimates, similar to the proof of Lemma (2.3)–(2.5), we can obtain

Combining (4.10) and (4.11) yields

so we complete the proof of the lemma. □

So far, we complete the proof of local existence.

Lemma 4.3. Let $(\rho_{1, 0}-1, u_{1, 0}, \rho_{2, 0}-1, u_{2, 0}, \nabla\Phi_{0}) \in H^{s+1}(s\geq4),$ and

where $E_0$ is sufficiently small, then there exists a time $T > 0$, such that the Cauchy problem (1.1) and (1.2) admits a unique classical solution in [0, T) that satisfies

4.2.   The global existence of solutions

In this subsection, we will establish the global solution to the systems (1.1) and (1.2) by using the bootstrap argument if the initial data satisfies

where $E_{0}$ is sufficiently small. However, for the derivatives of $\rho_{1, 0}, \rho_{2, 0}, u_{1, 0}, u_{2, 0},$ and $\nabla\Phi_{0}$, we only assume that they are bounded. Now, we first give the following abstract bootstrap argument.

Lemma 4.4. ^[32] Let $T > 0.$ Assume that two statements C(t) and H(t) with $t\in [0, T]$ satisfy the following conditions:

1) If H(t) holds for some $t\in [0, T]$, then C(t) holds for the same t;

2) If C(t) holds for some $t_{0}\in [0, T]$, then H(t) holds for t in a neighborhood of $t_{0};$

3) If C(t) holds for $t_{m}\in [0, T]$ and $t_{m}\rightarrow t,$ then C(t) holds;

4) C(t) holds for at least one $t_{1}\in [0, T].$

Then, C(t) holds on $[0, T].$

For any fixed $0 < T < \infty,$ $t\in[0, T],$ through the regularity criterion in the Section 2, we know that if $(\rho_{1}, \rho_{2}, u_{1}, u_{2}, \nabla\Phi)(x, t)$ satisfies

and

where $\varepsilon_{1} > 0$ is small enough, then

where $s\geq 4.$ From (3.1) and (3.2), we get

is equal to

and

is equal to

as well as

is equal to

For convenience, in this subsection, we use the condition (4.13) and the assumption (4.14). Let $\delta$ be a fixed positive number, say,

where $V_{0}$ is defined as $V_{0} = (n_{0}, v_{0}, m_{0}, w_{0}, \nabla\Phi_{0})^T.$ For any fixed $0 < T < \infty, t\in[0, T],$ let us denote

and

Based on the local existence of solutions, we only need to verify the condition $1$ in Lemma (4.4) under the condition (4.12), i.e., given $H(T)$ as the condition, to derive $C(T)$ is valid. Before we check the condition 1, we need some lemmas.

Lemma 4.5. Under the assumption (4.15) and the condition (4.13), the following estimate holds:

Proof. By (4.15) and Lemma 2.2, we complete the proof of the lemma. □

Lemma 4.6. Under the assumption (4.15) and the condition (4.13), we have

Proof. By the Gagliardo-Nirenberg inequality, we know

and

where $\theta = \frac{2}{2\epsilon+5}, \beta = \frac{1}{k}.$ Then, from (4.17) and (4.18), we have

Taking $k = \frac{1}{\epsilon}, \ \epsilon = \frac{1}{22}$ in the above inequality, we obtain

The same computation also gives

By the interpolation inequality, it holds

Thus we complete the proof of the lemma. □

Lemma 4.7. Under the condition (4.13), we have

Proof. By $(3.3)_{1}$ and $(3.3)_{3}$, and using the condition (4.13), we can obtain the result. □

Now let us use Green's function to prove some lemmas. To begin, by Duhamel's principle, we know

where

where the $Q_{i}(i = 1, 2, 3, 4)$ are defined by $(3.4)$–$(3.7).$ From (4.19), for the component $v$ and $w$ of $V$, we have

where $\mathbb{G}(t)*V_{0}, \ \mathbb{G}(t-s)*N(V)(s)$ obey matrix multiplication.

Lemma 4.8. Under the assumption (4.15) and the condition (4.13), it holds that

Proof. Because we can't directly get the estimate of $\|(v(t), w(t))\|_{L^1}$, we first obtain the boundedness of $\|(v(t), w(t))\|_{L^r}$ for some $r\in(1, 2).$ In the following, we take $r = \frac{4}{3}.$

We first consider $\|(v, w)(t)\|_{L^r}.$ By (4.21) and the representation (3.15) of $S,$ we have

For the estimate of the linear part, it is easy to check that

For the estimate of the nonlinear part, utilizing the definitions (3.4)–(3.7) of $Q_i$, we have

and

Combining (4.24)–(4.26), we obtain

The same procedure gives

So far, we can obtain

Now, we can obtain the estimate of $\|(v, w)\|_{L^1}$ by using $\|(v, w)\|_{L^{\frac{4}{3}}}.$ For $M_{1},$

A simple check gives us that

Then, by using Lemma 4.6 and Young's inequality for convolution, we have

and

Combining the estimate of each $F_{i}$, we obtain

The same procedure gives

So far, we can obtain

and we complete the proof of the lemma. □

Lemma 4.9. Under the assumption (4.15) and the condition (4.13), we have

Proof. Applying Duhamel's principle, we have

Then,

Now we estimate the righthand side of the inequality. Simple computation yields

and

Now we turn to estimate each $O_i.$ By the definition (4.20) of $N(V)$, we easily check that

Using Sobolev's inequality, we obtain

then for $O_{1}$, we have

For $O_{2}$, we can get

Also by Sobolev's inequality, it holds

then we have

For $O_{3}$, we have

Sobolev's inequality gives

then we have

Combining (4.32), (4.34), and (4.37), we obtain

Using (4.30) and (4.38), we complete the proof of the lemma. □

With the above lemmas at hand, we now check the condition 1 of Lemma 4.4, that is, the following lemma.

Lemma 4.10. Under the assumption (4.15) and the condition (4.13), we have

Proof. Duhamel's principle gives rise to

Then

Similar to the proof of Lemma 4.9, it holds for the first term on the righthand side of the inequality that

For for the second term on the right side of the inequality, it is easy to get that

For the nonlinear part of lower frequency, Lemma 4.5 gives

For the terms of $\int_{0}^{t}\|\nabla(\mathbb{G}_{L}^{k, \{2, 3, 4\}}(t-s)*N(V)^{\{2, 3, 4\}}(s))\|_{L^{\infty}(\mathbb{R}^3)}ds,$ Lemma 4.5 gives

By Lemmas 3.2 and 4.5–4.9, we have

The same procedure gives

and

Combining (4.40), (4.41), (4.42), and (4.43), we have

For the nonlinear part of high frequency, Lemma 4.5 gives

and

(4.45) and (4.46) gives

For the nonlinear part estimate for the singular part, by Lemmas 4.6 and 4.9, we have

Combining (4.43), (4.47), and (4.48), we obtain

By (4.39) and (4.49), we complete the proof of the lemma. □

Under the condition (4.13), the bootstrap argument and the local existence of the solutions for the syestem (1.1) and (1.2) gives the following result.

Proposition 4.1. Let $(\rho_{1, 0}-1, u_{1, 0}, \rho_{2, 0}-1, u_{2, 0}, \nabla\Phi_{0}) \in H^{s+1}(s\geq4),$ and

where $E_0$ is sufficiently small, then the Cauchy problem (1.1) and (1.2) has a global solution in time that satisfies

and

5.   The decay rate of solutions

In this section, we would like to get the decay rate of solutions. The main result is stated as follows.

Proposition 5.1. Let $(\rho_{1, 0}-1, u_{1, 0}, \rho_{2, 0}-1, u_{2, 0}, \nabla\Phi_{0})\in H^{s+1}(s\geq4),$ and

where $E_0$ is sufficiently small, $(\rho_1, u_1, \rho_2, u_2)$ is the solutions for the Cauchy problem (1.1) and (1.2), then when $2\leq p\leq +\infty$ and $\alpha = (\alpha_{1}, \alpha_{2}, \alpha_{3}),$ $\alpha_i\geq 0,$ $|\alpha|\leq s-1,$ it holds

Proof. From (3.1) and (3.2), the decay rate of $(\rho_1-1, u_1, \rho_2-1, u_2)$ is equivalent to the decay rate of $(n, v, m, w)$. Therefore, we only need to consider the attenuation estimate of $(n, v, m, w)$. Note that if the $L^2$ decay rate of the higher-order spatial derivatives of the solution are obtained, then the general $L^q$ decay rate of the solution follows by the Sobolev interpolation. For instance, using the Sobolev embedding theorem, we have

where V(t) is defined as (3.10). So we only consider the decay of $\|D^{\alpha}V(t)\|_{L^2}.$ Below we will prove the following assertion by induction,

When $|\alpha| = 0,$ we have

For the terms $\int_{0}^{t}\|\mathbb{G}_{L}(t-s)*N(V)(s))\|_{L^2}ds$ and $\int_{0}^{t}\|\mathbb{G}_{RH}(t-s)*N(V)(s))\|_{L^2}ds$, we have

Then, from Proposition (4.1), using Sobolev's inequality, we obtain

and

For the term $\int_{0}^{t}\|S(t-s)*N(V)(s)\|_{L^2}ds$, from Proposition (4.1), we have

We assume that (5.1) holds when $|\alpha| = k-1,$ and later we shall prove that (5.1) holds when $|\alpha| = k.$ To get the estimate of $\|D^{k}V(t)\|_{L^2},$ we perform high and low frequency decomposition of the solution itself. The low frequency part of the solution V is

and the high frequency part of the solution V is

where $\chi(D)$ is a pseudo-differential operator with symbol $\chi(\xi),$ we can see (3.14) for the definition of $\chi(\xi).$ Then, we can decompose $D^{k}V(t)$ as follows:

where $\mathbb{G}$, $\mathbf{G}$, and $N(V)$ are defined in (3.12), (3.13), and (4.20), respectively. We define $\widetilde{N(V)}$ as follows:

Here, $\widetilde{Q_{3}} = Q_{3}+2\nabla\triangle^{-1}m,$ and $Q_{1}, Q_{2}, Q_{3}$, and $Q_{4}$ are defined in (3.4)–(3.7). For $D^{k}V(t)$, we have

Then, (5.2) and (5.3) are equal to

Now let us estimate (5.5) and (5.6) separately. For the low frequency part, we have

For $H_1,$ we have

For $H_2,$ we first estimate $\|(N(V))_L\|_{L^1},$

Then, for $H_2,$ we have

For $H_3,$ we have

where we use the Gagliardo-Nirenberg inequality to obtain the estimate for $\|(n, v, m, w)(t)\|_{L^\infty}$ and $\|\nabla^2(v, w)(t)\|_{L^{\infty}}$. Combining the estimate of $H_1$, $H_2$, and $H_3$, we obtain the low frequency part estimate of $D^{k}V(t)$

For the high frequency part of $D^{k}V(t)$, by Lemma 3.5, we have

By the definition of $\chi_2(\xi)$ in (3.16), we know (5.7) is equal to

Now we turn to estimate each $I_i.$ To start, for $I_1,$

From the definition of $N(V)$ and $\widetilde{N(V)}$ in (4.20), (5.4), we have

For the nonlinear item, we can check without difficulty that

From the conclusion of Proposition (4.1), we know (4.14) holds, i.e.,

and by the Sobolev embedding theorem, we have

Now for $I_2,$ we can obtain

For $K_{1}$, we have

With the help of (5.11)–(5.13), we can get the estimates of $K_{1}$ as follows. By using Lemma 3.4, we can get the estimate for $K_{1}$,

where we used the (5.1) when $\alpha = k-1$ in the last inequality of the above. The estimate of $K_{3}$ is parallel to $K_{1},$ i.e.,

By integrating by parts, the estimate of $K_{2}$ is similar to $K_{1},$ then we have

As for $K_{4},$ by the defination of $\mathbf{G}_{\widetilde{S}}$, and using (5.12) and (5.13), we have

By the Gagliardo-Nirenberg inequality, we have

Then, for $K_5,$ it holds

By the same analysis as $K_{5},$ it holds

By the definition of $\mathbf{G}_{\widetilde{S}}$ in (3.17) and $\widetilde{N(V)}_2$ in (5.10), we have

Combining the estimate for each $K_i$, and from (5.7), it holds,

To close the estimate, we take our attention to the estimate of $\|D^{k+1}(v, m)(t)\|_{L^2}.$ From $(4.14),$ we know

For the linear partition, we have

For the nonlinear partition, we still adopt the method of high and low frequency decomposition of the solution. Here we omit the details of the calculation since the analysis is parallel to the estimate of $D^kV(t).$

For $\|D^{k+1}v(t)\|_{L^2},$ we have

The same way also gives

Combing (5.14)–(5.16), since the $E_{0}$ is small enough, we have

Now we proved (5.1) when $\alpha = k.$ The proposition is proved. □

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Nos.12271357, 12161141004) and Shanghai Science and Technology Innovation Action Plan (No.21JC1403600).

Conflict of interest

The authors declare there is no conflicts of interest.