Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in $\mathbb R^3$

1.   Introduction

We consider the following 3D compressible Navier-Stokes- Poisson (NSP) system:

Here, the density $\rho>0$, $m$ represents the momentum, $u = \frac{m}{\rho}$ stands for the velocity, and $\phi$ represents the electrostatic potential. The viscosity coefficients $\mu>0$ and $\nu$ satisfy $\frac{2}{3}\mu+\nu\geq0$, the Debye length $\lambda>0$ and the pressure functions $p = p(\rho)$. $\bar{\rho}$ is the background doping profile, which is regarded as a positive constant for the sake of simplicity.

1.1.   History of the problem

To go directly to the theme of this paper, let us give some explanations about the above model. For the existence and long time behavior of the solution to the compressible Navier–Stokes (NS) system (i.e., $\phi\equiv 0$ in $(1.1)$), one can refer to [10,11,12,18,19,6] and references therein. For the compressible NSP system, Li–Matsumura–Zhang [14] proved that the density of the NSP system converges to its equilibrium state at the same $L^2$–rate $(1+t)^{-\frac34}$ as the compressible Navier–Stokes (NS) system, but the momentum of the NSP system decays at the $L^2$–rate $(1+t)^{-\frac14}$, which is slower than the $L^2$–rate $(1+t)^{-\frac34}$ for the compressible NS system due to the effect of the electric field. And then they extended similar result to the non–isentropic case [27]. Wang [25] obtained the optimal asymptotic decay of solutions just by pure energy estimates, and particularly he proved that the density of the compressible NSP system decays at the $L^2$–rate $(1+t)^{-\frac54}$, which is faster than the $L^2$–rate $(1+t)^{-\frac34}$ for the NS system due to the effect of the electric field. Hao–Li [9], Tan–Wu [22], Chikami–Danchin [4], Bie–Wang–Yao [2] and Shi-Xu [21] also established the unique global solvability and the optimal decay rates in critical spaces. We mention that there are many results on the existence and long time behavior of the weak solutions or non–constant stationary solutions, see, for example [1,5,8,28] and the references therein. For the compressible NSP system with external force, [16] investigated the existence and zero-electron-mass limit of strong solutions to the stationary with large external force and [17] proved that the strong solution existence to the boundary value problem in a bounded domain. For the bipolar Navier–Stokes–Poisson(BNSP) system, we refer to [7], [15], [24], [26], [29], [30], and references therein.

We remark that this paper is strongly motivated by Li–Matsumura–Zhang [14]. Their main results can be outlined as follows: Assume that $(\rho_{0}-\bar{\rho},m_{0})\in{H^{l}\cap L^{1}}$, $l\geq4$, and $\|(\rho_{0}-\bar{\rho},m_{0})\|_{H^{l}\cap L^{1}}$ is sufficiently small. Then, the Cauchy problem $(1.1)$ admits a global smooth solution $(\rho,m,\phi)$ satisfying:

and

However, it is clear that $(1.2)-(1.4)$ give no information on the decay rates of higher order spatial derivatives ($k\geq 2$) of the solution. The main purpose of this article is to give a clear answer to this issue.

1.2.   Main results

In this paper, we first introduce some notations and conventions. We use $H^l(\mathbb R^3)$ to denote the usual Sobolev spaces with norm $\|\cdot\|_{H^l}$ and use $L^p$, $1\leq p\leq \infty$ to denote the usual $L^p(\mathbb R^3)$ spaces with norm $\|\cdot\|_{L^p}$. The notation $a\lesssim b$ means that $a\leq Cb$ for a universal positive constant which is only dependent on the parameters of the problem. For a radial function $\phi\in C_0^\infty(\mathbb R^3_{{\bf \xi}})$ such that $\phi({\bf \xi}) = 1$ when $|{\bf \xi}|\leq 1$ and $\phi({\bf \xi}) = 0$ when $|{\bf\xi}|\geq 2$, we define the low–frequency part and the high–frequency part of $f$ by

If the Fourier transform of $f$ exists, then $f = f^{h}+f^{l}$.

Now, we state out our main results in the following theorem:

Theorem 1.1. Let $p'(\rho)>0$ for $\rho>0$. Assume that $(\rho_{0}-\bar{\rho},m_{0})\in H^{4}(\mathbb{R}^{3})\cap L^{1}(\mathbb{R}^{3})$, with $\delta_0 = :\|(\rho_{0}-\bar{\rho},m_{0})\|_{H^{3}(\mathbb{R}^{3})\cap L^{1}(\mathbb{R}^{3})}$ small. Then for any $t\geq 0$, the Cauchy problem $(1.1)$ has a unique global solution $(\rho,m,\phi)$ satisfying the following optimal time decay rates

and

for $2 \leq k \leq 4$.

Remark 1.2. Compared to the result (1.2)-(1.4) in [14], the $L^{2}$ decay rates of higher spatial derivatives from 2-order to 4-order of the solution $(\rho,m,\nabla\phi)$ in (1.5)-(1.7) are totally new. Furthermore, we only need the smallness of $H^3\cap L^1$-norm of the initial data, while $H^4$-norm of the initial data may be arbitrarily large. Finally, by noticing that, for $2 \leq k \leq 4$, we can obtain the optimal decay rates of $\Vert \nabla^k(m^h,\nabla\phi^h)\Vert_{L^{2}}$ in (1.7), which are the same as those of the compressible Navier–Stokes equations and heat equation, and particularly faster than one of $\Vert \nabla^k(m,\nabla\phi)\Vert_{L^{2}}$ in (1.6). This implies that the electric field has no effect on the decay rates of the density and the high-frequency part of the momentum, but reduces the decay rate of the momentum.

Now, let us sketch the strategy of proving Theorem 1.1. Motivated by the results in (1.2)-(1.4), we will focus on deriving the $L^{2}$ time decay rates of higher spatial derivatives from 2-order to 4-order of the solution $(\rho,m,\nabla\phi)$. Our strategy can be outlined as follows. First, when we prove the Theorem 1.1, we encounter such a difficulty: we need the optimal decay rate of the second-order spatial derivative of the solution to deduce the decay rate of the third-order spatial derivative of the solution. In the same way, we also need the decay rate of the third-order spatial derivative of the solution to derive the decay rate of the fourth-order spatial derivative of the solution. Therefore, we will prove Theorem 1.1 through three cases. Notice that the optimal decay rates on low-frequency part of $(\rho,m,\nabla\phi)$ has been established by [3]. Therefore, we only need to derive the optimal decay rates on $\|\nabla^k(\rho^h,m^h,\nabla\phi^h)\|$, $k = 2,3,4$. For case 1, we hope to establish the $L^2$-time decay rates of second-order spatial derivative of $(\rho,m,\nabla\phi)$. We rewrite system (1.1) into (2.1), and deduce the energy inequality as follows:

However, it should be mentioned that in the process of deducing the energy inequality (1.8), we encounter the trouble term $I_{2} = -\langle\nabla^2(\nabla n\cdot u)^h,\nabla^2 n^h\rangle$ in (2.7). If we estimate it directly, we will meet the norm $\|\nabla^3 n\|_{L^2}$ which however can not be controlled in our framework. To overcome this obstacle, we need to reduce the order of the density by splitting $I_{2}$ into three parts as follows:

and then make full use of the benefit of the low–frequency and high–frequency decomposition technique to bound the terms $I_{21}$, $I_{22}$ and $I_{23}$ one by one. For $I_{21}$, it is clear that

and using integration by parts, we have

Thus we can deduce

as in (2.13). Now, note that (1.8) only gives the dissipation estimate for $u^h$. In order to explore the dissipation estimates for $n^h$ and $\nabla\phi^h$, we will employ the new interactive energy functional between $u^h$ and $\rho^h$ by using critical $L^2$ estimates, low-frequency and high-frequency decomposition. Particularly, we can get

Next, we choose a sufficiently large positive constant $D_1$ and then define the temporal energy functional as

Noting that (1.12) is equivalent to $\|\nabla^{2}n^h\|_{L^2}^2+\|\nabla^{2}u^h\|_{L^2}^2+\|\nabla^{2}E^h\|_{L^2}^2$. Multiplying (1.8) by $D_{1}$, adding the resulting inequality with (1.11), we obtain

then the optimal $L^2$ time decay rates of $\Vert \nabla^2((\rho-\bar{\rho})^{h},m^h,\nabla\phi^h)\Vert_{L^{2}}$ in (1.7) for $k = 2$ can be deduced by virtue of Lemma 2.1, Lemma 2.2, Gronwall's argument. Furthermore, combining with the 2-order low-frequency decay estimates in Lemma 2.2, we get the decay rates of 2-order spatial derivatives of $(\rho,m,\nabla\phi)$ immediately. This implies that we complete the proof of Theorem 1.1 for $k = 2$. We use similar methods to prove the Theorem 1.1 for $k = 3$ and $k = 4$ in case 2 and case 3 respectively.

2.   Proof of theorem 1.1

This section is devoted to prove the optimal $L^2$ time decay rates of solution stated in Theorem 1.1. First, we rewrite the system. Denoting $(\rho,u) = (1+n,\frac{m}{\rho})$ and $\nabla\phi = E$, then the Cauchy problem (1.1) can be reformulated as

where

and as in [14], we have taken $\bar{\rho} = 1$, $p'(1) = 1$, $\mu = \mu_{1}$, $(\mu+\nu) = \mu_{2}$ and $d = 1$ for simplicity. Notice that

and

Second, the decay rates on the linearized system of (2.1) are given in the following lemma:

Lemma 2.1 Assume that $U_{0} = (n_{0},m_{0}) \in H^{l}(\mathbb{R}^3)\cap L^{1}(\mathbb{R}^3)$, $l \geq 4$, and denote $(\bar{n}(t),\bar{m}(t)) = :\bar{U}(t)$. Then, for $0 \leq k \leq l$, the solution $(\bar{n},\bar{m},\bar{E})$ of the linearized system of (1.1) with $\bar{E} = \nabla(-\Delta)^{-1}\bar{n}$ satisfies that

Proof. See [14].

Third, we state the $L^{2}$-time decay estimates on the low-frequency part of the solution in the nonliear system (1.1).

Lemma 2.2. Assume that the assumptions of Theorem 1.1 are in force, Then for $0\leq k \leq l$, the following decay rates hold:

Proof. See [3].

Next, we will prove Theorem 1.1 in three cases. Before giving the precise proofs of these three cases, we must emphasize that under the assumption of Theorem 1.1, [14] proved that the following estimate holds

More details of $(2.6)$, please refer to (3.62)-(3.64) on page 697 of reference [14].

Case 1. Proof of Theorem 1.1 for $k = 2$. Theorem 1.1 will be proved by the good properties of the low-frequency and high-frequency decomposition. The proof involves the following steps.

Step 1. High–frequency $L^2$ energy estimate. Taking

and using integration by parts, yields directly

The right-hand side of (2.7) can be estimated one by one. For term $I_{1}$, due to Hölder's inequality, Young inequality, Sobolev interpolation theorem, it holds that

The second term $I_2$ can be rewritten as follows

It is easy to obtain

Following from integration by parts, we have

thus

For the term $I_{22}$, we have

For the term $I_{23}$, by Lemma A.4, Hölder's inequality, Young inequality, we have

Substituting (2.10)-(2.12) into (2.9), we can conclude that

For the term $I_{3}$ and $I_{4}$, similar to the proofs of (2.8) and (2.13), it holds that

For the term $I_{5}$, making use of integration by parts, we obtain

For the term $I_{51}$, we have

For the term $I_{52}$, it holds that

For the term $I_{53}$, we get

For the term $I_{54}$, we have

Thus, one may deduce that

Substituting (2.8), (2.13)–(2.14) and (2.20) into (2.7) and using the smallness of $\delta_0$, yield directly

Step 2. Dissipation of $\nabla^{2} n^h$. Taking $\langle\mathfrak{F}^{-1}(1-\phi({\bf \xi}))\nabla(2.1)_2,\nabla^{2}n^h\rangle$, it holds that

We should estimate the right-hand side of (2.22) one by one. For $I_{6}$, using a similar argument like (2.8), we have

For $I_{7}$, similar to the proof of (2.13), it holds that

For $I_{8}$, as the proof of (2.19), we have

For $I_{9}$, we obtain

Putting (2.23)–(2.26)into (2.22) and using the smallness of $\delta_0$, we conclude that

Step 3. Closing the estimates. Now, we are in a position to close the estimates. To do this, we choose sufficiently large time $T_1$ and positive constant $D_1$, and then define the temporary energy functional

for $t\geq T_1$, it is noticed that $\mathfrak{E_{1}(t)}$ is equivalent to $\|\nabla^{2}n^h\|_{L^2}^2+\|\nabla^{2}u^h\|_{L^2}^2+\|\nabla^{2}E^h\|_{L^2}^2$ since $D_1$ is large enough. Substituting (2.21) and (2.27) into

which together with the smallness of $\delta_0$, for $t\geq T_1$, it holds that

where we have used the fact that $T_1$ is large enough. On other hand, it is clear that

Hence, by virtue of (2.4)-(2.5), (2.29)-(2.30) and Gronwall's inequality, we can arrive at

Furthermore, combining with (2.4)-(2.5) and (2.31), the following estimates can be obtained:

and

The proof of Theorem 1.1 for $k = 2$ is completed.

Case 2. Proof of Theorem 1.1 for $k = 3$. Theorem 1.1 for $k = 3$ will be proved as in Case 1 by the good properties of the low-frequency and high-frequency decomposition. The proof involves the following steps.

Step 1. High–frequency $L^2$ energy estimate. Taking

and using integration by parts, which implies

The five terms in the above equation can be estimated as follows. Firstly, for term $J_{1}$, using Hölder's inequality, Young inequality, Sobolev interpolation theorem, we obtain

The term $J_2$ can be rewritten as follows

It is obvious that

By virtue of integration by parts, we arrive at

thus

For the term $J_{22}$, we have

For the term $J_{23}$, with the help of LemmaA.1-Lemma A.3, Hölder's inequality, (1.3), (2.32)-(2.33) and Young inequality, we have

Substituting (2.37)-(2.39) into (2.36), we can achieve

For the term $J_{3}$ and $J_{4}$, similarly to the proof of (2.35) and (2.38), it holds that

For the term $J_{5}$, by integration by parts, we have

For the term $J_{51}$, we have

For the term $J_{52}$, we have

For the term $J_{53}$, we have

For the term $J_{54}$, we have

Thus, we can immediately to obtain

Substitute (2.35), (2.40)–(2.41) and (2.47) into (2.34) and use the smallness of $\delta_0$, we deduce that in fact

Step 2. Dissipation of $\nabla^{3} n^h$. Applying the operator $\nabla^2\mathfrak{F}^{-1}(1-\phi({\bf \xi}))$ to (2.1)$_2$, multiplying the resulting equality by $\nabla^{3} n^h$, integrating over $\mathbb R^3$, it holds that

For the term $J_{6}$, we have

For the term $J_{7}$, similarly to the proof of (2.40), we obtain

For the term $J_{8}$, similarly to the proof of (2.42), it holds that

For $J_{9}$, we obtain

Substituting (2.50)–(2.53)into (2.49) and making use of the smallness of $\delta_0$, we conclude that

Step 3. Closing the estimates. Now, let us close the estimates. For any $t\geq0$, we define the temporary energy functional as follows

for $t\geq T_2$, where it is noticed that $\mathfrak{E_{2}(t)}$ is equivalent to $\|\nabla^{3}n^h\|_{L^2}^2+\|\nabla^{3}u^h\|_{L^2}^2+\|\nabla^{3}E^h\|_{L^2}^2$ if we choose sufficiently large time $T_2$ and positive constant $D_2$. Putting (2.48) and (2.54) into

for $t\geq T_2$, $\delta_0$ is small enough implies

where we have used the fact that $T_2$ is large enough. On other hand, it is clear that

Thus, in view of (2.4)-(2.5), (2.56)-(2.57) and Gronwall's argument, we have

Furthermore, utilizing (2.4)-(2.5) and (2.58), it is easy to have

This complete the proof of Theorem 1.1 for $k = 3$.

Case 3. Proof of Theorem 1.1 for $k = 4$. taking the similar argument as in Case 2, we can prove Theorem 1.1 for $k = 4$ by the following steps.

Step 1. High–frequency $L^2$ energy estimate. Taking

and applying integration by parts, we have

We will estimate the right-hand side of the above equation term by term. First, for term $K_{1}$, from Hölder's inequality, Young inequality, Sobolev interpolation theorem, it leads to

For the term $K_2$, we can rewrite it as

Notice that

From integration by parts, one has

Note that

and

Thus, we can have the following estimate

For the term $K_{22}$, we have

For the term $K_{23}$, we achieve

Substituting (2.64)-(2.66) into (2.63), we can arrive at

For the term $K_{3}$ and $K_{4}$, similarly to the proof of (2.62) and (2.65), it holds that

For the term $K_{5}$, we have

For the term $K_{51}$, we have

For the term $K_{52}$, we have

For the term $K_{53}$, we obtain

For the term $K_{54}$, we have

Thus, we can arrive at

Substituting (2.62), (2.67)–(2.68) and (2.74) into (2.61) and using the smallness of $\delta_0$, we conclude that

Step 2. Dissipation of $\nabla^{4} n^h$. Applying the operator $\nabla^3\mathfrak{F}^{-1}(1-\phi({\bf \xi}))$ to (2.1)$_2$, multiplying the resulting equality by $\nabla^{4} n^h$, integrating over $\mathbb R^3$, we have

For the term $K_{6}$, we have

For the term $K_{7}$, similarly to the proof of (2.67), we obtain

For the term $K_{8}$, similarly to the proof of (2.69), it holds that

For $J_{9}$, we have

Substituting (2.77)–(2.80) into (2.76) and using the smallness of $\delta_0$, we dedeuce that

Step 3. Closing the estimates. Now, we are in a position to close the estimates. To do this, we define the temporal energy functional

for $t\geq T_3$, where it is noticed that $\mathfrak{E_{3}(t)}$ is equivalent to $\|\nabla^{4}n^h\|_{L^2}^2+\|\nabla^{4}u^h\|_{L^2}^2+\|\nabla^{4}E^h\|_{L^2}^2$ if we choose sufficiently large time $T_3$ and positive constant $D_3$. Substituting (2.75) and (2.81) into

and using the smallness of $\delta_0$, for $t\geq T_3$, we have

since $T_3$ is large enough. On other hand, it is clear that

Therefore, together with (2.4)-(2.5), (2.83)-(2.84) and Gronwall's argument, we have

Consequently, we have from (2.4)-(2.5) and (2.85) that

This completes the proof of Theorem 1.1 for $k = 4$.

Therefore, we complete the proof of Theorem 1.1.

Appendix A. Analytic tools. We will use the Sobolev interpolation of Gagliardo-Nirenberg inequality.

Lemma A.1. Let $0 \leq i,j \leq k$; then we have

where $\theta$ satisfies

Lemma A.2. Let $k\ge 1$ be an integer; then one has

where $p,p_2,p_3\in[1,+\infty]$ and

Finally, we introduce the lemma concerning the estimate for the low-frequency part and the high-frequency part of $f$.

Lemma A.3. If $2\leq p \leq \infty$, $f\in L^{p}(\mathbb{R}^3)$, then we have

Proof. For $2\leq p \leq \infty$, by Young inequality's for convolutions, for the low-frequency, we have

and hence

Acknowledgments

Guochun Wu's research was in part supported by National Natural Science Foundation of China (No. 11701193, 11671086), Natural Science Foundation of Fujian Province (No. 2018J05005, 2017J01562), Program for Innovative Research Team in Science and Technology in Fujian Province University Quanzhou High–Level Talents Support Plan (No. 2017ZT012). Yinghui Zhang' research is partially supported by Guangxi Natural Science Foundation (No. 2019JJG110003, 2019AC20214), and National Natural Science Foundation of China (No. 11771150, 11571280, 11301172 and 11226170.)