Regularity of weak solution of the compressible Navier-Stokes equations with self-consistent Poisson equation by Moser iteration

1.   Introduction

In this article, we will investigate the regularity of weak solutions of the compressible Navier-Stokes-Poisson (NSP) equations. The compressible (NSP) system is used to simulate the motion of charged particles. It consists of the compressible Navier-Stokes (NS) equations with the electrostatic potential controlled by the self-consistent Poisson equation. A carrier type of NSP system is represented as the following form in $\mathbb{R}^3$:

Here, $\rho = \rho(t, x) > 0$ is the density, $u(t, x) = (u_1, u_2, u_3)(t, x)$ is the velocity, $p(\rho)$ denotes the pressure and $\Phi$ denotes the electrostatic potential. $\overline{\rho} > 0$ is a constant. $\mu > 0$ and $\nu$ are constant viscous coefficients that satisfy $2\mu +3\nu \geq 0$. The term $f$ denotes a given external force.

There are many conclusions about NS equations on the condition that $\nabla \Phi = 0$ in (1.1). In the past, people have studied the existence of weak solution to the compressible NS equations. P. L. Lions ^[12] proved the global existence of finite energy weak solution to NS equations when $p(\rho) = R\rho^{\gamma}, \gamma > \frac{9}{5}$ in 1998; Feireisl ^[6] claimed the existence of the weak solution when $p(\rho) = R\rho^{\gamma}, \gamma > \frac{3}{2}$ in 2001. Afterwards, Choe, Kim and Cho ^[2,3] showed the local existence of the strong solution to NS equations. In 2011, Sun, Wang and Zhang ^[19] proved a blow-up criterion for a strong solution to the compressible viscous heat-conductive flows in $\mathbb{R}^3$ on condition that $\mu > \frac{1}{7}\nu$ in accordance with the upper bound of $(\rho, \rho^{-1}, \theta)$. Valli ^[24] proved the uniqueness of the solution if the data $\rho_0, u_0$ and $f$ satisfy some condition in bounded smooth domain $\Omega \subseteq \mathbb{R}^3$. In 2013, Wen and Zhu ^[25] established a blow-up criterion for strong solutions to the Cauchy problem of compressible isentropic NS equations with vacuum provided $\mu > \frac{3}{29}\nu$ in the light of the integrability of the density. Though Choe ^[4] proved the regularity to the weak solution, which is defined by D. Hoff ^[9], it is indeterminate whether the weak solutions is the same to the finite energy weak solution.

However, there are not many conclusions related to the solution to NSP equation. In a series of articles ^[14,15,18], M. Okada et al. ^[17,18] studied the vacuum problem of NSP equations which are spherically symmetrical with solid core. They ^[17] considered a free boundary problem for the equation of the one-dimensional isentropic motion with density-dependent viscosity and proved that there exists a unique weak solution globally in time; Then, Kobayashi and Suzuki ^[11] proved the existence of weak solution to NSP equations of which the method is similar to the method of ^[6]. In 2007, Zhang and Tan ^[27] proved the existence of the global weak solution to the compressible NSP equations where $p(\rho) = a\rho\log ^{d} \rho$ for sufficiently large densities by using suitable Orlicz spaces in $\mathbb{R}^2$. We may refer to ^{[1,7,20,21,22,26]} for more about NS or NSP equations. As far as we are concerned, to the author's best knowledge, there are not many studies on the regularity for the corresponding weak solution to the compressible NSP equations.

For simplicity, let $f \equiv 0$ of (1.1) in the paper. Thus, the problem we are concerned with in our study is the regularity of weak solutions to the isentropic compressible NSP equations in the periodic domain $\Omega = \mathbb{R}^3-\mathbb{Z}^3$ as

with the following initial conditions:

where $\overline{\rho} > 0$ is a constant, $p(\rho) = R\rho ^{\gamma}, R > 0$, $\gamma > 1$ and $m$ and $M$ both are constants.

Definition 1.1. $(\rho, u)$ is called the finite energy weak solution of the Eq (1.2) for $(t, x)\in\Omega_T = (0, T) \times \Omega$, if $(\rho, u)$ satisfies

and the energy inequality

holds in $\mathcal{D}'((0, \infty))$ with the finite total energy, where

All the priori estimates of this paper depend on the assumption that $\rho$ and $u$ are $C^{\infty}$ for the time interval as mentioned in ^[4,5].

Theorem 1.1. Suppose $(\rho, u)$ is the weak solutions denoted by the Definition 1.1 to the Eqs (1.2) and (1.3). If $\rho \in L^{\infty}\big(0, T;L^{q_0}(\Omega)\big)$ for time $T > 0$ and for $q_0$ depending only on $\gamma$, then

Furthermore, we show that $\rho\in C(0, T;L^{q}(\Omega))\cap L^{\infty}(\Omega_T)$ for all $q\in [1, \infty)$, and $\sqrt{\rho}u_t\in L^{2}(\Omega_{T}), \mathcal{P}u\in L^{2}(0, T;H^{2}(\Omega)), G \in L^{2}(0, T;H^{2}(\Omega)), \nabla u\in L^{\infty}(0, T;L^{2}(\Omega))$, where $\mathcal{P}u$ refers to the divergence-free part of $u$, $G = (2\mu+\nu) \text{div}\ u-p(\rho)$.

The paper is structured as follows. In Section 2, we give some preliminaries to study the regularity of the weak solution to the NSP equations. In Section 3, one develops the priori estimate of $\rho$ under some condition by many techniques; In Section 4, we shall perfect the proof of Section 3; Finally, in Section 5, we present a prior $L^{\infty}$ estimate of $u$ by Moser iteration.

2.   Preliminary

In this section, we collect some auxiliary results which will be used to investigate subsequent studies.

First, we define the divergence-free part of any vector field $u$ as $\mathcal{P} u$, and the gradient part of $u$ as $\mathcal{Q}u$ which means $\mathcal{Q}u = \nabla \Delta^{-1}\text{div} u$. Naturally, the following propositions hold.

Proposition 2.1.

Proposition 2.2. (Page 67 of ^[13]). Suppose that $f\in L^2(\Omega)$, and

then $v\in W^{2, 2}(\Omega)$, and $\nabla p \in L^2(\Omega)$. Furthermore,

Proposition 2.3. Suppose that $\rho\in L^{\infty}(0, T;L^{p_0}(\Omega))$ for $p_{0} = \max \{\frac{(2\gamma-1)\alpha+4\gamma}{2}, 4\}$, where $\alpha$ is a nonnegative real number. Then, we obtain

Proof. Let $\vert u\vert^{\alpha}u$ as a test function to (1.2), then one obtain

Applying Hölder's inequality and Young's inequality to (2.1), implies that

Similarly, we obtain

Noting that $\Delta \Phi = \rho-\overline{\rho}$, and using Hölder's inequality and Sobolev inequality, we could have

Combining (2.2)–(2.4) with (2.1), and letting $U(t) = \int_{\Omega}\rho \vert u\vert^{\alpha+2}dx$, one could have

This reduces to

□

3.   A priori estimates for density $\rho$

3.1.   The estimates for $\mathcal{P}(H)$, $\mathcal{Q}(H)$ and $G$

Now, applying the operator "divergence-free fields" $\mathcal{P}$ to (1.2), we consequently obtain that

Denote $H = \rho u_{t}+\rho u \cdot \nabla u-\rho \nabla \Phi$, then (3.1) will be reformulated equivalently as

Then, by taking divergence operator to the Eq (1.2), one deduces that

Denote $G = (2\mu+\nu) \text{div} u-p(\rho)$, and we have again by formula (3.3) that

Note that the definition of $\mathcal{Q}u = \nabla \Delta^{-1}\text{div} u$, then (3.4) will be reformulated equivalently as

By virtue of (3.2) and (3.5), we get

Note that $H = \mathcal{P}(H)+\mathcal{Q}(H)$, and from Proposition 2.2 and the Stokes problem (see 2.2 The stokes problem and the operator A of Chapter 2 in Temam ^[23]), it follows that

Observe that $H = \rho u_{t}+\rho u \cdot \nabla u-\rho \nabla \Phi$, then clearly we have

where $c = c(N)$, and $N$ denotes the dimension.

By virtue of Proposition 2.1, we can get

From the Gagliardo-Nirenberg-Sobolev (G-N-S) inequality, we have

From $G = (2\mu+\nu) \text{div} u-p(\rho)$, we obtain

Combining (3.8)–(3.10), immediately we have

3.2.   The estimates for $\sup\limits_{\Omega_{T}}\rho(x, t)$, and $\inf\limits_{\Omega_{T}}\rho(x, t)$

Lemma 3.1. Suppose that $\rho_{0}\in L^{\infty}(\Omega)$ and $u_{0}\in H^{1}(\Omega)$, and assume that $\nabla^{2} \mathcal{P}u$ and $\nabla G$ is bounded, and $\rho \in L^{\infty}\big(0, T;L^{p_{1}}(\Omega)\big)$, where $p_{1} = \max\{p_{0}, 39, 5\gamma, 40\gamma-19 \}$. Then we could obtain that

Proof. From $\rho_{t}+\text{div}(\rho u) = 0$, we get the following relations

We can rewrite it in differential notation and integrate the equality above from $0$ to $t$, then we have

Using the equation $G = (2\mu+\nu)\text{div} u-p(\rho)$, one deduces that

Note that $G = \Delta ^{-1}\text{div}\Big((\rho u)_{t}+\text{div}(\rho u \otimes u)-\rho \nabla \Phi\Big)$. Now, considering the term with respect to $G$ of (3.12), we can infer that

Note that

Then, one obtains

where $A_0 = \Delta ^{-1}\text{div}(\rho u), a_0 = \Delta ^{-1}\text{div}(\rho_0 u_0), A_1 = \Delta ^{-1}\text{div}(\rho\nabla \Phi), A_2 = \Delta^{-1}\text{div}^{2}(\rho u\otimes u)-u \cdot \nabla \Delta ^{-1}\text{div}(\rho u)$, and $\text{div}^{2}$ is an operator defined by $\text{div}^{2}\; M = \partial_{ij}M_{ij}$ for a $3\times 3$ matrix $M = (M_{ij})$.

Therefore, we have

Using (3.14), one could obtain that

Hence, one yields that

From above inequality, (3.15) implies that

Therefore

Obviously, once one proves $\sup\limits_{t\in[0, T]}\mathcal{A}(t) < \infty$, we may obtain

Therefore, it's necessary to study the estimates of $\mathcal{A}(t)$.

3.3.   The estimates for $\mathcal{A}(t)$

First, observe $A_0 = \Delta ^{-1}\text{div}(\rho u), A_1 = \Delta ^{-1}\text{div}(\rho\nabla \Phi), A_2 = \Delta^{-1}\text{div}^{2}(\rho u\otimes u)-u \cdot \nabla \Delta ^{-1}\text{div}(\rho u)$, then we obtain

Meanwhile, we have

Using the Calderon-Zygmund theorem (Theorem 9.9 of ^[8]), we get

where $1 < p < \infty$, $c$ depends on $p$. For all $v\in W^{1, 4}(\Omega)$, one yields

Hence, we have

Applying the same method of estimating $\Vert A_0\Vert_{L^{\infty}(\Omega)}$ to estimate $\Vert A_1\Vert_{L^{\infty}(\Omega)}$, we obtain that

According to $\Delta \Phi = \rho-\overline{\rho}$ and the Calderon-Zygmung inequality ^[16], we have

Moreover, we estimate that

Similarly for $A_2$, we could obtain

Applying Hölder's inequality to $B_1$, we obtain

Similarly, by using the Calderon-Zygmung inequality and Hölder's inequality, one has

Hence, we get the estimate of $A_2$ by Young inequality,

Therefore, combining (3.16), (3.17) and (3.19), one could have

Observe Proposition 2.3 and that if $\nabla^{2} \mathcal{P}u$ and $\nabla G$ is bounded which we will prove in the Lemma 4.3 of the next section, then we could obtain that $\sup\limits_{t\in [0, T]}\mathcal{A}(t) < \infty$. □

4.   A priori bounds for $\nabla ^2 \mathcal{P}u$ and $\nabla G$

Lemma 4.1. Let $1 < r < 2$, $\rho \in L^{\infty}(0, T;L^{p_{2}}(\Omega)),$ and $p_{2}(r) = \max\{p_{0}, p_{1}, \frac{5r-2}{2-r}, \frac{r}{2-r}, 4\},$ then

Proof. Using (3.6) and (3.7), we obtain

where $1 < r < \infty$, $c = c(N)$ and $N$ is the dimension. For $1 < r < 2$, it follows that

According to $\Delta \Phi = \rho-\overline{\rho}$, we obtain

Hence, if $1 < r < 2$, we obtain

Hence, integrating (4.1) over $[0, T]$, we could get

□

Lemma 4.2. Let $\frac{6}{5} < r < 2$, and $p_{3}(r) = \max \big\{4\gamma, \frac{(2\gamma-1)r+1}{r-1}, \frac{6\gamma r}{6-5r}, \frac{3\gamma r}{3-r}, \frac{3}{2r-3}, \frac{2\gamma-1}{r-1}+2\gamma, \frac{3(2\gamma-1)}{2r-3}+2\gamma\big\}$. Here we assume $\rho_0\in L^{\infty}(0, T;L^{p_3}(\Omega))$. Then, one chould obtain that

where $\mathcal{C} = \mathcal{C}(\mu, \nu, R, \Vert u_{0}\Vert_{H^{1}(\Omega)}, \Vert\rho_{0}\Vert_{L^{2\gamma}(\Omega)}, \sup\limits_{t\in[0, T]}\Vert\rho\Vert_{L^{p_3}(\Omega)}).$

Proof. Multiply $u_{t}$ to (1.2)$_2$ and integrate over $\Omega_T$, then we could obtain that

Therefore, we have

Set

Noting that $\rho_{t}+\text{div}(\rho u) = 0$, one obtains that

Note that $G = (2\mu+\nu)\text{div} u-p(\rho)$, we deduce that

Further, one obtains

Hence, we have

Integrate $\mathcal{I}$ over $[0, T]$, then one could obtain

Observing that $\Delta \Phi = \rho-\bar{\rho}$, estimate the term $\mathcal{J}$, then one obtains

Hence, we obtain

Combining (4.5) and (4.6), (4.3) can be written as

In what follows, estimate $i_1-i_9$. First, based on Cauchy's inequality, we have

Similarly, we obtain

Observe that $\Vert\rho\Vert_{L^{2\gamma}(\Omega)}^{2\gamma}\lesssim \Vert\rho\Vert_{L^{\frac{6\gamma r}{6-5r}}(\Omega)}^{2 \gamma}$, and make use of the same method as $i_4$, then one obtains that

Note that $\Vert\text{div} u\Vert_{L^{q}}\leq \frac{1}{2\mu+\nu}(\Vert G\Vert_{L^{q}}+R||\rho||_{L^{\gamma q}}^{\gamma})$ on account of $G = (2\mu+\nu)\text{div} u-p(\rho)$. Make use of (3.11), Cauchy inequality and the interpolation inequality, then one yields

where $\frac{3}{2} < r < 2$. Similarly, we have

Combining ${i_1-i_9}$ with (4.7), we arrive at the conclusion of the lemma easily. □

Combining Lemma 4.1 with Lemma 4.2, let $\varepsilon$ be small sufficiently, then the following lemma is obtained naturally.

Lemma 4.3. Suppose that $\rho_0$ and $u_{0}$ satisfy the assumptions of Lemma 4.2. Assume $\rho\in L^{\infty}\big(0, T; L^{p_{4}}(\Omega)\big)$, then we obtain that

where $C = C(\mu, R, T, \Vert u_0\Vert_{H^{1}(\Omega)}, \Vert\rho_{0}\Vert_{L^{2\gamma}(\Omega)}, \sup\limits_{t\in[0, T]}\Vert\rho\Vert_{L^{p_{4}}(\Omega)}),$ and $p_{4}(r) = \max\{p_{0}(r), p_{1}(r), p_{2}(r), p_{3}(r)\}.$

Moreover, combining the results of the Lemma 4.3 with (3.20), we could obtain $\sup\limits_{0 < t < T}\mathcal{A}(t) < \infty$ easily, which is obvious that we prove that Lemma 3.1 completely.

5.   The estimate of $\Vert u\Vert_{L^{\infty}}$

In the following, we shall present the $L^{\infty}$ bound of $u$ by the Moser iteration ^[4,10] provided that the result of the Lemma 3.1 holds on. Furthermore, we could get the local $L^{\infty}$ estimate of $u$ by Gronwall inequality.

Lemma 5.1. Let all conditions satisfy the assumptions of Proposition 2.3 and Lemma 3.1, then we have

where the constant $C = C(T, \sup\limits_{\Omega_{T}}\rho(x, t), \inf\limits_{\Omega_{T}}\rho(x, t), \Vert u_{0}\Vert_{L^{\infty}(\Omega)}) > 0$.

Proof. Integrating the equality (2.1) on $(0, T)$, one could arrive at

Now, we consider the estimates of the right terms of (5.1).

where $c_1 = p(\hat{\rho})$ and $\hat{\rho} = \sup\limits_{\Omega_{T}}\rho(t, x)$. Similarly, we have

Note that $\Delta \Phi = \rho-\overline{\rho}$ and (2.4), then we have

Hence, one could get

where $\tilde{\rho} = \inf\limits_{\Omega_{T}}\rho(x, t)$. Observe

Hence, we could obtain that

Combining (5.2) and (5.3), we could get

where $C = C(T, M, \sup\limits_{\Omega_{T}}\rho, \inf\limits_{\Omega_{T}}\rho, \Vert u_0\Vert_{L^{\infty}(\Omega)})$.

Let $r = \frac{5}{3}, \ r^k = 2+\alpha_{k},$ where $\ k\geq 2$, then the inequality (5.3) is rearranged to be

where $C_1 = C_1(T, M, \sup\limits_{\Omega_{T}}\rho, \inf\limits_{\Omega_{T}}\rho, \Vert u_0\Vert_{L^{\infty}(\Omega)}) > 1$, and $C_2 = C_2(T, M, \sup\limits_{\Omega_{T}}\rho, \inf\limits_{\Omega_{T}}\rho, \Vert u_0\Vert_{L^{\infty}(\Omega)}) > 1$.

Make use of the inequality (5.5), then we obtain

Next, we will consider the following inequality

Denote $a = 2^{(r-1)+(r^2-1)+\cdots+(r^{k-1}-1)}$, then we have the inequality

Observe that $b = \sum\limits_{l = 1}^{k-1}r^{l-k-2} < +\infty$, $c = \sum\limits_{l = 1}^{k-1}(k-l)r^{l-k-2} < +\infty$, $d = (k-1)^{r^{-(k+1)}} < +\infty$. Moreover, we have $2^{\big((r-1)+\cdots+(r^{k-2}-1)\big)\cdot r^{-(k+1)}} < \infty$.

Obviously when $k$ goes to $\infty$, we could obtain that

Hence, we obtain the inequality

Combining the above result and Proposition 2.3, we could obtain that

where $C = C(T, \sup\limits_{\Omega_{T}}\rho, \inf\limits_{\Omega_{T}}\rho, \Vert u_0\Vert_{L^{\infty}(\Omega)})$. □

Finally, let us give the proof of Theorem 1.1.

Proof. Under the assumptions of Lemmas 4.1–4.3, we employ the result in Lemma 3.1, Proposition 2.3 and the a prior estimates in Lemma 5.1. Therefore, we could achieve the results of Theorem 1.1 easily. □

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 supported by the National Natural Science Foundation of China (42274166), the Fundamental Research Funds for the Central Universities (3132021195, 3132022202), Basic Scientific Research Project of The Educational Department of Liaoning Province (LJKMZ20220368).

Conflict of interest

The authors declare that they have no conflict of interest.