Global classical solution of the fractional Nernst-Planck-Poisson-Navier- Stokes system in $\mathbb{R}^{3}$

1.   Introduction

1.1.   The drift-diffusion system

In order to describe the charge transport in the semiconductor device, Roosbroeck ^[21] first proposed the drift-diffusion (DD) system. We briefly review some known results on the DD system. As early as the last century, Mock ^[14] considered the following system:

on the bounded domain $\Omega\subseteq\mathbb{R}^{N}(N\geqslant 1)$ with the Neumann boundary condition used to describe charge transport in semiconductor devices. Here, $n$ and $v$ represent the densities of electrons and holes. $\phi$ stands for the electrostatic potential. Fang and Ito ^[5] proved the global existence of weak solutions to system (1.1) with the Neumann boundary condition. Afterwards, Jüngel ^[10] obtained the asymptotic behavior of the global solution under the Neumann boundary condition. The global existence of the solution for the Cauchy problem of system (1.1) was studied by Kurokiba and Ogawa ^[13]. Then, Kamashima and Kobayashi ^[12] deduced the optimal dacay estimate by the weighted energy method and obtained an asymptotic result as $t\rightarrow \infty$.

Then, some scientists have focused on the dopant anomalous diffusion in a semiconductor. The anomalous diffusion is shown as the result of two competing phenomena: dopant trapping on the defects and enhanced diffusivity at the edges of the defect-rich region ^[3]. The classical Laplace operator cannot describe this anomalous diffusion phenomenon. Scholars have found that the fractional Laplace operator can well describe the process involving anomalous diffusion. Owaga and Yamamoto ^[16] investigated the fractional DD system

for $\alpha = \beta\in(\frac{1}{2}, 1)$. They proved the global existence and asymptotic behavior of the mild solution in $\Omega = \mathbb{R}^{N}$ for $N\geqslant 2$. Here, the fractional Laplace operator is defined by

where $\hat{f}$ denotes the Fourier transform of the function $f$. Then Granero-Belinchón ^[7] considered a fractional DD system with different diffusion orders $\alpha\neq\beta\in(0, 1)$, and obtained the global existence of the classical solution to system (1.2) and several decay estimates for the Lebesgue and Sobolev norms in $\Omega = \mathbb{R}^{N}$ for $N = 1, 2, 3$.

1.2.   The Nernst-Planck-Poisson-Navier-Stokes system

A similar phenomenon occurs for the ions in a solution. The Nernst-Planck-Poisson(NPP) system was originally conceived to describe the motion of ions in a solution in the context of electrochemistry. It should be emphasized that, although both the DD system and the NPP system share a common structure, their physical meanings are different. Ions in the NPP system are charged particles, electrons and holes in the DD system enjoy a duality between quantum waves and particles. In order to reveal the interaction between the movement of the macroscopic fluid and the transport of the microscopic charge, Rubinstein ^[17] proposed the classical Nernst-Planck-Poisson-Navier-Stokes(NPPNS) system

to show the electrokinetic effects consisting of the interplay between charges and the flow field. Here $n$ and $v$ represent the densities of negatively and positively charged particles, $u$ represents the velocity field of the fluid, and $P$ denotes the pressure function. System (1.3) describes an isothermal, incompressible, viscous Newtonian fluid of uniform and homogeneous composition of a high number of positively and negatively charged particles ranging from colloidal to nanosize. It is further assumed to be a dilute fluid so that the electromagnetic forces can be neglected. Readers can refer to ^[1] and the references therein for more discussion on the physical background of the system (1.3).

In the past few decades, a number of scientists have developed great interest in and conducted indepth research on the NPPNS system (1.3). Based on Kato's semigroup framework, Jerome ^[8] established the local existence and uniqueness of strong solutions to system (1.3) in $\mathbb{R}^{N}(N\geqslant 2)$. Jerome and Sacco ^[9] obtained the global existence of weak solutions under the mixed Dirichlet boundary condition. By using the energy inequalities and Schauder's fixed point theorem, Schmuck ^[18] obtained the global existence of weak solutions to system (1.3) in a bounded domain $\Omega\subseteq\mathbb{R}^{N}$($N$ = 2 or 3) with the Neumann boundary condition. Deng, Zhao and Cui ^[23,24] obtained the global well-posedness of system (1.3) with small initial data in negative-order Besov space and critical Lebesgue spaces in $\mathbb{R}^{N}(N\geqslant 2)$. Zhao, Zhang, and Liu ^[25] showed the global well-posedness of system (1.3) in the critical Besov space with a large vertical velocity component in $\mathbb{R}^{3}$. Fan, Li, and Nakamura ^[4] also proved some regularity criteria for the strong solutions to the Cauchy problem of system (1.3) in $\mathbb{R}^{3}$. Zhang and Yin ^[22] proved the global existence of the classical solution to the Cauchy problem of system (1.3) in $\mathbb{R}^{2}$ and established the $L^{2}$ decay estimates by using the Fourier splitting method. Recently, Gong, Wang, and Zhang ^[6] proved the existence of suitable weak solutions to system (1.3) in $\mathbb{R}^{3}$. Based on the spectral analysis and the energy method, Tong and Tan ^[19] obtained the lower bound and upper bound decay rates of the solution to a generalized NPPNS system in $\mathbb{R}^{3}$.

As for the classical chemotaxis-Navier-Stokes system proposed by Tuval et al.^[20], Chae, Kang and Lee ^[2] established the global existence of smooth solutions in $\mathbb{R}^{N}$($N$ = 2 or 3). Afterwards, Zhu, Liu and Zhou ^[26] considered the fractional chemotaxis-Navier-Stokes system in $\mathbb{R}^{3}$ and established the global existence and uniqueness of classical solution with small initial data. Research shows that macroscopic diffusion has been used to describe the random walk of particles; however, Ley's flights can effectively describe the anomalous diffusion phenomenon. Instead of the classical Laplace operator, the fractional Laplace operator can well describe this anomalous diffusion phenomenon. Considering the anomalous diffusion phenomenon of the ions in the solution, we couple the NPPNS system with the fractional Laplace operator and consider the following fractional NPPNS system:

where $s\in(0, 1)$. The aim of this paper is to obtain the global existence of a classical solution to system (1.4) under the condition of small initial data. Firstly, we obtain a priori estimates of the solution to system (1.4) through the energy method. Then we construct a solution sequence by iterative methods and prove the local existence of the solution sequence of the iterative system (3.25). It should be emphasized that different from ^[26], firstly, the physical background is different. The chemotaxis-Navier-Stokes system was proposed to study the interaction between cells and fluids. However, the NPPNS system was originally conceived to reveal the interaction between the movement of the macroscopic fluid and the transport of the microscopic charge. Secondly, due in the difference of the external force in the Navier-Stokes equation, the iterative solution sequence of the approximate problem is different from ^[26] in the process of proving the existence of solutions to system (1.4). Besides, we also prove that the solution sequence of (3.25) is a Cauchy sequence. Thus, according to the convergence of the solution sequence, we can obtain the local existence and uniqueness of the classical solution to the system (1.4). Finally, we prove the global existence of the classical solution to system (1.4) by combining the priori estimates with the local existence. Furthermore, since the global existence of the classical solution for the three-dimensional Navier-Stokes equation is still open, compared with the condition of large initial data in ^[22], we study the global existence and uniqueness of the classical solution to system (1.4) under the condition of small initial data in $\mathbb{R}^{3}$.

The main theorem of this paper states as follows:

Theorem 1.1. Assume that $s\in(\frac{1}{2}, 1)$, $(n_{0}, v_{0}, u_{0})\in (L^{1}\cap H^{m})\times(L^{1}\cap H^{m})\times H^{m}(m\geqslant 3)$, $n_{0}, v_{0}\geqslant 0$. If there exists a small enough constant $\varepsilon_{0} > 0$ such that $\|n_{0}\|_{H^{3}}+\|v_{0}\|_{H^{3}}+\|u_{0}\|_{H^{3}}\leqslant\varepsilon_{0}$, then system (1.4) possesses a unique global solution $(n, v, \phi, u)$ satisfying that for all $t\geqslant 0$,

where $C$ is a time-independent positive constant.

The rest of this paper is organized as follows: In Section 2, we state some useful lemmas that will be used throughout the paper. In Section 3, we first give a priori estimates and prove the local existence of classical solutions to system (1.4) by iterative methods; then, combining the local existence and a priori estimates, we can obtain the global existence of classical solutions to system (1.4).

2.   Preliminary

In this section, we give the following notations: $C > 0$ is a time-independent constant. We reduce $L^{q}(\mathbb{R}^{3})$ and $H^{s}(\mathbb{R}^{3})$ to $L^{q}$ and $H^{s}$, respectively. Moreover, we use $a\lesssim b$ to denote $a\leqslant Cb$. Next, we will give some useful lemmas.

Lemma 2.1. (^[15]) Assume that $0\leqslant k, m\leqslant l$ and $1\leqslant p, q, r\leqslant\infty$. It follows that

where $\theta\in[0, 1]$, and $k, m, l$ satisfy

Especially, when $p = \infty$, we require that $\theta\in(0, 1)$, $k\leqslant m+1$, and $l\geqslant m+2$.

Lemma 2.2. (^[11]) Assume that $0\leqslant k < \infty$, then we have

where $p, p_{2}, p_{3}\in(1, \infty)$, $p_{1}, p_{4}\in(1, \infty]$ with $\frac{1}{p} = \frac{1}{p_{1}}+\frac{1}{p_{2}} = \frac{1}{p_{3}}+\frac{1}{p_{4}}$.

3.   Proof of Theorem 1.1

3.1.   A priori estimates

Lemma 3.1. Assume that $s\in(\frac{1}{2}, 1)$, $(n_{0}, v_{0}, u_{0})\in (L^{1}\cap H^{m})\times(L^{1}\cap H^{m})\times H^{m}(m\geqslant 3)$, $n_{0}, v_{0}\geqslant 0$. Suppose that $(n, v, \phi, u)$ is a solution of (1.4). Then $n, v\in L^{1}$ and $n, v\geqslant 0$ hold for any $t\geqslant 0$. Furthermore, there exists a small enough constant $\varepsilon_{0}$ such that if $\|n\|_{H^{3}}+\|v\|_{H^{3}}+\|u\|_{H^{3}}\leqslant\varepsilon_{0}$,

holds for any $t\geqslant 0$, where $C_{1}$ is a time-independent constant.

Proof. First, integrating over $\mathbb{R}^{3}\times[0, t]$ for the first and second equation of (1.4), we conclude that $n, v\in L^{1}$. Then, we briefly prove that $n$ is nonnegative (the situation of $v$ is similar to $n$). For this, we introduce the following auxiliary problem:

Here $n_{+}$ = max$\{n, 0\}$, $n_{-}$ = max$\{-n, 0\}$ and $n = n_{+}-n_{-}$. Multiplying the first equation of (3.2) by $n_{-}$ and integrating by parts, we have

Therefore, $\|n_{-}(t)\|_{L^{2}}^{2}\leqslant\|n_{-}(0)\|_{L^{2}}^{2}$. Since $n_{0}$ is nonnegative, $n$ is nonnegative. Similarly, $v$ is also nonnegative. We conclude that system (3.2) is equivalent to system (1.4), the nonnegativity of $n$ and $v$ is proved. Finally, we will prove (3.1).

$(i)$ The estimate of $n$. Applying $\partial^{\alpha}(0\leqslant\alpha\leqslant m)$ to the first equation of (1.4) and multiplying by $\partial^{\alpha}n$, then integrating over $\mathbb{R}^{3}$ by parts, we get

As for the term $I_{1}$, for $\alpha = 0$, recalling that $\nabla\cdot u = 0$, we get

For $\alpha = 1$, recalling that $\nabla\cdot u = 0$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we obtain

For $2\leqslant\alpha\leqslant m$, we have

where $\theta = \frac{2\alpha-2l+2s-1}{2\alpha+4s-3}\in(0, 1).$

In a word,

As for the term $I_{2}$, for $\alpha = 0$, by Lemma 2.1 and Hölder's inequality, we have

For $\alpha = 1$, it follows that

For $2\leqslant\alpha\leqslant m$, we deduce

where $\theta = \frac{2\alpha-2l+2s-1}{2\alpha+2s-3}\in(0, 1).$

In a word,

As for the term $I_{3}$, for $\alpha = 0$, by Lemma 2.1 and Hölder's inequality, we get

For $1\leqslant\alpha\leqslant m$, we obtain that

In a word,

Inserting (3.8), (3.12), and (3.15) into (3.4) and summing up with respect to $\alpha$ from 0 to $m$, we obtain

$(ii)$ The estimate of $v$. The estimate of $v$ is similar to the estimate of $n$. We get

$(iii)$ The estimate of $u$. Applying $\partial^{\alpha}(0\leqslant\alpha\leqslant m)$ to the fourth equation of (1.4) and multiplying by $\partial^{\alpha}u$, then integrating over $\mathbb{R}^{3}$ by parts, we get

For $I_{4}$, by Lemma 2.2, Hölder's inequality, and Young's inequality, we have

As for the term $I_{5}$, for $\alpha = 0$, we get

For $1\leqslant\alpha\leqslant m$, it follows that

In a word,

Therefore, inserting (3.19) and (3.22) into (3.18) and summing up with respect to $\alpha$ from 0 to $m$, we get

Eventually, combining (3.16) and (3.17) with (3.23), we obtain

Then, integrating (3.24) from 0 to $t$ and according to the fifth equation of (1.4) immediately implies (3.1). □

3.2.   The local existence and uniqueness of the solution

Next, we will study the local existence and uniqueness of the solution to system (1.4). We construct the solution sequence $(n^{j}, v^{j}, \phi^{j}, u^{j})_{j\geqslant 0}$ by iteratively solving the Cauchy problem:

where $(n^{j+1}, v^{j+1}, \phi^{j+1}, u^{j+1})|_{t = 0}$ = ($n_{0}, v_{0}, \phi_{0}, u_{0}$) for $j\geqslant 0$ and $\Delta\phi_{0} = n_{0}-v_{0}$. It should be emphasized that the iterative sequence is different from ^[26] due to the difference in the external force in the Navier-Stokes equation. We first set $(n^{0}, v^{0}, u^{0}) = (0, 0, 0)$. Then we solve (3.25) with the initial data to get $(n^{1}, v^{1}, \phi^{1}, u^{1})$, respectively. Similarly, we can define $(n^{j}, v^{j}, \phi^{j}, u^{j})$ iteratively.

Lemma 3.2. Assume that $s\in(\frac{1}{2}, 1)$, $(n_{0}, v_{0}, u_{0})\in (L^{1}\cap H^{m})\times(L^{1}\cap H^{m})\times H^{m}(m\geqslant 3)$, $n_{0}, v_{0}\geqslant 0$. Then, there exists a constant $T_{1} > 0$ such that system (1.4) possesses a unique classical solution $(n, v, \phi, u)$ satisfying

Proof. In order to prove the conclusion, we divide the proof into several steps.

Step 1. (Uniform boundedness)

$(i)$ The estimate of $n^{j+1}$. Applying $\partial^{\alpha}(0\leqslant\alpha\leqslant m)$ to the first equation of (3.25), multiplying by $\partial^{\alpha}n^{j+1}$ and integrating over $\mathbb{R}^{3}$ by parts, we get

First, we deal with the term $J_{1}$. For $\alpha = 0$, recalling that $\nabla\cdot u^{j} = 0$, we get

For $\alpha = 1$, recalling that $\nabla\cdot u^{j} = 0$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we have

For $2\leqslant\alpha\leqslant m$, recalling that $\nabla\cdot u^{j} = 0$ and employing Lemma 2.1, Hölder's inequality, and Young's inequality, we obtain

In a word,

As for $J_{2}$, for $\alpha = 0$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we get

For $\alpha = 1$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we have

For $2\leqslant\alpha\leqslant m$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we obtain

In a word,

For $J_{3}$, by Lemmas 2.1 and 2.2, Hölder's inequality and Young's inequality, we get

Therefore, by inserting (3.30), (3.34), and (3.35) into (3.26) and summing up with respect to $\alpha$ from 0 to $m$, we get

$(ii)$ The estimate of $v^{j+1}$. The estimate of $v^{j+1}$ is similar to the estimate of $n^{j+1}$; we get

$(iii)$ The estimate of $u^{j+1}$. Applying $\partial^{\alpha}(0\leqslant\alpha\leqslant m)$ to the fourth equation of (3.25), multiplying by $\partial^{\alpha}u^{j+1}$ and integrating over $\mathbb{R}^{3}$ by parts, we get

First, we deal with the term $J_{4}$. For $\alpha = 0$, recalling that $\nabla\cdot u^{j} = 0$, we get

For $1\leqslant\alpha\leqslant m$, by Lemma 2.2, Hölder's inequality, and Young's inequality, we have

In a word,

As for $J_{5}$, for $\alpha = 0$, recalling that $\nabla\cdot u^{j} = 0$, we have

For $1\leqslant\alpha\leqslant m$, by Lemma 2.2, Hölder's inequality, and Young's inequality, we obtain

In a word,

Thus, inserting (3.41) and (3.44) into (3.38) and summing up with respect to $\alpha$ from 0 to $m$, we get

Next, we show that there exists a constant $M > 0$ such that for any $j$, the following inequality holds in a small time interval $[0, T_{1}]$ ($T_{1}$ will be specified later):

Here $M$ satisfies $M\geqslant 4(\|n_{0}\|_{H^{m}}^{2}+\|v_{0}\|_{H^{m}}^{2}+\|u_{0}\|_{H^{m}}^{2})$.

We prove (3.46) by mathematical induction. Suppose (3.46) holds for $j\leqslant i-1$. Combining (3.36), (3.37) and (3.45) to obtain

We choose $T_{1}$ such that $CMT_{1}\leqslant\frac{1}{4}$ and $e^{CMT_{1}}\leqslant 2$. Then, according to Gronwall's inequality, we have (3.46). Thus, we get $(n^{j+1}, v^{j+1}, \phi^{j+1}, u^{j+1})\in$ $L^{\infty}(0, T_{1}$$;H^{m}\times H^{m}\times H^{m+2}\times H^{m})$ and $(\partial^{s}n^{j+1}, \partial^{s}v^{j+1}, \partial^{s}\phi^{j+1}, \partial u^{j+1})$$\in L^{2}(0, T_{1};$$H^{m}\times H^{m}\times H^{m+2}\times H^{m})$ for $T_{1} > 0$.

Step 2. (Convergence)

The estimate in this part is similar to that in the previous part. Since both $(n^{j}, v^{j}, \phi^{j}, u^{j})$ and $(n^{j+1}, v^{j+1}, \phi^{j+1}, u^{j+1})$ satisfy (3.25), we get the following equations:

$(i)$ The estimate of $n^{j+1}-n^{j}$. Applying $\partial^{\alpha}(0\leqslant\alpha\leqslant m-1)$ to the first equation of (3.47), multiplying by $\partial^{\alpha}(n^{j+1}-n^{j})$, and integrating over $\mathbb{R}^{3}$ by parts, we get

First, we deal with $R_{1}$. For $\alpha = 0$, recalling that $\nabla\cdot u^{j} = 0$, we have

For $\alpha = 1$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we get

For $2\leqslant\alpha\leqslant m-1$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we obtain

In a word,

For $R_{2}$, by Lemmas 2.1 and 2.2, Hölder's inequality, and Young's inequality, we get

Then we deal with the term $R_{3}$. For $\alpha = 0$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we have

For $\alpha = 1$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we obtain

For $2\leqslant\alpha\leqslant m-1$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we get

In a word,

As for $R_{4}$, for $\alpha = 0$, we obtain

For $1\leqslant\alpha\leqslant m-1$, it follows that

In a word,

For $R_{5}$, by Lemmas 2.1 and 2.2, Hölder's inequality, and Young's inequality, we have

For $R_{6}$, by Lemmas 2.1 and 2.2, Hölder's inequality, and Young's inequality, we obtain

Therefore, inserting (3.52), (3.53), (3.57), and (3.60)–(3.62) into (3.48) and summing up with respect to $\alpha$ from 0 to $m-1$, we have

$(ii)$ The estimate of $v^{j+1}-v^{j}$. The estimate of $v^{j+1}-v^{j}$ is similar to the estimate of $n^{j+1}-n^{j}$. We get

$(iii)$ The estimate of $u^{j+1}-u^{j}$. Applying $\partial^{\alpha}(0\leqslant\alpha\leqslant m-1)$ to the fourth equation of (3.47), multiplying by $\partial^{\alpha}(u^{j+1}-u^{j})$ and integrating over $\mathbb{R}^{3}$ by parts, we get

First, we deal with the term $R_{7}$. For $\alpha = 0$, recalling that $\nabla\cdot u^{j} = 0$, we have

For $1\leqslant\alpha\leqslant m-1$, by Lemma 2.2, Hölder's inequality, and Young's inequality, we get

In a word,

For $R_{8}$, by Lemmas 2.1 and 2.2, Hölder's inequality, and Young's inequality, we obtain

We next deal with the term $R_{9}$. For $\alpha = 0$, we have

For $1\leqslant\alpha\leqslant m-1$, it follows that

In a word,

As for $R_{10}$, for $\alpha = 0$, we have

For $1\leqslant\alpha\leqslant m-1$, we obtain

In a word,

Thus, inserting (3.68), (3.69), (3.72), and (3.75) into (3.65) and summing up with respect to $\alpha$ from $0$ to $m-1$, we get

Combining (3.63), (3.64), and (3.76), we obtain

where $C$ depends on the $H^{m}\times H^{m}\times H^{m}\times H^{m}\times H^{m}$ norm of $(n^{j}, n^{j-1}, v^{j}, v^{j-1}, u^{j})$.

Through Gronwall's inequality, we obtain

Therefore, we conclude that $(n^{j}, v^{j}, \phi^{j}, u^{j})$ is a Cauchy sequence in the Banach space $C(0, T_{1}; H^{m-1}\times H^{m-1}\times H^{m+1}\times H^{m-1})$ for small $T_{1} > 0$. So, we can take the limit to get $(n, v, \phi, u)$, which is the solution of system (1.4) and satisfies $(n, v, \phi, u)\in L^{\infty}(0, T;H^{m}\times H^{m}\times H^{m+2}\times H^{m}), (\partial^{s}n, \partial^{s}v, \partial^{s}\phi, \partial u)\in L^{2}(0, T;H^{m}\times H^{m}\times H^{m+2}\times H^{m})$.

Step 3. (Uniqueness)

In order to show the uniqueness of the solution, we assume that there are two solutions $(n_{1}(x, t), v_{1}(x, t), \phi_{1}(x, t), u_{1}(x, t))$ and $(n_{2}(x, t), v_{2}(x, t), \phi_{2}(x, t), u_{2}(x, t))$ of (1.4) with the same initial data in the time interval $[0, T_{1}]$, where $T_{1}$ is any time before the maximal time of existence. Then $(n_{1}(x, t)$$-n_{2}(x, t), v_{1}(x, t)$$-v_{2}(x, t), \phi_{1}(x, t)$$-\phi_{2}(x, t), u_{1}(x, t)$$-u_{2}(x, t))$ satisfies

Multiplying $n_{1}-n_{2}$ to both sides of the first equation of (3.79), then integrating over $R^{3}$ by parts, by Hölder's inequality, we have

Therefore,

Similarly,

Multiplying $u_{1}-u_{2}$ to both sides of the fourth equation of (3.79), then integrating over $R^{3}$ by parts, by Hölder's inequality, we have

Therefore,

Summing (3.81), (3.82), and (3.84), we have

Since $(n_{1}(x, t)-n_{2}(x, t), v_{1}(x, t)-v_{2}(x, t), u_{1}(x, t)-u_{2}(x, t))|_{t = 0} = (0, 0, 0)$, according to Gronwall's inequality, for $0\leqslant t\leqslant T_{1}$,

Therefore, the uniqueness of local classical solutions is proved. □

3.3.   The global existence of the solution

First, we prove that the solution of (3.25) satisfies the following lemma under the condition of small initial data.

Lemma 3.3. Assume that $s\in(\frac{1}{2}, 1)$, $(n_{0}, v_{0}, u_{0})\in (L^{1}\cap H^{3})\times(L^{1}\cap H^{3})\times H^{3}$, $n_{0}, v_{0}\geqslant 0$. If there is a small constant $\varepsilon_{1} > 0$ such that $\|n_{0}\|_{H^{3}}+\|v_{0}\|_{H^{3}}+\|u_{0}\|_{H^{3}}\leqslant\varepsilon_{1}$, then there exist small constants $\varepsilon_{2} > 0$ and $T_{2} > 0$ such that the solution of (3.25) satisfies

Proof. We will prove the conclusion by induction. The assumption $(n^{0}, v^{0}, u^{0}) = (0, 0, 0)$ implies that (3.86) holds for $j = 0$. Assuming that (3.86) is true for any $j > 0$, we will prove that the conclusion is also true for $j+1$.

$(i)$ The estimate of $n^{j+1}$. Applying $\partial^{\alpha}(0\leqslant\alpha\leqslant 3)$ to the first equation of (3.25), multiplying by $\partial^{\alpha}n^{j+1}$ and integrating over $\mathbb{R}^{3}$ by parts, we get

First, we deal with the term $Q_{1}$. For $\alpha = 0$, noting that $\nabla\cdot u^{j} = 0$, we get

For $\alpha = 1$, recalling that $\nabla\cdot u^{j} = 0$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we have

For $2\leqslant\alpha\leqslant 3$, recalling that $\nabla\cdot u^{j} = 0$, by Lemma 2.1, Hölder's inequality, and Young's inequality, it follows that

where $\theta = \frac{4s+2l-5}{2\alpha}\in(0, 1)$.

In a word,

As for $Q_{2}$, for $\alpha = 0$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we obtain

For $\alpha = 1$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we have

For $2\leqslant\alpha\leqslant 3$, it follows that

where $\theta = \frac{4s+2l-5}{2\alpha}\in(0, 1)$.

In a word,

We next deal with the term $Q_{3}$. For $\alpha = 0$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we obtain

For $1\leqslant\alpha\leqslant 3$, we obtain

In a word,

Thus, inserting (3.91), (3.95), and (3.98) into (3.87) and summing up with respect to $\alpha$ from 0 to 3, we have

$(ii)$ The estimate of $v^{j+1}$. The estimate of $v^{j+1}$ is similar to the estimate of $n^{j+1}$; we have

$(iii)$ The estimate of $u^{j+1}$. Applying $\partial^{\alpha}(0\leqslant\alpha\leqslant 3)$ to the fourth equation of (3.25), multiplying by $\partial^{\alpha}u^{j+1}$ and integrating over $\mathbb{R}^{3}$ by parts, we obtain

We first deal with $Q_{4}$. For $\alpha = 0$, recalling that $\nabla\cdot u^{j} = 0$, we have

For $1\leqslant\alpha\leqslant 3$, by Lemma 2.1, Hölder's inequality, and Young's inequality, we have

In a word,

As for $Q_{5}$, for $\alpha = 0$, noting that $\nabla\cdot u^{j} = 0$, we get

For $1\leqslant\alpha\leqslant 3$, by Lemma 2.2, Hölder's inequality, and Young's inequality, we obtain

In a word,

Thus, inserting (3.104) and (3.107) into (3.101) and summing up with respect to $\alpha$ from 0 to 3, we have

Summing (3.99), (3.100), and (3.108), we obtain

Integrating (3.109) from 0 to $t$, we conclude that for any $t\in[0, T_{2}]$,

Taking properly $\varepsilon_{1}$, $\varepsilon_{2}$, $T_{2}$ such that $(\|n^{j+1}\|_{H^{3}}^{2}+\|v^{j+1}\|_{H^{3}}^{2}+\|u^{j+1}\|_{H^{3}}^{2}) +C\int_{0}^{t}(\|\partial^{s}n^{j+1}\|_{H^{3}}^{2}+\|\partial^{3}v^{j+1}\|_{H^{3}}^{2}+\|\partial u^{j+1}\|_{H^{3}}^{2}) \text{d}\tau \leqslant\varepsilon_{2}^{2}$, we can conclude that (3.86) holds for any $j\geqslant 0$. □

Next, we prove the main theorem.

Proof of Theorem 1.1. Let $T^{\ast}$ = min$\{T_{1}, T_{2}\}$, where $T_{1}$ and $T_{2}$ are given in Lemmas 3.2 and 3.3. According to Lemmas 3.2 and 3.3, we obtain that if $\|n_{0}\|_{H^{3}}+\|v_{0}\|_{H^{3}}+\|u_{0}\|_{H^{3}}\leqslant\varepsilon_{1}$, then

Next, we prove $T^{\ast} = \infty$ by contradiction. Let $M_{1}$ = min$\{\varepsilon_{0}, \varepsilon_{1}, \varepsilon_{2}\}$. Assume that $\|n_{0}\|_{H^{3}}+\|v_{0}\|_{H^{3}}+\|u_{0}\|_{H^{3}}\leqslant\frac{M_{1}}{2\sqrt{1+C_{1}}}$, where $C_{1}$ is given in Lemma 3.1. We define $T: = sup\{t:\sup\limits_{0\leqslant s\leqslant t}(\|n\|_{H^{3}}+\|v\|_{H^{3}}+\|u\|_{H^{3}}\leqslant M_{1})\}$ as the lifespan of solutions to (1.4). Because

recalling Lemma 3.2, we find that $T > 0$. If $T$ is finite, according to the definition of $T$, we obtain that

Besides, by Lemma 3.1, we know that

which is a contradiction to $\sup\limits_{0\leqslant s\leqslant T}(\|n\|_{H^{3}}+\|v\|_{H^{3}}+\|u\|_{H^{3}}) = M_{1}$. Thus, $\|n\|_{H^{3}}+\|v\|_{H^{3}}+\|u\|_{H^{3}}\leqslant\varepsilon_{2}$ for any $t > 0$. Thus, the global existence and uniqueness of classical solutions to (1.4) have been obtained. □

Author contributions

Zihang Cai: writing-original draft, writing-review and editing; Chao Jiang: funding acquisition, writing-review and editing, supervision; Yuzhu Lei: writing-review and editing, supervision; Zuhan Liu: methodology, writing-review and editing, supervision. All authors have read and approved the final version of the manuscript for publication.

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 is supported by the National Natural Science Foundation of China (11771380) and the Innovation Projects of Jiangsu Province, China (KYCX23-3498).

Conflict of interest

The authors declare that they have no conflicts of interest.