1. Introduction

The compressible Navier–Stokes–Poisson equations are used to describe the motion of charged particles (i.e. the electrons and the holes or, the electrons and the ions) under the influence of the self-consistent electrostatic potential force arising from semiconductors and plasmas. In this model, the heavy holes and ions are assumed to be immobile and uniformly distributed in space, providing as a background of positive charge. The light electrons are modelled as a charged compressible fluid moving against the ionic forces. Neglecting magnetic effect and heat-conductive effect, the governing dynamics of the electron fluid are given by the following viscous isentropic compressible Navier–Stokes–Poisson equations (see [Reference Degond7, Reference Jüngel21, Reference Sitnko and Malnev35]):

(1.1)\begin{equation} \left\{\begin{aligned} & \partial_t n+\mathrm{div}(n u)=0,\\ & m_e[\partial_t(n u)+\mathrm{div}(n u\otimes u)] +\nabla p(n)=\mathrm{div}\,S(u)+\rho\nabla\phi,\\ & \lambda^2\Delta\phi=n-N, \end{aligned}\right.\end{equation}

for $(t,\,x)\in [0,\,\infty )\times \mathbb {R}^d(d\geq 1)$. The unknown variables $n,\, u,\,\phi$ are the electron density, the velocity and the electrostatic potential, $p(n)$ is the pressure function, usually given by $p(\rho )=A\rho ^{\gamma }$ with the constants $A>0$ and $\gamma \geq 1$. In this work, we assume that $p(n)$ is smooth and strictly increasing. $S(u)$ denotes the viscous stress tensor with the form

\[ \mathrm{div}\,S(u)=\mu'\Delta u+\nu'\nabla \mathrm{div}\,u, \]

here $\mu '$ and $\nu '$ are viscosity coefficients satisfying $\mu '>0$ and $\mu '+\nu '>0$ for the sake of simplicity. The constant $m_e$ is the ratio of the electron/ions mass, and $\lambda >0$ is the Debye length. $N$ stands for a given (constant) density of positively charged ions. Moreover, we also mention that, without viscous terms, (1.1) becomes the compressible isentropic Euler–Poisson system, which is another model describing the motion of charged particles. Finally, if the electrostatic potential $\phi$ is neglected, the compressible Navier–Stokes–Poisson equations (1.1) is then reduced to the classical compressible Navier–Stokes equations.

Recently, some important progress has been made for the compressible Navier–Stokes–Poisson system. Here we only refer to some results about the isentropic compressible Navier–Stokes–Poisson system. The local and/or global existence of renormalized weak solutions to the Cauchy problem of the multi-dimensional compressible Navier–Stokes–Poisson system are proved in [Reference Ducomet14, Reference Zhang and Tan43]. The existence of non-trivial stationary solutions with compact support and their stability related to a free-boundary value problem for the three-dimensional Navier–Stokes–Poisson system are discussed in [Reference Ducomet15]. Some nonexistence result of global weak solutions is obtained in [Reference Chae5]. Large-time behaviour of the spherically symmetric Navier–Stokes–Poisson system with degenerate viscosity coefficients and with vacuum in $\mathbb {R}^3$ is shown in [Reference Zhang and Fang42]. The global existence of spherically symmetric weak solutions, and the regularity and long-time behaviour of global solution for free boundary value problem to three-dimensional spherically symmetric compressible Navier–Stokes–Poisson equations are shown in [Reference Kong and Li25]. The linear and nonlinear dynamical stability for the Lane–Emden solutions to the compressible Navier–Stokes–Poisson system is studied in [Reference Jang and Tice20]. The global strong solutions of the initial value problem for the multi-dimensional compressible Navier–Stokes–Poisson system with the strictly positive background profile in Besov spaces are investigated in [Reference Hao and Li19]. The global existence and $L^2$-decay rate of the smooth solution of the initial value problem for the compressible Navier–Stokes–Poisson system in $\mathbb {R}^3$ are proved in [Reference Li, Matsumura and Zhang27]. The pointwise estimates of the smooth solutions for the three-dimensional isentropic compressible Navier–Stokes–Poisson equation are obtained in [Reference Wang and Wu38]. The asymptotic stability of the nonlinear wave such as the rarefaction wave, the viscous shock wave and the stationary wave of the one-dimensional compressible Navier–Stokes–Poisson equation is studied in [Reference Duan and Liu12, Reference Duan, Liu and Zhang13, Reference Jiang, Lai, Yin and Zhu23, Reference Li and Zhu28, Reference Wang, Zhang and Zhang39].

Moreover, the zero-electron-mass limit $m_e \to 0$ and the quasi-neutral limit $\lambda \to 0$ of the compressible Navier–Stokes–Poisson equations are the important problems in the theory of the compressible fluid of semiconductors and plasmas. Li and Liao [Reference Li and Liao26] showed the existence and zero-electron-mass limit of strong solutions to the stationary compressible Navier–Stokes–Poisson equation with large external force. Donatelli et al. [Reference Donatelli, Feireisl and Novotný10] discussed the vanishing electron-mass limit of weak solution for the plasma hydrodynamics in three-dimensional unbounded domain. Li et al. [Reference Li, Zhou and Liao29] investigated zero-electron-mass limit of the two-dimensional compressible Navier–Stokes–Poisson equations over bounded domain. The quasineutral limit of weak solution and smooth solutions of the compressible unipolar Navier–Stokes–Poisson system was studied in [Reference Donatelli and Marcati11, Reference Ju, Li and Li22, Reference Wang and Jiang37]. We also mention that many authors discussed the zero-electron-mass limit in the Euler–Poisson system for both well- and ill-prepared initial data, and we can refer to [Reference Ali and Chen3, Reference Ali, Chen, Jüngel and Peng4, Reference Xu and Yong40, Reference Xu and Zhang41] and some references therein. To our knowledge, there were no results on the zero-electron-mass limit of the classical local solutions to compressible Navier–Stokes–Poisson equations (1.1). The goal of this work is to fill in the void and study the zero-electron-mass limit of the classical solutions to compressible Navier–Stokes–Poisson equations (1.1) with well- and ill-prepared initial data in $\mathbb {R}^d(d\geq 1)$.

To study the zero-electron-mass limit of the Navier–Stokes–Poisson equations (1.1), we denote $\varepsilon ^2 = m_e$, and we assume $\mu '=\varepsilon ^2\mu,\, \ \nu '=\varepsilon ^2\nu$. Moreover, let us introduce the electrostatic field $E=\nabla \phi$ and define the enthalpy $h=h(n)$ by $h'(n) = p'(n) / n$ and $h(1)=0$. Then the viscous isentropic compressible Navier–Stokes–Poisson system (1.1) can be written as:

(1.2)\begin{equation} \left\{ \begin{aligned} & \partial_t n+\mathrm{div}(n u)=0,\\ & \varepsilon^2 ( \partial_t + u \cdot \nabla) u +\nabla h(n) = \displaystyle \frac{\varepsilon^2}{n} (\mu\Delta u+\nu\nabla \mathrm{div}\,u) + E,\\ & \displaystyle\lambda^2 E = \nabla\Delta^{{-}1} (n-N) := K * (n - N ), \quad K(x) = \frac{1}{{\rm d} \, \omega_{d-1}} \frac{x}{|x|^d}, \end{aligned}\right. \end{equation}

where $\omega _{d-1}$ is the volume of the unit ball in $\mathbb {R} ^d$. From the analysis of [Reference Ali and Chen3], the operator $\Delta ^{-1}$ represents symbolically the fundamental solution of the Poisson equation, and $\Delta ^{-1}f$ might not be well defined for not good enough functions $f$ when the problem is considered on the whole space. However, $\nabla \Delta ^{-1} f$ is well defined for $f \in H^s( \mathbb {R} ^d)$ as long as $s>0$. Since we will prove the zero-electron-mass limit of the isentropic compressible Navier–Stokes–Poisson equations (1.2) with well- and ill-prepared initial data, we now supply the system (1.2) with the following initial data:

(1.3)\begin{equation} (n, u)({\cdot}, 0)= (n^\varepsilon_I, u^\varepsilon_I ). \end{equation}

Moreover, for smooth solutions, the initial value problem (1.2)–(1.3) is equivalent to the following hyperbolic–parabolic type system:

(1.4)\begin{equation} \left\{ \begin{aligned} & ( \partial_t + u \cdot \nabla) h + p'(n) \mathrm{div} \, u =0,\\ & \varepsilon^2 ( \partial_t + u \cdot \nabla) u +\nabla h = \displaystyle \frac{\varepsilon^2}{n(h)} (\mu\Delta u+\nu\nabla \mathrm{div}\,u ) + E,\\ & \lambda^2 E = K * (n - N ), \end{aligned}\right. \end{equation}

with initial conditions

(1.5)\begin{equation} (h, u) ({\cdot}, 0)= ( h^\varepsilon_I , u^\varepsilon_I ) , \quad h^\varepsilon_I = h(n_I^\varepsilon) . \end{equation}

Here $n(h)$ is a smooth reversible function of $h(n)$, which can be assured by the assumption of $p(n)$.

The zero-electron-mass limit $\varepsilon \rightarrow 0$ in the problem (1.2)–(1.3) or (1.4)–(1.5) is reminiscent of the low Mach number (incompressible) limit of the compressible fluid equation, which has been investigated in a number of recent studies, see the monograph [Reference Danchin6] and the survey papers [Reference Alazard2, Reference Feireisl and Novotny16, Reference Masmoudi30, Reference Schochet32], and the references cited therein. The objective of this paper is to perform the limit as $\varepsilon \to 0$ in (1.2). Then as in [Reference Ali and Chen3, Reference Desjardins and Grenier8, Reference Dasjardins, Grenier, Lions and Masmoudi9, Reference Masmoudi30], we introduce the new variables:

\[ ( \tilde n, \tilde E) = \left(\frac{n-N}{\varepsilon}, \frac{E}{\varepsilon}\right), \quad \tilde h = \frac{h(n) - h^0}{\varepsilon} , \quad h^0 = h(N) , \]

then systems (1.2) and (1.4) are rewritten as

(1.6)\begin{equation} \left\{ \begin{aligned} & \partial_t \tilde n+ \frac{1}{\varepsilon} \mathrm{div}((N+ \varepsilon \tilde n) u)=0,\\ & (\partial_t + u \cdot \nabla) u + \frac{1}{\varepsilon} h' (N + \varepsilon \tilde n ) \nabla \tilde n = \displaystyle \frac{1}{N + \varepsilon \tilde n} (\mu \Delta u+\nu \nabla \mathrm{div}\,u) + \frac{1}{\varepsilon} \tilde{E},\\ & \lambda^2 \tilde E =K *\tilde n , \end{aligned}\right. \end{equation}

and

(1.7)\begin{equation} \left\{ \begin{aligned} & A(\varepsilon \tilde h ) ( \partial_t + u \cdot \nabla) \tilde h + \frac{1}{\varepsilon} \mathrm{div} \, u =0,\\ & \displaystyle ( \partial_t + u \cdot \nabla) u + \frac{1}{\varepsilon} \nabla \tilde h = \displaystyle \frac{1}{n(h^0 + \varepsilon \tilde h )} (\mu\Delta u+\nu\nabla \mathrm{div}\,u) + \frac{1}{\varepsilon} \tilde{E},\\ & \lambda^2 \tilde E = K *\tilde n, \end{aligned}\right. \end{equation}

respectively, where $A( \varepsilon \tilde h ) = 1/ p' (n(h^0 + \varepsilon \tilde h ))$. For smooth solutions, we use the variable $\tilde h$ instead of $\tilde n$, because the system (1.7) with some symmetric part is more convenient for standard energy estimates.

Notations. First, $L^p(\mathbb {R}^d)$ with $1\leq p<\infty$ denotes the space of measurable functions whose $p$-powers are integrable on $\mathbb {R}^d$, with norm $\|\cdot \|_{L^p}=(\int _{\mathbb {R}^d}|\cdot |^p{\rm d}x)^{1/p}$, and $L^\infty (\mathbb {R}^d)$ is the space of bounded measurable functions on $\mathbb {R}^d$, with the norm $\|\cdot \|_{L^\infty }=\mathrm {ess sup}_{x\in \mathbb {R}^d}|\cdot |$. Without confusion, we also denote the norm of $L^2(\mathbb {R}^d)$ by $\|\cdot \|$ for brevity. Next, for a nonnegative integer $k$, $H^k=H^k(\mathbb {R}^d)$ denotes the usual $L^2$-type Sobolev space of order $k$. We also write $\|\cdot \|_k$ for the standard norm of $H^k(\mathbb {R}^d)$. Moreover, we denote ${\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert \cdot \right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }_{s,T} = \sup \nolimits _{0< t< T} \| \cdot \|_s$ for $s\geq 0$. In addition, we denote by $C([0,\, T],\, {\bf X})$ (resp. $L^2([0,\, T],\, {\bf X})$) the space of continuous (resp. square integrable) functions on $[0,\, T]$ with values in a Banach space ${\bf X}$. Finally, the symbols $c_i\ (i=1,\,2,\,\ldots )$ or $C_j\ (j=0,\,1,\,2,\,\ldots )$ are always used to denote generic positive constants independent of $\varepsilon$, $c(\cdot )$ and $C(\cdot )$ denote some positive smooth functions which may vary from line to line.

The first result in this paper is the following uniform local existence.

Theorem 1.1 (Uniform-in-$\varepsilon$ local-in-time existence)

Let $d\geq 1,\, s>d/2+2$ and $N>0,$ assume that the initial data $(n_I^\varepsilon,\, u_I^\varepsilon )$ satisfy

(1.8)\begin{equation} n^\varepsilon_I - N \in L^{1} (\mathbb{R}^d), \quad \Bigg\| \left(\frac{n^\varepsilon_I - N}{\varepsilon} , u^\varepsilon_I \right) \Bigg\|_s \leq M_0, \end{equation}

where $M_0$ a given constant independent of $\varepsilon$. Then there exist constants $T_0>0$ and $M_0'>0$ independent of $\varepsilon$, and $\varepsilon _0(M_0)>0$, such that, for all $\varepsilon$ with $0< \varepsilon < \varepsilon _0(M_0)$, the problem (1.2)–(1.3) has a classical solution $(n^\varepsilon,\, u^\varepsilon,\, E^\varepsilon )$ in $[0,\, T_0]$ satisfying

(1.9)\begin{equation} {\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \left(\frac{n^\varepsilon - N}{\varepsilon} , u^\varepsilon, \frac{E^\varepsilon}{\varepsilon} \right)\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}_{s, T_0}\leq M_0' . \end{equation}

Next, we have the following zero-electron-mass limit for the problem (1.2)–(1.3) with well-prepared initial data, that is, the initial data is prepared to make the ‘initial time-derivatives’ uniformly bounded. Here the ‘initial time-derivative’ is understood by compatibility condition using the differential equation that time-derivative can be replaced by spatial-derivatives. The term ill-prepared initial data refers to that is not well-prepared.

Theorem 1.2 (Limit for well-prepared initial data)

Let the assumption of theorem 1.1 be held. Assume that the initial data are well-prepared, that is, $u_I^\varepsilon =u_I^0+\varepsilon u_I^1$ with $\nabla \cdot u_I^0 = 0,$ and

(1.10)\begin{equation} \left\|\frac{n^\varepsilon_I - N}{\varepsilon^2}\right\|_{s} \leq M_1, \end{equation}

where $M_1$ is a constant. Let $(n^\varepsilon,\, u^\varepsilon,\, E^\varepsilon )$ be a classical solution to (1.2)–(1.3) in $[0,\, T_0]$ with $T_0>0$ independent of $\varepsilon$ as obtained in theorem 1.1. Then we have the limit as $\varepsilon \to 0$ that

\[ \begin{array}{l} ( n^\varepsilon, E^\varepsilon ) \to(N, 0) \quad \text{strongly in}\quad L^\infty( [0, T_0] ; H^\alpha (\mathbb{R}^d) ) \cap C^{0,1} ( [0, T_0] ; L^2 (\mathbb{R}^d) ),\\ u^\varepsilon \to u^0, \quad \text{strongly in}\quad C^{0} ( [0, T_0] ; H^\alpha (\mathbb{R}^d) ) \quad \text{for all}\quad \alpha< s, \end{array} \]

where $u^0$ is the unique classical solution of the incompressible Navier–Stokes equations

(1.11)\begin{equation} \begin{array}{l} \nabla \cdot u^0 = 0, \quad ( \partial_t + u^0 \cdot \nabla) u^0 = \displaystyle \dfrac{\mu}{N } \Delta u^0 + \nabla \pi \quad \text{for}\quad t>0,\\ u^0({\cdot}, 0 ) = u_I^0, \end{array} \end{equation}

and $\pi$ is the limit of

\[ \frac{E^\varepsilon - \nabla h (n^\varepsilon)}{\varepsilon} \rightharpoonup \nabla \pi \quad \text{weakly* in }\quad L^\infty( [0, T_0] ; L^2 (\mathbb{R}^d) ) . \]

Finally, for the ill-prepared initial data of the problem (1.2)–(1.3), we have

Theorem 1.3 (Limit for ill-prepared initial data)

Let the assumptions of theorem 1.1 hold and let $d\geq 3,$ suppose that the initial data $u^\varepsilon _I \to u^0_I$ in $H^s (\mathbb {R}^d)$ as $\varepsilon \to 0,$ and let $(n^\varepsilon,\, u^\varepsilon,\, E^\varepsilon )$ be a classical solution to (1.2)–(1.3) in $[0,\, T_0]$ with $T_0>0$ independent of $\varepsilon$. Then, as $\varepsilon \to 0,$

\[ \begin{array}{l} ( n^\varepsilon , E^\varepsilon) \to (N, 0) \quad \text{strongly in}\quad L^\infty( [0, T_0] ; H^s (\mathbb{R}^d) ),\\ u^\varepsilon \rightharpoonup u_*^0 \quad \text{weakly* in}\quad L^\infty( [0, T_0] ; H^s (\mathbb{R}^d) ),\\ u^\varepsilon \to u_*^0 \quad \text{strongly in}\quad C^{0}_{loc} ( (0, T_0] \times \mathbb{R}^d ), \end{array} \]

where $u^0_* \in L^\infty ( [0,\, T_0] ; H^s (\mathbb {R}^d) )$ is the solution of the incompressible Navier–Stokes equations

(1.12)\begin{equation} \begin{array}{l} \nabla \cdot u^0_* = 0, \quad ( \partial_t + u^0_* \cdot \nabla) u^0_* = \displaystyle \dfrac{\mu}{N } \Delta u^0_* + \nabla \pi\quad \text{for}\quad t>0,\\ u^0_*({\cdot}, 0 ) = P u_I^0, \end{array} \end{equation}

for some $\pi \in L^\infty ( [0,\, T_0] ; H^s (\mathbb {R}^d) ),$ and $P$ is the orthogonal projection of $H^s$ onto the subspace $\{v\in H^s: \, \nabla \cdot v = 0 \}$.

The ideas and outlines of proving theorems 1.1, 1.2 and 1.3 are as follows. As a first step, we will get the uniform-in-$\varepsilon$ estimate in short time using a similar idea as in [Reference Alazard1, Reference Ali and Chen3, Reference Ali, Chen, Jüngel and Peng4]. Then applying the local existence of [Reference Kawashima24] and standard continuation argument, we can show theorem 1.1. To get the uniform estimate, we need to treat the singular terms with $\frac {1}{\varepsilon }$. Note that the terms $\frac {1}{\varepsilon } \mathrm {div} \, u$ and $\frac {1}{\varepsilon } \nabla \tilde h$ in (1.6) are symmetric, the key ingredient of the uniform local estimate is the control of the term $\frac {1}{\varepsilon } \tilde E$ in (1.6), which can be similarly estimated as in [Reference Ali and Chen3]. Then, after obtaining the estimate of time derivatives with a similar argument as the one used in [Reference Ali and Chen3, Reference Ali, Chen, Jüngel and Peng4], the zero-electron-mass limit for the well-prepared initial data on $\mathbb {R}^d(d\geq 1)$ can be achieved by using Aubin's lemma. We will prove theorem 1.2 in § 4.

For the proof of theorem 1.3, we still follow the approach of [Reference Grenier17, Reference Grenier18, Reference Ukai36], which was used in [Reference Ali and Chen3]. That is, we first consider the linearized Navier–Stokes–Poisson equations, and next study the properties of the semigroup $\mathcal {L}^\varepsilon (t)$ generated by the linear operator. Then we can decompose the solution $(\tilde {n},\,u)=(0,\,u_1)+(\tilde {n},\,u_2)$ with $u_1$ being the divergence free part. Finally, by further use of the properties of $\mathcal {L}^\varepsilon (t)$, it is possible to get estimates for $\partial _tu_1$, which help the discussion of the convergence away from ${t = 0}$. Therefore, we can prove the limit for ill-prepared initial data in § 5. It is worth noting that the eigenvalues of the linear compressible Navier–Stokes–Poisson equations are different from that of the linear compressible Euler equations in [Reference Grenier17, Reference Grenier18, Reference Ukai36] and that of the linear compressible Euler–Poisson equations in [Reference Ali and Chen3]. The solution of the linearized equation in [Reference Grenier17, Reference Grenier18, Reference Ukai36] has an algebra decay rate, and the solution of the linearized equation in [Reference Ali and Chen3] has an exponential decay rate due to the damping. However, the dissipation mechanism for the linear compressible Navier–Stokes–Poisson system is different here, which can be seen from its eigenvalues

\[ \begin{array}{l} \lambda_* = \mu |\xi|^2 , \qquad \text{($d-1$ multiple)}\\ \displaystyle\lambda_\pm{=} \dfrac{\mu +\nu }{ 2 } |\xi|^2 \pm \dfrac{i}{ 2 \varepsilon} \sqrt{ 4 (1+a|\xi|^2) - (\mu +\nu )^2 |\xi|^4 \varepsilon^2}, \end{array} \]

this is somehow similar to that of heat equation, and the fundamental solution contains some part like the heat kernel, and a big difficulty is the singularity of the heat kernel at $t=0$. Next, we have the characteristic decomposition of the solution operator in the form (5.3), with each mode corresponding to an eigenvalue, and then analyse carefully the uniform-in-$\varepsilon$ estimates of each mode. See lemma 5.1 for details. Due to the singularity of the heat kernel at $t=0$ and good decay away from $t=0$, we need to have more precise local in time estimate of the solution operator such that the result is integrable in time. Thus we need two sets of estimates: $t<\delta$ and $t\geq \delta$, for any given positive constant $\delta$. Without loss of generality, we consider $\delta =1$, that is, we derive the estimate (5.6) for $t<1$, and, estimate (5.7) for $t\geq 1$. The solution of the full linearized problem (5.1) can be represented by Duhamel's principle using the solution operator of the linear part, which essentially relies on the time integrability of the estimate near $t=0$. See also remark 5.2. Another difficulty in the analysis is due to the complicated structure of the eigenvalues $\lambda _\pm$. More specifically, the term inside the square-root in $\lambda _\pm$, $F(|\xi |^2) := 4 (1+a|\xi |^2) - (\mu +\nu )^2 |\xi |^4 \varepsilon ^2$, is not monotone with respect to $|\xi |^2$, unlike $\tilde F(|\xi |^2)= 4 (1+a|\xi |^2) - \varepsilon ^2$ for the Euler–Poisson case which was strictly increasing, will bring difficulty to our analysis. To deal with this problem, we properly decompose the solution operator with respect to frequencies, say, the low, medium and high frequency parts, and estimate each part respectively. See (5.13) and subsequent computations for details.

This paper is organized as follows. In the next section we state some useful lemmas which will be used later, then we give the uniform local estimate of the terms involving the term $\frac {1}{\varepsilon } \tilde E$. We will prove the uniform local existence of the compressible Navier–Stokes–Poisson equation in § 3. Finally, the limit for the solutions of the compressible Navier–Stokes–Poisson equation with well- and ill-prepared initial data will be considered in § 4 and § 5, respectively.