Global zero-relaxation limit of the non-isentropic Euler

Abstract

This paper is concerned with smooth solutions of the non-isentropic Euler–Poisson system for ion dynamics. The system arises in the modeling of semi-conductor, in which appear one small parameter, the momentum relaxation time. When the initial data are near constant equilibrium states, with the help of uniform energy estimates and compactness arguments, we rigorously prove the convergence of the system for all time, as the relaxation time goes to zero. The limit system is the drift-diffusion system.

Keywords

Non-isentropic Euler–Poisson system global smooth solution global zero-relaxation limit

1. Introduction and main results

The Euler–Poisson systems for semi-conductor have been intensively studied in the literature both analytically and numerically [5,7,25,28]. Since the electron mass is much smaller than the ion mass, the electron density $n_{e}$ can be approximated by a function of the electric potential ϕ through the Boltzmann relation $n^{e} = e^{ϕ}$ . In the zero-relaxation-time limit, the model equations can be approximated by the drift-diffusion equations, which are much simpler to treat than the hyperbolic problem. Here we give a rigorous proof of the zero-relaxation-time limit for the non-isentropic Euler–Poisson system for ion dynamics [4,15] $\begin{matrix} (1) & \{\begin{matrix} \partial_{t} n + div (n u) = 0, \\ \partial_{t} (n u) + div (n u \otimes u) + \nabla p = - n \nabla ϕ - \frac{n u}{τ}, \\ \partial_{t} (n E + n e) + div ((n E + n e) u + p u) = - n u \cdot \nabla ϕ - \frac{n E + n e - n e_{l}}{τ}, \\ - λ^{2} Δ ϕ + e^{ϕ} = n, in R^{+} \times R^{3}, \end{matrix} \end{matrix}$ where the unknowns $n, u = {(u_{1}, u_{2}, u_{3})}^{T}$ , θ and ϕ are the density, the velocity, the absolute temperature and the electric potential of the ion. Functions $p = n θ$ , $E = \frac{1}{2} | u |^{2}$ , $e = C_{*} θ$ denote the pressure, the kinetic energy and the internal energy, respectively. The constants $C_{*} > 0$ , $θ_{l} > 0$ and $e_{l} = C_{*} θ_{l}$ stand for the coefficient of heat conduction, the background temperature and the background internal energy, respectively. The physical parameters τ, $λ \in (0, 1)$ stand for the momentum relaxation time and the Debye length, respectively. In what follows, we investigate the zero-relaxation limit $τ \to 0$ with $λ = 1$ .

The Boltzmann relation is always used to simplify mathematical models from semi-conductor physics [3,9–11,16,17]. This relation can be formally derived from isothermal Euler–Poisson systems as the electron mass tends to zero. A rigorous justification was recently given for smooth solutions on a time interval independent of the electron mass [20].

For convenience, we set $C_{*} = θ_{l} = 1$ . This is not an essential assumption in the study of global convergence of smooth solutions. Then for smooth solutions in any non-vacuum field, system (1) is equivalent to $\begin{matrix} (2) & \{\begin{matrix} \partial_{t} n + div (n u) = 0, \\ \partial_{t} u + (u \cdot \nabla) u + \nabla θ + θ \nabla ln n = - \nabla ϕ - \frac{u}{τ}, \\ \partial_{t} θ + u \cdot \nabla θ + θ \nabla \cdot u = - \frac{1}{2 τ} | u |^{2} - \frac{1}{τ} (θ - 1), \\ - Δ ϕ + e^{ϕ} = n, in R^{+} \times R^{3} . \end{matrix} \end{matrix}$ supplemented by the following initial condition depending on the parameter $\begin{matrix} (3) & (n, u, θ) |_{t = 0} = (n_{0}^{τ}, u_{0}^{τ}, θ_{0}^{τ}), x \in R^{3} . \end{matrix}$

It’s easy to see that $(n, u, θ, ϕ) = (1, 0, 1, 0)$ is a constant steady-state of system (2). Let $s ⩾ 3$ be an integer. It follows from [13] that for a function $n ⩾ const. > 0$ satisfying $n - 1 \in H^{s} (R^{3})$ , the Poisson equation in system (2) has an unique solution ϕ such that $e^{ϕ} - n \in H^{s} (R^{3})$ . Moreover, it also holds $‖ ϕ ‖_{H^{1} (R^{3})} ⩽ C ‖ n - 1 ‖_{L^{2} (R^{3})}$ . This estimate together with (32) gives $‖ ϕ ‖_{H^{s + 1} (R^{3})} ⩽ C ‖ n - 1 ‖_{H^{s} (R^{3})}$ , which implies that $ϕ \in H^{s + 1} (R^{3})$ . Thus $\nabla ϕ$ can be regarded as a zero-order term in system (2). We deduce that system (2) is symmetrizable hyperbolic.Then, according to the classical result of Kato [18], the Cauchy problem (2)–(3) has an unique local smooth solution when the initial data are smooth.

Proposition 1.1 (Local existence of smooth solutions, see [18,24]).

Let $s ⩾ 3$ be an integer. Suppose $(n_{0}^{τ} - 1, u_{0}^{τ}, θ_{0}^{τ} - 1) \in H^{s} (R^{3})$ with $n_{0}^{τ}, θ_{0}^{τ} ⩾ 2 κ$ for some given constant $κ > 0$ . Then there is $T > 0$ such that problem ( 2 )–( 3 ) has an unique smooth solution $(n, u, θ, ϕ)$ which satisfies $n, θ ⩾ κ$ and $\begin{array}{l} (n - 1, u, θ - 1) \in C ([0, T]; H^{s} (R^{3})) \cap C^{1} ([0, T]; H^{s - 1} (R^{3})), \\ ϕ \in C ([0, T]; H^{s + 1} (R^{3})) \cap C^{1} ([0, T]; H^{s} (R^{3})), \end{array}$ where $T > 0$ may depend on parameter τ.

There are many interesting works on both the asymptotic limits of small physical parameters and the global stability problem around steady-states for Euler–Poisson systems. When $λ \to 0$ , the quasineutral limit of system (1) leads to compressible Euler equations. It was proved for local smooth solutions in one space dimension by Cordier and Grenier in [6] and in several space dimensions by Varet, Kwan and Rousset [9]. For the Euler–Poisson system for electrons in which the Poisson equation in (1) is replaced by $- λ^{2} Δ ϕ = n - b (x)$ with $b (x) > 0$ being a given smooth function, the quasineutral limit leads to incompressible Euler equations. The zero-relaxation limit would be investigated in a slow time $ξ = τ t$ . It converges to drift-diffusion equations. For smooth solutions, these limits were proved locally-in-time in [2,21,30,32] and globally-in-time by Peng in [26] when the solutions are close to constant equilibrium states. We also recall that the zero-relaxation limit was also proved for global entropy solutions [14,16,17,25]. On the other hand, we talk about the global stability problem around steady-states of the Euler–Poisson system for electrons. In the case that b is a positive constant, the global existence of smooth solutions was shown in [13,23,30] and the solutions decay exponentially to the constant steady-state for large time. These results were extended to the case in which b is a small perturbation of a constant [1,31]. In the case that b is no longer small, the stability problems were solved by Guo and Strauss in [12], and Peng in [27], respectively.

Recently, Liu and Peng consider the zero-relaxation limit for system (1) in isentropic case [22]. However, there is no result about the zero-relaxation limit of system (1). Therefore, we try to rigorously prove the global-in-time convergence of the non-isentropic Euler–Poisson system (2) for ion dynamics in the whole space $R^{3}$ as $τ \to 0$ .

The main results of this article can be stated as follows.

Theorem 1.1 (Global existence of smooth solutions with respect to τ).

Let $s ⩾ 3$ be an integer. There are constants $δ_{0} > 0$ and $C > 0$ such that for all $τ \in (0, \frac{2}{3})$ , if $\begin{matrix} (4) & \begin{matrix} {‖ n_{0}^{τ} - 1 ‖}_{s} + {‖ u_{0}^{τ} ‖}_{s} + {‖ θ_{0}^{τ} - 1 ‖}_{s} ⩽ δ_{0}, \end{matrix} \end{matrix}$ then problem ( 2 )–( 3 ) has an unique global smooth solution $(n, u, θ, ϕ)$ which satisfies $\begin{array}{l} (5) & (n - 1, u, θ - 1) \in C (R^{+}; H^{s}) \cap C^{1} (R^{+}; H^{s - 1}), n, θ ⩾ \frac{1}{2}, \\ (6) & ϕ \in C (R^{+}; H^{s + 1}) \cap C^{1} (R^{+}; H^{s}) . \end{array}$ Furthermore, for all $t > 0$ , $\begin{matrix} (7) & \begin{matrix} {‖ n (t) - 1 ‖}_{s}^{2} + {‖ u (t) ‖}_{s}^{2} + {‖ θ (t) - 1 ‖}_{s}^{2} + {‖ ϕ (t) ‖}_{s + 1}^{2} \\ + \int_{0}^{t} (\frac{1}{τ} {‖ u (ξ) ‖}_{s}^{2} + \frac{2}{3 τ} {‖ θ (ξ) - 1 ‖}_{s}^{2} + \frac{τ}{9} {‖ \nabla n (ξ) ‖}_{s - 1}^{2}) d ξ \\ ⩽ C ({‖ n_{0}^{τ} - 1 ‖}_{s}^{2} + {‖ u_{0}^{τ} ‖}_{s}^{2} + {‖ θ_{0}^{τ} - 1 ‖}_{s}^{2}), \end{matrix} \end{matrix}$ and $\begin{matrix} (8) & \begin{matrix} {‖ n (t) - e^{ϕ (t)} ‖}_{s - 1} ⩽ C ({‖ n_{0}^{τ} - 1 ‖}_{s} + {‖ u_{0}^{τ} ‖}_{s} + {‖ θ_{0}^{τ} - 1 ‖}_{s}) . \end{matrix} \end{matrix}$

Remark 1.1.
It should be pointed out that both the velocity dissipative term and the temperature dissipative term of the non-isentropic Euler–Poisson system (2) play a key role in the proof of the global existence.
Remark 1.2.
Note that in Theorem 1.1, the background temperature $θ_{l}$ and the heat conduct $C_{*}$ is assumed to take the same value 1. In fact, this assumption is not necessary. That is, if we remove this assumption, similar results as in Theorem 1.1 can be obtained even though the different values could give rise to the extra analytical difficulty. We omit this.
Theorem 1.2 (Global convergence of smooth solutions as $τ \to 0$ ).

Let $(n, u, θ, ϕ)$ be the unique global solution given by Theorem 1.1 . Set $\begin{matrix} (9) & \begin{matrix} (ρ^{τ}, v^{τ}, θ^{τ}, ψ^{τ}) = (n, \frac{u}{τ}, θ, ϕ) (\frac{t}{τ}, x) . \end{matrix} \end{matrix}$

If the initial data satisfy $\begin{matrix} (10) & \begin{matrix} n_{0}^{τ} - 1 ⇀ n_{0} - 1, θ_{0}^{τ} - 1 ⇀ θ_{0} - 1 weakly in H^{s}, as τ \to 0, \end{matrix} \end{matrix}$ then there exist functions $(\bar{n}, \bar{v}, \bar{θ}, \bar{ϕ})$ which satisfy $\begin{matrix} (11) & \begin{matrix} \bar{n} - 1 \in L^{\infty} (R^{+}; H^{s}), \bar{v} \in L^{2} (R^{+}; H^{s}), \\ \bar{θ} - 1 \in L^{\infty} (R^{+}; H^{s}), \bar{ϕ} \in L^{\infty} (R^{+}; H^{s + 1}), \end{matrix} \end{matrix}$ such that, $\begin{array}{l} (12) & \begin{matrix} ρ^{τ} - 1 ⇀ \bar{n} - 1 weakly* in L^{\infty} (R^{+}; H^{s}), \end{matrix} \\ (13) & \begin{matrix} v^{τ} ⇀ \bar{v} weakly in L^{2} (R^{+}; H^{s}), \end{matrix} \\ (14) & \begin{matrix} θ^{τ} - 1 ⇀ \bar{θ} - 1 weakly* in L^{\infty} (R^{+}; H^{s}), \end{matrix} \\ (15) & \begin{matrix} ψ^{τ} ⇀ \bar{ϕ} weakly* in L^{\infty} (R^{+}; H^{s + 1}), as τ \to 0 . \end{matrix} \end{array}$ Furthermore, for all $T > 0$ and $s_{*} \in [0, s)$ , it holds $\begin{array}{l} (16) & \begin{matrix} ρ^{τ} - 1 \to \bar{n} - 1 strongly in C ([0, T]; H_{loc}^{s_{*}}), \end{matrix} \\ (17) & \begin{matrix} ψ^{τ} \to \bar{ϕ} strongly in C ([0, T]; H_{loc}^{s_{*} + 1}), as τ \to 0, \end{matrix} \end{array}$ where $(\bar{n}, \bar{ϕ})$ is the global solution of the drift-diffusion equations $\begin{matrix} (18) & \{\begin{matrix} \partial_{t} \bar{n} - div (\bar{n} \nabla (\bar{n} + \bar{ϕ})) = 0, \\ - Δ \bar{ϕ} + e^{\bar{ϕ}} = \bar{n}, \end{matrix} \end{matrix}$ with the initial condition $\begin{matrix} (19) & \begin{matrix} \bar{n} |_{t = 0} = n_{0} . \end{matrix} \end{matrix}$

We complete the proof of Theorems 1.1–1.2 by using the careful energy estimates and dissipation estimates together with compactness arguments. It should be pointed out that the non-isentropic case is much more complex than the isentropic case. For example, for the isentropic Euler–Poisson equations, Liu and Peng [22] introduce the energy conservation of the Euler equations and the Taylor formula to get the $L^{2}$ energy estimate. But this is not true for the non-isentropic Euler–Poisson system due to the complexity of non-isentropic case caused by the coupled energy equations. Here the standard technique of symmetrizer is used here to remove this difficulty when we deal with the Euler part of the Euler–Poisson system so as to obtain the $L^{2}$ energy estimates for the velocity and the temperature. Moreover, we should overcome the difficulty caused by the Boltzmann’s relation in the Poisson equation when we deal with the density dissipation estimate. This can be done by introduce a new function $h (ϕ)$ . Now, we list the steps for solving the problem. Firstly, we introduce the perturbation variables $(N, u, Θ, ϕ) = (n - 1, u, θ - 1, ϕ)$ which satisfy the reformulated system (20) and the Poisson equation (21). Then system (20) is turned into the matrix form which is symmetrizable hyperbolic. Since we consider small solution for which n and θ are near 1, we suppose that n, θ are bounded in $[\frac{1}{2}, \frac{3}{2}]$ which implies that both $\frac{n}{θ}$ and $\frac{θ}{n}$ are bounded in $[\frac{1}{3}, 3]$ , this fact will be used in estimate (58). Secondly, we establish the estimates on $‖ \partial_{t} ϕ ‖_{s}$ , $‖ \partial^{α} \nabla ϕ ‖$ , $‖ \nabla ϕ ‖_{s - 1}$ , $⟨ N \partial^{α} u - \partial^{α} (N u), \partial^{α} \nabla ϕ ⟩$ , $‖ \partial_{t} N ‖$ , $‖ \partial_{t} N ‖_{L^{\infty}}$ , $‖ \partial_{t} Θ ‖$ and $‖ \partial_{t} Θ ‖_{L^{\infty}}$ (see Lemma 3.1), these estimates would be very useful in the establishment of uniform energy estimates. Next, we begin to construct detailed energy estimates, they conclude $L^{2}$ estimate (see Lemma 3.2), higher order estimate (see Lemma 3.3) and the dissipation estimate (see Lemma 3.4). Due to the fact that the non-isentropic Euler–Poisson system (1) does’t fulfill the Kawashima’s stability condition, we use the careful energy methods and the skew-symmetric dissipative structure of Euler–Poisson system, which concludes the global existence results in Theorem 1.1. Following this idea, we define the total energy $E (t)$ and the dissipation energy $D (t)$ , respectively, by $\begin{array}{l} E (t) \sim {‖ N (t) ‖}_{s}^{2} + {‖ u (t) ‖}_{s}^{2} + {‖ Θ (t) ‖}_{s}^{2} + {‖ ϕ (t) ‖}_{s + 1}^{2}, \\ D (t) \sim ‖ \nabla N ‖_{s - 1}^{2} + \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{1}{τ} ‖ Θ ‖_{s}^{2}, \end{array}$ and the rest term $R (t)$ satisfying $C R (t) ⩽ \frac{1}{2} D (t)$ . After that, by combining these above estimates, we obtain the energy inequality $\begin{matrix} \frac{d}{d t} E (t) + \frac{1}{2} D (t) ⩽ 0, \end{matrix}$ which gives the uniformly global existence of solutions by a standard continuous argument(see the proof of Theorem 1.1 in Section 3.2). Finally, with the help of a classical compactness theorem [29] and the elaborate estimates, the global convergence results in Theorem 1.2 are achieved.

We conclude this section by stating the arrangement of the rest of this paper. In the next section, we give some notations, inequalities and other preliminary works for proving Theorems 1.1 to 1.2. Section 3 is devoted to detailed energy estimates and the proof of Theorem 1.1. The proof of the second main result is given in Section 4, and the global convergence results are obtained by compactness methods.

2. Preliminaries and reformulation of system (2)

2.1. Preliminaries

We first introduce some notations. For a multi-index $α = (α_{1}, α_{2}, α_{3}) \in N^{3}$ , we denote $\begin{matrix} \partial^{α} = \partial_{x_{1}}^{α_{1}} \partial_{x_{2}}^{α_{2}} \partial_{x_{3}}^{α_{3}} = \partial_{1}^{α_{1}} \partial_{2}^{α_{2}} \partial_{3}^{α_{3}}, with | α | = α_{1} + α_{2} + α_{3} . \end{matrix}$ For $α = (α_{1}, α_{2}, α_{3})$ and $β = (β_{1}, β_{2}, β_{3}) \in N^{3}$ , $β ⩽ α$ stands for $β_{j} ⩽ α_{j}$ for $j = 1, 2, 3$ , and $β < α$ stands for $β ⩽ α$ and $β \neq α$ .

For all integer $s \in N$ , we denote by $H^{s}$ , $L^{2}$ and $L^{\infty}$ the usual Sobolev spaces $H^{s} (R^{3})$ , $L^{2} (R^{3})$ and $L^{\infty} (R^{3})$ , and by $‖ \cdot ‖_{s}$ , $‖ \cdot ‖$ and $‖ \cdot ‖_{\infty}$ the corresponding norms, respectively. $f \sim g$ denotes $γ f ⩽ g ⩽ \frac{1}{γ} f$ for a constant $0 < γ < 1$ . $⟨ \cdot, \cdot ⟩$ stands for the inner product of $L^{2} (R^{3})$ . We also denote by $C > 0$ a generic constant independent of any time and parameters.

The following inequalities in Sobolev space will be used in the proof of Theorems 1.1–1.2.

Lemma 2.1 (Moser-type calculus inequalities, see [19,24]).

Let $s ⩾ 3$ be an integer. Suppose $u \in H^{s}$ , $\nabla u \in L^{\infty}$ , $v \in H^{s - 1} \cap L^{\infty}$ and f is a smooth function. Then for all multi-indexes α with $1 ⩽ | α | ⩽ s$ , we have $\partial^{α} (u v) - u \partial^{α} v \in L^{2}$ and $\begin{array}{l} ‖ \partial^{α} (u v) - u \partial^{α} v ‖ ⩽ C (‖ \nabla u ‖_{\infty} ‖ D^{| α | - 1} v ‖ + ‖ D^{| α |} u ‖ ‖ v ‖_{\infty}), \\ ‖ \partial^{α} f (u) ‖ ⩽ C {(1 + {‖ u ‖}_{s})}^{s - 1} ‖ u ‖_{s}, \end{array}$ where the constant C may depend on $‖ u ‖_{\infty}$ and s, and $\begin{matrix} ‖ D^{s^{'}} u ‖ = \sum_{| α | = s^{'}} ‖ \partial^{α} u ‖ . \end{matrix}$

2.2. Reformulation of system (2)

Now, let us set $\begin{matrix} N = n - 1, Θ = θ - 1, U = {(N, u, Θ)}^{T}, W = {(U, ϕ)}^{T} . \end{matrix}$ It follows from the Euler equations in (2) that $\begin{matrix} (20) & \{\begin{matrix} \partial_{t} N + u \cdot \nabla N + n div u = 0, \\ \partial_{t} u + (u \cdot \nabla) u + \nabla Θ + \frac{θ}{n} \nabla N = - \nabla ϕ - \frac{u}{τ}, \\ \partial_{t} Θ + u \cdot \nabla Θ + θ div u + \frac{1}{2 τ} | u |^{2} = - \frac{1}{τ} Θ, \end{matrix} \end{matrix}$ coupled with the Poisson equation $\begin{matrix} (21) & - Δ ϕ + h (ϕ) ϕ = N, \end{matrix}$ where h is the function defined by $\begin{matrix} h (ϕ) = \{\begin{matrix} \frac{e^{ϕ} - 1}{ϕ}, & if ϕ \neq 0, \\ 1, & if ϕ = 0 . \end{matrix} \end{matrix}$ Obviously, h satisfies $\begin{matrix} h \in C^{\infty} (R; R), h (ϕ) = 1 + O (ϕ) for | ϕ | small enough . \end{matrix}$

We denote $\begin{array}{l} (22) & A_{j} (n, u, θ) = (\begin{matrix} u_{j} & n e_{j}^{T} & 0 \\ \frac{θ}{n} e_{j} & u_{j} I_{3} & e_{j} \\ 0 & θ e_{j}^{T} & u_{j} \end{matrix}), j = 1, 2, 3, \\ (23) & K (u, Θ, \nabla ϕ) = - (\begin{matrix} 0 \\ \nabla ϕ + \frac{u}{τ} \\ \frac{| u |^{2}}{2 τ} + \frac{1}{τ} Θ \end{matrix}), \end{array}$ where $I_{3}$ is the unit matrix of order 3 and $(e_{1}, e_{2}, e_{3})$ is the canonical basis of $R^{3}$ , and $e_{j}^{T}$ is the transpose of $e_{j}$ . Then we can rewrite (20) in the matrix form $\begin{matrix} (24) & \partial_{t} U + \sum_{j = 1}^{3} A_{j} (n, u, θ) \partial_{j} U = K (u, Θ, \nabla ϕ) . \end{matrix}$

We introduce $\begin{matrix} (25) & A_{0} (n, θ) = (\begin{matrix} \frac{θ}{n} & 0 & 0 \\ 0 & n I_{3} & 0 \\ 0 & 0 & \frac{n}{θ} \end{matrix}), \end{matrix}$ which is symmetric and positive definite when $n > 0$ and $θ > 0$ . It is easy to check that $\begin{matrix} (26) & {\tilde{A}}_{j} = A_{0} A_{j} = (\begin{matrix} \frac{θ}{n} u_{j} & θ e_{j}^{T} & 0 \\ θ e_{j} & n u_{j} I_{3} & n e_{j} \\ 0 & n e_{j}^{T} & \frac{n}{θ} u_{j} \end{matrix}) \end{matrix}$ is a symmetric matrix which implies that system (24) is symmetrizable hyperbolic.

3. Global existence of solutions to problem (2)–(3)

In this section, we shall prove Theorem 1.1 for the global existence and uniqueness of solutions to the Cauchy problem (2)–(3). In the first subsection, we obtain some uniform-in-time a priori estimates for any smooth solution. In the second subsection, we combine those a priori estimates with the local existence of solutions to extend the local solution up to infinite time with the help of the continuity argument.

In what follows, we set $T > 0$ and $W$ be a smooth solution to the Cauchy problem (2)–(3) defined on time interval $[0, T]$ . Let $\begin{matrix} ω_{T} = sup_{0 ⩽ t ⩽ T} {‖ U (t) ‖}_{s} . \end{matrix}$ Throughout this paper, we suppose that $s ⩾ 3$ and $ω_{T}$ is small enough with respect to T. Then from the continuous embedding $H^{s - 1} ↪ L^{\infty}$ (see [8]), there is a constant $C_{e p} > 0$ such that $\begin{matrix} ‖ f ‖_{L^{\infty}} ⩽ C_{e p} ‖ f ‖_{s - 1}, \forall f \in H^{s - 1} . \end{matrix}$ Since we consider small solution for which n and θ are near 1, we may assume that $\begin{matrix} (27) & \frac{1}{2} ⩽ n ⩽ \frac{3}{2}, \frac{1}{2} ⩽ θ ⩽ \frac{3}{2}, \end{matrix}$ and then $\begin{matrix} (28) & \frac{1}{3} ⩽ \frac{n}{θ}, \frac{θ}{n} ⩽ 3 . \end{matrix}$

The following lemma is concerned the estimates for the density and the temperature equations in (20) and the Poisson equation (21).

Lemma 3.1.
For all $t \in [0, T]$ , we have $\begin{array}{l} (29) & ‖ \partial_{t} N ‖ + ‖ \partial_{t} N ‖_{L^{\infty}} ⩽ C ‖ u ‖_{s}, \\ (30) & ‖ \partial_{t} Θ ‖ + ‖ \partial_{t} Θ ‖_{L^{\infty}} ⩽ C (‖ u ‖_{s} + \frac{1}{τ} ‖ Θ ‖_{s} + \frac{1}{τ} ‖ u ‖_{s}^{2}), \\ (31) & \begin{matrix} ‖ \partial_{t} ϕ ‖_{s} ⩽ C ‖ u ‖_{s}^{2}, \end{matrix} \\ (32) & \begin{matrix} {‖ \partial^{α} \nabla ϕ ‖}^{2} + ‖ \nabla ϕ ‖_{s - 1}^{2} ⩽ C ‖ \nabla N ‖_{s - 1}^{2}, 1 ⩽ | α | ⩽ s, \end{matrix} \end{array}$ and $\begin{matrix} (33) & | ⟨ N \partial^{α} u - \partial^{α} (N u), \partial^{α} \nabla ϕ ⟩ | ⩽ C ‖ \nabla N ‖_{s - 1}^{2} ‖ u ‖_{s}, 1 ⩽ | α | ⩽ s . \end{matrix}$
Proof.
By the density equation, the temperature equation in (20), Lemma 2.1, the continuous embedding inequality in Sobolev space and the smallness condition of $ω_{T}$ , we obtain $\begin{array}{l} \begin{matrix} ‖ \partial_{t} N ‖ ⩽ ‖ u ‖_{\infty} ‖ \nabla N ‖ + C ‖ \nabla \cdot u ‖ ⩽ C ‖ u ‖_{s}, \end{matrix} \\ \begin{matrix} ‖ \partial_{t} N ‖_{\infty} ⩽ ‖ u ‖_{\infty} ‖ \nabla N ‖_{\infty} + C ‖ \nabla \cdot u ‖_{\infty} ⩽ C ‖ u ‖_{s}, \end{matrix} \\ \begin{matrix} ‖ \partial_{t} Θ ‖ ⩽ ‖ u ‖_{\infty} ‖ \nabla Θ ‖ + C ‖ \nabla \cdot u ‖ + \frac{C}{τ} ‖ u ‖_{\infty} ‖ u ‖ + \frac{1}{τ} ‖ Θ ‖ ⩽ C (‖ u ‖_{s} + \frac{1}{τ} ‖ Θ ‖_{s} + \frac{1}{τ} ‖ u ‖_{s}^{2}), \end{matrix} \end{array}$ and $\begin{matrix} ‖ \partial_{t} Θ ‖_{\infty} ⩽ ‖ u ‖_{\infty} ‖ \nabla Θ ‖_{\infty} + C ‖ \nabla \cdot u ‖_{\infty} + \frac{C}{τ} ‖ u ‖_{\infty}^{2} + \frac{1}{τ} ‖ Θ ‖_{\infty} ⩽ C (‖ u ‖_{s} + \frac{1}{τ} ‖ Θ ‖_{s} + \frac{1}{τ} ‖ u ‖_{s}^{2}), \end{matrix}$ these estimates imply (29)–(30). Moreover, the proof of (31)–(33) can be found in Lemma 3.1 with $λ = 1$ in [22]. This completes the proof of Lemma 3.1. □

3.1. Energy estimates

Lemma 3.2 ( $L^{2}$ estimates).

For all $t \in [0, T]$ , it holds $\begin{matrix} (34) & \begin{matrix} \frac{d}{d t} (⟨ A_{0} U, U ⟩ + ‖ \nabla ϕ ‖^{2} + \int_{R^{3}} H (ϕ) d x) + \frac{2}{τ} ⟨ n, | u |^{2} ⟩ + \frac{2}{τ} ⟨ \frac{n}{θ}, | Θ |^{2} ⟩ \\ ⩽ C (‖ \nabla N ‖_{2}^{2} + \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{1}{τ} ‖ Θ ‖_{s}^{2}) (‖ N ‖ + ‖ u ‖ + ‖ Θ ‖), \end{matrix} \end{matrix}$ where $\begin{matrix} H (ϕ) = 2 (ϕ e^{ϕ} - e^{ϕ} + 1) . \end{matrix}$

Proof.
Taking the inner product of (24) with $2 A_{0} U$ in $L^{2}$ yields the classical energy equality $\begin{matrix} \frac{d}{d t} ⟨ A_{0} U, U ⟩ = ⟨ \partial_{t} A_{0} U, U ⟩ + \sum_{j = 1}^{3} ⟨ \partial_{j} {\tilde{A}}_{j} U, U ⟩ + 2 ⟨ A_{0} K, U ⟩ . \end{matrix}$ In view of the expression (25) and (22), applying Lemma 2.1 yields $\begin{matrix} ⟨ \partial_{t} A_{0} U, U ⟩ \\ = ⟨ \frac{1}{n} \partial_{t} Θ - \frac{θ}{n^{2}} \partial_{t} N, N^{2} ⟩ + ⟨ u \partial_{t} N, u ⟩ + ⟨ \frac{1}{θ} \partial_{t} N - \frac{N}{θ^{2}} \partial_{t} Θ, | Θ |^{2} ⟩ \\ ⩽ C (‖ \partial_{t} Θ ‖ + ‖ \partial_{t} N ‖) ‖ N ‖_{L^{\infty}} ‖ N ‖ + C ‖ \partial_{t} N ‖_{L^{\infty}} ‖ u ‖^{2} + C (‖ \partial_{t} Θ ‖_{L^{\infty}} + ‖ \partial_{t} N ‖_{L^{\infty}}) ‖ Θ ‖^{2} \\ ⩽ C (‖ u ‖_{s} + \frac{1}{τ} ‖ Θ ‖_{s} + \frac{1}{τ} ‖ u ‖_{s}^{2}) (‖ \nabla N ‖_{1} ‖ N ‖ + ‖ Θ ‖^{2}) + C ‖ u ‖_{s} ‖ u ‖^{2} \\ ⩽ C (‖ \nabla N ‖_{1}^{2} + \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{1}{τ} ‖ Θ ‖_{s}^{2}) (‖ N ‖ + ‖ u ‖ + ‖ Θ ‖), \end{matrix}$ and $\begin{matrix} ⟨ \partial_{j} {\tilde{A}}_{j} U, U ⟩ \\ = ⟨ \frac{θ}{n} \partial_{j} u_{j} + \frac{u_{j}}{n} \partial_{j} Θ - \frac{θ u_{j}}{n^{2}} \partial_{j} N, N^{2} ⟩ + 2 ⟨ \partial_{j} Θ, N u ⟩ + ⟨ u_{j} \partial_{j} N + n \partial_{j} u_{j}, u^{2} ⟩ \\ + 2 ⟨ \partial_{j} N, Θ u ⟩ + ⟨ \frac{n}{θ} \partial_{j} u_{j} + \frac{u_{j}}{θ} \partial_{j} N - \frac{n u_{j}}{θ^{2}} \partial_{j} Θ, Θ^{2} ⟩ \\ ⩽ C (‖ \nabla N ‖_{2}^{2} + ‖ u ‖_{3}^{2} + ‖ Θ ‖_{2}^{2}) (‖ N ‖ + ‖ u ‖ + ‖ Θ ‖) . \end{matrix}$ Using (23) and (25), it is easy to get $\begin{matrix} 2 ⟨ A_{0} K, U ⟩ \\ = - 2 ⟨ n u, \nabla ϕ ⟩ - \frac{2}{τ} ⟨ n, u^{2} ⟩ - \frac{2}{τ} ⟨ \frac{n}{θ}, Θ^{2} ⟩ - \frac{1}{τ} ⟨ \frac{n}{θ} u^{2}, Θ ⟩ \\ ⩽ - 2 ⟨ n u, \nabla ϕ ⟩ - \frac{2}{τ} ⟨ n, u^{2} ⟩ - \frac{2}{τ} ⟨ \frac{n}{θ}, Θ^{2} ⟩ + \frac{C}{τ} ({‖ u ‖}^{2} + ‖ Θ ‖_{2}^{2}) ‖ u ‖, \end{matrix}$ with the estimate $\begin{matrix} 2 ⟨ n u, \nabla ϕ ⟩ & = - 2 ⟨ \nabla \cdot (n u), ϕ ⟩ \\ = 2 ⟨ \partial_{t} N, ϕ ⟩ \\ = - 2 ⟨ Δ \partial_{t} ϕ, ϕ ⟩ + 2 ⟨ e^{ϕ} \partial_{t} ϕ, ϕ ⟩ \\ = \frac{d}{d t} ‖ \nabla ϕ ‖^{2} + \frac{d}{d t} \int_{R^{3}} H (ϕ) d x . \end{matrix}$

Therefore, (34) follows from by combining the above formulas. The proof of Lemma 3.2 is complete. □

Next, we establish the higher order energy estimates. Let $α \in N^{3}$ with $1 ⩽ | α | ⩽ s$ . Applying $\partial^{α}$ to (13) gives $\begin{matrix} (35) & \begin{matrix} \partial_{t} \partial^{α} U + \sum_{j = 1}^{3} A_{j} \partial_{j} \partial^{α} U = \partial^{α} K + g^{α}, \end{matrix} \end{matrix}$ where $\begin{matrix} (36) & \begin{matrix} g^{α} = \sum_{j = 1}^{3} (A_{j} \partial_{j} \partial^{α} U - \partial^{α} (A_{j} \partial_{j} U)) . \end{matrix} \end{matrix}$
Lemma 3.3 (Higher order energy estimates).

For all $t \in [0, T]$ and $α \in N^{3}$ with $1 ⩽ | α | ⩽ s$ , there is a constant $C_{0} > 0$ such that $\begin{matrix} (37) & \begin{matrix} \frac{d}{d t} (⟨ A_{0} \partial^{α} U, \partial^{α} U ⟩ + {‖ \partial^{α} \nabla ϕ ‖}^{2} + ⟨ e^{ϕ}, {| \partial^{α} ϕ |}^{2} ⟩) \\ + \frac{2}{τ} ⟨ n, {| \partial^{α} u |}^{2} ⟩ + \frac{2}{τ} ⟨ \frac{n}{θ}, {| \partial^{α} Θ |}^{2} ⟩ \\ ⩽ C (‖ \nabla N ‖_{s - 1}^{2} + \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{1}{τ} ‖ Θ ‖_{s}^{2}) (‖ N ‖_{s} + ‖ u ‖_{s} + ‖ Θ ‖_{s}) . \end{matrix} \end{matrix}$

Proof.
Taking the inner product of (32) with $2 A_{0} (p, θ) \partial^{α} W$ in $L^{2}$ , we obtain $\begin{matrix} (38) & \begin{matrix} \frac{d}{d t} ⟨ A_{0} \partial^{α} U, \partial^{α} U ⟩ \\ = ⟨ (\partial_{t} A_{0} + \sum_{j = 1}^{3} \partial_{j} {\tilde{A}}_{j}) \partial^{α} U, \partial^{α} U ⟩ + 2 ⟨ A_{0} \partial^{α} U, g^{α} ⟩ + 2 ⟨ A_{0} \partial^{α} U, \partial^{α} K ⟩ \\ = I_{1} + I_{2} + I_{3} . \end{matrix} \end{matrix}$ with the natural correspondence for $I_{1}$ , $I_{2}$ , $I_{3}$ . Let us estimate them in the following.

Estimates of $I_{1}$ . Noticing (25)–(26), applying Lemmas 2.1, 3.1 to $I_{1}$ yields $\begin{array}{l} \begin{matrix} | ⟨ \partial_{t} A_{0} \partial^{α} U, \partial^{α} U ⟩ | ⩽ C (‖ \nabla N ‖_{s - 1}^{2} + \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{1}{τ} ‖ Θ ‖_{s}^{2}) (‖ N ‖_{s} + ‖ u ‖_{s} + ‖ Θ ‖_{s}), \end{matrix} \\ \begin{matrix} | \sum_{j = 1}^{3} ⟨ \partial_{j} {\tilde{A}}_{j} \partial^{α} U, \partial^{α} U ⟩ | ⩽ C (‖ \nabla N ‖_{s - 1}^{2} + ‖ u ‖_{s}^{2} + ‖ Θ ‖_{s}^{2}) (‖ N ‖_{s} + ‖ u ‖_{s} + ‖ Θ ‖_{s}), \end{matrix} \end{array}$ and then $\begin{matrix} (39) & \begin{matrix} | I_{1} | ⩽ C (‖ \nabla N ‖_{s - 1}^{2} + \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{1}{τ} ‖ Θ ‖_{s}^{2}) (‖ N ‖_{s} + ‖ u ‖_{s} + ‖ Θ ‖_{s}) . \end{matrix} \end{matrix}$

Estimate of $I_{2}$ . In view of the expressions of (22) and (36), we have $\begin{matrix} (40) & \begin{matrix} ‖ g^{α} ‖ & = ‖ A_{j} (n, u, θ) \partial^{α} \partial_{j} U - \partial^{α} (A_{j} (n, u, θ) \partial_{j} U) ‖ \\ ⩽ C {‖ \partial_{x} A_{j} (n, u, θ) ‖}_{L^{\infty}} ‖ \partial^{α} U ‖ + ‖ \partial^{α} A_{j} (n, u, θ) ‖ ‖ \partial_{x} U ‖_{L^{\infty}} \\ ⩽ C {‖ \partial_{x} A_{j} (n, u, θ) ‖}_{H^{2}} ‖ \partial^{α} U ‖ + ‖ \partial^{α} A_{j} (n, u, θ) ‖ ‖ \partial_{x} U ‖_{H^{2}} \\ ⩽ C ‖ \partial^{α} U ‖ ‖ \nabla U ‖_{H^{2}}, \end{matrix} \end{matrix}$ which implies that $\begin{matrix} (41) & \begin{matrix} | I_{2} | & ⩽ C ‖ g^{α} ‖ ‖ A_{0} ‖_{L^{\infty}} ‖ \partial^{α} U ‖ ⩽ C ‖ \nabla U ‖_{H^{2}} {‖ \partial^{α} U ‖}^{2} \\ ⩽ C (‖ \nabla N ‖_{s - 1}^{2} + ‖ u ‖_{| s}^{2} + ‖ Θ ‖_{s}^{2}) ({‖ N ‖}_{s} + {‖ u ‖}_{s} + {‖ Θ ‖}_{s}) . \end{matrix} \end{matrix}$

Estimate of $I_{3}$ . In view of the expressions of (23) and (25), we have $\begin{array}{l} 2 ⟨ A_{0} \partial^{α} K, \partial^{α} U ⟩ \\ = - \frac{2}{τ} ⟨ n, {| \partial^{α} u |}^{2} ⟩ - \frac{2}{τ} ⟨ \frac{n}{θ}, {| \partial^{α} Θ |}^{2} ⟩ - \frac{1}{τ} ⟨ \frac{n}{θ} \partial^{α} Θ, \partial^{α} (u^{2}) ⟩ - 2 ⟨ n \partial^{α} u, \partial^{α} \nabla ϕ ⟩ \\ (42) & ⩽ - \frac{2}{τ} ⟨ n, {| \partial^{α} u |}^{2} ⟩ - \frac{2}{τ} ⟨ \frac{n}{θ}, {| \partial^{α} Θ |}^{2} ⟩ - 2 ⟨ n \partial^{α} u, \partial^{α} \nabla ϕ ⟩ + C ‖ u ‖_{s}^{2} ‖ Θ ‖_{s} . \end{array}$ For the third term on the right-hand side of (42), by an integration by parts, Lemmas 2.1, 3.1, the density equation and the Poisson equation, we get $\begin{matrix} (43) & \begin{matrix} - 2 ⟨ n \partial^{α} u, \partial^{α} \nabla ϕ ⟩ & = - 2 ⟨ \partial^{α} \partial_{t} N, \partial^{α} ϕ ⟩ + 2 ⟨ \partial^{α} (N u) - N \partial^{α} u, \partial^{α} \nabla ϕ ⟩ \\ ⩽ - 2 ⟨ \partial^{α} \partial_{t} N, \partial^{α} ϕ ⟩ + C ‖ \nabla N ‖_{s - 1}^{2} ‖ u ‖_{s} \\ ⩽ 2 ⟨ \partial^{α} Δ \partial_{t} ϕ, \partial^{α} ϕ ⟩ - 2 ⟨ \partial^{α} \partial_{t} (e^{ϕ}), \partial^{α} ϕ ⟩ + C ‖ \nabla N ‖_{s - 1}^{2} ‖ u ‖_{s} \\ ⩽ - \frac{d}{d t} {‖ \partial^{α} \nabla ϕ ‖}^{2} - 2 ⟨ \partial^{α} \partial_{t} (e^{ϕ}), \partial^{α} ϕ ⟩ + C ‖ \nabla N ‖_{s - 1}^{2} ‖ u ‖_{s} \\ ⩽ - \frac{d}{d t} {‖ \partial^{α} \nabla ϕ ‖}^{2} - \frac{d}{d t} ⟨ e^{ϕ} \partial^{α} ϕ, \partial^{α} ϕ ⟩ + ⟨ e^{ϕ} \partial_{t} ϕ \partial^{α} ϕ, \partial^{α} ϕ ⟩ \\ + 2 ⟨ e^{ϕ} \partial^{α} \partial_{t} ϕ - \partial^{α} (e^{ϕ} \partial_{t} ϕ), \partial^{α} ϕ ⟩ + C ‖ \nabla N ‖_{s - 1}^{2} ‖ u ‖_{s} \\ ⩽ - \frac{d}{d t} {‖ \partial^{α} \nabla ϕ ‖}^{2} - \frac{d}{d t} ⟨ e^{ϕ} \partial^{α} ϕ, \partial^{α} ϕ ⟩ + C (‖ \nabla N ‖_{s - 1}^{2} + ‖ \nabla ϕ ‖_{s - 1}^{2}) ‖ u ‖_{s} \\ ⩽ - \frac{d}{d t} {‖ \partial^{α} \nabla ϕ ‖}^{2} - \frac{d}{d t} ⟨ e^{ϕ} \partial^{α} ϕ, \partial^{α} ϕ ⟩ + C ‖ \nabla N ‖_{s - 1}^{2} ‖ u ‖_{s} . \end{matrix} \end{matrix}$ Then, we have $\begin{matrix} (44) & \begin{matrix} I_{3} + \frac{2}{τ} ⟨ n, {| \partial^{α} u |}^{2} ⟩ + \frac{2}{τ} ⟨ \frac{n}{θ}, {| \partial^{α} Θ |}^{2} ⟩ \\ ⩽ - \frac{d}{d t} {‖ \partial^{α} \nabla ϕ ‖}^{2} - \frac{d}{d t} ⟨ e^{ϕ}, {| \partial^{α} ϕ |}^{2} ⟩ + C ‖ \nabla N ‖_{s - 1}^{2} ‖ u ‖_{s} . \end{matrix} \end{matrix}$ Thus, (37) follows from (38)–(39), (41) and (44). The proof of Lemma 3.3 is finished. □

Next, in order to complete the proof of global existence of solutions, we establish the dissipation estimates for the perturbation of the density $N$ .
Lemma 3.4 (Dissipation estimates for $N$ ).

For all $t \in [0, T]$ , $β \in N^{3}$ with $| β | ⩽ s - 1$ , it holds $\begin{matrix} (45) & \begin{matrix} \frac{d}{d t} ⟨ \partial^{β} u, \partial^{β} \nabla N ⟩ + \frac{1}{9} {‖ \partial^{β} \nabla N ‖}^{2} \\ ⩽ C ‖ \nabla Θ ‖_{s - 1}^{2} + \frac{C}{τ^{2}} ‖ u ‖_{s}^{2} + C (‖ \nabla N ‖_{s - 1}^{2} + ‖ u ‖_{| s}^{2}) ({‖ N ‖}_{s} + {‖ Θ ‖}_{s}) . \end{matrix} \end{matrix}$

Proof.
For $β \in N^{3}$ with $| β | ⩽ | α | - 1$ , applying $\partial^{β}$ to the second equation in (20) and taking the inner product with $\partial^{β} \nabla N$ in $L^{2}$ yields $\begin{matrix} (46) & \begin{matrix} ⟨ \frac{θ}{n} \partial^{β} \nabla N, \partial^{β} \nabla N ⟩ + ⟨ \partial^{β} \nabla ϕ, \partial^{β} \nabla N ⟩ \\ = - ⟨ \partial_{t} \partial^{β} u, \partial^{β} \nabla N ⟩ - ⟨ \partial^{β} ((u \cdot \nabla) u), \partial^{β} \nabla N ⟩ - ⟨ \partial^{β} \nabla Θ, \partial^{β} \nabla N ⟩ \\ - \frac{1}{τ} ⟨ \partial^{β} u, \partial^{β} \nabla N ⟩ + ⟨ \frac{θ}{n} \partial^{β} \nabla N - \partial^{β} (\frac{θ}{n} \nabla N), \partial^{β} \nabla N ⟩ . \end{matrix} \end{matrix}$ Let us estimate each term on the right-hand side of (46). For the first term, by an integration by parts and the density equation in (20), it gives $\begin{matrix} (47) & \begin{matrix} - ⟨ \partial_{t} \partial^{β} u, \partial^{β} \nabla N ⟩ \\ = - \frac{d}{d t} ⟨ \partial^{β} u, \partial^{β} \nabla N ⟩ + ⟨ \partial^{β} u, \partial^{β} \nabla \partial_{t} N ⟩ \\ = - \frac{d}{d t} ⟨ \partial^{β} u, \partial^{β} \nabla N ⟩ + ⟨ \partial^{β} div u, \partial^{β} div (n u) ⟩ \\ ⩽ - \frac{d}{d t} ⟨ \partial^{β} u, \partial^{β} \nabla N ⟩ + C ‖ u ‖_{s}^{2} . \end{matrix} \end{matrix}$ For the second term, we get $\begin{array}{l} (48) & - ⟨ \partial^{β} ((u \cdot \nabla) u), \partial^{β} \nabla N ⟩ ⩽ C ‖ \nabla N ‖_{s - 1} ‖ u ‖_{s}^{2} . \end{array}$ For the third and the fourth terms, applying Lemma 2.1 and the Young inequality, we obtain $\begin{array}{l} (49) & \begin{matrix} - ⟨ \partial^{β} \nabla Θ, \partial^{β} \nabla N ⟩ ⩽ C ‖ \nabla Θ ‖_{s - 1}^{2} + \frac{1}{9} {‖ \partial^{β} \nabla N ‖}^{2}, \end{matrix} \\ (50) & \begin{matrix} - \frac{1}{τ} ⟨ \partial^{β} u, \partial^{β} \nabla N ⟩ ⩽ \frac{C}{τ^{2}} ‖ u ‖_{s}^{2} + \frac{1}{9} {‖ \partial^{β} \nabla N ‖}^{2}, \end{matrix} \end{array}$ and $\begin{matrix} (51) & \begin{matrix} ⟨ \frac{θ}{n} \partial^{β} \nabla N - \partial^{β} (\frac{θ}{n} \nabla N), \partial^{β} \nabla N ⟩ ⩽ C ‖ \nabla N ‖_{s - 1}^{2} ({‖ N ‖}_{s} + {‖ Θ ‖}_{s}) . \end{matrix} \end{matrix}$

Next, we estimate $⟨ \partial^{β} \nabla ϕ, \partial^{β} \nabla N ⟩$ on the left hand side of (46). Similarly as that in [22], by the Poisson equation, an integration by parts and Lemmas 2.1–3.1, we have $\begin{matrix} (52) & \begin{matrix} ⟨ \partial^{β} \nabla ϕ, \partial^{β} \nabla N ⟩ & ⩾ ⟨ \partial^{β} \nabla ϕ, \partial^{β} \nabla ((h (ϕ) - 1) ϕ) ⟩ \\ ⩾ - C ‖ \nabla N ‖_{s - 1}^{2} ‖ ϕ ‖_{s} \\ ⩾ - C ‖ \nabla N ‖_{s - 1}^{2} ‖ N ‖_{s}, \end{matrix} \end{matrix}$ Thus, by noting (28), combining (46)–(52), we get (45). The proof of Lemma 3.4 is complete. □

Now we define the total energy $E (t)$ , the dissipation energy $D (t)$ and the rest term $R (t)$ by $\begin{array}{l} (53) & \begin{matrix} E (t) & = \sum_{0 ⩽ | α | ⩽ s} ⟨ A_{0} \partial^{α} U, \partial^{α} U ⟩ + \int_{R^{3}} H (ϕ) d x + \sum_{0 ⩽ | α | ⩽ s} {‖ \partial^{α} \nabla ϕ ‖}^{2} \\ + \sum_{1 ⩽ | α | ⩽ s} ⟨ e^{ϕ}, {| \partial^{α} ϕ |}^{2} ⟩ + k τ \sum_{0 ⩽ | β | ⩽ s - 1} ⟨ \partial^{β} u, \partial^{β} \nabla N ⟩, \end{matrix} \\ (54) & \begin{matrix} D (t) = \frac{2}{τ} \sum_{0 ⩽ | α | ⩽ s} ⟨ n, {| \partial^{α} u |}^{2} ⟩ + \frac{2}{τ} \sum_{0 ⩽ | α | ⩽ s} ⟨ \frac{n}{θ}, {| \partial^{α} Θ |}^{2} ⟩ + \frac{k τ}{9} \sum_{0 ⩽ | β | ⩽ s - 1} {‖ \partial^{β} \nabla N ‖}^{2}, \end{matrix} \end{array}$ and $\begin{matrix} (55) & \begin{matrix} R (t) & = \frac{k}{τ} ‖ u ‖_{s}^{2} + k τ ‖ \nabla Θ ‖_{s - 1}^{2} + k τ (‖ \nabla N ‖_{s - 1}^{2} + ‖ u ‖_{s}^{2}) ({‖ N ‖}_{s} + {‖ Θ ‖}_{s}) \\ + (‖ \nabla N ‖_{s - 1}^{2} + \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{1}{τ} ‖ Θ ‖_{s}^{2}) ({‖ N ‖}_{s} + {‖ u ‖}_{s} + {‖ Θ ‖}_{s}), \end{matrix} \end{matrix}$ where $k > 0$ is a small constant to be determined later.

By Lemmas 3.2–3.4, summing (34), (37) for all indexes $1 ⩽ | α | ⩽ s$ and (45) multiplying by $k τ$ for all indexes $| α | ⩽ s - 1$ , we obtain the following result. Lemma 3.5.
For all $t \in [0, T]$ , it holds that $\begin{matrix} (56) & \begin{matrix} \frac{d}{d t} E (t) + D (t) ⩽ C R (t) . \end{matrix} \end{matrix}$

3.2. Proof of Theorem 1.1

When $ω_{T}$ is small enough, we want to prove that $\begin{matrix} (57) & \begin{matrix} C R (t) ⩽ \frac{1}{2} D (t) . \end{matrix} \end{matrix}$ It follows from (27)–(28) that $\begin{matrix} (58) & \begin{matrix} D (t) ⩾ \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{2}{3 τ} ‖ Θ ‖_{s}^{2} + \frac{k τ}{9} ‖ \nabla N ‖_{s - 1}^{2} . \end{matrix} \end{matrix}$

Firstly, we choose small constant $k > 0$ such that $\begin{matrix} (59) & \begin{matrix} C k ⩽ \frac{1}{4}, \end{matrix} \end{matrix}$ which implies that $\begin{matrix} (60) & \begin{matrix} C R (t) & ⩽ \frac{1}{4 τ} ‖ u ‖_{s}^{2} + \frac{1}{4} τ ‖ \nabla Θ ‖_{s - 1}^{2} + \frac{1}{4} τ (‖ \nabla N ‖_{s - 1}^{2} + ‖ u ‖_{s}^{2}) ({‖ N ‖}_{s} + {‖ Θ ‖}_{s}) \\ + \frac{1}{4 k} (‖ \nabla N ‖_{s - 1}^{2} + \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{1}{τ} ‖ Θ ‖_{s}^{2}) ({‖ N ‖}_{s} + {‖ u ‖}_{s} + {‖ Θ ‖}_{s}) . \end{matrix} \end{matrix}$ Next, for fixed $k > 0$ , we use the condition that $ω_{T}$ is small enough with respect to T to obtain $\begin{matrix} (61) & \begin{matrix} \frac{1}{4 k} (‖ \nabla N ‖_{s - 1}^{2} + \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{1}{τ} ‖ Θ ‖_{s}^{2}) ({‖ N ‖}_{s} + {‖ u ‖}_{s} + {‖ Θ ‖}_{s}) \\ ⩽ \frac{k τ}{36} ‖ \nabla N ‖_{s - 1}^{2} + \frac{1}{8 τ} ‖ u ‖_{s}^{2} + \frac{1}{4 τ} ‖ Θ ‖_{s}^{2}, \end{matrix} \end{matrix}$ and $\begin{matrix} (62) & \begin{matrix} \frac{1}{4} τ (‖ \nabla N ‖_{s - 1}^{2} + ‖ u ‖_{s}^{2}) ({‖ N ‖}_{s} + {‖ Θ ‖}_{s}) ⩽ \frac{k τ}{36} ‖ \nabla N ‖_{s - 1}^{2} + \frac{1}{8 τ} ‖ u ‖_{s}^{2} . \end{matrix} \end{matrix}$ Then, from (60)–(62), we have $\begin{matrix} (63) & \begin{matrix} C R (t) ⩽ \frac{1}{2 τ} ‖ u ‖_{s}^{2} + \frac{1}{2 τ} ‖ Θ ‖_{s}^{2} + \frac{k τ}{18} ‖ \nabla N ‖_{s - 1}^{2}, \end{matrix} \end{matrix}$ which together with (58) imply (57) provided that $0 < τ ⩽ \frac{2}{3}$ .

We deduce from Lemma 3.5 that $\begin{matrix} (64) & \begin{matrix} \frac{d}{d t} E (t) + \frac{1}{2} D (t) ⩽ 0, \forall t \in [0, T] . \end{matrix} \end{matrix}$ Integrating this inequality over $[0, t]$ yields $\begin{matrix} (65) & \begin{matrix} E (t) + \frac{1}{2} \int_{0}^{t} D (ξ) d ξ ⩽ E (0), \forall t \in [0, T] . \end{matrix} \end{matrix}$ It is obviously that $\begin{matrix} D (t) \sim ‖ \nabla N ‖_{s - 1}^{2} + \frac{1}{τ} ‖ u ‖_{s}^{2} + \frac{1}{τ} ‖ Θ ‖_{s}^{2} . \end{matrix}$ On the other hand, in view of the expression of $H (ϕ)$ in Lemma 3.2, we get $\begin{matrix} H (ϕ) = ϕ^{2} + O (ϕ^{3}), \end{matrix}$ which implies that $\begin{matrix} H (ϕ) \sim ϕ^{2}, for | ϕ | small enough . \end{matrix}$ Thus, due to the fact that $A_{0}$ is positive definite and the assumption that k is small enough, we have $\begin{matrix} E (t) \sim {‖ n (t) - 1 ‖}_{s}^{2} + {‖ u (t) ‖}_{s}^{2} + {‖ θ (t) - 1 ‖}_{s}^{2} + {‖ ϕ (t) ‖}_{s + 1}^{2} . \end{matrix}$ Thus, from (65) we obtain estimate (7), which gives the uniformly global existence of solutions by a standard continuous argument. Moreover, (65) together with the Poisson equation in (2) implies (8). The proof of Theorem 1.1 is finished.

4. Global convergence of smooth solutions as $τ \to 0$

4.1. Proof of Theorem 1.2

By (2) and (9), $(ρ^{τ}, v^{τ}, θ^{τ}, ψ^{τ})$ satisfy $\begin{matrix} (66) & \{\begin{matrix} \partial_{t} ρ^{τ} + div (ρ^{τ} v^{τ}) = 0, \\ τ^{2} (\partial_{t} v^{τ} + (v^{τ} \cdot \nabla) v^{τ}) + \nabla (ρ^{τ} θ^{τ} + ψ^{τ}) = - v^{τ}, \\ τ^{2} (\partial_{t} θ^{τ} + v^{τ} \cdot \nabla θ^{τ} + θ^{τ} div v^{τ} - \frac{1}{2} {| v^{τ} |}^{2}) + θ^{τ} - 1 = 0, \\ - Δ ψ^{τ} + e^{ψ^{τ}} = ρ^{τ} . \end{matrix} \end{matrix}$ By the estimate (7) in Theorem 1.1, it holds that $\begin{matrix} (67) & \begin{matrix} sup_{t ⩾ 0} ({‖ ρ^{τ} (t) - 1 ‖}_{s}^{2} + τ^{2} {‖ v^{τ} (t) ‖}_{s}^{2} + {‖ θ^{τ} (t) - 1 ‖}_{s}^{2} + {‖ ψ^{τ} (t) ‖}_{s + 1}^{2}) \\ + \int_{0}^{t} {‖ v^{τ} (ξ) ‖}_{s}^{2} d ξ \\ ⩽ C ({‖ n_{0}^{τ} - 1 ‖}_{s}^{2} + {‖ u_{0}^{τ} ‖}_{s}^{2} + {‖ θ_{0}^{τ} - 1 ‖}_{s}^{2}) . \end{matrix} \end{matrix}$ Hence, the sequence ${(ρ^{τ} - 1)}_{τ > 0}$ , ${(θ^{τ} - 1)}_{τ > 0}$ , ${(ψ^{τ})}_{τ > 0}$ , and ${(v^{τ})}_{τ > 0}$ are bounded in $L^{\infty} (R^{+}, H^{s} (R^{3}))$ , $L^{\infty} (R^{+}, H^{s} (R^{3}))$ , $L^{\infty} (R^{+}, H^{s + 1} (R^{3}))$ , and $L^{2} (R^{+}, H^{s} (R^{3}))$ , respectively. It follows that there exist functions $\bar{n}$ , $\bar{v}$ , $\bar{θ}$ , and $\bar{ϕ}$ such that, up to subsequences, as $τ \to 0$ , (12)–(15) hold, $\begin{matrix} τ^{2} (\partial_{t} v^{τ} + (v^{τ} \cdot \nabla) v^{τ}) ⇀ 0 in D^{'} (R^{+} \times R^{3}), \end{matrix}$ and $\begin{matrix} τ^{2} (\partial_{t} θ^{τ} + v^{τ} \cdot \nabla θ^{τ} + θ^{τ} div v^{τ} - \frac{1}{2} {| v^{τ} |}^{2}) ⇀ 0 in D^{'} (R^{+} \times R^{3}) . \end{matrix}$ Moreover, by the first equation in (66), ${(\partial_{t} ρ^{τ})}_{τ > 0}$ is bounded in $L^{2} (R^{+}, H^{s - 1} (R^{3}))$ . Let $T > 0$ . By a classical compactness theorem [29], for all $s_{*} \in [0, s)$ , ${(ρ^{τ} - 1)}_{τ > 0}$ , is relatively compact in $C ([0, T]; H_{loc}^{s_{*}})$ , which implies the strong convergence (16). It follows from (31) in Lemma 3.1 that $\begin{matrix} {‖ \partial_{t} ψ^{τ} ‖}_{s} ⩽ C {‖ v^{τ} ‖}_{s} . \end{matrix}$ Thus, the sequence ${(\partial_{t} ψ^{τ})}_{τ > 0}$ is bounded in $L^{2} (R^{+}, H^{s} (R^{3}))$ . We deduce that, for all $s_{*} \in [0, s)$ , it is relatively compact in $C ([0, T]; H_{loc}^{s_{*} + 1})$ , which gives the strong convergence (17). These convergence imply that, up to subsequences, $\begin{array}{l} \partial_{t} ρ^{τ} ⇀ \partial_{t} \bar{n}, weakly in L^{2} (R^{+}, H^{s - 1} (R^{3})), \\ \nabla (ρ^{τ} θ^{τ}) ⇀ \nabla (\bar{n} \bar{θ}), weakly* in L^{\infty} (R^{+}, H^{s - 1} (R^{3})), \\ \nabla ψ^{τ} ⇀ \nabla \bar{ϕ}, weakly* in L^{\infty} (R^{+}, H^{s} (R^{3})), \\ Δ ψ^{τ} ⇀ Δ \bar{ϕ}, weakly* in L^{\infty} (R^{+}, H^{s - 1} (R^{3})), \\ e^{ψ^{τ}} - 1 ⇀ e^{\bar{ϕ}} - 1, weakly* in L^{\infty} (R^{+}, H^{s + 1} (R^{3})) . \end{array}$ The above convergence allow us to pass to the limit in (66) to obtain $\begin{matrix} \{\begin{matrix} \partial_{t} \bar{n} + div (\bar{n} \bar{v}) = 0, \\ - Δ \bar{ϕ} + e^{\bar{ϕ}} = \bar{n}, \end{matrix} \end{matrix}$ with $\begin{matrix} \bar{v} = - \nabla (\bar{n} \bar{θ} + \bar{ϕ}), \bar{θ} = θ_{l} = 1 . \end{matrix}$ Finally, the initial condition (19) follows from the uniform convergence (16) and (10). The uniqueness of solutions to (18)–(19) with small smooth initial data yields the convergence of the whole sequence ${(ρ^{τ}, θ^{τ}, ψ^{τ})}_{τ > 0}$ and then ${(v^{τ})}_{τ > 0}$ . This complete the proof of Theorem 1.2.

Footnotes

Acknowledgements

The authors are grateful to the referee for the comments. The authors are supported by the BNSF (1164010, 1132006), NSFC (11771031, 11371042, 11831003), NSF of Qinghai Province (2017-ZJ-908), NSF of Henan Province (162300410084), the Key Research Fund of Henan Province (16A110019), the general project of scientific research project of the Beijing education committee of China, the 2016 Beijing project of scientific activities for the excellent students studying abroad.

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Global zero-relaxation limit of the non-isentropic Euler–Poisson system for ion dynamics

Abstract

Keywords

1. Introduction and main results

Proposition 1.1 (Local existence of smooth solutions, see [18,24]).

Theorem 1.1 (Global existence of smooth solutions with respect to τ).

2. Preliminaries and reformulation of system (2)

2.1. Preliminaries

Lemma 2.1 (Moser-type calculus inequalities, see [19,24]).

2.2. Reformulation of system (2)

3. Global existence of solutions to problem (2)–(3)

Lemma 3.2 ( L 2 estimates).

4. Global convergence of smooth solutions as τ → 0

4.1. Proof of Theorem 1.2

Footnotes

Acknowledgements

References

Lemma 3.2 ( $L^{2}$ estimates).

4. Global convergence of smooth solutions as $τ \to 0$