Global Convergence to Compressible Full Navier–Stokes Equations by Approximation with Oldroyd-Type Constitutive Laws

Abstract

We consider smooth solutions to a relaxed Euler system with Oldroyd-type constitutive laws. This system is derived from the one-dimensional compressible full Navier-Stokes equations for a Newtonian fluid by using the Cattaneo–Christov model and the Oldroyd-B model. In a neighborhood of equilibrium states, we construct an explicit symmetrizer and show that the system is symmetrizable hyperbolic with partial dissipation. Moreover, by establishing uniform estimates with respect to the relaxation times, we prove the uniform global existence of smooth solutions and the global-in-time convergence of the system towards the full Navier–Stokes equations.

1 Introduction

Euler and Navier-Stokes equations are two important models in fluid dynamics. There are rich literatures on the mathematical analysis around these equations. We refer to [11, 23,24,25,26] and references therein for mathematical results. There are deep relationships between Euler and Navier-Stokes equations. It is well known that the Euler equations can be derived from Navier-Stokes equations through the vanishing viscosity limit. Meanwhile, the Navier-Stokes equations can also be approximated by first-order partial differential equations using different kinds of constitutive laws for non-Newtonian fluids. These approximate equations are referred to as relaxed Euler systems or hyperbolic Navier-Stokes equations, see for instance [10, 16, 30, 33, 34].

In the paper, we study the global-in-time convergence from relaxed Euler-type equations with Oldroyd’s constitutive laws to compressible full (non-isentropic) Navier-Stokes equations by letting relaxation times tend to zero. Let $t\ge 0$ be the time variable and $x=(x_1,\cdots , x_d)\in {\mathbb {R}}^d$ be the space variable. The compressible full Navier-Stokes equations are of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho +\mathrm {div}(\rho u)=0,\\ \partial _t(\rho u)+\mathrm {div}(\rho u\otimes u)+\nabla p= \mathrm {div}\tau ,\\ \partial _t(\rho E)+\mathrm {div}(\rho uE+up)+ \mathrm {div}q=\mathrm {div}(u\tau ), \end{array}\right. } \end{aligned}$$

(1.1)

in ${\mathbb {R}}^+\times {\mathbb {R}}^d$, where $\rho >0$ is the density, $u=(u_1, \cdots , u_d)^\top \in {\mathbb {R}}^d$ is the velocity, $\theta$ is the temperature, p is the pressure function, $q\in {\mathbb {R}}^d$ is the heat flux, $\tau$ is the stress tensor and $E=e+\dfrac{1}{2}|u|^2$ is the total energy per unit mass with e being the specific internal energy. The symbols $\top$ and $\otimes$ represent the transpose and the tensor product, respectively. In (1.1), $\rho$, u and $\theta$ are independent variables, e and p are functions of $(\rho ,\theta )$. In particular, for the ideal fluid, we have

$$\begin{aligned} e=c_v\theta , \qquad p=R\rho \theta , \end{aligned}$$

(1.2)

which satisfy the thermodynamic equation

$$\begin{aligned} \rho ^2 e_\rho =p-\theta p_\theta , \end{aligned}$$

(1.3)

where R and $c_v$ are positive constants, and $e_\rho =\frac{\partial e}{\partial \rho }$ etc. Generally speaking, the heat flux q satisfies the following Fourier’s law

$$\begin{aligned} q=-\kappa \nabla \theta , \end{aligned}$$

(1.4)

where $\kappa >0$ is the heat conduction constant. For Newtonian viscous fluids, the stress tensor $\tau$ takes the form

$$\begin{aligned} \tau = \mu \sigma (u)+\lambda (\mathrm {div}u)I_d, \end{aligned}$$

(1.5)

with

$$\begin{aligned} \sigma (u)=\nabla u+(\nabla u)^\top -\dfrac{2}{d}(\mathrm {div}u)I_d. \end{aligned}$$

Here $I_d$ denotes the unit matrix of order d. The parameters $\mu >0$ and $\lambda >0$ are the shear and bulk viscosity coefficients, respectively, which are all assumed to be constants.

For the full Navier-Stokes equations (1.1) with constitutive laws (1.4) and (1.5), the construction of the corresponding relaxed Euler systems depends on the way how to decompose the second-order derivatives $\mathrm {div}\tau , \mathrm {div}(u\tau )$ and $\mathrm {div}q$ into first-order derivative terms. Clearly, there are lots of ways to do it, among which the most natural one is to replace (1.4) and (1.5) by the following Maxwell’s constitutive laws [28]

$$\begin{aligned} \varepsilon _1^2 \partial _tq+q= & {} -\kappa \nabla \theta , \end{aligned}$$

(1.6)

$$\begin{aligned} \varepsilon _2^2 \partial _t\tau +\tau= & {} \mu \sigma (u)+\lambda (\mathrm {div}u)I_d, \end{aligned}$$

(1.7)

where $\varepsilon _1>0$ and $\varepsilon _2>0$ are the relaxation times. Equations in (1.1) together with (1.6)-(1.7) form a relaxed Euler system. Formally, letting $(\varepsilon _1,\varepsilon _2)\rightarrow 0$ recovers the Navier-Stokes equations (1.1) with (1.4) and (1.5). This idea is not recent. It dates back to 1860s, see for instance [3, 4, 28]. These approximations have not only the mathematical sense but also physical interpretations. Relation (1.6), also known as the Cattaneo’s law, gives rise to heat waves with finite propagation speed. Relation (1.7) describes motions of viscoelastic fluids. The laws are combinations of the Newtonian’s law of viscosity and the Hooke’s law of elasticity.

The existence of smooth solutions to system (1.1) with Maxwell’s constitutive laws (1.6) and (1.7) and their convergence to the classical non-isentropic Navier-Stokes equations (1.1) with (1.4) and (1.5) have been studied in previous works. In the case where $\varepsilon _1=0$, the authors of [15] proved the global existence of smooth solutions near constant equilibrium states for fixed $\varepsilon _2>0$ and the local-in-time convergence towards the Navier-Stokes equations as $\varepsilon _2\rightarrow 0$. Similar results are obtained in [14] in the case where $\varepsilon _2=0$ and $\varepsilon _1>0$. In these results, only one of the constitutive laws within (1.6) and (1.7) is used. Hence the systems studied in [14, 15] are of mixed hyperbolic-parabolic type in the sense of Shizuta-Kawashima [19, 39]. The local-in-time convergence is based on the error estimates between the original system and the limiting system. For the isentropic Navier-Stokes equations with constitutive law (1.7), the author of [42] obtained the local existence and the local convergence to the classical isentropic Navier-Stokes equations under condition $\text {tr}(\tau )=0$, where $\text {tr}(\tau )$ means the trace of matrix $\tau$. In [30], the first author of the present paper constructed approximate systems with vector variables instead of tensor variables by using Hurwitz-Radon matrices in both compressible and incompressible cases. He proved the uniform (with respect to $\varepsilon _1$ and $\varepsilon _2$) global existence of smooth solutions near constant equilibrium state and the global-in-time convergence of the systems towards classical isentropic Navier-Stokes equations. He also obtained similar results for the isentropic Navier-Stokes equations with constitutive law (1.7) without condition $\text {tr}(\tau )=0$. For the approximation of incompressible isentropic Navier-Stokes equations with constitutive law (1.7), we also refer to [10, 33, 34, 37, 38].

However, these two constitutive laws (1.6) and (1.7) have drawbacks as they do not ensure Galilean invariance. In other words, these laws lead to paradoxical evolution of thermal waves in a moving frame, see [7]. To overcome it, the Oldroyd’s upper-convected time derivative (or simply Oldroyd derivative) should be considered. In this paper, we consider the following two constitutive laws. The first one is the Cattaneo-Christov model introduced in [6],

$$\begin{aligned} \varepsilon _1^2\big (\partial _tq+u\cdot \nabla q-q\cdot \nabla u+(\mathrm {div}u)q\big ) =-q-\kappa \nabla \theta , \end{aligned}$$

(1.8)

in which the terms on the left-hand side are the Oldroyd derivative. It is proved in [6] that the constitutive law (1.8) is Galilean invariant. The second one is the following Oldroyd-B model for the tensor variable $\tau$ (see for instance [2, 29, 35, 36] and the references therein)

$$\begin{aligned}&\varepsilon _2^2(\partial _t\tau +u\cdot \nabla \tau +g(\tau ,\nabla u))+\tau =\mu \sigma (u)+\lambda (\mathrm {div}u)I_d, \end{aligned}$$

(1.9)

where

$$\begin{aligned} g(\tau ,\nabla u)=\tau W(u)-W(u)\tau , \quad \text {with} \quad W(u) =\dfrac{1}{2}\big (\nabla u-(\nabla u)^\top \big ). \end{aligned}$$

Hence, the relaxed Euler system for (1.1) with constitutive laws (1.8) and (1.9) is of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho +\mathrm {div}(\rho u)=0,\\ \partial _t(\rho u)+\mathrm {div}(\rho u\otimes u)+\nabla p= \mathrm {div}\tau ,\\ \partial _t(\rho E)+\mathrm {div}(\rho uE+up)+ \mathrm {div}q=\mathrm {div}(u\tau ),\\ \varepsilon _1^2\big (\partial _tq +u\cdot \nabla q-q\cdot \nabla u+(\mathrm {div}u) q\big ) =-q-\kappa \nabla \theta , \\ \varepsilon _2^2(\partial _t\tau +u\cdot \nabla \tau +g(\tau ,\nabla u))+\tau \\ \qquad \qquad \qquad =\mu \Big (\nabla u+\nabla u^\top -\dfrac{2}{d}(\mathrm {div}u)I_d\Big )+\lambda (\mathrm {div}u)I_d, \end{array}\right. } \end{aligned}$$

(1.10)

in ${\mathbb {R}}^+\times {\mathbb {R}}^d$.

System (1.10) is very complicated. One can observe that it contains $d^2+2d+2$ equations for fluids in space ${\mathbb {R}}^d$. So far the symmetrizable hyperbolicity for (1.10) is unknown in cases $d\ge 2$. This makes it hard to establish the existence results. See classical theories [18, 21, 24]. Moreover, when considering the constitutive laws (1.8) and (1.9) at the same time, there are no apparent dissipative structures for $\nabla u$ and $\nabla \theta$ due to the loss of the elliptic structures for u and $\theta$. This is different from the situation for the systems treated in [14, 15].

In a recent paper [16], the authors considered (1.10) in one space dimension. In this case, system (1.10) is reduced to the following form

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho +\partial _x(\rho u)=0,\\ \partial _t(\rho u)+\partial _x(\rho u^2+ p-\tau )=0,\\ \partial _t(\rho E)+\partial _x(\rho uE+up+q-u\tau )=0,\\ \varepsilon _1^2 (\partial _tq +u\partial _xq)+\kappa \partial _x\theta =-q, \\ \varepsilon _2^2(\partial _t\tau +u\partial _x\tau )-\lambda \partial _xu=-\tau , \end{array}\right. } \end{aligned}$$

(1.11)

in ${\mathbb {R}}^+\times {\mathbb {R}}$, with the initial data

$$\begin{aligned} (\rho , u, \theta , q, \tau )|_{t=0} =(\rho ^\varepsilon _0, u^\varepsilon _0, \theta ^\varepsilon _0, q^\varepsilon _0, \tau ^\varepsilon _0)(x), \end{aligned}$$

(1.12)

where $\varepsilon =(\varepsilon _1,\varepsilon _2)$. In the limit as $\varepsilon \rightarrow 0$, we have formally

$$\begin{aligned} q=-\kappa \partial _x\theta , \quad \tau =\lambda \partial _xu. \end{aligned}$$

Substituting these relations into (1.11), we recover the following one-dimensional non-isentropic Navier-Stokes equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho +\partial _x(\rho u)=0,\\ \partial _t(\rho u)+\partial _x(\rho u^2)+\partial _xp= \lambda \partial _{xx} u,\\ \partial _t(\rho E)+\partial _x(\rho uE+up)=\kappa \partial _{xx}\theta +\lambda \partial _x(u\partial _xu), \end{array}\right. } \end{aligned}$$

(1.13)

in ${\mathbb {R}}^+\times {\mathbb {R}}$. Equations (1.13) have been widely studied. We refer to [13, 17, 20, 27] for the global existence of smooth solutions. See also [11, 23] for related topics and references therein. In (1.11), $\rho$, u, $\theta$, q and $\tau$ are independent variables, and both the internal energy e and the pressure function p are functions of $(\rho ,\theta ,q,\tau )$. In [16], the authors used the following state equations for e and p :

$$\begin{aligned} e= & {} c_v\theta +\dfrac{\varepsilon _1^2}{\kappa \rho \theta }q^2 +\dfrac{\varepsilon _2^2}{2\lambda \rho }\tau ^2, \end{aligned}$$

(1.14)

$$\begin{aligned} p= & {} R\rho \theta -\dfrac{\varepsilon _1^2}{2\kappa \theta }q^2 -\dfrac{\varepsilon _2^2}{2\lambda }\tau ^2. \end{aligned}$$

(1.15)

Equation (1.14) is based on the results in [8], where the authors rigorously proved that the constitutive law (1.8) is consistent with the second law of thermodynamics if and only if the dependence of e on q is quadratic. Similarly, the quadratic dependence of e on $\tau$ also implies the compatibility with the second law of thermodynamics. The choice of equation (1.15) makes it consistent with (1.3) and the state equations (1.14)-(1.15) yield formally those for the ideal fluid (1.2) as $\varepsilon \rightarrow 0$. For more explanations, see [5, 9, 16, 41].

Let $V_e=(1,0,1,0,0)^\top$ be an equilibrium state for (1.11). Due to the explicit expressions (1.14) and (1.15), the authors of [16] constructed a strictly convex entropy for system (1.11), see Lemma 3.1 in [16] or Lemma 3.1 below. Based on this, they established the global existence of smooth solutions near $V_e$ for fixed $\varepsilon _1>0$ and $\varepsilon _2>0$, and the local convergence of the system towards the classical full Navier-Stokes equations as $\varepsilon _1=\varepsilon _2\rightarrow 0$. However, the convergence for large time has not been investigated.

The purpose of this paper is to study the global convergence of system (1.11) with state equations (1.14) and (1.15). The main results of the paper are contained in the following two theorems. The first theorem shows the uniform global existence of smooth solutions to the Cauchy problem (1.11)-(1.12) near $V_e$. The second one concerns the global convergence of the solution of (1.11) to that of the one-dimensional full Navier-Stokes equations (1.13) for the ideal fluid (1.2) as $\varepsilon \rightarrow 0$. Remark that in these results condition $\varepsilon _1=\varepsilon _2$ is not needed and system (1.11) is not included in the class of systems studied in [22, 31, 32, 43]. The proof of the results is based on uniform estimates with respect to the time and the relaxation parameters. We use the strictly convex entropy given in [16] for the $L^2$ estimate. A key step is to find an appropriate symmetrizer of system (1.11) for higher-order estimates. It is well-known that the second-order derivative of a strictly convex entropy provides a symmetrizer for a system of conservation laws [1, 12]. However, this result cannot be applied to (1.11) because it is a non-conservative system.

It is worth mentioning that the global existence result in the present paper is different from that obtained in [16], which is not uniform with respect to $\varepsilon _1$ and $\varepsilon _2$. More precisely, there are terms $\partial _tq$ and $\partial _t\tau$ in the definition of the energy in [16]. Because of boundary layers in the limit as $\varepsilon \rightarrow 0$, such an energy cannot be uniformly bounded with respect to the relaxation parameters. In order to avoid this situation, in the proof of our results, the energy contains only terms of derivative of solutions with respect to x.

Theorem 1.1

(Uniform global existence). Let $s\ge 2$ be an integer and $(\rho _0^\varepsilon -1, u_0^\varepsilon , \theta _0^\varepsilon -1, q_0^\varepsilon , \tau _0^\varepsilon )\in H^s({\mathbb {R}})$. There exist two positive constants $\delta$ and C, independent of $\varepsilon _1$ and $\varepsilon _2$, such that if

$$\begin{aligned} \Vert \rho _0^\varepsilon -1\Vert _s+\Vert u_0^\varepsilon \Vert _s+\Vert \theta _0^\varepsilon -1\Vert _s +\varepsilon _1\Vert q_0^\varepsilon \Vert _s+\varepsilon _2\Vert \tau _0^\varepsilon \Vert \le \delta , \end{aligned}$$

then for all $\varepsilon _1,\varepsilon _2\in (0,1]$, the Cauchy problem (1.11)-(1.12) together with (1.14)-(1.15) admits a unique global smooth solution $(\rho ^\varepsilon , u^\varepsilon ,\theta ^\varepsilon , q^\varepsilon , \tau ^\varepsilon )$ satisfying

$$\begin{aligned} \rho ^\varepsilon -1, u^\varepsilon , \theta ^\varepsilon -1, q^\varepsilon , \tau ^\varepsilon \in C({\mathbb {R}}^+;H^s({\mathbb {R}}))\cap C^1({\mathbb {R}}^+;H^{s-1}({\mathbb {R}})), \end{aligned}$$

and

$$\begin{aligned}&\Vert \rho ^\varepsilon (t)-1\Vert _s^2+\Vert u^\varepsilon (t)\Vert _s^2+\Vert \theta ^\varepsilon (t)-1\Vert _s^2 +\varepsilon _1^2 \Vert q^\varepsilon (t)\Vert _s^2+\varepsilon _2^2\Vert \tau ^\varepsilon (t)\Vert _s^2\nonumber \\&\qquad +\int _0^t\big (\Vert \partial _x\rho ^\varepsilon (t^\prime )\Vert _{s-1}^2+\Vert \partial _xu^\varepsilon (t^\prime )\Vert _{s-1}^2+\Vert \partial _x\theta ^\varepsilon (t^\prime )\Vert _{s-1}^2 +\Vert q^\varepsilon (t^\prime )\Vert _s^2+\Vert \tau ^\varepsilon (t^\prime )\Vert _s^2\big )dt^\prime \nonumber \\&\quad \le C\left( \Vert \rho _0^\varepsilon -1\Vert _s^2+\Vert u_0^\varepsilon \Vert _s^2+\Vert \theta _0^\varepsilon -1\Vert _s^2 +\varepsilon _1^2\Vert q_0^\varepsilon \Vert _s^2+\varepsilon _2^2\Vert \tau _0^\varepsilon \Vert _s^2\right) , \quad \forall ~ t\ge 0, \end{aligned}$$

(1.16)

where $\Vert \cdot \Vert _k$ denotes the usual norm of $H^k({\mathbb {R}})$.

Theorem 1.2

(Global convergence). Let $\varepsilon =(\varepsilon _1,\varepsilon _2)$ and $(\rho ^\varepsilon , u^\varepsilon , \theta ^\varepsilon , q^\varepsilon , \tau ^\varepsilon )$ be the global solution obtained in Theorem 1.1, then there exist functions $({\bar{\rho }},{\bar{u}},{\bar{\theta }})\in L^\infty ({\mathbb {R}}^+;H^s({\mathbb {R}}))$ and $({\bar{q}},{\bar{\tau }})\in L^2({\mathbb {R}}^+; H^s({\mathbb {R}}))$, such that, as $\varepsilon \rightarrow 0$ up to subsequences,

$$\begin{aligned} (\rho ^\varepsilon ,u^\varepsilon ,\theta ^\varepsilon )&\rightharpoonup&({\bar{\rho }},{\bar{u}},{\bar{\theta }}) \quad \text {weakly-}* \; \text {in} \;\; L^\infty ({\mathbb {R}}^+;H^s({\mathbb {R}})), \end{aligned}$$

(1.17)

$$\begin{aligned} (q^\varepsilon , \tau ^\varepsilon )&\rightharpoonup&({\bar{q}},{\bar{\tau }}) \quad \text {weakly in} \;\; L^2({\mathbb {R}}^+;H^s({\mathbb {R}})), \end{aligned}$$

(1.18)

where $({\bar{\rho }},{\bar{u}},{\bar{\theta }})$ is the solution to the one-dimensional full compressible Navier-Stokes equations (1.13) for the ideal fluid (1.2), with initial value $({\bar{\rho }}_0,{\bar{u}}_0,{\bar{\theta }}_0)$ which is the weak limit of $(\rho _0^\varepsilon , u_0^\varepsilon , \theta _0^\varepsilon )$ up to subsequences. Moreover,

$$\begin{aligned} {\bar{q}}=-\kappa \partial _x{\bar{\theta }}, \quad {\bar{\tau }}=\lambda \partial _x{\bar{u}}. \end{aligned}$$

The rest of the paper is organized as follows. In the next section, we prove results on the hyperbolic structure for system (1.11). These results are crucial in the proof of the above theorems. Section 3 is devoted to uniform global estimates. The proof of the theorems are completed in the last section.

2 Symmetrizable Hyperbolicity

In what follows, $s\ge 2$ denotes an integer. Let $\varepsilon =(\varepsilon _1,\varepsilon _2)$ and C be a generic positive constant independent of $\varepsilon$ and any time. We assume that $\varepsilon _1,\varepsilon _2\in (0,1]$. For a integer $k\ge 1$, we denote by $\Vert \cdot \Vert _k$, $\Vert \cdot \Vert$ and $\Vert \cdot \Vert _\infty$ the norms of the usual Sobolev spaces $H^k({\mathbb {R}})$, $L^2({\mathbb {R}})$ and $L^\infty ({\mathbb {R}})$, respectively. The inner product in $L^2\left( {\mathbb {R}}\right)$ is denoted by $\big \langle \cdot ,\,\cdot \big \rangle$. In the proof we will frequently use the fact that the embedding from $H^l({\mathbb {R}})$ to $L^\infty ({\mathbb {R}})$ is continuous for all integers $l\ge 1$.

For simplicity, the dependence of solution on the parameters $\varepsilon _1$ and $\varepsilon _2$ is not expressed explicitly. We want to write (1.11) into a first-order quasilinear system of variables $(\rho ,u,\theta ,q,\tau )$. First, it is clear that by using (1.15) and the first equation in (1.11), the second equation in (1.11) is equivalent to

$$\begin{aligned} \rho (\partial _tu+u\partial _xu)+p_\rho \partial _x\rho +p_\theta \partial _x\theta +p_q\partial _xq+(p_\tau -1)\partial _x\tau =0. \end{aligned}$$

Similarly, by using the first two equations in (1.11), the third equation in (1.11) is equivalent to

$$\begin{aligned} \rho \partial _te+\rho u\partial _xe+(p-\tau )\partial _xu+\partial _xq=0. \end{aligned}$$

(2.1)

This equation can be further treated by using equations (1.3), (1.14) and (1.15). Indeed,

$$\begin{aligned} \rho \partial _te= & {} \rho e_\theta \partial _t\theta + \rho e_\rho \partial _t\rho +\rho e_q \partial _tq +\rho e_\tau \partial _t\tau \\= & {} \rho e_\theta \partial _t\theta + \rho e_\rho \partial _t\rho +\rho \dfrac{2q}{\kappa \theta \rho }(-\varepsilon _1^2 u\partial _xq-q-\kappa \partial _x\theta )+\rho \dfrac{\tau }{\lambda \rho }(-\varepsilon _2^2 u\partial _x\tau -\tau +\lambda \partial _xu)\\= & {} \rho e_\theta \partial _t\theta + \rho e_\rho \partial _t\rho -\dfrac{2q}{\theta }\partial _x\theta -\dfrac{2\varepsilon _1^2uq}{\kappa \theta }\partial _xq-\dfrac{2q^2}{\kappa \theta }-\dfrac{\varepsilon _2^2u\tau }{\lambda }\partial _x\tau -\dfrac{\tau ^2}{\lambda }+\tau \partial _xu, \end{aligned}$$

and

$$\begin{aligned} \rho u\partial _xe= & {} \rho ue_\theta \partial _x\theta +\rho ue_\rho \partial _x\rho +\rho ue_q \partial _xq +\rho ue_\tau \partial _x\tau \\= & {} \rho u e_\theta \partial _x\theta +\rho u e_\rho \partial _x\rho +\dfrac{2\varepsilon _1^2uq}{\kappa \theta }\partial _xq+\dfrac{\varepsilon _2^2u\tau }{\lambda }\partial _x\tau , \end{aligned}$$

hence,

$$\begin{aligned} \rho \partial _t e+\rho u\partial _xe=\rho e_\theta \partial _t\theta +\Big (\rho u e_\theta -\frac{2q}{\theta }\Big )\partial _x\theta +\rho e_\rho (\partial _t\rho +u\partial _x\rho )-\frac{2q^2}{\kappa \theta }-\frac{\tau ^2}{\lambda }+\tau \partial _xu. \end{aligned}$$

Combining the last equation with (1.3), (2.1) and the first equation in (1.11), we have

$$\begin{aligned} \rho e_\theta \partial _t\theta +\theta p_\theta \partial _xu +\Big (\rho u e_\theta -\frac{2q}{\theta }\Big )\partial _x\theta +\partial _xq =\frac{2q^2}{\kappa \theta }+\frac{\tau ^2}{\lambda }. \end{aligned}$$

Therefore, system (1.11) is equivalent to

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho +\partial _x(\rho u)=0,\\ \rho (\partial _tu+u\partial _xu)+p_\rho \partial _x\rho +p_\theta \partial _x\theta +p_q\partial _xq +(p_\tau -1)\partial _x\tau =0,\\ \rho \theta e_\theta \partial _t\theta +\theta ^2p_\theta \partial _xu+(\rho u\theta e_\theta -2q)\partial _x\theta +\theta \partial _xq =\frac{2q^2}{\kappa }+\frac{\theta \tau ^2}{\lambda },\\ \varepsilon _1^2(\partial _tq+u\partial _xq)+\kappa \partial _x\theta =-q,\\ \varepsilon _2^2(\partial _t\tau +u\partial _x\tau )-\lambda \partial _xu=-\tau . \end{array}\right. } \end{aligned}$$

(2.2)

Let $D_0(\varepsilon )$ be a diagonal matrix defined by

$$\begin{aligned} D_0(\varepsilon )=\text {diag}\left( 1,1,1,\varepsilon _1^2,\varepsilon _2^2\right) . \end{aligned}$$

Then system (2.2) is written as

$$\begin{aligned} D_0(\varepsilon )V_t+A(V)\partial _xV+BV=F(V), \end{aligned}$$

(2.3)

where

$$\begin{aligned} V=(\rho ,u,\theta ,q,\tau )^\top , \qquad F(V)=\left( 0,0,\frac{1}{\rho \theta e_\theta } \Big (\frac{2q^2}{\kappa }+\frac{\theta \tau ^2}{\lambda }\Big ),0,0\right) ^\top , \end{aligned}$$

and

$$\begin{aligned} A(V)=\left( \begin{matrix} u&{}\rho &{}0&{}0&{}0\\ \frac{p_\rho }{\rho }&{}u&{}\frac{p_\theta }{\rho }&{}\frac{p_q}{\rho }&{}\frac{p_\tau -1}{\rho }\\ 0&{}\frac{\theta p_\theta }{\rho e_\theta } &{}u-\frac{2q}{\rho \theta e_\theta }&{}\frac{1}{\rho e_\theta } &{}0\\ 0&{}0&{}\kappa &{}\varepsilon _1^2u&{}0\\ 0&{}-\lambda &{}0&{}0&{}\varepsilon _2^2u \end{matrix}\right) ,\quad B=\left( \begin{matrix} 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}1&{}0\\ 0&{}0&{}0&{}0&{}1 \end{matrix} \right) . \end{aligned}$$

We define

$$\begin{aligned} A_0(V)={\tilde{A}}_0(V)D_0(\varepsilon ), \qquad {\tilde{A}}(V)={\tilde{A}}_0(V)A(V), \end{aligned}$$

with

$$\begin{aligned} {\tilde{A}}_0(V)=\left( \begin{matrix} \theta ^2 p_\rho p_\theta &{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad \rho ^2\theta ^2 p_\theta &{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad \rho ^2\theta e_\theta p_\theta &{}-\frac{1}{\kappa }(\rho ^2q e_\theta )&{}\quad 0\\ 0&{}\quad 0&{}\quad -\frac{\varepsilon ^2_1}{\kappa }\rho ^2q e_\theta &{}\quad \frac{1}{\kappa }(\rho \theta p_\theta +2\rho qp_q)&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad \frac{1}{\lambda }\big (\rho \theta ^2p_\theta (1-p_\tau )\big ) \end{matrix} \right) . \end{aligned}$$

Then straightforward calculations give

$$\begin{aligned} A_0(V)= & {} \left( \begin{matrix} \theta ^2 p_\rho p_\theta &{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad \rho ^2\theta ^2 p_\theta &{}0&{}0&{}0\\ 0&{}\quad 0&{}\quad \rho ^2\theta e_\theta p_\theta &{}\quad -\frac{\varepsilon ^2_1}{\kappa }(\rho ^2 q e_\theta )&{}0\\ 0&{}\quad 0&{}\quad -\frac{\varepsilon ^2_1}{\kappa }(\rho ^2 q e_\theta )&{}\quad \frac{\varepsilon ^2_1}{\kappa }\big (\rho \theta p_\theta +2\rho qp_q\big )&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad \frac{\varepsilon ^2_2}{\lambda }\big (\rho \theta ^2 p_\theta (1-p_\tau )\big ) \end{matrix}\right) , \\ {\tilde{A}}(V)= & {} \left( \begin{matrix} u\theta ^2 p_\rho p_\theta &{}\rho \theta ^2 p_\rho p_\theta &{} \quad 0&{}\quad 0&{}\quad 0\\ \rho \theta ^2 p_\rho p_\theta &{} \quad \rho ^2u\theta ^2p_\theta &{}\quad \rho \theta ^2p^2_\theta &{} \quad \rho \theta ^2 p_\theta p_q &{}\quad \rho \theta ^2 p_\theta (p_\tau -1) \\ 0&{}\quad \rho \theta ^2 p^2_\theta &{}\quad J_1&{}\quad \rho \theta p_\theta +\rho ^2u\theta e_\theta p_q&{}\quad 0\\ 0&{}\quad \rho \theta ^2p_\theta p_q&{}\quad \rho \theta p_\theta +\rho ^2u\theta e_\theta p_q&{} \quad J_2&{}\quad 0\\ 0&{}\quad \rho \theta ^2 p_\theta (p_\tau -1) &{}\quad 0&{}\quad 0&{}\quad \frac{\varepsilon _2^2}{\lambda }\big (\rho u\theta ^2p_\theta (1-p_\tau )\big )\end{matrix}\right) , \end{aligned}$$

where

$$\begin{aligned} J_1=\rho ^2u\theta e_\theta p_\theta -2\rho qp_\theta -\rho ^2 qe_\theta , \qquad J_2=\frac{\varepsilon _1^2}{\kappa }\big (\rho u(\theta p_\theta +2qp_q)-\rho q\big ). \end{aligned}$$

It is clear that $A_0(V)$ and ${\tilde{A}}(V)$ are symmetric. Let $T>0$ and

$$\begin{aligned} N=\rho -1, \quad \Theta =\theta -1, \quad W=\big (N,u,\Theta ,\varepsilon _1q,\varepsilon _2\tau \big )^\top . \end{aligned}$$

We denote

$$\begin{aligned} {\mathscr {E}}(t)=\Vert W(t,\cdot )\Vert _s^2, \qquad {\mathscr {E}}_T=\sup _{0\le t\le T}{\mathscr {E}}(t). \end{aligned}$$

(2.4)

In Theorem 1.1, estimate (1.16) implies that ${\mathscr {E}}_T$ is uniformly sufficiently small when ${\mathscr {E}}(0)$ is, although the $L^\infty (0,T;H^s({\mathbb {R}}))$ norm of q and $\tau$ may not be uniformly small with respect to $\varepsilon _1$ and $\varepsilon _2$. The following result shows that $A_0(V)$ is positive definite when ${\mathscr {E}}_T$ is sufficiently small.

Lemma 2.1

Let $W\in C([0,T];H^s({\mathbb {R}}))$ be the smooth solution to (2.2) with (1.14)-(1.15) and (1.12). Then there exist constants $\delta >0$ and $c_1>0$, independent of $\varepsilon _1$ and $\varepsilon _2$, such that if ${\mathscr {E}}_T^{1/2}\le \delta$, we have

$$\begin{aligned}&c_1\le \rho ,\theta , p_\rho ,p_\theta ,e_\theta \le C, \end{aligned}$$

(2.5)

$$\begin{aligned}&\Vert p_q\Vert _s\le C\delta \varepsilon _1,\quad \Vert p_\tau \Vert _s\le C\delta \varepsilon _2, \quad \Vert qp_q\Vert _s\le C\delta , \end{aligned}$$

(2.6)

and $A_0(V)$ is positive definite and then system (2.2) is symmetrizable hyperbolic.

Proof

When ${\mathscr {E}}_T$ is sufficiently small, $\rho$ and $\theta$ are sufficiently close to 1, then they have uniform positive upper and lower bounds. From the state equations (1.14) and (1.15), we have

$$\begin{aligned} p_\rho =R\theta , \quad p_\theta =R\rho +\dfrac{\varepsilon _1^2}{2\kappa \theta ^2}q^2, \quad e_\theta =c_v-\dfrac{\varepsilon _1^2}{\kappa \rho \theta ^2}q^2, \end{aligned}$$

(2.7)

which imply (2.5) since $\Vert \varepsilon _1 q\Vert _\infty \le {\mathscr {E}}_T$ which is sufficiently small. Besides,

$$\begin{aligned} p_q=-\dfrac{\varepsilon _1^2}{\kappa \theta }q, \quad p_\tau =-\dfrac{\varepsilon _2^2}{\lambda }\tau , \quad qp_q=-\dfrac{\varepsilon _1^2}{\kappa \theta }q^2, \end{aligned}$$

(2.8)

which imply (2.6).

Moreover,

$$\begin{aligned} \left| \begin{matrix} \rho ^2\theta e_\theta p_\theta &{}-\frac{\varepsilon ^2_1}{\kappa }(\rho ^2 q e_\theta )\\ -\frac{\varepsilon ^2_1}{\kappa }(\rho ^2 q e_\theta )&{}\frac{\varepsilon ^2_1}{\kappa }\big (\rho \theta p_\theta +2\rho qp_q\big )\\ \end{matrix}\right| =\dfrac{\varepsilon _1^2\rho ^3e_\theta }{\kappa }\left( \theta ^2p_\theta ^2-\dfrac{\varepsilon _1^2q^2}{\kappa }(2p_\theta +\rho e_\theta )\right) . \end{aligned}$$

Then (2.5) implies that the above determinate is positive for sufficiently small ${\mathscr {E}}_T$. From (2.6) and $\varepsilon _2\le 1$, we also have $\Vert p_\tau \Vert _s\le C\delta$. Therefore, $A_0(V)$ is positive definite from its explicit expression. $\square$

Applying the theory on the symmetrizable hyperbolic system, Lemma 2.1 implies the local existence of smooth solutions to the Cauchy problem (2.2) with (1.14)-(1.15) and (1.12) from the classical iteration technique and fixed point theorems, see for instance [18, 21, 24].

3 Uniform Global Estimates

Let $T>0$ and $W=(N,u,\Theta ,\varepsilon _1q,\varepsilon _2\tau )^\top$ be the unique local smooth solution to (2.2) with (1.14)-(1.15) and (1.12), defined on [0, T]. We assume that ${\mathscr {E}}_T$ defined in (2.4) is sufficiently small. This gives rise to the rational assumption that

$$\begin{aligned} |\rho -1|\le \frac{1}{2}, \quad |\theta -1| \le \frac{1}{2}, \quad |p_\tau |\le \dfrac{1}{2}. \end{aligned}$$

(3.1)

We want to establish the uniform global estimate for W with respect to the parameters $\varepsilon _1$ and $\varepsilon _2$ and the time T. For this purpose, we introduce the following dissipative energy

$$\begin{aligned} {\mathscr {D}}(t)=\Vert \partial _xN(t)\Vert _{s-1}^2+\Vert \partial _xu(t)\Vert _{s-1}^2 +\Vert \partial _x\Theta (t)\Vert _{s-1}^2+\Vert q(t)\Vert _s^2+\Vert \tau (t)\Vert _s^2. \end{aligned}$$

3.1 $L^2$-estimate

The $L^2$ estimate relies on the existence of a strictly convex entropy and its corresponding entropy flux.

Lemma 3.1

For system (2.2), there exists a strictly convex entropy $\eta$ and its corresponding entropy flux $\Psi$ satisfying

$$\begin{aligned} \partial _t\eta (W)+\partial _x\Psi (W)+\frac{q^2}{\kappa \theta ^2} +\frac{\tau ^2}{\lambda \theta }=0, \end{aligned}$$

(3.2)

where

$$\begin{aligned} \eta (W)=R(\rho \ln \rho -\rho +1)+\dfrac{1}{2}\rho u^2 +c_v\rho (\theta -\ln \theta -1) +\dfrac{\varepsilon _1^2}{\kappa \theta }\left( 1-\dfrac{1}{2\theta }\right) q^2 +\dfrac{\varepsilon _2^2}{2\lambda }\tau ^2, \end{aligned}$$

and

$$\begin{aligned} \Psi (W)= & {} R\rho (\ln \rho -1)u+\frac{1}{2}\rho u^3 +c_v\rho u(\theta -\ln \theta -1)+(p-\tau )u+\Big (1-\frac{1}{\theta }\Big )q\\&+\,\frac{\varepsilon _1^2u}{\kappa \theta }\Big (1-\frac{1}{2\theta }\Big )q^2 +\frac{\varepsilon _2^2}{2\lambda }u\tau ^2. \end{aligned}$$

In addition, if ${\mathscr {E}}_T$ is sufficiently small, the following $L^2$-energy estimate holds

$$\begin{aligned}&\Vert (\rho -1,u,\theta -1,\varepsilon _1 q,\varepsilon _2 \tau )(t)\Vert ^2 +\int _0^t\big (\Vert q(t^\prime )\Vert ^2+\Vert \tau (t^\prime )\Vert ^2\big )dt^\prime \nonumber \\&\quad \le C\left( \Vert \rho _0-1\Vert ^2+\Vert u_0\Vert ^2+\Vert \theta _0-1\Vert ^2+\varepsilon _1^2\Vert q_0\Vert ^2 +\varepsilon _2^2\Vert \tau _0\Vert ^2\right) ,\quad \forall \, t\in [0,T]. \end{aligned}$$

(3.3)

Proof

The entropy-entropy flux identity (3.2) was established in [16]. We now prove (3.3). By the Taylor’s expansion at $(\rho ,\theta )=(1,1)$ and (3.1), we have

$$\begin{aligned} c_v\rho (\theta -\ln \theta -1) =\dfrac{c_v\rho }{2{\hat{\theta }}^2}\Theta ^2\ge \dfrac{1}{9}c_v \Theta ^2, \quad R(\rho \ln \rho -\rho +1)=\dfrac{R}{2{\hat{\rho }}}N^2\ge \dfrac{R}{3}N^2, \end{aligned}$$

where ${\hat{\rho }}$ is between $\rho$ and 1, and ${\hat{\theta }}$ is between $\theta$ and 1. By Lemma 2.1, this implies that there exists a constant $c_2>0$, such that

$$\begin{aligned} c_2|W|^2\le \eta (W) \le C|W|^2, \end{aligned}$$

where $|\cdot |$ is the Euclidean norm in ${\mathbb {R}}^5$. Integrating (3.2) over [0, t] for $t\in [0,T]$ implies (3.3). $\square$

3.2 Higher Order Estimates

Let l be an integer with $1\le l\le s$. Applying $\partial _x^l$ to both sides of the equation (2.3) yields

$$\begin{aligned} D_0(\varepsilon )\partial _t(\partial _x^l V)+A(V)\partial _x(\partial _x^l V)+\partial _x^l(BV)=\partial _x^l F(V)+K_l, \end{aligned}$$

where

$$\begin{aligned} K_l=A(V)\partial _x^{l+1}V-\partial _x^l(A(V)\partial _xV). \end{aligned}$$

Then,

$$\begin{aligned} A_0(V)\partial _t(\partial _x^l V)+{\tilde{A}}(V)\partial _x(\partial _x^l V)+{\tilde{A}}_0(V)\partial _x^l(BV) ={\tilde{A}}_0(V)\partial _x^l F(V)+{\tilde{A}}_0(V)K_l. \end{aligned}$$

Taking the inner product of the above system with $\partial _x^l V$ in $L^2({\mathbb {R}})$, we have

$$\begin{aligned}&\dfrac{d}{dt}\big<A_0(V)\partial _x^l V,\partial _x^l V\big> +2\big<{\tilde{A}}_0(V)\partial _x^l(B V),\partial _x^l V\big> \nonumber \\&\quad =\big<\partial _tA_0(V)\partial _x^l V,\partial _x^l V\big> +\big<\partial _x{\tilde{A}}(V)\partial _x^l V,\partial _x^l V\big> +2\big<{\tilde{A}}_0(V)\partial _x^l F(V),\partial _x^l V\big> +2\big <{\tilde{A}}_0(V)K_l,\partial _x^l V\big >. \end{aligned}$$

(3.4)

We deal with (3.4) term by term in a series of lemmas as follows.

Lemma 3.2

There exists positive constants $c_3$ and $c_4$ such that

$$\begin{aligned} 2\big |\big <{\tilde{A}}_0(V)\partial _x^l(B V),\partial _x^l V\big >\big | \ge c_3\Vert \partial _x^l q\Vert ^2+c_4\Vert \partial _x^l \tau \Vert ^2-C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

(3.5)

Proof

By the definition of ${\tilde{A}}_0(V)$ and B together with (2.8), we have

$$\begin{aligned} {\tilde{A}}_0(V)\partial _x^l(BV)= & {} \left( 0,0,0,-\dfrac{\rho ^2qe_\theta }{\kappa }\partial _x^lq, \frac{\rho \theta p_\theta +2\rho qp_q}{\kappa }\partial _x^lq,\frac{\rho \theta ^2p_\theta (1-p_\tau ) }{\lambda }\partial _x^l\tau \right) ^\top , \end{aligned}$$

which implies that

$$\begin{aligned} \big <{\tilde{A}}_0(V)\partial _x^l(B V),\partial _x^l V\big >= & {} -\left\langle \dfrac{\rho ^2qe_\theta }{\kappa }\partial _x^l q,\partial _x^l \Theta \right\rangle +\left\langle \frac{\rho \theta p_\theta +2\rho qp_q}{\kappa }\partial _x^lq,\partial _x^l q\right\rangle \\&+\left\langle \frac{\rho \theta ^2p_\theta (1-p_\tau )}{\lambda }\partial _x^l\tau ,\partial _x^l\tau \right\rangle . \end{aligned}$$

Obviously,

$$\begin{aligned} \left| \left\langle \dfrac{\rho ^2qe_\theta }{\kappa }\partial _x^l q,\partial _x^l\Theta \right\rangle \right| \le C\Vert \partial _x^l\Theta \Vert _\infty \Vert q\Vert \Vert \partial _x^l q\Vert \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

For the second term, noticing that $\dfrac{\rho \theta p_\theta }{\kappa }$ has a uniform lower bound when ${\mathscr {E}}_T$ is sufficiently small. Besides, the last estimate in (2.6) implies that $\dfrac{2\rho qp_q}{\kappa }$ is small and consequently, there exists a positive constant $c_3$ such that

$$\begin{aligned} 2\left\langle \frac{\rho \theta p_\theta +2\rho qp_q}{\kappa }\partial _x^lq,\partial _x^l q\right\rangle \ge c_3\Vert \partial _x^l q\Vert ^2. \end{aligned}$$

Similarly, (3.1) implies that $1-p_\tau \ge \dfrac{1}{2}$. Therefore, there exists a positive constant $c_4$ such that

$$\begin{aligned} 2\left\langle \frac{\rho \theta ^2p_\theta (1-p_\tau )}{\lambda }\partial _x^l\tau ,\partial _x^l \tau \right\rangle \ge c_4\Vert \partial _x^l \tau \Vert ^2. \end{aligned}$$

Combining all these estimates yields (3.5). $\square$

Lemma 3.3

It holds

$$\begin{aligned} \big |\big <\partial _tA_0(V)\partial _x^l V,\partial _x^l V\big >\big | \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

(3.6)

Proof

We denote

$$\begin{aligned} V=\left( \begin{matrix} V_1\\ V_2 \end{matrix}\right) , \quad V_1=\left( \begin{matrix} N\\ u\\ \Theta \end{matrix}\right) , \quad V_2=\left( \begin{matrix} q\\ \tau \end{matrix}\right) , \end{aligned}$$

and

$$\begin{aligned} A_0(V)=\left( \begin{matrix} A^{11}_0(V)&{} A^{12}_0(V)\\ A^{21}_0(V)&{} A^{22}_0(V) \end{matrix}\right) , \qquad {\tilde{A}}(V)=\left( \begin{matrix} {\tilde{A}}^{11}(V)&{} {\tilde{A}}^{12}(V)\\ {\tilde{A}}^{21}(V)&{} {\tilde{A}}^{22}(V) \end{matrix}\right) , \end{aligned}$$

where

$$\begin{aligned} \left\{ \begin{aligned}&A^{11}_0(V)=\left( \begin{array}{lll} \theta ^2 p_\rho p_\theta &{}0&{}0\\ 0&{}\rho ^2\theta ^2 p_\theta &{}0\\ 0&{}0&{}\rho ^2\theta e_\theta p_\theta \end{array}\right) , \\&A^{12}_0(V)=\big (A^{21}_0(V)\big )^\top =\left( \begin{array}{lll} 0&{}0\\ 0&{}0\\ -\frac{\varepsilon ^2_1}{\kappa }\big (\rho ^2qe_\theta \big )&{}0 \end{array}\right) , \\&A^{22}_0(V)=\left( \begin{array}{lll} \frac{\varepsilon ^2_1}{\kappa }(\rho \theta p_\theta +2\rho qp_q)&{}0\\ 0&{}\frac{\varepsilon ^2_2}{\lambda }\big (\rho \theta ^2 p_\theta (1-p_\tau )\big ) \end{array}\right) . \end{aligned}\right. \end{aligned}$$

Obviously,

$$\begin{aligned} \left\langle \partial _tA_0(V)\partial _x^l V,\partial _x^l V\right\rangle =\left\langle \partial _tA^{11}_0(V)\partial _x^l V_1,\partial _x^l V_1\right\rangle +2\left\langle \partial _tA^{12}_0(V)\partial _x^l V_2,\partial _x^l V_1\right\rangle +\left\langle \partial _tA^{22}_0(V)\partial _x^l V_2,\partial _x^l V_2\right\rangle . \end{aligned}$$

From (1.14) and (1.15), we see that $A^{11}_0(V)$ is a smooth function of $(N,\Theta ,\varepsilon ^2_1q^2)$. Moreover, from (2.2) together with the Cauchy-Schwarz inequality, we have

$$\begin{aligned} \Vert \partial _tN\Vert _\infty +\Vert \partial _t\Theta \Vert _\infty\le & {} C{\mathscr {E}}_T^{1/2} +C\big (\Vert q\Vert _s+\Vert q\Vert ^2_s+\Vert \tau \Vert ^2_s\big ),\\ \varepsilon ^2_1\Vert \partial _tq\Vert _\infty\le & {} C{\mathscr {E}}_T^{1/2}+C\Vert q\Vert _s,\\ \varepsilon ^2_1\Vert \partial _t(q^2)\Vert _\infty\le & {} C{\mathscr {E}}_T^{1/2}+C\Vert q\Vert _s+C\Vert q\Vert ^2_s. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert \partial _tA^{11}_0(V)\Vert _\infty \le C{\mathscr {E}}_T^{1/2}+C\big (\Vert q\Vert _s+\Vert q\Vert ^2_s+\Vert \tau \Vert ^2_s\big ). \end{aligned}$$

Moreover,

$$\begin{aligned} 2\Vert q\Vert _s\Vert \partial _x^l V_1\Vert ^2\le \big (\Vert \partial _x^l V_1\Vert ^2+\Vert q\Vert ^2_s\big )\Vert \partial _x^l V_1\Vert \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

Since ${\mathscr {E}}_T$ is uniformly small, we also have

$$\begin{aligned} \big (\Vert q\Vert ^2_s+\Vert \tau \Vert ^2_s\big )\Vert \partial _x^l V_1\Vert ^2 \le C{\mathscr {E}}_T{\mathscr {D}}(t)\le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \left\langle \partial _tA^{11}_0(V)\partial _x^l V_1,\partial _x^l V_1\right\rangle \right| \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

Next, a straightforward calculation yields

$$\begin{aligned} \big<\partial _tA^{12}_0(V)\partial _x^l V_2,\partial _x^l V_1\big> =-\frac{\varepsilon ^2_1}{\kappa }\big <\partial _t(\rho ^2qe_\theta )\partial _x^l q,\partial _x^l\Theta \big >. \end{aligned}$$

From (2.7), we have

$$\begin{aligned} \rho ^2qe_\theta =c_\nu \rho ^2q-\frac{\varepsilon ^2_1\rho q^3}{\kappa \theta ^2}. \end{aligned}$$

Then (2.2) together with the bound of ${\mathscr {E}}_T$ yields

$$\begin{aligned} \frac{\varepsilon ^2_1}{\kappa }\Vert \partial _t\big (\rho ^2qe_\theta \big )\Vert _\infty \le C{\mathscr {E}}_T^{1/2}+C\Vert q\Vert _s+C\varepsilon _1\Vert \tau \Vert _s^2. \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \big <\partial _tA^{12}_0(V)\partial _x^l V_2,\partial _x^l V_1\big >\right|\le & {} \frac{\varepsilon ^2_1}{\kappa }\Vert \partial _t\big (\rho ^2qe_\theta \big )\Vert _\infty \Vert \partial _x^l q\Vert \Vert \partial _x^l\Theta \Vert \\\le & {} C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t)+C{\mathscr {E}}_T^{1/2}\Vert q\Vert _s^2+C(\varepsilon _1\Vert \partial _x^l q\Vert \Vert \partial _x^l\Theta \Vert )\Vert \tau \Vert _s^2\\\le & {} C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

Finally,

$$\begin{aligned} \big<\partial _tA^{22}_0(V)\partial _x^l V_2,\partial _x^l V_2\big> =\frac{\varepsilon ^2_1}{\kappa }\big<\partial _t(\rho \theta p_\theta +2\rho qp_q)\partial _x^lq,\partial _x^lq\big>+\frac{\varepsilon ^2_2}{\lambda }\big <\partial _t\big (\rho \theta ^2p_\theta (1-p_\tau )\big )\partial _x^l\tau ,\partial _x^l\tau \big >. \end{aligned}$$

Similarly to the estimates above, we have

$$\begin{aligned} \varepsilon ^2_1\big \Vert \partial _t(\rho \theta p_\theta +2\rho qp_q)\big \Vert _\infty \le C{\mathscr {E}}_T^{1/2}+C\varepsilon ^2_1\big (\Vert q\Vert _s+\Vert q\Vert ^2_s+\Vert \tau \Vert ^2_s\big ), \end{aligned}$$

and

$$\begin{aligned} \varepsilon ^2_2\big \Vert \partial _t\big (\rho \theta ^2p_\theta (1-p_\tau )\big )\big \Vert _\infty \le C{\mathscr {E}}_T^{1/2}+C\varepsilon ^2_2\big (\Vert q\Vert _s+\Vert q\Vert ^2_s+\Vert \tau \Vert _s+\Vert \tau \Vert ^2_s\big ), \end{aligned}$$

which imply that

$$\begin{aligned} \big |\big <\partial _tA^{22}_0(V)\partial _x^l V_2,\partial _x^l V_2\big >\big | \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

This proves the lemma. $\square$

Lemma 3.4

It holds

$$\begin{aligned} \big |\big <\partial _x{\tilde{A}}(V)\partial _x^l V,\partial _x^l V\big >\big | \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

(3.7)

Proof

From (1.14) and (1.15) and the explicit expression of ${\tilde{A}}(V)$, we see that all elements of ${\tilde{A}}(V)$ are smooth functions of $(\rho ,u,\theta ,\varepsilon ^2_1q^2,\varepsilon _2^2\tau ^2)$ except $J_1$. Noticing that $J_1$ appears at the position of the third line and the third column of ${\tilde{A}}(V)$ and

$$\begin{aligned} \Vert \partial _x J_1\Vert _\infty \le C(\Vert \partial _x W\Vert _{s-1}+\Vert q\Vert _s), \end{aligned}$$

we obtain

$$\begin{aligned} \big |\big <\partial _x{\tilde{A}}(V)\partial _x^l V,\partial _x^l V\big >\big |\le & {} C\Vert \partial _xW\Vert _{\infty }\Vert \partial _x^l V\Vert ^2 +\left| \left\langle \partial _x J_1 \partial _x^l\Theta , \partial _x^l\Theta \right\rangle \right| \\\le & {} C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t)+C\Vert \partial _x J_1\Vert _\infty \Vert \partial _x^l\Theta \Vert ^2\\\le & {} C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t)+C\Vert q\Vert _s\Vert \partial _x^l\Theta \Vert ^2\\\le & {} C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

This proves (3.7). $\square$

Lemma 3.5

It holds

$$\begin{aligned} \big |\big <{\tilde{A}}_0(V)\partial _x^l F(V),\partial _x^l V\big >\big | \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

(3.8)

Proof

By the definition of ${\tilde{A}}_0(V)$ and F, we have

$$\begin{aligned} {\tilde{A}}_0(V)\partial _x^l F(V)=\left( 0,0,\rho ^2\theta e_\theta p_\theta \partial _x^l\left( \frac{1}{\rho \theta e_\theta }\left( \frac{2q^2}{\kappa }+\frac{\theta \tau ^2}{\lambda }\right) \right) , -\dfrac{\varepsilon _1^2\rho ^2qe_\theta }{\kappa } \partial _x^l\left( \frac{1}{\rho \theta e_\theta }\left( \frac{2q^2}{\kappa }+\frac{\theta \tau ^2}{\lambda }\right) \right) ,0\right) ^\top . \end{aligned}$$

It follows that

$$\begin{aligned}&\big |\big <{\tilde{A}}_0(V)\partial _x^l F(V),\partial _x^l V\big >\big |\\&\quad \le \left| \left\langle \rho ^2\theta e_\theta p_\theta \partial _x^l\left( \frac{1}{\rho \theta e_\theta }\left( \frac{2q^2}{\kappa }+\frac{\theta \tau ^2}{\lambda }\right) \right) ,\partial _x^l\Theta \right\rangle \right| +\left| \left\langle \dfrac{\varepsilon _1^2\rho ^2qe_\theta }{\kappa } \partial _x^l\left( \frac{1}{\rho \theta e_\theta }\left( \frac{2q^2}{\kappa }+\frac{\theta \tau ^2}{\lambda }\right) \right) ,\partial _x^l q\right\rangle \right| \\&\quad \le C(\Vert \partial _x^l\Theta \Vert _\infty +\Vert \varepsilon _1\partial _x^l q\Vert \Vert \varepsilon _1 q\Vert _\infty )(\Vert q\Vert _s^2+\Vert \tau \Vert _s^2)\\&\quad \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t), \end{aligned}$$

which proves (3.8). $\square$

Lemma 3.6

It holds

$$\begin{aligned} \big |\big <{\tilde{A}}_0(V)K_l,\partial _x^lV\big >\big | \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

(3.9)

Proof

Recall that

$$\begin{aligned} K_l=A(V)\partial _x^{l+1}V-\partial _x^l(A(V)\partial _xV). \end{aligned}$$

Similarly to the proof of Lemma 3.4, it is easy to see that all elements of A(V) are smooth functions of $(\rho ,u,\theta ,\varepsilon ^2_1q^2,\varepsilon _2^2\tau ^2)$ except the element at the position of the third line and the third column. We denote by $J_3$ this element. Then

$$\begin{aligned} J_3=u-\frac{2q}{\rho \theta e_\theta }, \end{aligned}$$

and

$$\begin{aligned} \Vert \partial _x J_3\Vert _{s-1}\le C(\Vert \partial _x W\Vert _{s-1}+\Vert q\Vert _s). \end{aligned}$$

The only two terms that contain $J_3$ are the following

$$\begin{aligned} \left\langle \partial _x^l \Theta , \rho ^2 \theta e_\theta p_\theta \left( \partial _x^l\left( J_3\partial _x\Theta \right) -J_3\partial _x^{l+1}\Theta \right) \right\rangle \quad \text {and}\quad -\left\langle \partial _x^l q,\dfrac{\varepsilon _1^2\rho ^2 q e_\theta }{\kappa }\left( \partial _x^l\left( J_3\partial _x\Theta \right) -J_3\partial _x^{l+1}\Theta \right) \right\rangle , \end{aligned}$$

which can be treated similarly to the proof of Lemma 3.4 and are obviously bounded by $C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t)$.

On the other hand, each element of ${\tilde{A}}_0(V)$ is uniformly bounded in $L^\infty ([0,T]\times {\mathbb {R}})$ except for the element at the position of the third line and the fourth column. This element is $-\frac{1}{\kappa }(\rho ^2q e_\theta )$ which only touches the fourth line of $K_l$ in the product ${\tilde{A}}_0(V)K_l$. Now the only nonzero element on the fourth line of $K_l$ is $\varepsilon ^2_1(u\partial _x^{l+1}q-\partial _x^l(u\partial _xq))$. Then the fourth component of ${\tilde{A}}_0(V)K_l$ is $-\frac{\varepsilon ^2_1}{\kappa }(\rho ^2q e_\theta )(u\partial _x^{l+1}q-\partial _x^l(u\partial _xq))$. By the Moser-type calculus inequalities (see [24] for instance), we have

$$\begin{aligned} \varepsilon ^2_1\big |\big <\rho ^2q e_\theta (u\partial _x^{l+1}q-\partial _x^l(u\partial _xq)),\partial _x^l\Theta \big >\big | \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

The estimates for the other terms can be easily obtained. This proves (3.9). $\square$

3.3 Dissipative Estimates for $\partial _xN$, $\partial _xu$ and $\partial _x\Theta$

Lemma 3.7

(Dissipative estimates for $\partial _x\Theta$). It holds

$$\begin{aligned} \varepsilon _1^2\sum _{m=0}^{s-1}\dfrac{d}{dt} \big <\partial _x^mq,\partial _x^{m+1}\Theta \big >+\frac{\kappa }{4}\Vert \partial _x\Theta \Vert _{s-1}^2 \le C\nu \Vert \partial _xu\Vert _{s-1}^2+C\Vert q\Vert _s^2+C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t), \end{aligned}$$

(3.10)

where $\nu >0$ is a small positive constant to be determined later.

Proof

Let m be an integer with $0\le m\le s-1$. Applying $\partial _x^m$ to the fourth equation in (2.2) and making the inner product of the resulting equation with $\partial _x^{m+1}\Theta$ in $L^2({\mathbb {R}})$, we have

$$\begin{aligned} \kappa \Vert \partial _x^{m+1}\Theta \Vert ^2= & {} -\varepsilon _1^2\dfrac{d}{dt}\big<\partial _x^m q,\partial _x^{m+1}\Theta \big> +\varepsilon _1^2\big<\partial _t\partial _x^{m+1}\Theta ,\partial _x^m q\big>\\&-\,\varepsilon _1^2\big<\partial _x^{m+1}\Theta ,\partial _x^m(u\partial _xq)\big> -\big <\partial _x^m q,\partial _x^{m+1} \Theta \big >. \end{aligned}$$

Obviously,

$$\begin{aligned} \big |\varepsilon _1^2\big<\partial _x^{m+1}\Theta ,\partial _x^m(u\partial _xq)\big> +\big <\partial _x^m q,\partial _x^{m+1} \Theta \big >\big | \le \dfrac{\kappa }{2}\Vert \partial _x^{m+1}\Theta \Vert ^2+C\Vert q\Vert _s^2. \end{aligned}$$

Therefore,

$$\begin{aligned} \varepsilon _1^2\dfrac{d}{dt}\big<\partial _x^m q,\partial _x^{m+1}\Theta \big> +\dfrac{\kappa }{2}\Vert \partial _x^{m+1}\Theta \Vert ^2 \le \varepsilon _1^2\big <\partial _t\partial _x^{m+1}\Theta ,\partial _x^m q\big >+C\Vert q\Vert _s^2. \end{aligned}$$

(3.11)

By using the third equation in (2.2) and an integration by parts, we have

$$\begin{aligned} \varepsilon ^2_1\left| \left\langle \partial _t\partial _x^{m+1}\Theta ,\partial _x^m q\right\rangle \right|\le & {} \varepsilon _1\left| \left\langle \partial _x^m\left( u\partial _x\theta +\dfrac{\theta p_\theta }{\rho e_\theta }\partial _xu-\dfrac{2q}{\rho \theta e_\theta }\partial _x\theta +\dfrac{1}{\rho e_\theta }\partial _xq\right) ,\partial _x^{m+1}(\varepsilon _1q)\right\rangle \right| \\&+\,\varepsilon _1\left| \left\langle \partial _x^m\left( \dfrac{2q^2}{\kappa \rho \theta e_\theta } +\dfrac{\tau ^2}{\lambda \rho e_\theta }\right) ,\partial _x^{m+1}(\varepsilon _1q)\right\rangle \right| . \end{aligned}$$

Obviously,

$$\begin{aligned}&\varepsilon _1\left| \left\langle \partial _x^m\left( u\partial _x\theta +\dfrac{\theta p_\theta }{\rho e_\theta }\partial _xu-\dfrac{2q}{\rho \theta e_\theta }\partial _x\theta +\dfrac{1}{\rho e_\theta }\partial _xq\right) ,\partial _x^{m+1}(\varepsilon _1q)\right\rangle \right| \\&\quad \le \dfrac{\kappa }{4}\Vert \partial _x\Theta \Vert _{s-1}^2+\nu \Vert \partial _xu\Vert _{s-1}^2+C\Vert q\Vert _s^2+C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t), \end{aligned}$$

and

$$\begin{aligned} \varepsilon _1\left| \left\langle \partial _x^m\left( \dfrac{2q^2}{\kappa \rho \theta e_\theta } +\dfrac{\tau ^2}{\lambda \rho e_\theta }\right) ,\partial _x^{m+1}(\varepsilon _1q)\right\rangle \right|\le & {} C\varepsilon _1\Vert \varepsilon _1 q\Vert _s\Vert q\Vert _s^2+C\varepsilon _1\Vert \varepsilon _1 q\Vert _s\Vert \tau \Vert _s^2\\\le & {} C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

Combining these estimates, we arrive at

$$\begin{aligned} \varepsilon ^2_1\left| \left\langle \partial _t\partial _x^{m+1}\Theta ,\partial _x^m q\right\rangle \right| \le \dfrac{\kappa }{4}\Vert \partial _x\Theta \Vert _{s-1}^2+\nu \Vert \partial _xu\Vert _{s-1}^2+C\Vert q\Vert _s^2+C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

This together with (3.11) yields

$$\begin{aligned} \varepsilon _1^2\dfrac{d}{dt}\big <\partial _x^m q,\partial _x^{m+1}\Theta \big > +\dfrac{\kappa }{4}\Vert \partial _x^{m+1}\Theta \Vert ^2 \le \nu \Vert \partial _xu\Vert _{s-1}^2+C\Vert q\Vert _s^2+C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

Adding this inequality for all $0\le m\le s-1$ yields (3.10). $\square$

Lemma 3.8

(Dissipative estimates for $\partial _xu$). It holds

$$\begin{aligned} -\varepsilon _2^2\sum _{m=0}^{s-1}\dfrac{d}{dt}\big <\partial _x^m\tau ,\partial _x^{m+1}u\big > +\dfrac{\lambda }{2}\Vert \partial _xu\Vert ^2_{s-1}\le & {} C\nu (\Vert \partial _xN\Vert _{s-1}^2+\Vert \partial _xu\Vert _{s-1}^2+\Vert \partial _x\Theta \Vert _{s-1}^2)\nonumber \\&+\,C\big (\Vert q\Vert _s^2+\Vert \tau \Vert _s^2\big )+C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t), \end{aligned}$$

(3.12)

where $\nu >0$ is a small positive constant to be determined later.

Proof

For $0\le m\le s-1$, applying $\partial _x^m$ to the fifth equation in (2.2) and making the inner product of the resulting equation with $\partial _x^{m+1}u$ in $L^2({\mathbb {R}})$, we have

$$\begin{aligned} \lambda \Vert \partial _x^{m+1}u\Vert ^2= & {} \varepsilon _2^2\dfrac{d}{dt}\big<\partial _x^m\tau ,\partial _x^{m+1}u\big> -\varepsilon _2^2\big<\partial _t\partial _x^{m+1} u,\partial _x^m\tau \big>\\&+\,\varepsilon _2^2\big<\partial _x^{m+1}u,\partial _x^m(u\partial _x\tau )\big>+\big <\partial _x^{m+1}u,\partial _x^m\tau \big >. \end{aligned}$$

Obviously,

$$\begin{aligned} \big |\varepsilon _2^2\big<\partial _x^{m+1}u,\partial _x^m(u\partial _x\tau )\big> +\big <\partial _x^{m+1}u,\partial _x^m\tau \big >\big | \le \dfrac{\lambda }{2}\Vert \partial _x^{m+1}u\Vert ^2+C\Vert \tau \Vert _s^2 +C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

Therefore,

$$\begin{aligned} -\varepsilon _2^2\dfrac{d}{dt}\big<\partial _x^m\tau ,\partial _x^{m+1}u\big> +\dfrac{\lambda }{2}\Vert \partial _x^{m+1} u\Vert ^2 \le -\varepsilon _2^2\big <\partial _t\partial _x^{m+1} u,\partial _x^m\tau \big > +C\Vert \tau \Vert _s^2+C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

(3.13)

By using the second equation in (2.2) and an integration by parts, we have

$$\begin{aligned} \big |\varepsilon _2^2\big <\partial _t\partial _x^{m+1}u,\partial _x^m\tau \big >\big |\le & {} \varepsilon _2^2\left| \left\langle \partial _x^m\left( \dfrac{\rho u\partial _xu+p_\rho \partial _x\rho +p_\theta \partial _x\theta }{\rho }\right) ,\partial _x^{m+1} \tau \right\rangle \right| \\&+\varepsilon _2^2\left| \left\langle \partial _x^m\left( \dfrac{p_q\partial _xq+(p_\tau -1)\partial _x\tau }{\rho }\right) ,\partial _x^{m+1} \tau \right\rangle \right| \\\le & {} \nu \big (\Vert \partial _xN\Vert _{s-1}^2+\Vert \partial _xu\Vert _{s-1}^2+\Vert \partial _x\Theta \Vert _{s-1}^2\big ) +C\big (\Vert q\Vert _s^2+\Vert \tau \Vert _s^2\big ). \end{aligned}$$

Substituting this estimate into (3.13) and adding the resulting equation for all $0\le m\le s-1$ yield (3.12). $\square$

Lemma 3.9

(Dissipative estimates for $\partial _xN)$. It holds

$$\begin{aligned}&\sum _{m=0}^{s-1}\dfrac{d}{dt}\big <\partial _x^m u,\partial _x^{m+1}N\big > +\dfrac{R}{6}\Vert \partial _xN\Vert _{s-1}^2\nonumber \\&\quad \le C\big (\Vert \partial _xu\Vert _{s-1}^2+\Vert \partial _x\Theta \Vert _{s-1}^2\big ) +C\big (\Vert q\Vert _s^2+\Vert \tau \Vert ^2_s\big )+C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

(3.14)

Proof

We first write the second equation in (2.2) as

$$\begin{aligned} \partial _tu+\dfrac{R\theta }{\rho }\partial _xN+\frac{1}{\rho } \big (\rho u\partial _xu+p_\theta \partial _x\theta +p_q\partial _xq+(p_\tau -1)\partial _x\tau \big )=0. \end{aligned}$$

For $m\le s-1$, applying $\partial _x^m$ to the above equation and making the inner product of the resulting equation with $\partial _x^{m+1}N$ in $L^2({\mathbb {R}})$, we have

$$\begin{aligned} \left\langle \partial _x^{m+1}N, \dfrac{R\theta }{\rho } \partial _x^{m+1}N\right\rangle= & {} -\dfrac{d}{dt}\big<\partial _x^m u,\partial _x^{m+1}N\big>+\big <\partial _x^m u,\partial _t\partial _x^{m+1}N\big >\\&-\left\langle \partial _x^m\left( \frac{\rho u\partial _xu+p_\theta \partial _x\theta +p_q\partial _xq+(p_\tau -1)\partial _x\tau }{\rho }\right) ,\partial _x^{m+1}N\right\rangle \\&-\left\langle \partial _x^m\left( \dfrac{R\theta }{\rho }\partial _xN\right) -\dfrac{R\theta }{\rho }\partial _x^{m+1}N,\partial _x^{m+1}N\right\rangle . \end{aligned}$$

Noticing (3.1), we have

$$\begin{aligned} \left\langle \partial _x^{m+1}N, \dfrac{R\theta }{\rho } \partial _x^{m+1}N\right\rangle \ge \dfrac{R}{3}\Vert \partial _x^{m+1} N\Vert ^2. \end{aligned}$$

It is clear that

$$\begin{aligned}&\left| \left\langle \partial _x^m\left( \dfrac{\rho u\partial _xu+p_\theta \partial _x\theta +p_q\partial _xq+(p_\tau -1)\partial _x\tau }{\rho }\right) ,\partial _x^{m+1} N\right\rangle \right| \\&\quad \le \dfrac{R}{6}\Vert \partial _x^{m+1} N\Vert ^2 +C\big (\Vert \partial _xu\Vert _{s-1}^2+\Vert \partial _x\Theta \Vert _{s-1}^2\big ) +C\big (\Vert q\Vert _s^2+\Vert \tau \Vert ^2_s\big ), \end{aligned}$$

and by the Moser-type calculus inequalities,

$$\begin{aligned} \left| \left\langle \partial _x^m\left( \dfrac{R\theta }{\rho }\partial _xN\right) -\dfrac{R\theta }{\rho }\partial _x^{m+1}N,\partial _x^{m+1}N\right\rangle \right|\le & {} C\Vert \partial _x\theta \Vert _{s-1}\Vert \partial _xN\Vert _{s-1}^2+C\Vert \partial _xN\Vert _{s-1}^3\\\le & {} C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

Moreover, by the mass equation in (2.2) and the integration by parts, we obtain

$$\begin{aligned} \left| \left\langle \partial _x^m u,\partial _t\partial _x^{m+1}N\right\rangle \right|= & {} \left| \left\langle \partial _x^{m+1}u,\partial _x^{m+1}(\rho u)\right\rangle \right| \\\le & {} C\Vert \partial _xu\Vert _{s-1}(\Vert \partial _x^m(\partial _xN u+\rho \partial _xu)\Vert )\\\le & {} C\Vert \partial _xu\Vert _{s-1}^2+C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

Combining all these estimates, we arrive at

$$\begin{aligned} \dfrac{d}{dt}\left\langle \partial _x^m u,\partial _x^{m+1}N\right\rangle +\dfrac{R}{6}\Vert \partial _x^{m+1} N\Vert ^2\le & {} C\big (\Vert \partial _xu\Vert _{s-1}^2+\Vert \partial _x\Theta \Vert _{s-1}^2\big ) +C\big (\Vert q\Vert _s^2+\Vert \tau \Vert ^2_s\big )+C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

Adding the above estimate for all $m\le s-1$ yields (3.14). $\square$

4 Proof of Theorems 1.1-1.2

Lemma 4.1

(Final energy estimate). If ${\mathscr {E}}_T$ is sufficiently small, then

$$\begin{aligned} {\mathscr {E}}(t)+\int _0^t{\mathscr {D}}(t^\prime )dt^\prime \le C{\mathscr {E}}(0), \quad \forall \, t\in [0,T]. \end{aligned}$$

(4.1)

Proof

Combining (3.4) and Lemmas 3.2-3.6, and adding for all $1\le l\le s$, we arrive at

$$\begin{aligned} \sum _{l=1}^s\dfrac{d}{dt}\big <A_0(V)\partial _x^l V,\partial _x^l V\big > +c_3\Vert \partial _xq\Vert _{s-1}^2+c_4\Vert \partial _x\tau \Vert _{s-1}^2 \le C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

(4.2)

Since $A_0(V)$ is positive definite with respect to W, there exists a constant $c_5>0$, independent of $\varepsilon _1$ and $\varepsilon _2$, such that

$$\begin{aligned} \big <A_0(V)\partial _x^l V,\partial _x^l V\big >\ge c_5\Vert \partial _x^l W\Vert ^2. \end{aligned}$$

Integrating (4.2) over [0, T] together with (3.3) yields

$$\begin{aligned} \Vert W(t)\Vert _s^2+\int _0^t\big (\Vert q(t')\Vert _s^2+\Vert \tau (t')\Vert _s^2\big )dt' \le C{\mathscr {E}}(0)+C{\mathscr {E}}_T^{1/2}\int _0^t{\mathscr {D}}(t')dt'. \end{aligned}$$

(4.3)

On the other hand, combining the estimates in Lemmas 3.7-3.9, we have

$$\begin{aligned}&\sum _{m=0}^{s-1}\frac{d}{dt} \left( \varepsilon ^2_1\big<\partial _x^m q,\partial _x^{m+1}\Theta \big> -\varepsilon _2^2\big<\partial _x^m\tau ,\partial _x^{m+1}u\big> +\alpha \big <\partial _x^m u,\partial _x^{m+1}N\big >\right) \\&\qquad +\,\frac{R\alpha }{6}\Vert \partial _xN\Vert _{s-1}^2+\frac{\lambda }{2}\Vert \partial _xu\Vert _{s-1}^2 +\frac{\kappa }{4}\Vert \partial _x\Theta \Vert _{s-1}^2\\&\quad \le C\nu \Vert \partial _xN\Vert _{s-1}^2+C(\nu +\alpha )(\Vert \partial _xu\Vert _{s-1}^2 +\Vert \partial _x\Theta \Vert _{s-1}^2)+C\big (\Vert q\Vert _s^2+\Vert \tau \Vert _s^2\big ) +C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t), \end{aligned}$$

where $\alpha >0$ is a constant. We choose $\nu >0$ and $\alpha >0$ sufficiently small such that

$$\begin{aligned} 2C\nu \le \frac{R\alpha }{6},\quad 8C(\nu +\alpha )\le \min (\lambda ,\kappa ). \end{aligned}$$

It follows that there is a constant $a_0>0$, independent of $\varepsilon _1$ and $\varepsilon _2$, such that

$$\begin{aligned}&\sum _{m=0}^{s-1}\frac{d}{dt} \big (\varepsilon ^2_1\big<\partial _x^m q,\partial _x^{m+1}\Theta \big> -\varepsilon _2^2\big<\partial _x^m\tau ,\partial _x^{m+1}u\big> +\alpha \big <\partial _x^m u,\partial _x^{m+1}N\big >\big )\nonumber \\&\qquad +\,a_0\big (\Vert \partial _xN\Vert _{s-1}^2+\Vert \partial _xu\Vert _{s-1}^2 +\Vert \partial _x\Theta \Vert _{s-1}^2\big )\nonumber \\&\quad \le C\big (\Vert q\Vert _s^2+\Vert \tau \Vert _s^2\big )+C{\mathscr {E}}_T^{1/2}{\mathscr {D}}(t). \end{aligned}$$

(4.4)

It is clear that

$$\begin{aligned} \big |\varepsilon ^2_1\big<\partial _x^m q,\partial _x^{m+1}\Theta \big> -\varepsilon _2^2\big<\partial _x^m\tau ,\partial _x^{m+1}u\big>+\alpha \big <\partial _x^m u,\partial _x^{m+1}N\big >\big | \le C\Vert W\Vert ^2_s. \end{aligned}$$

Integrating (4.4) over [0, T] yields

$$\begin{aligned}&-C\Vert W(t)\Vert ^2_s+a_0\int _0^t\big (\Vert \partial _xN(t')\Vert _{s-1}^2+\Vert \partial _xu(t')\Vert _{s-1}^2 +\Vert \partial _x\Theta (t')\Vert _{s-1}^2\big )dt'\\&\quad \le C{\mathscr {E}}(0)+C\int _0^t\big (\Vert q(t')\Vert _s^2+\Vert \tau (t')\Vert _s^2\big ) +C{\mathscr {E}}_T^{1/2}\int _0^t{\mathscr {D}}(t')dt'. \end{aligned}$$

This inequality together with (4.3) yields

$$\begin{aligned} {\mathscr {E}}(t)+\int _0^t{\mathscr {D}}(t')dt' \le C{\mathscr {E}}(0)+C{\mathscr {E}}_T^{1/2}\int _0^t{\mathscr {D}}(t')dt', \end{aligned}$$

which implies (4.1) since ${\mathscr {E}}_T$ is sufficiently small. $\square$

Proof of Theorem 1.1

The estimate in Lemma 4.1 shows that the smooth solution W is uniformly bounded in $L^\infty ([0,T];H^s({\mathbb {R}}))$ with respect to $\varepsilon$ and T. By the bootstrap principle, it yields uniformly global solution. In particular, this estimate gives (1.16). $\square$

Proof of Theorem 1.2

From (1.11), $(\rho ^\varepsilon , u^\varepsilon , \theta ^\varepsilon , q^\varepsilon , \tau ^\varepsilon )$ satisfies the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho ^\varepsilon +\partial _x(\rho ^\varepsilon u^\varepsilon )=0,\\ \partial _t(\rho ^\varepsilon u^\varepsilon )+\partial _x(\rho ^\varepsilon (u^\varepsilon )^2+ p^\varepsilon -\tau ^\varepsilon )=0,\\ \partial _t(\rho ^\varepsilon E^\varepsilon )+\partial _x(\rho ^\varepsilon u^\varepsilon E^\varepsilon +u^\varepsilon p^\varepsilon +q^\varepsilon -u^\varepsilon \tau ^\varepsilon )=0,\\ \varepsilon _1^2 (\partial _tq^\varepsilon +u^\varepsilon \partial _xq^\varepsilon )+\kappa \partial _x\theta ^\varepsilon =-q^\varepsilon , \\ \varepsilon _2^2(\partial _t\tau ^\varepsilon +u^\varepsilon \partial _x\tau ^\varepsilon )-\lambda \partial _xu^\varepsilon =-\tau ^\varepsilon , \end{array}\right. } \end{aligned}$$

(4.5)

in ${\mathbb {R}}^+\times {\mathbb {R}}$, where

$$\begin{aligned} {\left\{ \begin{array}{ll} e^\varepsilon =c_v\theta ^\varepsilon +\dfrac{\varepsilon _1^2}{\kappa \rho ^\varepsilon \theta ^\varepsilon }(q^\varepsilon )^2 +\dfrac{\varepsilon _2^2}{2\lambda \rho ^\varepsilon }(\tau ^\varepsilon )^2, \\ p^\varepsilon =R\rho ^\varepsilon \theta ^\varepsilon -\dfrac{\varepsilon _1^2}{2\kappa \theta ^\varepsilon }(q^\varepsilon )^2 -\dfrac{\varepsilon _2^2}{2\lambda }(\tau ^\varepsilon )^2, \\ E^\varepsilon =e^\varepsilon +\dfrac{1}{2}(u^\varepsilon )^2. \end{array}\right. } \end{aligned}$$

(4.6)

The uniform estimate (1.16) implies that the sequences $(\rho ^\varepsilon -1)_{\varepsilon }$, $(u^\varepsilon )_{\varepsilon }$ and $(\theta ^\varepsilon -1)_{\varepsilon }$ are bounded in $L^\infty ({\mathbb {R}}^+;H^s({\mathbb {R}}))$ and the sequence $(q^\varepsilon )_{\varepsilon }$ and $(\tau ^\varepsilon )_{\varepsilon }$ are bounded in $L^2({\mathbb {R}}^+;H^s({\mathbb {R}}))$. It follows that there exist functions $({\bar{\rho }},{\bar{u}},{\bar{\theta }})\in L^\infty ({\mathbb {R}}^+;H^s({\mathbb {R}}))$ and $({\bar{q}},{\bar{\tau }})\in L^2({\mathbb {R}}^+; H^s({\mathbb {R}}))$, such that (1.17)-(1.18) hold. In addition, as $\varepsilon \rightarrow 0$,

$$\begin{aligned} \varepsilon _1^2 (\partial _tq^\varepsilon +u^\varepsilon \partial _xq^\varepsilon ) \rightharpoonup 0 \quad \text {in} \,\, {\mathcal {D}}^\prime ({\mathbb {R}}^+\times {\mathbb {R}}), \end{aligned}$$

and

$$\begin{aligned} \varepsilon _2^2(\partial _t\tau ^\varepsilon +u^\varepsilon \partial _x\tau ^\varepsilon ) \rightharpoonup 0 \quad \text {in} \,\, {\mathcal {D}}^\prime ({\mathbb {R}}^+\times {\mathbb {R}}). \end{aligned}$$

Moreover, from the first three equations in (4.5), it is easy to see that $(\partial _t\rho ^\varepsilon )_{\varepsilon }$, $(\partial _tu^\varepsilon )_{\varepsilon }$ and $(\partial _t\theta ^\varepsilon )_{\varepsilon }$ are bounded in $L^2({\mathbb {R}}^+;H^{s-1}({\mathbb {R}}))$. Hence, by a classical compactness theorem [40], for all $T>0$, $(\rho ^\varepsilon )_{\varepsilon },(u^\varepsilon )_{\varepsilon }$ and $(\theta ^\varepsilon )_{\varepsilon }$ are relatively compact in $C([0,T];H_{loc}^{s-1}({\mathbb {R}}))$. As a consequence, as $\varepsilon \rightarrow 0$, up to subsequences,

$$\begin{aligned} (\rho ^\varepsilon ,u^\varepsilon ,\theta ^\varepsilon )\rightarrow ({\bar{\rho }},{\bar{u}},{\bar{\theta }}) \quad \text {strongly in } \; C([0,T];H_{loc}^{s-1}({\mathbb {R}})). \end{aligned}$$

This is sufficient to pass the limit in (4.5)-(4.6) in the sense of distributions and to obtain the Navier-Stokes equations for the ideal fluid. This ends the proof of Theorem 1.2. $\square$