Asymptotics Toward Viscous Contact Waves for Solutions of the Landau Equation

Abstract

In the paper, we are concerned with the large time asymptotics toward the viscous contact waves for solutions of the Landau equation with physically realistic Coulomb interactions. Precisely, for the corresponding Cauchy problem in the spatially one-dimensional setting, we construct the unique global-in-time solution near a local Maxwellian whose fluid quantities are the viscous contact waves in the sense of hydrodynamics and also prove that the solution tends toward such local Maxwellian in large time. The result is proved by elaborate energy estimates and seems the first one on the dynamical stability of contact waves for the Landau equation. One key point of the proof is to introduce a new time-velocity weight function that includes an exponential factor of the form $\exp (q(t)\langle \xi \rangle ^2)$ with

$$\begin{aligned} q(t):=q_1-q_2\int _0^tq_3(s)\,ds, \end{aligned}$$

where $q_1$ and $q_2$ are given positive constants and $q_3(\cdot )$ is defined by the energy dissipation rate of the solution itself. The time derivative of such weight function is able to induce an extra quartic dissipation term for treating the large-velocity growth in the nonlinear estimates due to degeneration of the linearized Landau operator in the Coulomb case. Note that in our problem the explicit time-decay of solutions around contact waves is unavailable but no longer needed under the crucial use of the above weight function, which is different from the situation in Duan (Ann Inst H Poincaré Anal Non Linéaire 31:751–778, 2014) and Duan and Yu (Adv Math 362:106956, 2020).

Appendix

1.1 Burnett functions

In this appendix, we will give some basic estimates used in the previous energy estimates. To overcome some difficulties due to the term involving $L^{-1}_{M}$ and ${\overline{G}}$, we need to consider the integrality about the velocity. To this end, we first list some properties of the Burnett functions and then give the fast decay about the velocity $\xi$ of the Burnett functions. Recall the Burnett functions, cf. [2, 3, 8, 19, 47]:

$$\begin{aligned} {\hat{A}}_{j}(\xi )=\frac{|\xi |^{2}-5}{2}\xi _{j}\quad \text{ and } \quad {\hat{B}}_{ij}(\xi )=\xi _{i}\xi _{j}-\frac{1}{3}\delta _{ij}|\xi |^{2} \quad \text{ for } \quad i,j=1,2,3. \end{aligned}$$

(6.1)

Noting that ${\hat{A}}_{j}M$ and ${\hat{B}}_{ij}M$ are orthogonal to the null space ${\mathcal {N}}$ of $L_{M}$, we can define functions $A_{j}(\frac{\xi -u}{\sqrt{R\theta }})$ and $B_{ij}(\frac{\xi -u}{\sqrt{R\theta }})$ such that $P_{0}A_{j}=0$, $P_{0}B_{ij}=0$ and

$$\begin{aligned} A_{j}(\frac{\xi -u}{\sqrt{R\theta }})=L^{-1}_{M}[{\hat{A}}_{j}(\frac{\xi -u}{\sqrt{R\theta }})M]\quad \text{ and } \quad B_{ij}(\frac{\xi -u}{\sqrt{R\theta }})=L^{-1}_{M}[{\hat{B}}_{ij}(\frac{\xi -u}{\sqrt{R\theta }})M].\nonumber \\ \end{aligned}$$

(6.2)

We shall list some elementary but important properties of the Burnett functions summarized in the following lemma, cf. [2, 3, 19, 47].

Lemma 6.1

The Burnett functions have the following properties:

• $-\langle {\hat{A}}_{i}, A_{i}\rangle$   is positive and independent of i;

• $\langle {\hat{A}}_{i}, A_{j}\rangle =0$   for  any  $i\ne j$;   $\langle {\hat{A}}_{i}, B_{jk}\rangle =0$  for  any  i, j, k;

• $\langle {\hat{B}}_{ij},B_{kj}\rangle =\langle {\hat{B}}_{kl},B_{ij}\rangle =\langle {\hat{B}}_{ji},B_{kj}\rangle$,   which is independent of  i, j, for fixed  k, l;

• $-\langle {\hat{B}}_{ij}, B_{ij}\rangle$   is positive and independent of i, j when $i\ne j$;

• $-\langle {\hat{B}}_{ii}, B_{jj}\rangle$   is positive and independent of i, j when $i\ne j$;

• $-\langle {\hat{B}}_{ii}, B_{ii}\rangle$   is positive and independent of i;

• $\langle {\hat{B}}_{ij}, B_{kl}\rangle =0$   unless either $(i,j)=(k,l)$ or (l, k), or i=j and k=l;

• $\langle {\hat{B}}_{ii}, B_{ii}\rangle -\langle {\hat{B}}_{ii}, B_{jj}\rangle =2\langle {\hat{B}}_{ij}, B_{ij}\rangle$   holds for any  $i\ne j$.

In terms of Burnett functions, the viscosity coefficient $\mu (\theta )$ and heat conductivity coefficient $\kappa (\theta )$ in (2.8) can be represented by

$$\begin{aligned} \mu (\theta )=&- R\theta \int _{{\mathbb {R}}^{3}}{\hat{B}}_{ij}(\frac{\xi -u}{\sqrt{R\theta }}) B_{ij}(\frac{\xi -u}{\sqrt{R\theta }})\,d\xi>0,\quad i\ne j, \nonumber \\ \kappa (\theta )=&-R^{2}\theta \int _{{\mathbb {R}}^{3}}{\hat{A}}_{j}(\frac{\xi -u}{\sqrt{R\theta }}) A_{j}(\frac{\xi -u}{\sqrt{R\theta }})\,d\xi >0. \end{aligned}$$

(6.3)

Notice that these coefficients are positive smooth functions depending only on $\theta$.

The following lemma is borrowed from [16, Lemma 6.1], which is about the fast velocity decay of the Burnett functions.

Lemma 6.2

Suppose that $U(\xi )$ is any polynomial of $\frac{\xi -{\hat{u}}}{\sqrt{R}{\hat{\theta }}}$ such that $U(\xi ){\widehat{M}}\in (\ker {L_{{\widehat{M}}}})^{\perp }$ for any Maxwellian ${\widehat{M}}=M_{[1/{\widehat{v}},{\widehat{u}},{\widehat{\theta }}]}(\xi )$ as (1.13) where $L_{{\widehat{M}}}$ is as in (2.12) . For any $\varepsilon \in (0,1)$ and any multi-index $\beta$, there exists constant $C_{\beta }>0$ such that

$$\begin{aligned} |\partial _{\beta }L^{-1}_{{\widehat{M}}}(U(\xi ){\widehat{M}})|\le C_{\beta }({\widehat{v}},{\widehat{u}},{\widehat{\theta }}){\widehat{M}}^{1-\varepsilon }. \end{aligned}$$

In particular, under the assumptions of (3.8), there exists constant $C_{\beta }>0$ such that

$$\begin{aligned} |\partial _{\beta }A_{j}(\frac{\xi -u}{\sqrt{R\theta }})|+|\partial _{\beta }B_{ij}(\frac{\xi -u}{\sqrt{R\theta }})| \le C_{\beta }M^{1-\varepsilon }. \end{aligned}$$

(6.4)

1.2 Estimates on terms of ${\mathcal {L}}$ and $\Gamma$

Now, we shall turn to recall the refined estimates for the linearized operator ${\mathcal {L}}$ and the nonlinear collision terms $\Gamma (g_1,g_2)$ defined in (3.2). They can be proved by a straightforward modification of the arguments used in [45, Lemmas 9] and [50, Lemmas 2.2–2.3] and we thus omit their proofs for brevity.

Lemma 6.3

Assume $0\le q(t)\ll 1$ in $w=w(\beta )$ defined by (1.18). For any $\epsilon >0$ small enough, there exists $C_\epsilon >0$ such that

$$\begin{aligned} -\langle \partial ^\alpha _\beta {\mathcal {L}}g,w^2(\beta )\partial ^\alpha _\beta g\rangle \ge |w(\beta )\partial ^\alpha _\beta g|_\sigma ^2-\epsilon \sum _{|\beta _1|=|\beta |}|w(\beta _1)\partial ^\alpha _{\beta _1} g|_\sigma ^2 -C_\epsilon \sum _{|\beta _1|<|\beta |}|w(\beta _1)\partial ^\alpha _{\beta _1} g|_\sigma ^2.\nonumber \\ \end{aligned}$$

(6.5)

If $|\beta | = 0$, there exists $c_{4}>0$ such that

$$\begin{aligned} -\langle \partial ^\alpha {\mathcal {L}}g,w^2(0)\partial ^\alpha g\rangle \ge c_{4}|w(0)\partial ^\alpha g|_\sigma ^2-C_\epsilon |\chi _{\epsilon }(\xi )\partial ^\alpha g|_2^2, \end{aligned}$$

(6.6)

where $\chi _\epsilon (\xi )$ is a general cutoff function depending on $\epsilon$.

Lemma 6.4

Under the assumptions of Lemma 6.3. For any $\varepsilon >0$ small enough, one has

$$\begin{aligned} \langle \partial ^\alpha \Gamma (g_1,g_2),g_3\rangle \le C\sum _{\alpha _1\le \alpha }|\mu ^\varepsilon \partial ^{\alpha _1}g_1|_2| \partial ^{\alpha -\alpha _1}g_2|_\sigma | g_3|_\sigma , \end{aligned}$$

(6.7)

and

$$\begin{aligned} \langle \partial ^\alpha _\beta \Gamma (g_1,g_2), w^2(\beta ) g_3\rangle \le C\sum _{\alpha _1\le \alpha }\sum _{{\bar{\beta }}\le \beta _1\le \beta } |\mu ^\varepsilon \partial ^{\alpha _1}_{{\bar{\beta }}}g_1|_2|w(\beta ) \partial ^{\alpha -\alpha _1}_{\beta -\beta _1}g_2|_{\sigma }|w(\beta ) g_3|_{\sigma }.\nonumber \\ \end{aligned}$$

(6.8)

Next we prove some linear and nonlinear estimates, which are used in Sects. 3 and 4. We first consider the estimates of the terms $\Gamma ({\mathbf {g}},\frac{M-\mu }{\sqrt{\mu }})$ and $\Gamma (\frac{M-\mu }{\sqrt{\mu }},{\mathbf {g}})$.

Lemma 6.5

Let $|\alpha |+|\beta |\le 2$ and $0\le q(t)\ll 1$ in $w=w(\beta )$ defined by (1.18). Suppose that (3.6), (3.7) and (3.8) hold. If we choose $\eta _0>0$ in (3.8), $\varepsilon _{0}>0$ in (3.6) and $\delta >0$ in (2.18) small enough, one has

$$\begin{aligned}&|(\partial ^\alpha _\beta [v\Gamma (\frac{M-\mu }{\sqrt{\mu }},{\mathbf {g}})], w^2(\beta ) h)| +|(\partial ^\alpha _\beta [v\Gamma ({\mathbf {g}},\frac{M-\mu }{\sqrt{\mu }})],w^2(\beta ) h)| \nonumber \\&\le C(\eta _0+\delta +\varepsilon _{0})\Vert w(\beta )h\Vert _{\sigma }^2 +C(\eta _0+\delta +\varepsilon _{0}){\mathcal {D}}_{2,l,q}(t), \end{aligned}$$

(6.9)

and

$$\begin{aligned}&|(\partial ^\alpha [v\Gamma (\frac{M-\mu }{\sqrt{\mu }},{\mathbf {g}})],h)| +|(\partial ^\alpha [v \Gamma ({\mathbf {g}},\frac{M-\mu }{\sqrt{\mu }})],h)| \nonumber \\&\le C(\eta _0+\delta +\varepsilon _{0})\Vert h\Vert _{\sigma }^2 +C(\eta _0+\delta +\varepsilon _{0}){\mathcal {D}}_{2,l,q}(t). \end{aligned}$$

(6.10)

Proof

We only consider the first term on the left hand side of (6.9) while the second term can be handled in the same way. Notice that

$$\begin{aligned} \partial ^\alpha _\beta [v\Gamma (\frac{M-\mu }{\sqrt{\mu }},{\mathbf {g}})]=\sum _{\alpha _1\le \alpha } C^{\alpha _{1}}_{\alpha }\partial ^{\alpha -\alpha _{1}}v\partial ^{\alpha _{1}}_{\beta }\Gamma (\frac{M-\mu }{\sqrt{\mu }},{\mathbf {g}}). \end{aligned}$$

It follows from this and (6.8) that

$$\begin{aligned}&|(\partial ^\alpha _\beta [v\Gamma (\frac{M-\mu }{\sqrt{\mu }},{\mathbf {g}})], w^2(\beta ) h)| \nonumber \\&\le C\sum _{\alpha _2\le \alpha _1\le \alpha }\sum _{{\bar{\beta }}\le \beta _1\le \beta } \underbrace{\int _{\mathbb R}|\partial ^{\alpha -\alpha _{1}}v||\mu ^\varepsilon \partial ^{\alpha _2}_{{\bar{\beta }}}(\frac{M-\mu }{\sqrt{\mu }})|_2| w(\beta ) \partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}|_{\sigma }| w(\beta )h|_{\sigma }\,dx}_{I_{1}}. \end{aligned}$$

(6.11)

For any $\beta '\ge 0$ and any $b>0$, from (1.18), (1.22), (3.7) and (3.8), there exists a small $\varepsilon _{1}>0$ such that

$$\begin{aligned} | \langle \xi \rangle ^{b}\partial _{\beta '}(\frac{M-\mu }{\sqrt{\mu }})|_{\sigma ,w}^2+| \langle \xi \rangle ^{b}\partial _{\beta '}(\frac{M-\mu }{\sqrt{\mu }})|_{2,w}^2\le C_b\sum _{\beta '\le \beta ''\le \beta '+1}\int _{{\mathbb R}^3}\mu ^{-\varepsilon _1} |\partial _{\beta ''}(\frac{M-\mu }{\sqrt{\mu }})|^2\,d\xi . \end{aligned}$$

For $\eta _{0}>0$ in (3.8), there exists some large constant $R>0$ such that

$$\begin{aligned} \int _{|\xi |\ge R}\mu ^{-\varepsilon _1}|\partial _{\beta ''}(\frac{M-\mu }{\sqrt{\mu }})|^2 \,d\xi \le C(\eta _0+\varepsilon _{0})^2, \end{aligned}$$

and

$$\begin{aligned} \int _{|\xi |\le R}\mu ^{-\varepsilon _1}|\partial _{\beta ''}(\frac{M-\mu }{\sqrt{\mu }})|^2 \,d\xi \le C(|v-1|+|u|+|\theta -\frac{3}{2}|)^2\le C(\eta _0+\varepsilon _{0})^2. \end{aligned}$$

Thus, for any $\beta '\ge 0$ and $b>0$, we deduce from the above estimates that

$$\begin{aligned} | \langle \xi \rangle ^{b}\partial _{\beta '}(\frac{M-\mu }{\sqrt{\mu }})|_{\sigma ,w}^2+| \langle \xi \rangle ^{b}\partial _{\beta '}(\frac{M-\mu }{\sqrt{\mu }})|_{2,w}^2 \le C(\eta _0+\varepsilon _{0})^2. \end{aligned}$$

(6.12)

Note that $|\alpha _2|\le |\alpha _1|\le |\alpha |\le 2$ in (6.11) since we consider $|\alpha |+|\beta |\le 2$. If $|\alpha _2|=0$ and $|\alpha -\alpha _{1}|\le \frac{|\alpha |}{2}$, we have from (6.12) and (1.24) that

$$\begin{aligned} I_{1}=&\int _{\mathbb R}|\partial ^{\alpha -\alpha _{1}}v||\mu ^\varepsilon \partial ^{\alpha _2}_{{\bar{\beta }}} (\frac{M-\mu }{\sqrt{\mu }})|_2| w(\beta ) \partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}|_{\sigma }| w(\beta )h|_{\sigma }\,dx \\&\le C(\eta _0+\varepsilon _{0})(\Vert \partial ^{\alpha -\alpha _{1}}{\widetilde{v}}\Vert _{L_{x}^{\infty }} +\Vert \partial ^{\alpha -\alpha _{1}}{\bar{v}}\Vert _{L_{x}^{\infty }})\Vert w(\beta )\partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}\Vert _{\sigma }\Vert w(\beta )h\Vert _{\sigma } \\&\le C(\eta _0+\varepsilon _{0})(\Vert w(\beta ) h\Vert _{\sigma }^2+\Vert \partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}\Vert ^{2}_{\sigma ,w}) \le C(\eta _0+\varepsilon _{0})(\Vert w(\beta ) h\Vert _{\sigma }^2+{\mathcal {D}}_{2,l,q}(t)), \end{aligned}$$

where we have used the facts that $w(\beta )\le w(\beta -\beta _1)$ and

$$\begin{aligned} \Vert \partial ^{\alpha -\alpha _{1}}{\widetilde{v}}\Vert _{L_{x}^{\infty }} +\Vert \partial ^{\alpha -\alpha _{1}}{\bar{v}}\Vert _{L_{x}^{\infty }}\le C, \end{aligned}$$

due to the imbedding inequality, (2.20) and (3.6). If $|\alpha _2|=0$ and $|\alpha -\alpha _{1}|>\frac{|\alpha |}{2}$, we have

$$\begin{aligned} I_{1}&\le C(\eta _0+\varepsilon _{0})\int _{\mathbb R}|\partial ^{\alpha -\alpha _{1}}v|| w(\beta ) \partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}|_{\sigma }| w(\beta )h|_{\sigma }\,dx \\&\le C(\eta _0+\varepsilon _{0})\Vert \partial ^{\alpha -\alpha _{1}}v\Vert \Big \Vert |w(\beta )\partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}|_{\sigma }\Big \Vert _{L_{x}^{\infty }}\Vert w(\beta )h\Vert _{\sigma } \\&\le C(\eta _0+\varepsilon _{0})\Vert w(\beta )\partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}\Vert ^{\frac{1}{2}}_{\sigma } \Vert w(\beta )\partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}_{x}\Vert ^{\frac{1}{2}}_{\sigma }\Vert w(\beta )h\Vert _{\sigma } \\&\le (\eta _0+\varepsilon _{0})\Vert w(\beta ) h\Vert _{\sigma }^2+ C(\eta _0+\varepsilon _{0}){\mathcal {D}}_{2,l,q}(t). \end{aligned}$$

If $|\alpha _2|=1$, then $|\alpha -\alpha _{1}|\le 1$, we have from the imbedding inequality, (2.20), (1.23) and (3.6)that

$$\begin{aligned} I_{1}&\le C\Vert \partial ^{\alpha -\alpha _{1}}v\Vert _{L_{x}^{\infty }}\Vert \partial ^{\alpha _2}[v,u,\theta ]\Vert _{L_{x}^\infty } \Vert w(\beta )\partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}\Vert _{\sigma }\Vert w(\beta ) h\Vert _{\sigma } \\&\le C\big (\delta +\sqrt{{\mathcal {E}}_{2,l,q}(t)}\big )\Vert w(\beta )\partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}\Vert _{\sigma }\Vert w(\beta )h\Vert _{\sigma } \\&\le C(\delta +\varepsilon _{0})\Vert w(\beta ) h\Vert ^{2}_{\sigma }+C(\delta +\varepsilon _{0}){\mathcal {D}}_{2,l,q}(t). \end{aligned}$$

If $|\alpha _2|=2$, then $|\alpha _1|=|\alpha |=2$, we can obtain

$$\begin{aligned} I_{1}&\le C(\Vert \partial ^{\alpha _2}[v,u,\theta ]\Vert +\sum _{|\alpha '|=1}\Vert |\partial ^{\alpha '}[v,u,\theta ]|^{2}\Vert ) \Big \Vert |w(\beta )\partial ^{\alpha -\alpha _1}_{\beta _{1}-\beta _2}{\mathbf {g}}|_{\sigma }\Big \Vert _{L_{x}^{\infty }}\Vert w(\beta ) h\Vert _{\sigma } \\&\le C(\delta +\varepsilon _{0})\Vert w(\beta ) h\Vert ^{2}_{\sigma }+C(\delta +\varepsilon _{0}){\mathcal {D}}_{2,l,q}(t). \end{aligned}$$

Hence, for $\eta _0>0$, $\delta >0$ and $\varepsilon _{0}>0$ small enough, we deduce from the above estimates that

$$\begin{aligned} |(\partial ^\alpha _\beta [v \Gamma (\frac{M-\mu }{\sqrt{\mu }},{\mathbf {g}})],w^2(\beta ) h)| \le C(\eta _0+\delta +\varepsilon _{0})\big (\Vert w(\beta )h\Vert _{\sigma }^2+{\mathcal {D}}_{2,l,q}(t)\big ). \end{aligned}$$

Similar arguments as the above give

$$\begin{aligned} |(\partial ^\alpha _\beta [v\Gamma ({\mathbf {g}},\frac{M-\mu }{\sqrt{\mu }})], w^2(\beta ) h)| \le C(\eta _0+\delta +\varepsilon _{0})\big (\Vert w(\beta )h\Vert _{\sigma }^2+{\mathcal {D}}_{2,l,q}(t)\big ). \end{aligned}$$

Estimate (6.9) thus follows from the above two estimates. By (6.7) and the similar calculations as (6.9), we can prove that (6.10) holds and we omit the details for brevity. This completes the proof of Lemma 6.5. $\square$

The following estimates are concerned with the nonlinear term $\Gamma (\frac{G}{\sqrt{\mu }},\frac{G}{\sqrt{\mu }})$.

Lemma 6.6

Let $|\alpha |+|\beta |\le 2$ and $0\le q(t)\ll 1$ in $w=w(\beta )$ defined by (1.18). Suppose that (3.6), (3.7) and (3.8) hold. If we choose $\varepsilon _{0}>0$ in (3.6) and $\delta >0$ in (2.18) small enough, one has

$$\begin{aligned} |(\partial ^\alpha _\beta [v\Gamma (\frac{G}{\sqrt{\mu }},\frac{G}{\sqrt{\mu }})], w^2(\beta ) h)| \le C(\delta +\varepsilon _{0})\big (\Vert w(\beta )h\Vert _{\sigma }^2+{\mathcal {D}}_{2,l,q}(t)\big )+C\delta (1+t)^{-\frac{4}{3}},\nonumber \\ \end{aligned}$$

(6.13)

and

$$\begin{aligned} |(\partial ^\alpha [v\Gamma (\frac{G}{\sqrt{\mu }},\frac{G}{\sqrt{\mu }})],h)| \le C(\delta +\varepsilon _{0})\big (\Vert h\Vert _{\sigma }^2+{\mathcal {D}}_{2,l,q}(t)\big )+C\delta (1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.14)

Proof

Recalling that $G={\overline{G}}+\sqrt{\mu }{\mathbf {g}}$, we see

$$\begin{aligned} \Gamma (\frac{G}{\sqrt{\mu }},\frac{G}{\sqrt{\mu }})=\Gamma (\frac{{\overline{G}}}{\sqrt{\mu }},\frac{{\overline{G}}}{\sqrt{\mu }}) +\Gamma (\frac{{\overline{G}}}{\sqrt{\mu }},{\mathbf {g}})+\Gamma ({\mathbf {g}},\frac{{\overline{G}}}{\sqrt{\mu }})+\Gamma ({\mathbf {g}},{\mathbf {g}}). \end{aligned}$$

(6.15)

For the first term in (6.15), we have from the similar arguments as (6.11) that

$$\begin{aligned}&|(\partial ^\alpha _\beta [v\Gamma (\frac{{\overline{G}}}{\sqrt{\mu }},\frac{{\overline{G}}}{\sqrt{\mu }})], w^2(\beta )h)|\nonumber \\&\quad \le C\sum _{\alpha _2\le \alpha _1\le \alpha }\sum _{{\bar{\beta }}\le \beta _1\le \beta } \underbrace{\int _{\mathbb R}|\partial ^{\alpha -\alpha _{1}}v||\mu ^\varepsilon \partial ^{\alpha _2}_{{\bar{\beta }}}(\frac{{\overline{G}}}{\sqrt{\mu }})|_2| w(\beta ) \partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}(\frac{{\overline{G}}}{\sqrt{\mu }})|_{\sigma }| w(\beta )h|_{\sigma }\,dx}_{I_{2}}.\nonumber \\ \end{aligned}$$

(6.16)

By (6.1) and (6.2), we can rewrite ${\overline{G}}$ in (1.17) as

$$\begin{aligned} {\overline{G}}(t,x,\xi )=\frac{1}{v}\frac{\sqrt{R}\;{\overline{\theta }}_x}{\sqrt{\theta }}A_1(\frac{\xi -u}{\sqrt{R\theta }}) +\frac{1}{v}{\overline{u}}_{1x}B_{11}(\frac{\xi -u}{\sqrt{R\theta }}), \end{aligned}$$

(6.17)

which implies that for $\beta _1=(1,0,0)$,

$$\begin{aligned} \partial _{\beta _1}{\overline{G}}=\frac{1}{v}\frac{\sqrt{R}\;{\overline{\theta }}_x}{\sqrt{\theta }}\partial _{\xi _1}A_1(\frac{\xi -u}{\sqrt{R\theta }})(\frac{1}{\sqrt{R\theta }}) +\frac{1}{v}{\overline{u}}_{1x}\partial _{\xi _1}B_{11}(\frac{\xi -u}{\sqrt{R\theta }}) (\frac{1}{\sqrt{R\theta }}). \end{aligned}$$

Similarly, we also have

$$\begin{aligned} {\overline{G}}_x=&-\frac{v_{x}\,\sqrt{R}\;{\overline{\theta }}_x }{v^{2}\sqrt{\theta }}A_1(\frac{\xi -u}{\sqrt{R\theta }}) -\frac{v_{x}}{v^{2}}{\overline{u}}_{1x}B_{11}(\frac{\xi -u}{\sqrt{R\theta }}) \nonumber \\&+\frac{1}{v}\frac{\sqrt{R}\;{\overline{\theta }}_{xx}}{\sqrt{\theta }}A_1(\frac{\xi -u}{\sqrt{R\theta }}) -\frac{1}{v}\frac{\sqrt{R}\;{\overline{\theta }}_{x}{\theta }_{x}}{2\sqrt{\theta ^3}}A_1(\frac{\xi -u}{\sqrt{R\theta }}) \nonumber \\&-\frac{1}{v}\frac{\sqrt{R}\;{\overline{\theta }}_{x}}{\sqrt{\theta }}\nabla _\xi A_1(\frac{\xi -u}{\sqrt{R\theta }})\cdot \frac{u_x}{\sqrt{R\theta }} -\frac{1}{v}\frac{\sqrt{R}\; {\overline{\theta }}_{x}{\theta }_{x}}{\sqrt{\theta }}\nabla _\xi A_1(\frac{\xi -u}{\sqrt{R\theta }})\cdot \frac{\xi -u}{2\sqrt{R\theta ^3}} \nonumber \\&+\frac{1}{v}{\overline{u}}_{1xx}B_{11}(\frac{\xi -u}{\sqrt{R\theta }}) -\frac{1}{v}\frac{{\overline{u}}_{1x}u_x}{\sqrt{R\theta }}\cdot \nabla _\xi B_{11}(\frac{\xi -u}{\sqrt{R\theta }}) -\frac{1}{v}\frac{{\overline{u}}_{1x}\theta _x(\xi -u)}{2\sqrt{R\theta ^3}}\cdot \nabla _\xi B_{11}(\frac{\xi -u}{\sqrt{R\theta }}). \end{aligned}$$

(6.18)

And ${\overline{G}}_t$ has the similar expression as (6.18). For any $|{\bar{\alpha }}|\ge 1$ and $|{\bar{\beta }}|\ge 0$, we use the similar expansion as the above to get

$$\begin{aligned} | \langle \xi \rangle ^{b}\partial _{{\bar{\beta }}}(\frac{{\overline{G}}}{\sqrt{\mu }})|_{2,w}+|\langle \xi \rangle ^{b} \partial _{{\bar{\beta }}}(\frac{{\overline{G}}}{\sqrt{\mu }})|_{\sigma ,w} \le C|[{\overline{u}}_x,{\overline{\theta }}_x]|, \end{aligned}$$

(6.19)

and

$$\begin{aligned} |\langle \xi \rangle ^{b} \partial ^{{\bar{\alpha }}}_{{\bar{\beta }}}(\frac{{\overline{G}}}{\sqrt{\mu }})|_{2,w}+| \langle \xi \rangle ^{b} \partial ^{{\bar{\alpha }}}_{{\bar{\beta }}}(\frac{{\overline{G}}}{\sqrt{\mu }})|_{\sigma ,w} \le C(|\partial ^{{\bar{\alpha }}}[{\overline{u}}_x,{\overline{\theta }}_x]|+... +|[{\overline{u}}_x,{\overline{\theta }}_x]||\partial ^{{\bar{\alpha }}}[v,u,\theta ]|).\nonumber \\ \end{aligned}$$

(6.20)

Here we have used Lemma 6.2 and the fact that $|\langle \xi \rangle ^b w({\bar{\beta }})\mu ^{-\frac{1}{2}}M^{1-\varepsilon }|_2\le C$ for any $b\ge 0$ and any small $\varepsilon >0$ by (3.7) and (3.8).

Note that $|\alpha _2|\le |\alpha _1|\le |\alpha |\le 2$ in (6.16) due to the fact that $|\alpha |+|\beta |\le 2$. If $|\alpha -\alpha _{1}|\le 1$, by using (6.19), (6.20), (2.20), (3.6) and the imbedding inequality, one has from (6.16) that

$$\begin{aligned} I_{2}&\le C\Vert \partial ^{\alpha -\alpha _{1}}v\Vert _{L_{x}^{\infty }}\int _{{\mathbb {R}}} \big \{|\partial ^{\alpha _{2}}[{\overline{u}}_x,{\overline{\theta }}_x]|+... +|[{\overline{u}}_x,{\overline{\theta }}_x]||\partial ^{\alpha _{2}}[v,u,\theta ]|\big \} \\&\qquad \times \big \{|\partial ^{\alpha _{1}-\alpha _{2}}[{\overline{u}}_x,{\overline{\theta }}_x]|+... +|[{\overline{u}}_x,{\overline{\theta }}_x]||\partial ^{\alpha _{1}-\alpha _{2}}[v,u,\theta ]|\big \} |w(\beta ) h|_{\sigma }\,dx \\&\le C(\delta +\varepsilon _{0})\big (\Vert w(\beta )h\Vert _{\sigma }^2+{\mathcal {D}}_{2,l,q}(t)\big )+C\delta (1+t)^{-\frac{4}{3}}. \end{aligned}$$

If $|\alpha -\alpha _{1}|=2$, then $|\alpha |=2$ and $|\alpha _{1}|=|\alpha _{2}|=0$, we have

$$\begin{aligned} I_{2}&\le C\Vert [{\overline{u}}_x,{\overline{\theta }}_x]\Vert ^{2}_{L_{x}^{\infty }}\int _{\mathbb R}|\partial ^{\alpha -\alpha _{1}}v||w(\beta ) h|_{\sigma }\,dx \\&\le C\delta \Vert w(\beta ) h\Vert _{\sigma }^2 +C\delta {\mathcal {D}}_{2,l,q}(t)+C\delta (1+t)^{-\frac{4}{3}}. \end{aligned}$$

It follows from the above two estimates and (6.16) that

$$\begin{aligned} |(\partial ^\alpha _\beta [v\Gamma (\frac{{\overline{G}}}{\sqrt{\mu }},\frac{{\overline{G}}}{\sqrt{\mu }})], w^2(\beta )h)| \le C(\delta +\varepsilon _{0})\big (\Vert w(\beta )h\Vert _{\sigma }^2 +{\mathcal {D}}_{2,l,q}(t)\big )+C\delta (1+t)^{-\frac{4}{3}}.\nonumber \\ \end{aligned}$$

(6.21)

For the second term in (6.15), by (6.8), we can obtain

$$\begin{aligned}&|(\partial ^\alpha _\beta [v\Gamma (\frac{{\overline{G}}}{\sqrt{\mu }},{\mathbf {g}})], w^2(\beta )h)| \nonumber \\&\le C\sum _{\alpha _2\le \alpha _1\le \alpha }\sum _{{\bar{\beta }}\le \beta _1\le \beta } \underbrace{\int _{\mathbb R}|\partial ^{\alpha -\alpha _{1}}v||\mu ^\varepsilon \partial ^{\alpha _2}_{{\bar{\beta }}} (\frac{{\overline{G}}}{\sqrt{\mu }})|_2| w(\beta ) \partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}|_{\sigma }| w(\beta )h|_{\sigma }\,dx}_{I_{3}}. \end{aligned}$$

(6.22)

Notice that $|\alpha _2|\le |\alpha _1|\le |\alpha |\le 2$ in (6.22). If $|\alpha -\alpha _{1}|\le 1$ and $|\alpha _{2}|\le 1$, we can deduce from (6.19), (6.20), (2.20), (3.6) and the imbedding inequality that

$$\begin{aligned} I_{3}&\le C\Big \Vert |\partial ^{\alpha -\alpha _{1}}v|\big \{|\partial ^{\alpha _{2}}[{\overline{u}}_x,{\overline{\theta }}_x]| +|[{\overline{u}}_x,{\overline{\theta }}_x]||\partial ^{\alpha _{2}}[v,u,\theta ]|\big \}\Big \Vert _{L^{\infty }_{x}} \int _{\mathbb R}|w(\beta )\partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}|_{\sigma } |w(\beta )h|_{\sigma }\,dx \\&\le C(\delta +\varepsilon _{0})\Vert w(\beta )h\Vert _{\sigma }^2+C(\delta +\varepsilon _{0}){\mathcal {D}}_{2,l,q}(t), \end{aligned}$$

where we used the fact that $w(\beta )\le w(\beta -\beta _1)$ due to (1.18).

If $|\alpha -\alpha _{1}|\le 1$ and $|\alpha _{2}|=2$, then $|\alpha |=|\alpha _{1}|=|\alpha _{2}|=2$ and we have

$$\begin{aligned} I_{3}&\le C\Big \Vert |w(\beta )\partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}|_{\sigma }\Big \Vert _{L^{\infty }_{x}} \int _{\mathbb R}\big \{|\partial ^{\alpha _{2}}[{\overline{u}}_x,{\overline{\theta }}_x]|+\cdot \cdot \cdot +|[{\overline{u}}_x,{\overline{\theta }}_x]||\partial ^{\alpha _{2}}[v,u,\theta ]|\big \} |w(\beta )h|_{\sigma }\,dx \\&\le C(\delta +\varepsilon _{0})\Vert w(\beta )h\Vert _{\sigma }^2+C(\delta +\varepsilon _{0}){\mathcal {D}}_{2,l,q}(t). \end{aligned}$$

If $|\alpha -\alpha _{1}|=2$ , then $|\alpha |=2$ and $|\alpha _{1}|=|\alpha _{2}|=0$, it follows that

$$\begin{aligned} I_{3}&\le C\Big \Vert |[{\overline{u}}_x,{\overline{\theta }}_x]||w(\beta ) \partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}|_{\sigma }\Big \Vert _{L^{\infty }_{x}} \int _{{\mathbb {R}}}|\partial ^{\alpha -\alpha _{1}}v||w(\beta ) h|_{\sigma }\,dx \\&\le C(\delta +\varepsilon _{0})\Vert w(\beta )h\Vert _{\sigma }^2+ C(\delta +\varepsilon _{0}){\mathcal {D}}_{2,l,q}(t). \end{aligned}$$

Owing to these, we can derive that

$$\begin{aligned} |(\partial ^\alpha _\beta [v\Gamma (\frac{{\overline{G}}}{\sqrt{\mu }},{\mathbf {g}})], w^2(\beta )h)| \le C(\delta +\varepsilon _{0})\big (\Vert w(\beta )h\Vert _{\sigma }^2+{\mathcal {D}}_{2,l,q}(t)\big ). \end{aligned}$$

(6.23)

Similar arguments as (6.23) imply

$$\begin{aligned} |(\partial ^\alpha _\beta [v\Gamma ({\mathbf {g}},\frac{{\overline{G}}}{\sqrt{\mu }})], w^2(\beta )h)| \le C(\delta +\varepsilon _{0})\big (\Vert w(\beta )h\Vert _{\sigma }^2+{\mathcal {D}}_{2,l,q}(t)\big ). \end{aligned}$$

By (6.8) and the similar calculations as (6.23), we can arrive at

$$\begin{aligned}&|(\partial ^\alpha _\beta [v\Gamma ({\mathbf {g}},{\mathbf {g}})],w^2(\beta )h)| \nonumber \\&\le C\sum _{\alpha _2\le \alpha _1\le \alpha }\sum _{{\bar{\beta }}\le \beta _1\le \beta } \int _{\mathbb R}|\partial ^{\alpha -\alpha _{1}}v||\mu ^\varepsilon \partial ^{\alpha _2}_{{\bar{\beta }}}{\mathbf {g}}|_2| w(\beta ) \partial ^{\alpha _{1}-\alpha _2}_{\beta -\beta _1}{\mathbf {g}}|_{\sigma }| w(\beta )h|_{\sigma }\,dx \nonumber \\&\le C(\delta +\varepsilon _{0})\Vert w(\beta )h\Vert _{\sigma }^2+C(\delta +\varepsilon _{0}){\mathcal {D}}_{2,l,q}(t). \end{aligned}$$

(6.24)

By the estimates from (6.21) to (6.24), one gets (6.13). We can follow the similar calculations as (6.21)-(6.24) to get (6.14). Therefore, the proof of Lemma 6.6 is completed.

$\square$

1.3 A technical lemma for weighted macro estimates

Finally, we will deduce a crucial estimate to control the last term in (3.23) by using the system (3.4) and the properties of the viscous contact wave profiles. We first give the following lemma, which can be found in [25, Lemma 1].

Lemma 6.7

For $0<T\le +\infty$, suppose that h(t, x) satisfies

$$\begin{aligned} h_{x}\in L^{2}(0,T;L^{2}({\mathbb {R}})), \quad h_{t}\in L^{2}(0,T;H^{-1}({\mathbb {R}})). \end{aligned}$$

Then the following estimate holds

$$\begin{aligned} \int ^{T}_{0}\int _{{\mathbb {R}}} h^{2}\omega ^{2}\,dxdt\le 4\pi \Vert h(0)\Vert ^{2}+4\pi \lambda ^{-1}\int ^{T}_{0}\Vert h_{x}(t)\Vert ^{2}\,dt +8\lambda \int ^{T}_{0}( h_{t},h{\mathfrak {g}}^{2})\,dt, \end{aligned}$$

for some $\lambda >0$, where

$$\begin{aligned} \omega (t,x)=(1+t)^{-\frac{1}{2}}\exp \big (-\frac{\lambda x^{2}}{1+t}\big ), \quad {\mathfrak {g}}(t,x)=\int ^{x}_{-\infty }\omega (t,y)\,dy. \end{aligned}$$

(6.25)

The following lemma is used to deal with the last term in (3.23).

Lemma 6.8

For $\lambda \in (0,c_{1}/4]$ with $c_{1}$ in (2.18) and $\omega$ defined in (6.25), if (3.6) holds, there exists $C_{2}>0$ such that the following estimate holds

$$\begin{aligned} \int ^{t}_{0}\int _{{\mathbb {R}}}&({\widetilde{v}}^{2}+{\widetilde{u}}^{2}+{\widetilde{\theta }}^{2})\omega ^{2}\,dxds \nonumber \\&\le C_{2}+C_{2}\varepsilon _{0} \int ^{t}_{0} \Vert {\mathbf {g}}\Vert ^{2}_{\sigma }\, ds +C_{2}\sum _{|\alpha |=1}\int ^{t}_{0}\big (\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma } \big )\,ds. \end{aligned}$$

(6.26)

Proof

As in [25], we define

$$\begin{aligned} {\mathfrak {f}}(t,x)=\int _{-\infty }^{x}\omega ^{2}(t,y)\,dy. \end{aligned}$$

(6.27)

It is easy to check that

$$\begin{aligned} \Vert {\mathfrak {f}}(t,x)\Vert _{L_{x}^{\infty }}\le 2\lambda ^{-\frac{1}{2}}(1+t)^{-\frac{1}{2}}, \quad \Vert {\mathfrak {f}}_{t}(t,x)\Vert _{L_{x}^{\infty }}\le 4\lambda ^{-\frac{1}{2}}(1+t)^{-\frac{3}{2}}. \end{aligned}$$

(6.28)

Taking the inner product of (3.4)$_{2}$ with $(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}$ with respect to x over ${\mathbb {R}}$ and using the fact that $p-p_{+}=\frac{2{\widetilde{\theta }}-3p_{+}{\widetilde{v}}}{3v}$, the integration by parts and (6.27), we have

$$\begin{aligned} \frac{1}{2}\big ((\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})^{2},\omega ^{2}\big )&=\big ({\widetilde{u}}_{1t},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}\big )- \big (\frac{1}{v}(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})^{2},v_{x}{\mathfrak {f}}\big ) \nonumber \\&\qquad +\big (\frac{4}{3}\frac{\mu (\theta )}{v}u_{1x}, [(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}]_{x}\big ) +\big ({\bar{u}}_{1t},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}\big ) \nonumber \\&\qquad -\big (\int _{{\mathbb {R}}^{3}} \xi ^{2}_{1}L^{-1}_{M}\Theta _{1}\, d\xi ,[(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}]_{x}\big ) :=\sum _{i=4}^{8}I_{i}. \end{aligned}$$

(6.29)

By (6.29), the proof of (6.26) is similar to [25, Lemma 5] for the stability of viscous contact wave for the compressible Navier–Stokes system. Here the difference is that we need to estimate the terms involving $L^{-1}_{M}$ additionally. For completeness, we will estimate each term in (6.29). For the term $I_4$ in (6.29), we see

$$\begin{aligned} I_{4}&=\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}\big )_{t} -\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})_{t}v{\mathfrak {f}}\big )\nonumber \\&\qquad -\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v_{t}{\mathfrak {f}}\big ) -\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}_{t}\big ). \end{aligned}$$

(6.30)

By using (3.4)$_{1}$ and (3.4)$_{4}$, one has

$$\begin{aligned} ({\widetilde{\theta }}+p_{+}{\widetilde{v}})_{t}&=-(\frac{\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}}}{v})u_{1x} +\big (\frac{\kappa (\theta )}{v}\theta _{x}-\frac{\kappa ({\bar{\theta }})}{{\bar{v}}}{\bar{\theta }}_{x}\big )_{x} +Q_{1} \nonumber \\&\qquad +u\cdot \int _{{\mathbb {R}}^{3}} \xi \xi _{1}(L^{-1}_{M}\Theta _{1})_{x}\,d\xi -\frac{1}{2}\int _{{\mathbb {R}}^{3}}\xi _{1}|\xi |^{2}(L^{-1}_{M}\Theta _{1})_{x} \,d\xi . \end{aligned}$$

(6.31)

For the second term on the right hand side of (6.30), by this, (3.4)$_{1}$ and the integration by parts, one has

$$\begin{aligned}&-\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})_{t}v{\mathfrak {f}}\big )= \frac{5}{3}p_{+}\big ({\widetilde{u}}_{1},{\widetilde{v}}_{t}v{\mathfrak {f}}\big ) -\frac{2}{3}\big ({\widetilde{u}}_{1},({\widetilde{\theta }}+p_{+}{\widetilde{v}})_{t}v{\mathfrak {f}}\big ) \nonumber \\&\quad =\frac{5}{3}p_{+}\big ({\widetilde{u}}_{1},{\widetilde{v}}_{t}v{\mathfrak {f}}\big )+\frac{2}{3}\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})u_{1x}{\mathfrak {f}}\big ) +\frac{2}{3}\big (\frac{\kappa (\theta )}{v}\theta _{x}-\frac{\kappa ({\bar{\theta }})}{{\bar{v}}}{\bar{\theta }}_{x},({\widetilde{u}}_{1}v{\mathfrak {f}})_{x}\big ) -\frac{2}{3}\big ({\widetilde{u}}_{1}v{\mathfrak {f}},Q_{1}\big )\nonumber \\&\qquad +\frac{2}{3}\big ({\widetilde{u}}_{1}v{\mathfrak {f}},u_{x}\cdot \int _{{\mathbb {R}}^{3}} \xi \xi _{1}L^{-1}_{M}\Theta _{1}\, d\xi \big ) -\frac{2}{3}\big (({\widetilde{u}}_{1}v{\mathfrak {f}})_{x},\int _{{\mathbb {R}}^{3}}(\frac{1}{2}\xi _{1}|\xi |^{2}-u\cdot \xi \xi _{1})L^{-1}_{M}\Theta _{1} \,d\xi \big ). \end{aligned}$$

(6.32)

For the first term on the right hand side of (6.32), we have from (3.4)$_{1}$, (2.18), (6.27), (3.12) and the integration by parts that

$$\begin{aligned} \frac{5}{3}p_{+}\big ({\widetilde{u}}_{1},{\widetilde{v}}_{t}v{\mathfrak {f}}\big )&=\frac{5}{3}p_{+}\big ({\widetilde{u}}_{1},{\widetilde{u}}_{1x}v{\mathfrak {f}}\big ) =\frac{5}{6}p_{+}\big (({\widetilde{u}}_{1}^2)_x, v{\mathfrak {f}}\big )\\&= -\frac{5}{6}p_{+}\int _{{\mathbb {R}}}v{\widetilde{u}}_{1}^{2}\omega ^{2}\,dx -\frac{5}{6}p_{+}\int _{{\mathbb {R}}} {\widetilde{u}}_{1}^{2}({\bar{v}}_{x}+{\widetilde{v}}_{x}){\mathfrak {f}}\,dx \\&\le -\frac{5}{6}p_{+}\int _{{\mathbb {R}}}v{\widetilde{u}}_{1}^{2}\omega ^{2}\,dx +C\delta \int _{{\mathbb {R}}}{\widetilde{u}}_{1}^{2}\omega ^{2}\,dx +C(\Vert {\widetilde{v}}_{x}\Vert ^{2}+\Vert {\widetilde{u}}_{x}\Vert ^{2})+C(1+t)^{-\frac{3}{2}}. \end{aligned}$$

By using (3.6), (6.28) and (2.20), one has

$$\begin{aligned}&\frac{2}{3}\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})u_{1x}{\mathfrak {f}}\big ) +\frac{2}{3}\big (\frac{\kappa (\theta )}{v}\theta _{x}-\frac{\kappa ({\bar{\theta }})}{{\bar{v}}}{\bar{\theta }}_{x},({\widetilde{u}}_{1}v{\mathfrak {f}})_{x}\big ) \\&\quad \le C(1+t)^{-\frac{1}{2}}\Vert {\widetilde{u}}_{1}\Vert _{L_{x}^{\infty }} (\Vert {\widetilde{v}}\Vert +\Vert {\widetilde{\theta }}\Vert )\times (\Vert {\widetilde{u}}_{1x}\Vert +\Vert {\bar{u}}_{1x}\Vert ) \\&\qquad +C \{\Vert {\widetilde{\theta }}_{x}\Vert +(1+t)^{-\frac{1}{2}}(\Vert {\widetilde{\theta }}\Vert +\Vert {\widetilde{v}}\Vert ) \}\times \{(1+t)^{-\frac{1}{2}}(\Vert {\widetilde{u}}_{1x}\Vert +\Vert {\widetilde{u}}_{1}v_{x}\Vert )+(1+t)^{-1}\Vert {\widetilde{u}}_{1}\Vert \} \\&\quad \le C(\Vert {\widetilde{v}}_{x}\Vert ^{2}+\Vert {\widetilde{u}}_{x}\Vert ^{2}+\Vert {\widetilde{\theta }}_{x}\Vert ^{2})+C(1+t)^{-\frac{3}{2}}. \end{aligned}$$

By the expression of $Q_{1}$ in (3.5), one gets from (6.28) and (2.20) that

$$\begin{aligned} -\frac{2}{3}\big ({\widetilde{u}}_{1}v{\mathfrak {f}},Q_{1}\big ) \le C(1+t)^{-\frac{1}{2}}\Vert {\widetilde{u}}_{1}\Vert _{L_{x}^{\infty }} \big (\Vert {\widetilde{u}}_{x}\Vert ^{2}+\Vert {\bar{u}}_{x}\Vert ^{2}\big ) \le C\Vert {\widetilde{u}}_{x}\Vert ^{2}+C(1+t)^{-\frac{3}{2}}. \end{aligned}$$

For the last term in (6.32), we have from (3.13) that

$$\begin{aligned}&-\frac{2}{3}\big (({\widetilde{u}}_{1}v{\mathfrak {f}})_{x},\int _{{\mathbb {R}}^{3}}(\frac{1}{2}\xi _{1}|\xi |^{2}-u\cdot \xi \xi _{1})L^{-1}_{M}\Theta _{1}\,d\xi \big )\nonumber \\&\quad =-\frac{2}{3}\int _{{\mathbb {R}}}\Big \{({\widetilde{u}}_{1}v{\mathfrak {f}})_{x} (R\theta )^{\frac{3}{2}}\int _{{\mathbb {R}}^{3}}A_{1}(\frac{\xi -u}{\sqrt{R\theta }})\frac{\Theta _{1}}{M}\, d\xi \Big \}\,dx. \end{aligned}$$

(6.33)

Recall $\Theta _{1}$ in (2.13) that

$$\begin{aligned} \Theta _{1}=G_{t}-\frac{u_{1}}{v}G_{x}+\frac{1}{v}P_{1}(\xi _{1}G_{x})-Q(G,G). \end{aligned}$$

Recalling that $G={\overline{G}}+\sqrt{\mu }{\mathbf {g}}$, by using (6.18), (3.16), (2.20), (6.27) and the imbedding inequality, we have

$$\begin{aligned}&-\frac{2}{3}\int _{{\mathbb {R}}}\Big \{({\widetilde{u}}_{1}v{\mathfrak {f}})_{x} (R\theta )^{\frac{3}{2}}\int _{{\mathbb {R}}^{3}}A_{1}(\frac{\xi -u}{\sqrt{R\theta }})\frac{{\overline{G}}_{t}}{M}\, d\xi \Big \}\,dx \nonumber \\&\quad \le C\{(1+t)^{-\frac{1}{2}}(\Vert {\widetilde{u}}_{1x}\Vert +\Vert v_{x}\Vert )+(1+t)^{-1}\Vert {\widetilde{u}}_{1}\Vert \} \times \big (\int _{{\mathbb {R}}}\int _{{\mathbb {R}}^{3}}|\frac{{\overline{G}}_{t}}{\sqrt{\mu }}|^{2}\,d\xi dx\big )^{\frac{1}{2}} \nonumber \\&\quad \le C\{(1+t)^{-\frac{1}{2}}(\Vert {\widetilde{u}}_{x}\Vert +\Vert {\widetilde{v}}_{x}\Vert +\Vert {\bar{v}}_{x}\Vert )+(1+t)^{-1}\Vert {\widetilde{u}}\Vert \} \nonumber \\&\qquad \times \{\Vert [{\bar{u}}_{1xt},{\bar{\theta }}_{xt}]\Vert +\Vert [{\bar{u}}_{1x},{\bar{\theta }}_{x}]\cdot [v_{t},u_{t},\theta _{t}]\Vert \} \nonumber \\&\quad \le C\Vert [{\widetilde{v}}_{x},{\widetilde{u}}_{x},{\widetilde{\theta }}_{x}]\Vert ^{2} +C\Vert [{\widetilde{v}}_{t},{\widetilde{u}}_{t},{\widetilde{\theta }}_{t}]\Vert ^{2}+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.34)

Similarly, it holds that

$$\begin{aligned}&-\frac{2}{3}\int _{{\mathbb {R}}}\Big \{({\widetilde{u}}_{1}v{\mathfrak {f}})_{x} (R\theta )^{\frac{3}{2}}\int _{{\mathbb {R}}^{3}}A_{1}(\frac{\xi -u}{\sqrt{R\theta }})\frac{\sqrt{\mu }{\mathbf {g}}_{t}}{M} d\xi \Big \}\,dx \nonumber \\&\quad \le C\{(1+t)^{-\frac{1}{2}}(\Vert {\widetilde{u}}_{1x}\Vert +\Vert v_{x}\Vert )+(1+t)^{-1}\Vert {\widetilde{u}}_{1}\Vert \} \times \Vert \langle \xi \rangle ^{-\frac{1}{2}}{\mathbf {g}}_{t}\Vert \nonumber \\&\quad \le C\Vert [{\widetilde{v}}_{x},{\widetilde{u}}_{x},{\widetilde{\theta }}_{x}]\Vert ^{2}+C\Vert {\mathbf {g}}_{t}\Vert ^{2}_{\sigma } +C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.35)

It follows from (6.34) and (6.35) that

$$\begin{aligned}&-\frac{2}{3}\int _{{\mathbb {R}}}\Big \{({\widetilde{u}}_{1}v \mathfrak {{f}})_{x} (R\theta )^{\frac{3}{2}}\int _{{\mathbb {R}}^{3}}A_{1}(\frac{\xi -u}{\sqrt{R\theta }})\frac{G_{t}}{M} \,d\xi \Big \}\,dx \nonumber \\&\quad \le C\Vert [{\widetilde{v}}_{x},{\widetilde{u}}_{x},{\widetilde{\theta }}_{x}]\Vert ^{2} +C\Vert [{\widetilde{v}}_{t},{\widetilde{u}}_{t},{\widetilde{\theta }}_{t}]\Vert ^{2}+C\Vert {\mathbf {g}}_{t}\Vert ^{2}_{\sigma }+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.36)

Using the similar calculations as (6.34), (6.35) and (6.36), we can obtain

$$\begin{aligned}&-\frac{2}{3}\int _{{\mathbb {R}}}\Big \{({\widetilde{u}}_{1}v \mathfrak {{f}})_{x} (R\theta )^{\frac{3}{2}}\int _{{\mathbb {R}}^{3}}A_{1}(\frac{\xi -u}{\sqrt{R\theta }}) \{-\frac{u_{1}}{v}G_{x}+\frac{1}{v}P_{1}(\xi _{1}G_{x})\}\frac{1}{M}\,d\xi \Big \}\,dx \nonumber \\&\quad \le C\Vert [{\widetilde{v}}_{x},{\widetilde{u}}_{x},{\widetilde{\theta }}_{x}]\Vert ^{2}+C\Vert {\mathbf {g}}_{x}\Vert ^{2}_{\sigma }+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

By (6.7), (3.16) and the similar calculations as (3.21), we have

$$\begin{aligned}&-\frac{2}{3}\int _{{\mathbb {R}}}\Big \{({\widetilde{u}}_{1}v \mathfrak {{f}})_{x} (R\theta )^{\frac{3}{2}}\int _{{\mathbb {R}}^{3}}A_{1}(\frac{\xi -u}{\sqrt{R\theta }})\frac{Q(G,G)}{M}\, d\xi \Big \}\,dx \nonumber \\&\quad =-\frac{2}{3}\int _{{\mathbb {R}}}\Big \{({\widetilde{u}}_{1}v \mathfrak {{f}})_{x} (R\theta )^{\frac{3}{2}}\int _{{\mathbb {R}}^{3}}\frac{\sqrt{\mu }A_{1}(\frac{\xi -u}{\sqrt{R\theta }})}{M}\Gamma (\frac{G}{\sqrt{\mu }},\frac{G}{\sqrt{\mu }}) \,d\xi \Big \}\,dx \nonumber \\&\quad \le C\Vert [{\widetilde{v}}_{x},{\widetilde{u}}_{x},{\widetilde{\theta }}_{x}]\Vert ^{2}+C\Vert {\mathbf {g}}_{x}\Vert ^{2}_{\sigma } +C\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma }+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

By using (6.33) and the above estimates, we arrive at

$$\begin{aligned}&-\frac{2}{3}\big (({\widetilde{u}}_{1}v \mathfrak {{f}})_{x},\int _{{\mathbb {R}}^{3}}(\frac{1}{2}\xi _{1}|\xi |^{2}-u\cdot \xi \xi _{1})L^{-1}_{M}\Theta _{1}\, d\xi \big ) \nonumber \\&\quad \le C\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })+C\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma }+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.37)

Similar arguments as (6.37) imply

$$\begin{aligned}&\frac{2}{3}\big ({\widetilde{u}}_{1}v\mathfrak {{f}},u_{x}\cdot \int _{{\mathbb {R}}^{3}} \xi \xi _{1}L^{-1}_{M}\Theta _{1} \,d\xi \big ) =\sum ^{3}_{i=1}\frac{2}{3}\big ({\widetilde{u}}_{1}v\mathfrak {{f}},u_{ix} R\theta \int _{{\mathbb {R}}^{3}}B_{1i}(\frac{\xi -u}{\sqrt{R\theta }})\frac{\Theta _{1}}{M} \,d\xi \big ) \nonumber \\&\quad \le C\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })+C\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma }+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

For the second term on the right hand side of (6.30), by choosing a small $\delta >0$, we deduce from (6.32) and the above estimates that

$$\begin{aligned} -\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }} -p_{+}{\widetilde{v}})_{t}v{\mathfrak {f}}\big ) \le&-\frac{5}{12}p_{+}\int _{{\mathbb {R}}}v{\widetilde{u}}_{1}^{2}\omega ^{2}\,dx+C\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma }+C(1+t)^{-\frac{4}{3}} \nonumber \\&\quad + C\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma }). \end{aligned}$$

(6.38)

For the last two terms on the right hand side of (6.30), by using (6.28), (2.10)$_{1}$, (2.20), (3.6) and the imbedding inequality, one has

$$\begin{aligned}&-\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v_{t}{\mathfrak {f}}\big ) -\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}_{t}\big ) \nonumber \\&\quad \le C\Vert {\mathfrak {f}}\Vert _{L_{x}^{\infty }}\Vert {\widetilde{u}}_{1}\Vert _{L_{x}^{\infty }}(\Vert {\widetilde{v}}\Vert +\Vert {\widetilde{\theta }}\Vert )\Vert v_{t}\Vert +C\Vert {\mathfrak {f}}_{t}\Vert _{L_{x}^{\infty }} \Vert {\widetilde{u}}_{1}\Vert (\Vert {\widetilde{v}}\Vert +\Vert {\widetilde{\theta }}\Vert ) \nonumber \\&\quad \le C(1+t)^{-\frac{1}{2}}\Vert {\widetilde{u}}_{1}\Vert ^{\frac{1}{2}}\Vert {\widetilde{u}}_{1x}\Vert ^{\frac{1}{2}}(\Vert {\widetilde{v}}\Vert +\Vert {\widetilde{\theta }}\Vert )\Vert u_{1x}\Vert +C(1+t)^{-\frac{3}{2}} \nonumber \\&\quad \le C\Vert {\widetilde{u}}_{1x}\Vert ^{2}+C(1+t)^{-\frac{3}{2}}. \end{aligned}$$

(6.39)

It follows from (6.30), (6.38) and (6.39) that

$$\begin{aligned} I_{4}&\le \big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}\big )_{t} -\frac{5}{12}p_{+}\int _{{\mathbb {R}}}v{\widetilde{u}}_{1}^{2}\omega ^{2}\,dx+C\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma } \nonumber \\&\qquad +C\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.40)

By (6.29), (6.25), (6.28), (2.18), (3.6) and the imbedding inequality, we get

$$\begin{aligned} |I_{5}|&\le |\big (\frac{1}{v}(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})^{2},{\bar{v}}_{x}{\mathfrak {f}}\big )| +|\big (\frac{1}{v}(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})^{2},{\widetilde{v}}_{x}{\mathfrak {f}}\big )| \nonumber \\&\le C\delta \int _{{\mathbb {R}}}({\widetilde{v}}^{2}+{\widetilde{\theta }}^{2})\omega ^{2}\,dx+C(\Vert {\widetilde{v}}_{x}\Vert ^{2} +\Vert {\widetilde{\theta }}_{x}\Vert ^{2})+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.41)

By (6.29), (6.28) and (2.20), one has

$$\begin{aligned} |I_{6}|+|I_{7}|\le C(\Vert {\widetilde{v}}_{x}\Vert ^{2}+\Vert {\widetilde{u}}_{x}\Vert ^{2}+\Vert {\widetilde{\theta }}_{x}\Vert ^{2})+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.42)

By using (3.14), (3.16) and the similar arguments as (6.37), we can obtain

$$\begin{aligned} |I_{8}|&= |\int _{{\mathbb {R}}}\Big \{[(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}]_{x} R\theta \int _{{\mathbb {R}}^{3}}B_{11}(\frac{\xi -u}{\sqrt{R\theta }})\frac{\Theta _{1}}{M}\, d\xi \Big \}\,dx| \nonumber \\&\le C\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })+C\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma }+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.43)

By using (6.29), (6.40), (6.41), (6.42) and (6.43), we arrive at

$$\begin{aligned} \int _{{\mathbb {R}}}\big \{(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})^{2}+{\widetilde{u}}_{1}^{2}\big \}\omega ^{2}\,dx&\le C\big ({\widetilde{u}}_{1},(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})v{\mathfrak {f}}\big )_{t} +C\delta \int _{{\mathbb {R}}}({\widetilde{v}}^{2}+{\widetilde{\theta }}^{2})\omega ^{2}\,dx +C\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma } \nonumber \\&\qquad +C\sum _{|\alpha |=1}\big \{\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma }\big \}+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

Integrating it over (0, t), we have from this and (3.6) that

$$\begin{aligned} \int ^{t}_{0}\int _{{\mathbb {R}}}\big \{(\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}})^{2}+{\widetilde{u}}_{1}^{2}\big \}\omega ^{2}\,dxds&\le C+C\delta \int ^{t}_{0}\int _{{\mathbb {R}}}({\widetilde{v}}^{2}+{\widetilde{\theta }}^{2})\omega ^{2}\,dxds+C\varepsilon _{0}\int ^{t}_{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma }\,ds \nonumber \\&\quad +C\sum _{|\alpha |=1}\int ^{t}_{0}\big \{\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma }\big \}\,ds. \end{aligned}$$

(6.44)

On the other hand, we choose $h=\frac{2}{3}{\widetilde{\theta }}+\frac{2}{3}p_{+}{\widetilde{v}}$ in Lemma 6.7 and use (6.31) to deduce

$$\begin{aligned} ( h_{t},h {\mathfrak {g}}^{2})&=-\frac{2}{3}\big (\frac{\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}}}{v}({\widetilde{u}}_{1x}+{\bar{u}}_{1x}),h{\mathfrak {g}}^{2}\big ) +\frac{2}{3}\big ([\frac{\kappa (\theta )}{v}\theta _{x}-\frac{\kappa ({\bar{\theta }})}{{\bar{v}}}{\bar{\theta }}_{x}]_{x},h{\mathfrak {g}}^{2}\big ) +\frac{2}{3}(Q_{1},h{\mathfrak {g}}^{2}) \nonumber \\&\quad +\frac{2}{3}(\int _{{\mathbb {R}}^{3}}(\frac{1}{2}\xi _{1}|\xi |^{2}-u\cdot \xi \xi _{1})L^{-1}_{M}\Theta _{1} d\xi ,(h{\mathfrak {g}}^{2})_{x}) -\frac{2}{3}(u_{x}\int _{{\mathbb {R}}^{3}} \xi \xi _{1}L^{-1}_{M}\Theta _{1} d\xi ,h{\mathfrak {g}}^{2}). \end{aligned}$$

(6.45)

By using the facts that $\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}}=h-\frac{5}{3}p_{+}{\widetilde{v}}$ and ${\widetilde{u}}_{1x}={\widetilde{v}}_{t}$, we have

$$\begin{aligned}&-\frac{2}{3}\big (\frac{\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}}}{v}{\widetilde{u}}_{1x},h{\mathfrak {g}}^{2}\big ) \nonumber \\&\quad = -\frac{2}{3}\int _{{\mathbb {R}}}v^{-1}(h^{2}-\frac{5}{3}p_{+}{\widetilde{v}}h){\widetilde{v}}_{t}{\mathfrak {g}}^{2}\,dx =-\frac{1}{3}\int _{{\mathbb {R}}}\Big (2v^{-1}h^{2}{\mathfrak {g}}^{2}{\widetilde{v}}_{t} -\frac{5}{3}p_{+}v^{-1}h{\mathfrak {g}}^{2}({\widetilde{v}}^{2})_{t}\Big )\,dx \nonumber \\&\quad = -\frac{1}{3}\big (\int _{{\mathbb {R}}}v^{-1}h{\mathfrak {g}}^{2}{\widetilde{v}}(2h-\frac{5}{3}p_{+}{\widetilde{v}})dx\big )_{t} +\frac{2}{3}\int _{{\mathbb {R}}}v^{-1}h{\mathfrak {g}}{\widetilde{v}}(2h-\frac{5}{3}p_{+}{\widetilde{v}}){\mathfrak {g}}_{t}\,dx \nonumber \\&\qquad -\frac{1}{3}\int _{{\mathbb {R}}}v^{-2}v_{t}h{\mathfrak {g}}^{2}{\widetilde{v}}(2h-\frac{5}{3}p_{+}{\widetilde{v}})\,dx +\frac{1}{3}\int _{{\mathbb {R}}}v^{-1}{\mathfrak {g}}^{2}{\widetilde{v}}(4h-\frac{5}{3}p_{+}{\widetilde{v}})h_{t}\,dx :=\sum ^{12}_{i=9}I_{i}. \end{aligned}$$

(6.46)

We only estimate the last three terms in (6.46). In view of (6.25), it is easy to check that

$$\begin{aligned} 4\lambda {\mathfrak {g}}_{t}=\omega _{x}, \quad \Vert {\mathfrak {g}}(t,x)\Vert _{L_{x}^{\infty }}=\sqrt{\pi }\lambda ^{-\frac{1}{2}}. \end{aligned}$$

(6.47)

By using this and the facts that $h=\frac{2}{3}({\widetilde{\theta }}+p_{+}{\widetilde{v}})$, $|\omega _{x}|\le C(1+t)^{-1}$ and $v_{t}=u_{1x}$, one has

$$\begin{aligned} |I_{10}|+|I_{11}|&\le C(1+t)^{-1}\int _{{\mathbb {R}}}(|{\widetilde{v}}|^{3}+|{\widetilde{\theta }}|^{3})\,dx +C\int _{{\mathbb {R}}}|v_{t}|(|{\widetilde{v}}|^{3}+|{\widetilde{\theta }}|^{3})\,dx \\&\le C\Vert [{\widetilde{v}}_{x},{\widetilde{u}}_{x},{\widetilde{\theta }}_{x}]\Vert ^{2}+C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

By (6.31) and $h=\frac{2}{3}({\widetilde{\theta }}+p_{+}{\widetilde{v}})$, we have

$$\begin{aligned} I_{12}&=\frac{2}{9}\Big \{\int _{{\mathbb {R}}}v^{-1}{\mathfrak {g}}^{2}{\widetilde{v}}(4h-\frac{5}{3}p_{+}{\widetilde{v}}) \Big (-(\frac{\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}}}{v})u_{1x} +\big (\frac{\kappa (\theta )}{v}\theta _{x}-\frac{\kappa ({\bar{\theta }})}{{\bar{v}}}{\bar{\theta }}_{x}\big )_{x} +Q_{1} \Big )\,dx\Big \} \nonumber \\&\quad +\frac{2}{9}\Big \{\int _{{\mathbb {R}}}v^{-1}{\mathfrak {g}}^{2}{\widetilde{v}}(4h-\frac{5}{3}p_{+}{\widetilde{v}}) \Big (u\cdot \int _{{\mathbb {R}}^{3}} \xi \xi _{1}(L^{-1}_{M}\Theta _{1})_{x}\,d\xi -\frac{1}{2}\int _{{\mathbb {R}}^{3}}\xi _{1}|\xi |^{2}(L^{-1}_{M}\Theta _{1})_{x} d\xi \Big )dx\Big \} \nonumber \\&:=I^{1}_{12}+I^{2}_{12}. \end{aligned}$$

By using (6.47), (2.20), (3.6) and the expression of $Q_{1}$ in (3.5), one has

$$\begin{aligned} |I^{1}_{12}|\le C\Vert [{\widetilde{v}}_{x},{\widetilde{u}}_{x},{\widetilde{\theta }}_{x}]\Vert ^{2} +C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

Similar arguments as (6.37) imply

$$\begin{aligned} |I^{2}_{12}|\le C\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })+C\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma } +C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

It follows from the above two estimates that

$$\begin{aligned} |I_{12}|\le C\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })+C\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma } +C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

By using (6.46) and the above estimates, we can obtain

$$\begin{aligned}&-\frac{2}{3}\big (\frac{\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}}}{v}{\widetilde{u}}_{1x},h{\mathfrak {g}}^{2}\big ) +\frac{1}{3}\big (\int _{{\mathbb {R}}}v^{-1}h {\mathfrak {g}}^{2}{\widetilde{v}}(2h-\frac{5}{3}p_{+}{\widetilde{v}})dx\big )_{t} \nonumber \\&\quad \le C\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })+C\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma } +C(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.48)

By (6.45), (2.18), (6.47) and (3.12), we have

$$\begin{aligned}&-\frac{2}{3}\big (\frac{\frac{2}{3}{\widetilde{\theta }}-p_{+}{\widetilde{v}}}{v}{\bar{u}}_{1x},h{\mathfrak {g}}^{2}\big ) +\frac{2}{3}\big ((\frac{\kappa (\theta )}{v}\theta _{x}-\frac{\kappa ({\bar{\theta }})}{{\bar{v}}}{\bar{\theta }}_{x})_{x},h{\mathfrak {g}}^{2}\big ) +\frac{2}{3}(Q_{1},h{\mathfrak {g}}^{2}) \nonumber \\&\quad \le C(\epsilon +\delta )\int _{{\mathbb {R}}}({\widetilde{v}}^{2}+{\widetilde{\theta }}^{2})\omega ^{2}\,dx +C_{\epsilon }\Vert [{\widetilde{v}}_{x},{\widetilde{u}}_{x},{\widetilde{\theta }}_{x}]\Vert ^{2} +C_{\epsilon }(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.49)

For the last two terms of (6.45), by using (3.13), (3.14) and the similar arguments as (6.37), one has

$$\begin{aligned}&\big (\int _{{\mathbb {R}}^{3}}(\frac{1}{2}\xi _{1}|\xi |^{2}-u\cdot \xi \xi _{1})L^{-1}_{M}\Theta _{1} d\xi ,(h{\mathfrak {g}}^{2})_{x}\big ) -\big (u_{x}\cdot \int _{{\mathbb {R}}^{3}} \xi \xi _{1}L^{-1}_{M}\Theta _{1} d\xi ,h{\mathfrak {g}}^{2}\big ) \nonumber \\&\quad =\int _{{\mathbb {R}}}(R\theta )^{\frac{3}{2}}\int _{{\mathbb {R}}^{3}}A_{1}(\frac{\xi -u}{\sqrt{R\theta }})\frac{\Theta _{1}}{M} d\xi (h{\mathfrak {g}}^{2})_{x}\,dx -\sum ^{3}_{i=1}\int _{{\mathbb {R}}}R\theta \int _{{\mathbb {R}}^{3}}B_{1i}(\frac{\xi -u}{\sqrt{R\theta }})\frac{\Theta _{1}}{M} \,d\xi u_{ix}h{\mathfrak {g}}^{2}\,dx \nonumber \\&\quad \le C\epsilon \int _{{\mathbb {R}}}({\widetilde{v}}^{2}+{\widetilde{\theta }}^{2})\omega ^{2}\,dx+C_{\epsilon }\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma } +C_{\epsilon }\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })+C_{\epsilon }(1+t)^{-\frac{4}{3}}. \end{aligned}$$

(6.50)

By using (6.45), (6.48), (6.49) and (6.50), we arrive at

$$\begin{aligned} ( h_{t},h {\mathfrak {g}}^{2})&\le -\frac{1}{3}\big (\int _{{\mathbb {R}}}v^{-1}h{\mathfrak {g}}^{2}{\widetilde{v}}(2h-\frac{5}{3}p_{+}{\widetilde{v}})\,dx\big )_{t} +C(\epsilon +\delta )\int _{{\mathbb {R}}}({\widetilde{v}}^{2}+{\widetilde{\theta }}^{2})\omega ^{2}\,dx \nonumber \\&\quad +C_{\epsilon }\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma } +C_{\epsilon }\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })+C_{\epsilon }(1+t)^{-\frac{4}{3}}. \end{aligned}$$

Recalling that $h=\frac{2}{3}{\widetilde{\theta }}+\frac{2}{3}p_{+}{\widetilde{v}}$, by using this, (3.6) and Lemma 6.7, we have

$$\begin{aligned} \int ^{t}_{0}\int _{{\mathbb {R}}} (\frac{2}{3}{\widetilde{\theta }}+\frac{2}{3}p_{+}{\widetilde{v}})^{2}\omega ^{2}\,dxds&\le C_\epsilon +C(\epsilon +\delta )\int ^{t}_{0}\int _{{\mathbb {R}}}({\widetilde{v}}^{2}+{\widetilde{\theta }}^{2})\omega ^{2}\,dxds +C_{\epsilon }\varepsilon _{0}\int ^{t}_{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma }\,ds \nonumber \\&\quad +C_{\epsilon }\sum _{|\alpha |=1}\int ^{t}_{0}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })\,ds. \end{aligned}$$

(6.51)

Similarly, we take $h={\widetilde{u}}_{i}$ $(i=2,3)$ in Lemma 6.7 and use (3.4)$_{3}$ to deduce

$$\begin{aligned}&( h_{t},h{\mathfrak {g}}^{2}) =-\big (\frac{\mu (\theta )}{v}{\widetilde{u}}_{ix},(h{\mathfrak {g}}^{2})_{x}\big ) +\big (\int _{{\mathbb {R}}^{3}}\xi _{i}\xi _{1}L^{-1}_{M}\Theta _{1} d\xi ,(h{\mathfrak {g}}^{2})_{x}\big ) \\&\quad \le C\epsilon \int _{{\mathbb {R}}}({\widetilde{v}}^{2}+{\widetilde{\theta }}^{2})\omega ^{2}\,dx +C_{\epsilon }\varepsilon _{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma } +C_{\epsilon }\sum _{|\alpha |=1}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })+C_{\epsilon }(1+t)^{-\frac{4}{3}}. \end{aligned}$$

It follows from this, (3.6) and Lemma 6.7 that

$$\begin{aligned} \sum ^{3}_{i=2}\int ^{t}_{0}\int _{{\mathbb {R}}}{\widetilde{u}}_{i}^{2}\omega ^{2}\,dxds&\le C_\epsilon +C\epsilon \int ^{t}_{0}\int _{{\mathbb {R}}}({\widetilde{v}}^{2}+{\widetilde{\theta }}^{2})\omega ^{2}\,dxds +C_{\epsilon }\varepsilon _{0}\int ^{t}_{0}\Vert {\mathbf {g}}\Vert ^{2}_{\sigma }\,ds \nonumber \\&\quad +C_{\epsilon }\sum _{|\alpha |=1}\int ^{t}_{0}(\Vert \partial ^{\alpha }[{\widetilde{v}},{\widetilde{u}},{\widetilde{\theta }}]\Vert ^{2} +\Vert \partial ^{\alpha }{\mathbf {g}}\Vert ^{2}_{\sigma })\,ds. \end{aligned}$$

(6.52)

Therefore, the estimate (6.26) follows from (6.44), (6.51) and (6.52) by choosing both $\epsilon$ and $\delta$ small enough. This completes the proof of Lemma 6.8. $\square$