Proof. As in Lemma [lem:wave-interaction-L2], we split the real line and compute the wave interaction. Using Lemma [lem:bs] and ?? , we can obtain ?? and ?? . ◻

Proof. Let \(t=\tau/\kappa\) and \(x=y/\kappa\). Then \[d\tau\,dy = \kappa^2\,dt\,dx,\] and therefore \[\begin{align} & \int_0^T\int_{\mathbb{R}} \bigl\lVert f^\kappa-\bar f^{\,\kappa}\bigr\rVert_{M_\#}^2 \,dy\,d\tau \\ & \qquad = \kappa^2 \int_0^{T/\kappa}\int_{\mathbb{R}} \bigl\lVert f-\bar f\bigr\rVert_{M_\#}^2\,dx\,dt \\ & \qquad \le \kappa T \sup_{0\le t\le T/\kappa} \int_{\mathbb{R}} \bigl\lVert f-\bar f\bigr\rVert_{M_\#}\,dx \\ &\qquad \le \kappa T \sup_{0\le t\le T/\kappa} \Biggl[\int_{\mathbb{R}} \bigl\lVert M-\bar M\bigr\rVert_{M_\#}^2 \, dx+ \int_{\mathbb{R}} \bigl\lVert\widetilde{G}\bigr\rVert_{M_\#}^2 \, dx \Biggr]\, \qquad \qquad \qquad (\text{by }\eqref{eq:unvar}, \eqref{eq:pevar}) \\ &\qquad \le \kappa T \sup_{0\le t\le T/\kappa} \Biggl[ \bigl\lVert U-\bar U\bigr\rVert_{L^2_x}^2 + \int_{\mathbb{R}} \bigl\lVert\widetilde{G}_C\bigr\rVert_{M_\#}^2 \, dx + \int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{M_\#}^2 \, dx \Biggr] \quad (\text{by } \eqref{eq:mic2}, \eqref{eq:Maxwellian-Lipschitz-single})\\ &\qquad \le C\kappa T\,\bigl(\mathcal{E}(0)^2+\delta_0^{1/2}\bigr), \end{align}\] by 107 and ?? . The sharpened bound ?? follows from the well-preparedness hypothesis. ◻

Proof. By 52 , \[\frac{d}{d\tau}X_i^\kappa(\tau)=\dot{X}_i\!\left(\frac{\tau}{\kappa}\right),\] hence \[\begin{align} \mathrm{TV}(X_i^\kappa;[0,T]) &= \int_0^T \left|\dot{X}_i\!\left(\frac{\tau}{\kappa}\right)\right|\,d\tau = \kappa\int_0^{T/\kappa} |\dot{X}_i(t)|\,dt \\ &\leq \kappa^{1/2}T^{1/2} \left(\int_0^{T/\kappa}\left| \dot{X}_i\right|^2\right)^{1/2}\leq C_T\delta_i^{-1/2}, \end{align}\] since ?? gives \(\int_0^{T/\kappa}\delta_i|\dot{X}_i|^2 dt\le C\), and \(\delta_i>0\) is fixed. Thus, the family is bounded in \(BV(0,T)\), and the compactness of bounded sets in \(BV(0,T)\) into \(L^1(0,T)\) yields ?? . ◻

Proof of Theorem 2. Let \(f\) be the global normalized solution obtained above, and let \(f^\kappa\) be the rescaled solution 51 . Then \(f^\kappa\) exists uniquely on \([0,T]\) for every \(T>0\), since \(f\) exists globally on \([0,\infty)\).

By Lemma 11, after extraction of a subsequence if necessary, we may assume \[X_i^\kappa \to X_i^0 \qquad\text{in }L^1(0,T), \qquad i=1,3.\]

We first compare \(f^\kappa\) with the rescaled composite approximate wave. By ?? , \[\begin{align} \int_0^T\iint_{\mathbb{R}\times\mathbb{R}^3} \frac{|f^\kappa-\bar f^{\,\kappa}|^2}{M_\#}\,d\xi\,dy\,d\tau \le C\kappa\bigl((\varepsilon_1+\delta_0)^2+\delta_0^{1/2}\bigr)T . \label{eq:main-step-1} \end{align}\tag{53}\]

Next, we compare \(\bar f^{\,\kappa}\) with the shifted Euler Maxwellian profile. Using the exponential localization of the Boltzmann shock profiles(See Lemma 3) and the estimate for the viscous contact wave(See 24 ), we obtain \[\begin{align} \int_0^T\iint_{\mathbb{R}\times\mathbb{R}^3} \frac{|\bar f^{\,\kappa}-M_{X^\kappa}[U^E]|^2}{M_\#} \,d\xi\,dy\,d\tau \le C\delta_C\,\kappa^{1/2}T^{3/2}+ C(\delta_1+\delta_3)\kappa T. \label{eq:main-step-2} \end{align}\tag{54}\]

Finally, by observing that the height of the difference between the two shifted Euler shocks is of order the shock strength, while its width in \(x\) is the difference of the shifts for each shock wave, for each \(\tau\in[0,T]\), \[\begin{align} \iint_{\mathbb{R}\times\mathbb{R}^3} \frac{|M_{X^\kappa}[U^E] - M_{X^0}[U^E]|^2}{M_\#}\,d\xi\,dy \le C\delta_0^2\sum_{i=1,3}|X_i^\kappa(\tau)-X_i^0(\tau)|. \end{align}\] Integrating in \(\tau\) yields \[\begin{align} &\int_0^T\iint_{\mathbb{R}\times\mathbb{R}^3} \frac{|M_{X^\kappa}[U^E] - M_{X^0}[U^E]|^2}{M_\#}\,d\xi\,dy\,d\tau\nonumber\\ &\qquad \le C\delta_0^2\sum_{i=1,3}\|X_i^\kappa-X_i^0\|_{L^1(0,T)}. \label{eq:main-step-3} \end{align}\tag{55}\]

Combining 53 , 54 , and 55 , and using that \(M_\#\) is uniformly bounded from above by positive constants, we obtain \[\begin{align} &\int_0^T \iint_{\mathbb{R}\times\mathbb{R}^3} \left|f^\kappa-M_{X^0}[U^E]\right|^2 \,d\xi\,dy\,d\tau\\ &\quad \le C\kappa\bigl((\varepsilon_1+\delta_0)^2+\delta_0^{1/2}\bigr)T + C\delta_C\,\kappa^{1/2}T^{3/2} + C(\delta_1+\delta_2)\kappa T \\ &\qquad + C\delta_0^2\sum_{i=1,3}\|X_i^\kappa-X_i^0\|_{L^1(0,T)}. \end{align}\] This is exactly ?? . ◻

Proof. We next combine the modulation terms, the principal shock contribution \(\mathcal{B}^{\text{sh}}\), and the dissipative part \(\mathcal{D}_{\mathrm{mac}}(U)\) in order to recover the coercive shock functional \(\mathcal{G}_i^S\).

Estimate of \(\mathcal{B}_2-\mathfrak G\)↩︎

See (4.46) with \(R=\frac{2}{3}\) in [37]. By a similar argument, we obtain the following estimate. \[\label{eq:nbad2} \begin{align} \mathcal{B}_2(U)-\mathfrak G(U) \le\;& -\sum_{i\in\{1,3\}}\mathfrak G_{1i}(U) -\sum_{i\in\{1,3\}}\mathfrak G_{2i}(U) -\sum_{i\in\{1,3\}}\mathfrak G_{3i}(U) +B_{\mathrm{new}}(U)\\ &+ C\sum_{i\in\{1,3\}}\delta_i\sqrt{\delta_i}e^{-C\delta_i t} \int_{\mathbb{R}}\eta(U \mid \bar U)\,dx, \end{align}\tag{87}\] where \[\begin{align} \begin{aligned} & \mathfrak G_{1i}(U):=\frac{\theta_i\sigma_{i}^{\text{m}}}{3v_i^2}\int_{\mathbb{R}}|\partial_x(a_i^{-X_i})|\varphi_i\left(\phi+\frac{\psi_1}{\sigma_i^{\text{m}}}\right)^2dx,\\ & \mathfrak G_{2i}(U):=\frac{\sigma_{i}^{\text{m}}}{2\theta_i}\int_{\mathbb{R}}|\partial_x(a_i^{-X_i})|\varphi_i\left(\zeta-\frac{2\theta_i}{3\sigma_i^{\text{m}} v_i}\psi_1\right)^2dx,\\ & \mathfrak G_{3i}(U):=\frac{\sigma_{i}}{2}\sum_{j=2}^3\int_{\mathbb{R}}\partial_x(a_i^{-X_i})\varphi_i \psi_j^2dx=\frac{1}{2\sqrt{\delta_i}}\sum_{j=2}^3\int_\mathbb{R}\sigma_i\partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\varphi_i\psi_j^2 dx, \end{aligned} \end{align}\] and \[\begin{align} \begin{aligned} B_{\mathrm{new}}(U):=\;& C\sum_{i\in\{1,3\}}\delta_i\int_{\mathbb{R}}|\partial_x(a_i^{-X_i})||(\phi,\psi,\zeta)|^2dx + C\sum_{i\in\{1,3\}}\int_{\mathbb{R}}|\partial_x(a_i^{-X_i})||(\phi,\psi,\zeta)|^3dx\\ &+ C\int_{\mathbb{R}}|\partial_x(a_1^{-X_1})| \Bigl(|(v^{S_3}-v^*,\theta^{S_3}-\theta^*)|+|(v^C-v_*,\theta^C-\theta_*)|\Bigr) |(\phi,\psi_1,\zeta)|^2dx\\ &+ C\int_{\mathbb{R}}|\partial_x(a_3^{-X_3})| \Bigl(|(v^{S_1}-v_-,\theta^{S_1}-\theta_-)|+|(v^C-v_*,\theta^C-\theta_*)|\Bigr) |(\phi,\psi_1,\zeta)|^2dx. \end{aligned} \end{align}\] Indeed, \[\begin{align} \mathcal{B}_2(U)-\mathfrak G(U) &= \int_{\mathbb{R}} a_x \psi_1(p-\bar p)\,dx -\sum_{i\in\{1,3\}}\sigma_i \int_{\mathbb{R}} \partial_x(a_i^{-X_i}) \eta(U\mid \bar U)\,dx\\ &= \sum_{i\in\{1,3\}} \int_{\mathbb{R}}\partial_x(a_i^{-X_i}) \left[ \psi_1(p-\bar p) -\sigma_i\left( \frac{2}{3}\bar\theta\Phi\!\left(\frac{v}{\bar v}\right) +\bar\theta\Phi\!\left(\frac{\theta}{\bar\theta}\right) +\frac{|\psi|^2}{2} \right) \right]dx. \end{align}\] Using \[\begin{align} \Phi(z)=\frac{(z-1)^2}{2}+O(|z-1|^3), \quad p-\bar p=\frac{2}{3v}\zeta-\frac{\bar p}{v}\phi, \end{align}\] we complete squares separately for the \(1\)-shock and \(3\)-shock contributions and obtain 87 . Here, the good terms \(\mathfrak G_{3i}\) are the coercivity of the transverse velocities included in \(\mathcal{G}_i^S\).

It remains to estimate \(B_{\mathrm{new}}\). Using \(\varphi_1+\varphi_3=1\), Lemmas 3–6, the Sobolev inequality, and Young’s inequality, we obtain

\[\begin{align} \begin{aligned} B_{\mathrm{new}}(U)\le\;& C(\varepsilon+\sqrt{\delta_0}) \left( \sum_{i\in\{1,3\}}(\mathfrak G_{1i}+\mathfrak G_{2i})+\mathcal{D}_{\mathrm{mac}}(U)+\sum_{i=1,3}\mathcal{G}_i^S \right)\\ &+ C\sum_{i\in\{1,3\}}\varepsilon\sqrt{\delta_i}\delta_i e^{-C\delta_i t} \int_{\mathbb{R}}\eta(U \mid \bar U)\,dx\\ &+ C\delta_1^{\frac{4}{3}}\bigl(\delta_C^{\frac{4}{3}}+\delta_3^{\frac{4}{3}}\bigr)e^{-C\delta_1t}+ C\delta_3^{\frac{4}{3}}\bigl(\delta_C^{\frac{4}{3}}+\delta_1^{\frac{4}{3}}\bigr)e^{-C\delta_3t}. \end{aligned} \end{align}\]

Estimate of the shift terms↩︎

We now estimate the modulation term \[-\sum_{i=1,3}\frac{\delta_i}{2\mathfrak m_i}|\dot{X}_i|^2.\] Set \[\label{eq:newvar} \begin{align} & w_1:=\varphi_1\psi_1,\qquad w_3:=\varphi_3\psi_1,\\ & y_1:= -\frac{\left(v^{S_1}\right)^{-X_1}-v_-}{\delta_1},\qquad y_3:= \frac{\left(v^{S_3}\right)^{-X_3}-v^*}{\delta_3},\\ & z_1:=x-\sigma_1 t-X_1(t),\qquad z_3:=x-\sigma_3 t-X_3(t), \end{align}\tag{88}\] so that \[\frac{dy_1}{dz_1}=-\frac{\partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}}{\delta_1}>0, \qquad \frac{dy_3}{dz_3}=\frac{\partial_x\bigl(v^{S_{3}}\bigr)^{-X_{3}}}{\delta_3}>0.\]

Using the decomposition \(\mathcal{Y}_i=\sum_{j=1}^6\mathcal{Y}_{ij}\), we write the shift ODE in the form \[\label{eq:dsaf} \dot{X}_i(t)=-\frac{\mathfrak m_i}{\delta_i}(\mathcal{Y}_{i1}+\mathcal{Y}_{i2}+\mathcal{Y}_{i3}), \qquad i=1,3.\tag{89}\] For the \(1\)-shock, the terms \(\mathcal{Y}_{11},\mathcal{Y}_{12},\mathcal{Y}_{13}\) can be compared with \(\int_0^1 w_1\,dy_1\). More precisely, \[\begin{align} \left|\mathcal{Y}_{11}+\delta_1\sigma_{1}^{\text{m}}\int_0^1w_1\,dy_1\right| &\le C\delta_1(\sqrt{\delta_0}+\delta_0)\int_0^1|w_1|\,dy_1 \nonumber \\ &\quad + C\int_{\mathbb{R}}|\partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}|\varphi_3|\psi_1|\,dx, \\ \left|\mathcal{Y}_{12}+\frac{p_1\delta_1}{\sigma_{1}^{\text{m}}v_1}\int_0^1w_1\,dy_1\right| &\le C\delta_1(\delta_0+\sqrt{\delta_0})\int_0^1|w_1|\,dy_1 \nonumber\\ &\quad + C\sqrt{\delta_1}\int_{\mathbb{R}}|\partial_x(a_1^{-X_1})|\varphi_1\left|\phi-\frac{\psi_1}{\sigma_1}\right|dx \nonumber \\ &\quad + C\int_{\mathbb{R}}\left|\partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}\right|\varphi_3|\phi|\,dx, \\ \left|\mathcal{Y}_{13}+\frac{2}{3}\frac{p_1\delta_1}{\sigma_{1}^{\text{m}}v_1}\int_0^1w_1\,dy_1\right| &\le C\delta_1(\delta_0+\sqrt{\delta_0}+\varepsilon)\int_0^1|w_1|\,dy_1 \nonumber\\ &\quad + C\sqrt{\delta_1}\int_{\mathbb{R}}|\partial_x(a_1^{-X_1})|\varphi_1\left|\zeta-\frac{2\theta_-}{3\sigma_1v_-}\psi_1\right|dx\nonumber\\ &\quad + C\int_{\mathbb{R}}\varphi_3\left|\partial_x\bigl(\theta^{S_{1}}\bigr)^{-X_{1}}\right||\zeta|\,dx. \end{align}\] Hence, \[\begin{align} \left|\dot{X}_1-2\sigma_1^{\text{m}}\mathfrak m_1\int_0^1w_1\,dy_1\right|^2 \le\;& C(\delta_0+\sqrt{\delta_0}+\varepsilon)^2\int_0^1 w_1^2\,dy_1 +\frac{C}{\delta_1}(\mathfrak G_{11}+\mathfrak G_{21}) \nonumber\\ &+C\delta_1e^{-C\delta_1 t}\int_{\mathbb{R}}\eta(U \mid \bar U)\,dx. \end{align}\] Therefore, \[\begin{align} -\frac{\delta_1}{2\mathfrak m_1}|\dot{X}_1|^2 \le\;& -(\sigma_1^{\text{m}})^2\mathfrak m_1\delta_1\left(\int_0^1 w_1\,dy_1\right)^2 + C\delta_1(\delta_0+\sqrt{\delta_0}+\varepsilon)^2\int_0^1w_1^2\,dy_1 \\ &+ C\sqrt{\delta_1}(\mathfrak G_{11}+\mathfrak G_{21}) + C\delta_1^2e^{-C\delta_1 t}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx. \end{align}\] The \(3\)-shock case is treated in the same way.

Estimate of \(\mathcal{B}^{\text{sh}}\)↩︎

Recall that \[\mathcal{B}^{\text{sh}}(U) = \sum_{i\in\{1,3\}} \int_{\mathbb{R}} a\Bigl|\partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\Bigr| \left( \frac{4p_i\sigma_{i}^{\text{m}}}{3v_i^2}\phi^2 +\frac{\sigma_{i}^{\text{m}}}{3v_i\theta_i}\zeta^2 -\frac{\sigma_{i}^{\text{m}} p_i}{v_i\theta_i}\phi\zeta \right)\,dx.\] We decompose \[\mathcal{B}_1=\sum_{i\in\{1,3\}}(B_{11}^i+B_{12}^i+B_{13}^i)\] according to the three terms inside the bracket. By splitting \(\phi\) and \(\zeta\) into the favorable part on \(\varphi_i\) and the exponentially small interaction part on \(1-\varphi_i^2\), and using Lemma 6, we obtain \[\begin{align} B_{11}^1 &\le \frac{4p_1}{3v_1^2\sigma_{1}^{\text{m}}}(1+\sqrt{\delta_0}+\delta_1^{1/4})\delta_1\int_0^1|w_1|^2\,dy_1 \nonumber\\ &\quad + C\delta_1^{1/4}\mathfrak G_{11} + C\delta_1^2e^{-C\delta_1 t}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx, \\ B_{12}^1 &\le \frac{p_1}{3\sigma_{1}^{\text{m}}v_1^2}(1+\sqrt{\delta_0}+\delta_1^{1/4})\delta_1\int_0^1|w_1|^2\,dy_1 \nonumber\\ &\quad + C\delta_1^{1/4}\mathfrak G_{21} + C\delta_1^2e^{-C\delta_1 t}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx, \\ B_{13}^1 &\le \frac{\sigma_{1}^{\text{m}}p_1}{v_1\theta_1}(1+\sqrt{\delta_0}+\delta_1^{1/4})\delta_1\int_0^1|w_1|^2\,dy_1 \nonumber\\ &\quad + C\delta_1^{1/4}(\mathfrak G_{11}+\mathfrak G_{21}) + C\delta_1^2e^{-C\delta_1 t}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx. \end{align}\] The same estimate holds for the \(3\)-shock contribution. Consequently, \[\begin{align} \mathcal{B}^{\text{sh}} \le\;& \sum_{i\in\{1,3\}} \frac{20p_i}{9v_i^2\sigma_{i}^{\text{m}}}(1+\sqrt{\delta_0}+\delta_i^{1/4}) \delta_i\int_0^1|w_i|^2\,dy_i \\ &+ C\sum_{i\in\{1,3\}}\delta_i^{1/4}(\mathfrak G_{1i}+\mathfrak G_{2i}) + C\sum_{i\in\{1,3\}}\delta_i^2e^{-C\delta_i t} \int_{\mathbb{R}}\eta(U \mid \bar U)\,dx. \end{align}\]

Estimate of \(\mathcal{D}_{\mathrm{mac}}\)↩︎

Recall that \[\mathcal{D}_{\mathrm{mac}}(U) = \int_{\mathbb{R}}a \left( \frac{\alpha_{\rm{th}}(\bar\theta)}{v\theta}\zeta_x^2 + \frac{4}{3}\frac{\mu(\bar\theta)}{v}\psi_{1x}^2 + \frac{\mu(\theta)}{v}\sum_{k=2}^3\psi_{kx}^2 \right)dx.\] Using the cutoff decomposition and the change of variable \(x\mapsto y_i\), together with Lemmas 4 and 5, we obtain \[\begin{align}\label{eq:diffusion} -\mathcal{D}_{\mathrm{mac}}(U) \le\;& -\sum_{i\in\{1,3\}}2\alpha_i\bigl(1-C(\sqrt{\delta_0}+\varepsilon+\delta_i+\delta_*)-\delta_i^{1/4}\bigr)\delta_i\int_0^1 w_i^2\,dy_i \\ &+ \sum_{i\in\{1,3\}}\frac{20+12\gamma}{10+3\gamma}\alpha_i\delta_i\bigl( w_i^{\mathrm{avg}}\bigr)^2 + \sum_{i\in\{1,3\}}C\delta_i^{1/4}\mathfrak G_{2i} \\ &+ \frac{C}{\delta_*t^2}\int_{\mathbb{R}}\eta(U|\bar U)\,dx - \int_{\mathbb{R}}a\frac{\mu(\theta)}{v}\sum_{k=2}^3\psi_{kx}^2\,dx, \end{align}\tag{90}\] where \[\alpha_i=\frac{20}{9}\frac{p_i}{\sigma_{i}^{\text{m}} v_i^2}, \qquad w_i^{\mathrm{avg}}:=\int_0^1 w_i\,dy_i, \qquad \gamma=\frac{\alpha_{\rm{th}}(\bar\theta)}{\mu(\bar\theta)}.\]

More precisely, to investigate the diffusion term, we decompose as follows. \[\begin{align} \begin{aligned} \widetilde{D}_{u_1}(U)=\int_\mathbb{R} a \left(\frac{4}{3}\frac{\mu(\overline{\theta})}{v}\psi_{1x}^2\right)dx,\quad \widetilde{D}_{\theta}(U)=\int_\mathbb{R} a \left(\frac{\alpha_{\rm{th}}(\overline{\theta})}{v\theta}\zeta_x^2\right)dx. \end{aligned} \end{align}\]

We focus on \(\widetilde{D}_{u_1}\). Observe that

\[\begin{align} \begin{aligned} \widetilde{D}_{u_1}(U)\geq \sum_{i\in\{1,3\}}\int_\mathbb{R} a \varphi_i^2 \frac{4}{3}\frac{\mu(\overline{\theta})}{v}\psi_{1x}^2dx. \end{aligned} \end{align}\]

Using the Young inequality, we have the following.

\[\begin{align} \begin{aligned} \int_\mathbb{R} a \frac{\mu(\overline{\theta})}{v}\left| \left(\varphi_i \psi_1\right)_x\right|^2dx\leq (1+\delta_*)\int_\mathbb{R} a\frac{\mu(\overline{\theta})}{v}\varphi_i^2\psi_{1x}^2dx+\frac{C}{\delta_*}\int_\mathbb{R} a \frac{\mu(\overline{\theta})}{v}\left| \varphi_{ix}\right|^2\left| \psi_1\right|^2dx \end{aligned} \end{align}\]

for any sufficiently small \(\delta_*>0\) that is chosen later.

Since \(\varphi_1^2+\varphi_3^2\leq 1\), we get the following estimate,

\[\begin{align} \begin{aligned} -\widetilde{D}_{u_1}(U)\leq -\frac{1}{1+\delta_*}\frac{4}{3}\sum_{i\in\{1,3\}}\int_\mathbb{R} a \frac{\mu(\overline{\theta})}{v}\left| \left(\varphi_i\psi_1\right)_x\right|^2dx+\frac{4}{3}\frac{C}{\delta_*}\int_\mathbb{R} a\frac{\mu(\overline{\theta})}{v}\left(\varphi_{ix}\right)^2\left| \psi_1\right|^2dx. \end{aligned} \end{align}\]

For clarity, each term is defined as follows.

\[\begin{align} \begin{aligned} &J_1 := -\frac{1}{1+\delta_*}\sum_{i\in\{1,3\}}\int_\mathbb{R} a \frac{\mu(\overline{\theta})}{v}\left| \left(\varphi_i\psi_1\right)_x\right|^2dx,\\ &J_2:= \frac{C}{\delta_*}\int_\mathbb{R} a\frac{\mu(\overline{\theta})}{v}\left(\varphi_{ix}\right)^2\left| \psi_1\right|^2dx. \end{aligned} \end{align}\]

By the change of variable(cf. 88 ), we have

\[\begin{align} \begin{aligned} J_1 = -\frac{1}{1+\delta_*}\sum_{i\in\{1,3\}}\int_0^1 a \frac{\mu(\overline{\theta})}{v}\frac{dy_i}{dx}\left| (w_i)_{y_i}\right|^2dy_i. \end{aligned} \end{align}\]

From Lemmas 4 and 5, we have the following. \[\begin{align} \begin{aligned} \frac{dy_1}{dz_1}\geq \Biggl(\frac{1}{y_1(1-y_1)}\frac{4}{3}\frac{\mu\Bigl(\bigl(\theta^{S_{1}}\bigr)^{-X_{1}}\Bigr)}{\bigl(v^{S_{1}}\bigr)^{-X_{1}}}\Biggr)^{-1}(A\alpha_1\delta_1-C\delta_1^2)>0 \end{aligned} \end{align}\] where \(A := \frac{10}{10+\gamma}\).

Therefore, we can get the following estimate, \[\begin{align} \begin{aligned} J_1&\leq -\frac{1}{1+\delta_*}\sum_{i\in\{1,3\}}\int_0^1 a \frac{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}{v}\frac{\mu(\overline{\theta})}{\mu\left(\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}\right)}\frac{3}{4}y_i(1-y_i)(A\alpha_i\delta_i-C\delta_i^2)\left| \left(w_i\right)_{y_i}\right|^2dy_i\\ &\leq -\frac{1}{1+\delta_*}\sum_{i\in\{1,3\}}\int_0^1 \frac{3}{4}(1-2\sqrt{\delta_0})(1-C\varepsilon)(A\alpha_i-C\delta_i)\delta_iy_i(1-y_i)\left| \left(w_i\right)_{y_i}\right|^2dy_i. \end{aligned} \end{align}\] Since \(\frac{1}{1+\delta_*}\geq 1-\delta_*\) and \[\begin{align} \begin{aligned} (1-2\sqrt{\delta_0})(1-C\varepsilon)(A\alpha_i-C\delta_i)(1-\delta_*)\geq A\alpha_i(1-C(\sqrt{\delta_0}+\varepsilon+\delta_i+\delta_*)), \end{aligned} \end{align}\]

we organize the above estimate as follows.

\[\begin{align} \begin{aligned} J_1\leq -\sum_{i\in\{1,3\}}\int_0^1 \frac{3}{4}A\alpha_i(1-C(\sqrt{\delta_0}+\varepsilon+\delta_i+\delta_*))\delta_iy_i(1-y_i)\left| \left(w_i\right)_{y_i}\right|^2dy_i. \end{aligned} \end{align}\]

Now, we consider another term which is singular at time \(t=0\). By 46 , \[\frac{1}{2}\left(X_3(t)+\sigma_3t-X_1(t)-\sigma_1t\right)\geq \frac{\sigma_3-\sigma_1}{4}t>0, \quad T\geq t>0\] we have \[\begin{align} \begin{aligned}\label{eq:cutof1} &\left| \varphi_{ix}\right|\leq \frac{4}{\sigma_3-\sigma_1}\frac{1}{t},\quad \left| \varphi_{ix}\right|^2\leq \frac{16}{(\sigma_3-\sigma_1)^2}\frac{1}{t^2}, \quad T\geq t>0, \quad \forall x\in \mathbb{R}. \end{aligned} \end{align}\tag{91}\] From this, we can compute \[\begin{align} \begin{aligned} &J_2=\frac{C}{\delta_*}\int_\mathbb{R} a \frac{\mu(\overline{\theta})}{v}\left| \varphi_{ix}\right|^2\left| \psi_1\right|^2dx\leq \frac{C}{\delta_*t^2}\int_\mathbb{R} \eta(U\mid\bar{U})dx. \end{aligned} \end{align}\] Thus, we have the estimate of \(\widetilde{D}_{u_1}\) as follows: \[\begin{align} \begin{aligned} &-\widetilde{D}_{u_1}\leq-\sum_{i\in\{1,3\}}\int_0^1\frac{3}{4}A\alpha_i\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_i+\delta_*\right)\right)\delta_iy_i(1-y_i)\left| \left(w_i\right)_{y_i}\right|^2dy_i\\ &\qquad \qquad \qquad +\frac{C}{\delta_*t^2}\int_\mathbb{R}\eta(U\mid\bar{U})dx. \end{aligned} \end{align}\]

By the same manner, we perform the same procedure to \(\widetilde{D}_{\theta_1}(U)\). Take \(\mathfrak z_i = \varphi_i \zeta\) for \(i=1,3\). Then, we have the following estimate.

\[\begin{align} \begin{aligned}\label{eq:diffusionx} -\mathcal{D}_{\mathrm{mac}}(U)\leq&-\sum_{i\in\{1,3\}}\int_0^1A\alpha_i\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_i+\delta_*\right)\right)\delta_iy_i(1-y_i)\left| \left(w_i\right)_{y_i}\right|^2dy_i\\ &-\sum_{i\in\{1,3\}}\int_0^1\frac{3}{4}A\alpha_i\frac{\gamma}{\theta_i}\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_i+\delta_*\right)\right)\delta_iy_i(1-y_i)\left| \left(\mathfrak z_i\right)_{y_i}\right|^2dy_i\\ & +\frac{C}{\delta_*t^2}\int_\mathbb{R}\eta(U\mid\bar{U})dx-\int_\mathbb{R} a \left(\frac{\mu(\theta)}{v}\sum_{k=2}^3\psi_{kx}^2\right)dx. \end{aligned} \end{align}\tag{92}\]

Now, to replace \(\mathfrak z_i\) by \(w_i\), we compute these as follows.

\[\begin{align} \begin{aligned} &\int_0^1 \left| \mathfrak z_i\right|^2 dy_i \geq \left(\frac{2\theta_i}{3\sigma_i^{\text{m}}v_i}\right)^2\left(1-\delta_i^{\frac{1}{4}}\right)\int_0^1 \left| w_i\right|^2 dy_i-\frac{C}{\delta_i^{\frac{1}{4}}}\int_0^1 \left[\varphi_i\left(\zeta-\frac{2\theta_i}{3\sigma_i^{\text{m}}v_i}\psi_{1}\right)\right]^2dy_i,\\ &\left(\int_0^1 \mathfrak z_i dy_i\right)^2\leq 2\left[\int_0^1 \frac{2\theta_i}{3\sigma_i^{\text{m}}v_i}\varphi_i\psi_{1}dy_i\right]^2+2\left[\int_0^1 \varphi_i\left(\zeta-\frac{2\theta_i}{3\sigma_i^{\text{m}}v_i}\varphi_i\psi_{1}\right)dy_i\right]^2. \end{aligned} \end{align}\] Now, we introduce the following useful lemma for estimating 92 .

By the Poincare inequality ?? , we have \[\begin{align} \begin{aligned} &-\mathcal{D}_{\mathrm{mac}}(U)\\ &\quad \le -\sum_{i\in\{1,3\}}2A\alpha_i\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_i+\delta_*\right)\right)\delta_i\left[\int_0^1 w_i^2 dy_i-\bigl(w_i^{\mathrm{avg}}\bigr)^2\right]\\ &\quad \quad-\sum_{i\in\{1,3\}}\frac{3}{4}2A\alpha_i\frac{\gamma}{\theta_i}\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_i+\delta_*\right)\right)\delta_i\left[\int_0^1z_i^2 dy_i-\bigl(z_i^{\mathrm{avg}}\bigr)^2\right]\\ &\quad \quad+\frac{C}{\delta_*t^2}\int_\mathbb{R}\eta(U\mid\bar{U})dx-\int_\mathbb{R} a \left(\frac{\mu(\theta)}{v}\sum_{k=2}^3\psi_{kx}^2\right)dx \\ &\quad \le -\sum_{i\in\{1,3\}}2A\alpha_i\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_*\right)\right)\delta_i\left\{1+\frac{3}{4}\frac{\gamma}{\theta_i}\left(\frac{2\theta_i}{3\sigma_i^{\text{m}}v_i}\right)^2\left(1-\delta_i^{\frac{1}{4}}\right)\right\}\int_0^1 w_i^2 dy_i\\ &\quad \quad +\sum_{i\in\{1,3\}}2A\alpha_i\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_*\right)\right)\delta_i\left\{1+\frac{3}{4}2\frac{\gamma}{\theta_i}\left(\frac{2\theta_i}{3\sigma_i^{\text{m}}v_i}\right)^2\right\}\bigl(w_i^{\mathrm{avg}}\bigr)^2\\ &\quad \quad +\sum_{i\in\{1,3\}}\frac{3}{2}A\alpha_i\frac{\gamma}{\theta_i}\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_*\right)\right)\delta_i\left(\frac{C}{\delta_i^{\frac{1}{4}}}-2\right)\int_0^1 \left[\varphi_i\left(\zeta-\frac{2\theta_i}{3\sigma_i^{\text{m}}v_i}\psi_{1i}\right)\right]^2 dy_i\\ &\quad \quad +\frac{C}{\delta_*t^2}\int_\mathbb{R}\eta(U\mid\bar{U})dx-\int_\mathbb{R} a \left(\frac{\mu(\theta)}{v}\sum_{k=2}^3\psi_{kx}^2\right)dx. \end{aligned} \end{align}\] Also, we control the third term of right hand side of the above inequality as followed: \[\begin{align} \begin{aligned} & \sum_{i\in\{1,3\}}\frac{3}{2}A\alpha_i\frac{\gamma}{\theta_i}\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_*\right)\right)\delta_i\left(\frac{C}{\delta_i^{\frac{1}{4}}}-2\right)\int_0^1 \left[\varphi_i\left(\zeta-\frac{2\theta_i}{3\sigma_i^{\text{m}}v_i}\psi_{1}\right)\right]^2 dy_i \\ & \le \sum_{i\in\{1,3\}}\frac{3}{2}CA\alpha_i\frac{\gamma}{\theta_i}\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_*\right)\right)\delta_i^{\frac{3}{4}}\int_0^1\varphi_i \left[\left(\zeta-\frac{2\theta_i}{3\sigma_i^{\text{m}}v_i}\psi_{1}\right)\right]^2 dy_i\\ & \le \sum_{i\in\{1,3\}}C\delta_i^{\frac{1}{4}}\mathfrak G_{2i}. \end{aligned} \end{align}\] We organize the first term of right hand side of the above inequality as followed: \[\begin{align} \begin{aligned} &-\sum_{i\in\{1,3\}}2A\alpha_i\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_*\right)-\delta_i^{\frac{1}{4}}\right)\delta_i\left\{1+\frac{3}{4}\frac{\gamma}{\theta_i}\left(\frac{2\theta_i}{3\sigma_i^{\text{m}}v_i}\right)^2\right\}\int_0^1 w_i^2 dy_i\\ \le&-\sum_{i\in\{1,3\}}2\alpha_i\left(1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_*\right)-\delta_i^{\frac{1}{4}}\right)\delta_i\int_0^1 w_i^2 dy_i. \end{aligned} \end{align}\]

Therefore, we have \[\begin{align} \begin{aligned} & \sum_{i\in\{1,3\}}2A\alpha_i\left(1-C\left(\sqrt{\delta}+\varepsilon+\delta_*\right)\right)\delta_i\left\{1+\frac{3}{2}\frac{\gamma}{\theta_i}\left(\frac{2\theta_i}{3\sigma_iv_i}\right)^2\right\}\bigl(w_i^{\mathrm{avg}}\bigr)^2\\ \leq & \sum_{i\in\{1,3\}}\frac{20+12\gamma}{10+3\gamma} \alpha_i\delta_i\bigl(w_i^{\mathrm{avg}}\bigr)^2. \end{aligned} \end{align}\]

Then, we obtained the estimate 90 .

Conclusion of the dominant estimate↩︎

Choose \(\delta_*=O(1)>0\) such that \[\begin{align} 1-C\left(\sqrt{\delta_0}+\varepsilon+\delta_*\right)-\delta_i^{\frac{1}{4}}>\frac{1}{3} , \quad i=1,3. \end{align}\]

Collecting the estimates above, we find \[\begin{align} &-\sum_{i\in\{1,3\}}\frac{\delta_i}{2\mathfrak m_i}|\dot{X}_i|^2 +\mathcal{B}^{\text{sh}}+\mathcal{B}_2-\mathfrak G-\frac{3}{4}D\\ &\quad\le\; -\alpha\sum_{i\in\{1,3\}}\mathcal{G}_i^S +C\delta_1^{\frac{4}{3}}\bigl(\delta_C^{\frac{4}{3}}+\delta_3^{\frac{4}{3}}\bigr)e^{-C\delta_1t}+ C\delta_3^{\frac{4}{3}}\bigl(\delta_C^{\frac{4}{3}}+\delta_1^{\frac{4}{3}}\bigr)e^{-C\delta_3t} \\ &\qquad +\left(\sum_{i\in\{1,3\}}C\delta_i^2e^{-C\delta_i t}+\frac{C}{\delta_*t^2}\right)\int_{\mathbb{R}}\eta(U|\bar U)\,dx -\frac{3}{4}\int_{\mathbb{R}}a\frac{\mu(\theta)}{v}\sum_{k=2}^3\psi_{kx}^2\,dx, \end{align}\] for some positive constant \(\alpha=O(1)\). This proves Lemma 12. ◻

Proof. By the definition of the weight \(a\), we have \[|a_x| \le \frac{C}{\sqrt{\delta_1}}|\partial_x(v^{S_1})^{-X_1}| + \frac{C}{\sqrt{\delta_3}}|\partial_x(v^{S_3})^{-X_3}|.\] Since \(\delta_1,\delta_3\) are sufficiently small and the shock derivatives are exponentially localized, the factor \(a_x\) is supported only in the shock region and is controlled by the shock profiles.

Using the smoothness of \(\alpha_{\rm{th}}\) and \(\mu\), together with \[\alpha_{\rm{th}}(\theta)-\alpha_{\rm{th}}(\bar\theta)=O(|\zeta|), \qquad \mu(\theta)-\mu(\bar\theta)=O(|\zeta|),\] and Young’s inequality, we obtain \[\begin{align} &|\mathcal{B}_3| \le \lambda \mathcal{D}_{\mathrm{mac}}(U) + C_\lambda \int_{\mathbb{R}}\bigl(\delta_1|\partial_x(v^{S_1})^{-X_1}|+\delta_3|\partial_x(v^{S_3})^{-X_3}|\bigr)|(\phi,\psi,\zeta)|^2\,dx \\ &\qquad + C_\lambda \int_{\mathbb{R}} \bigl(|\partial_x(v^{S_1})^{-X_1}|^2+|\partial_x(v^{S_3})^{-X_3}|^2+|u_{1x}^C|^2+|\theta_x^C|^2\bigr) |(\phi,\psi,\zeta)|^2\,dx. \end{align}\] This proves ?? . ◻

Proof. All the terms in \(\mathcal{B}_4\), \(\mathcal{B}_5\), and \(\mathcal{B}_6\) are at least quadratic in the perturbation variables. Moreover, each coefficient depends smoothly on the background state \(\bar U\), hence is uniformly bounded under the bootstrap bound \(\mathcal{E}(T)^2\le \varepsilon^2\) in 44 and the smallness of \(\delta_0\).

For example, \[\left| \frac{\mu(\bar\theta)}{v\bar v}\bar u_{1x}\psi_{1x}\phi \right| \le \lambda \frac{\mu(\bar\theta)}{v}\psi_{1x}^2 + C_\lambda |\bar u_{1x}|^2\,\phi^2,\] and similarly \[\left| \frac{\alpha_{\rm{th}}(\bar\theta)}{v\bar v\theta}\bar\theta_x\zeta_x\phi \right| \le \lambda \frac{\alpha_{\rm{th}}(\bar\theta)}{v\theta}\zeta_x^2 + C_\lambda |\bar\theta_x|^2\,\phi^2.\] All remaining terms are treated in the same way, using \[|\mu(\theta)-\mu(\bar\theta)|+|\alpha_{\rm{th}}(\theta)-\alpha_{\rm{th}}(\bar\theta)| \le C|\zeta|\] and the fact that \[|\bar u_{1x}|^2+|\bar\theta_x|^2 \le C\bigl(|\partial_x(v^{S_1})^{-X_1}|^2+|\partial_x(v^{S_3})^{-X_3}|^2+|u_{1x}^C|^2+|\theta_x^C|^2\bigr).\] Therefore ?? follows after summing all contributions. ◻

Proof. This is immediate from the definition of \(\mathcal{B}^{\text{res}}\). ◻

Proof. Recall that \[\mathcal{S}_1=-\int_{\mathbb{R}}a\,\psi_1Q_1\,dx, \qquad \mathcal{S}_2=-\int_{\mathbb{R}}a\,\frac{\zeta}{\bar\theta}Q_2\,dx.\] Using the decomposition of \(Q_1\) and \(Q_2\) in 32 –33 , together with the Gaussian bound for the viscous contact wave and the exponential localization of the shock profiles, we obtain \[\begin{align} \left| \mathcal{S}_1\right|+\left| \mathcal{S}_2\right| &\leq \left(\bigl\lVert Q_1^I\bigr\rVert_{L^2}+\bigl\lVert Q_1^C\bigr\rVert_{L^2}+\bigl\lVert Q_2^I\bigr\rVert_{L^2}+\bigl\lVert Q_2^C\bigr\rVert_{L^2}\right)\bigl\lVert\left(\psi,\zeta\right)\bigr\rVert_{L^2}\\ &\leq C(\varepsilon+\delta_0)\left\{\frac{\delta_C}{\left(1+t\right)^{\frac{5}{4}}}+\delta_1\left(\delta_3+\delta_C\right)e^{-C\delta_1t}+\delta_3\left(\delta_1+\delta_C\right)e^{-C\delta_3t}\right\} \end{align}\] The desired estimate then follows from Young’s inequality. ◻

Proof. This follows immediately from Lemmas 14–17. ◻

Proof. By the definitions of \(Y_{i4}\), \(Y_{i5}\), and \(Y_{i6}\), together with \[\Phi\!\left(\frac{v}{\bar v}\right)\sim \phi^2, \qquad \Phi\!\left(\frac{\theta}{\bar\theta}\right)\sim \zeta^2,\] and the bootstrap bound \(\mathcal{E}(T)^2\le \varepsilon^2\) in 44 , we obtain

\[\begin{align} \left| \mathcal{Y}_{i4}\right| \le & C \int \left| \partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}\right| \left| \phi\right|^2 dx \\ \le & C \int \left| \varphi_i\partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}\right| \left| \phi\right|^2 dx + C \int \left| (1-\varphi_i^2)\partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}\right| \left| \phi\right|^2 dx \\ \le & C \mathcal{G}_i^S + C \int \left| (1-\varphi_i)\partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}\right|\left| \phi\right|^2 dx \\ \le & C \mathcal{G}_i^S + C(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it}, \end{align}\]

\[\begin{align} \frac{C}{\delta_i}\left| \mathcal{Y}_{i4}\right|^2 \le& \frac{C\left| \mathcal{Y}_{i4}\right|}{\delta_i}\left| \int a\partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}} \Phi\left(\frac{v}{\bar v}\right) dx\right|\\ \le& C\delta_i(\varepsilon+\delta_0)^2\left(\mathcal{G}_i^S +(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it}\right). \end{align}\]

By the similar argument for \(Y_{i5}\),

\[\label{eq:Y45-est} \frac{C}{\delta_i}\bigl(|\mathcal{Y}_{i4}|^2+|\mathcal{Y}_{i5}|^2\bigr) \le C\delta_i(\delta_0+\varepsilon)^2 \left(\mathcal{G}_i^S+(\delta_0+\varepsilon)^2\delta_i^2e^{-C\delta_i t}\right), \qquad i=1,3.\tag{93}\] Similarly, observe that \[\begin{align} \label{eq:Y6-est} \frac{C}{\delta_i}|\mathcal{Y}_{i6}|^2 \le & \frac{C}{\delta_i^2}\left(\int \left| \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\right|\left| (\phi,\psi,\zeta)\right|^2 dx\right)^2 \nonumber\\ \le & C(\delta_0+\varepsilon)^2\left(\mathcal{G}_i^S+(\delta_0+\varepsilon)^2\delta_i^2e^{-C\delta_i t}\right), \qquad i=1,3. \end{align}\tag{94}\]

Therefore, by Young’s inequality, \[\begin{align} \left| \sum_{i=1,3}\dot{X}_i\sum_{j=4}^6\mathcal{Y}_{ij} \right| &\le \sum_{i=1,3}\left( \frac{\delta_i}{8\mathfrak m_i}|\dot{X}_i|^2 + \frac{C}{\delta_i}\sum_{j=4}^6|\mathcal{Y}_{ij}|^2 \right)\\ &\le \sum_{i=1,3}\frac{\delta_i}{8\mathfrak m_i}|\dot{X}_i|^2 + C(\delta_0+\varepsilon)^2\sum_{i=1,3}\mathcal{G}_i^S \\ &\qquad + C (\delta_0+\varepsilon)^2 \sum_{i=1,3}\delta_i^2e^{-C\delta_i t}, \end{align}\] which proves ?? . ◻

Proof. Control of \(K_4\). By definition, \[K_4 = -\iint a\psi_1\,\xi_1^2 \widetilde{\Pi}_{1x}\,d\xi\,dx.\] Integrating by parts in \(x\), we obtain \[\label{eq:K4-ibp} K_4 = \iint (a\psi_1)_x\,\xi_1^2 \widetilde{\Pi}_1\,d\xi\,dx.\tag{98}\] Hence, for arbitrary \(\alpha,\beta>0\), \[\begin{align} |K_4| \le\;& \frac{\alpha}{2}\int_{\mathbb{R}}(a_x\psi_1)^2\,dx +\frac{\alpha^{-1}}{2}\int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1^2 \widetilde{\Pi}_1\,d\xi \right)^2dx \nonumber\\ &+ \frac{\beta}{2}\int_{\mathbb{R}}(a\psi_{1x})^2\,dx +\frac{\beta^{-1}}{2}\int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1^2 \widetilde{\Pi}_1\,d\xi \right)^2dx. \label{eq:K4-Young} \end{align}\tag{99}\]

We now decompose the perturbative microscopic flux. Recall that \[\begin{align} \mathcal{N}(\widetilde{G},\widetilde{G}) =&-\mathcal{N}\bigl((G^{S_1})^{-X_1},\widetilde{G}\bigr)-\mathcal{N}\bigl(\widetilde{G},(G^{S_1})^{-X_1}\bigr)-\mathcal{N}\bigl((G^{S_1})^{-X_1},(G^{S_1})^{-X_1}\bigr)+\mathcal{N}(G,G)\nonumber\\ &-\mathcal{N}\bigl(\widetilde{G},(G^{S_3})^{-X_3}\bigr)-\mathcal{N}\bigl((G^{S_1})^{-X_1},(G^{S_3})^{-X_3}\bigr)-\mathcal{N}\bigl((G^{S_3})^{-X_3},\widetilde{G}\bigr)\nonumber\\ &-\mathcal{N}\bigl((G^{S_3})^{-X_3},(G^{S_3})^{-X_3}\bigr)-\mathcal{N}\bigl((G^{S_3})^{-X_3},(G^{S_1})^{-X_1}\bigr). \label{eq:Q-tilde-decomp} \end{align}\tag{100}\] Accordingly, \[\begin{align} & \widetilde{\Pi}_1 = L_M^{-1} \left( \widetilde{G}_t-\frac{u_1}{v}\widetilde{G}_x+\frac{1}{v}P_1(\xi_1\widetilde{G}_x)-\mathcal{N}(\widetilde{G},\widetilde{G}) \right) \nonumber\\ &\qquad -L_M^{-1} \left[ \mathcal{N}\bigl(\widetilde{G},(G^{S_1})^{-X_1}+(G^{S_3})^{-X_3}\bigr) + \mathcal{N}\bigl((G^{S_1})^{-X_1}+(G^{S_3})^{-X_3},\widetilde{G}\bigr) \right] \nonumber\\ &\qquad -L_M^{-1} \left[ \mathcal{N}\bigl((G^{S_1})^{-X_1},(G^{S_3})^{-X_3}\bigr) + \mathcal{N}\bigl((G^{S_3})^{-X_3},(G^{S_1})^{-X_1}\bigr) \right] \nonumber\\ &\qquad -\dot{X}_1(L_{1}^{S})^{-1}\partial_x(G^{S_1})^{-X_1} -\dot{X}_3(L_{3}^{S})^{-1}\partial_x(G^{S_3})^{-X_3} +J, \label{eq:Pi-perturb} \end{align}\tag{101}\] where \[\begin{align} J:=\;& \sum_{i\in\{1,3\}} \bigl(L_M^{-1}-(L_{i}^{S})^{-1}\bigr) \left((G^{S_i})_t^{-X_i}-\mathcal{N}\bigl((G^{S_i})^{-X_i},(G^{S_i})^{-X_i}\bigr)\right) \nonumber\\ &+ \sum_{i\in\{1,3\}} \left( \frac{(u_1^{S_i})^{-X_i}}{(v^{S_i})^{-X_i}}(L_{i}^{S})^{-1} -\frac{u_1}{v}L_M^{-1} \right)\partial_x(G^{S_i})^{-X_i} \nonumber\\ &+ \sum_{i\in\{1,3\}} \left( \frac{1}{v}L_M^{-1}P_1\xi_1 -\frac{1}{(v^{S_i})^{-X_i}}(L_{i}^{S})^{-1}P_1^{S_i}\xi_1 \right)\partial_x(G^{S_i})^{-X_i}. \label{eq:J-term} \end{align}\tag{102}\]

Using the weighted inverse estimate for \(L_M^{-1}\) and the collision estimates from Appendix A, we have \[\begin{align} \int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1^2L_M^{-1}\widetilde{G}_t\,d\xi \right)^2dx &\le C\iint\frac{(1+|\xi|)^{-1}|\widetilde{G}_t|^2}{M_\#}\,d\xi\,dx, \tag{103} \\ \int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1^2L_M^{-1}\left(\frac{u_1}{v}\widetilde{G}_x\right)\,d\xi \right)^2dx &\le C\iint\frac{(1+|\xi|)^{-1}|\widetilde{G}_x|^2}{M_\#}\,d\xi\,dx, \tag{104} \\ \int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1^2L_M^{-1}\left(\frac{1}{v}P_1(\xi_1\widetilde{G}_x)\right)\,d\xi \right)^2dx &\le C\iint\frac{(1+|\xi|)^{-1}|\widetilde{G}_x|^2}{M_\#}\,d\xi\,dx. \tag{105} \end{align}\] Moreover, \[\begin{align} & \int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1^2L_M^{-1}\mathcal{N}(\widetilde{G},\widetilde{G})\,d\xi \right)^2dx \nonumber \\ &\qquad \le C(\delta_0+\varepsilon)\mathcal{D}_{\text{mic}} + C\delta_C^4 \left( \frac{1}{(1+t)^4}+\frac{1}{(1+t)^2} \right) \int_{\mathbb{R}}e^{-c_0x^2/(1+t)}\,dx. \label{eq:K4-QGG} \end{align}\tag{106}\]

Here we used \[\begin{align} \label{eq:GCe} \bigl\lVert\widetilde{G}_C\bigr\rVert_{\nu,M_\#}^2 \approx |(u_{1x}^C,\theta_x^C)|^2. \end{align}\tag{107}\]

Similarly, \[\begin{align} &\int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1^2L_M^{-1}\mathcal{N}\bigl(\widetilde{G},(G^{S_1})^{-X_1}\bigr)\,d\xi \right)^2dx \nonumber\\ &\qquad \le C\delta_1^2\mathcal{D}_{\text{mic}} + C\int |(u_{1x}^C,\theta_x^C)|^2 |\partial_x\bigl((v^{S_1})^{-X_1}\bigr)|^2\,dx, \label{eq:K4-mixed} \end{align}\tag{108}\] and the same estimate holds for the remaining mixed and interaction terms involving \((G^{S_3})^{-X_3}\).

Next, we estimate the term \(J\). Observe that \[L_M\bigl(L_M^{-1}-(L_{1}^{S})^{-1}\bigr)h = \mathcal{N}(M_1-M,(L_{1}^{S})^{-1}h)+\mathcal{N}((L_{1}^{S})^{-1}h,M_1-M)\] and \[\begin{align} &L_M (L_{1}^{S})^{-1} \bigl((G^{S_{1}})^{-X_{1}}\bigr)_x \\ &\quad = \mathcal{N}\Bigl(M-\bigl(M^{S_{1}}\bigr)^{-X_{1}},(L_{1}^{S})^{-1} \partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}}\Bigr) + \mathcal{N}\Bigl(\bigl(M^{S_{1}}\bigr)^{-X_{1}},(L_{1}^{S})^{-1} \partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}}\Bigr)\\ &\qquad + \mathcal{N}\Bigl((L_{1}^{S})^{-1} \partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}},M-\bigl(M^{S_{1}}\bigr)^{-X_{1}}\Bigr)+\mathcal{N}\Bigl((L_{1}^{S})^{-1} \partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}},\bigl(M^{S_{1}}\bigr)^{-X_{1}}\Bigr)\\ &\quad =\mathcal{N}\Bigl(M- \bigl(M^{S_{1}}\bigr)^{-X_{1}},(L_{1}^{S})^{-1} \partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}}\Bigr)\\ &\qquad +\mathcal{N}\Bigl((L_{1}^{S})^{-1} \partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}},M-\bigl((M^{S_{1}})^{-X_{1}}\bigr)\Bigr) +\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}}. \end{align}\] Using the identity \[M(a)-M(a') = \int_0^1 D_aM(a_s)\,(a-a')\,ds, \qquad a_s:=a'+s(a-a'),\] with \(a=(v,u,\theta)\) and \(a'=(v',u',\theta')\), we infer \[\bigl\lVert M-\bigl(M^{S_{1}}\bigr)^{-X_{1}}\bigr\rVert_{\nu,M_\#}^2 \le C\Bigl|\bigl(v-\bigl(v^{S_{1}}\bigr)^{-X_{1}},u-\bigl(u^{S_{1}}\bigr)^{-X_{1}},\theta-\bigl(\theta^{S_{1}}\bigr)^{-X_{1}}\bigr)\Bigr|^2.\]

Therefore, \[\begin{align} &\int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1^2\bigl(L_M^{-1}-(L_{1}^{S})^{-1}\bigr)\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}}\,d\xi \right)^2dx\nonumber\\ &\qquad \le C\delta_1 \int_{\mathbb{R}}\left|\partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}\right|^2 \left|\left(v-\bigl(v^{S_{1}}\bigr)^{-X_{1}},u-\bigl(u^{S_{1}}\bigr)^{-X_{1}},\theta-\bigl(\theta^{S_{1}}\bigr)^{-X_{1}}\right)\right|^2\,dx. \label{eq:J-est1} \end{align}\tag{109}\] Using the decomposition \[\begin{align} &\left(v-\bigl(v^{S_{1}}\bigr)^{-X_{1}},u-\bigl(u^{S_{1}}\bigr)^{-X_{1}} ,\theta-\bigl(\theta^{S_{1}}\bigr)^{-X_{1}}\right) \nonumber\\ &\qquad =\bigl(v-\overline{v},u-\overline{u},\theta-\overline{\theta}\bigr)+ \left(\bigl(v^{S_{3}}\bigr)^{-X_{3}}-v^*,\bigl(u^{S_{3}}\bigr)^{-X_{3}}-u^*,\bigl(\theta^{S_{3}}\bigr)^{-X_{3}}-\theta^*\right)\nonumber\\ &\qquad \quad +\bigl(v^C-v_*,u^C-u_*,\theta^C-\theta_*\bigr) \end{align}\] the first part is controlled by \(\mathcal{G}_1^S\) of ?? , while the second part is estimated by Lemma 7. Hence, \[\begin{align} &\int_{\mathbb{R}}\left|\partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}\right|^2 \left|\left(v-\bigl(v^{S_{1}}\bigr)^{-X_{1}},u-\bigl(u^{S_{1}}\bigr)^{-X_{1}},\theta-\bigl(\theta^{S_{1}}\bigr)^{-X_{1}}\right)\right|^2\,dx\nonumber\\ &\qquad \le\; C\delta_1\mathcal{G}_1^S + C\delta_1^3e^{-C\delta_1 t}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx +C(\delta_1^2\delta_3^2+\delta_1^2\delta_C^2)\delta_1e^{-C\delta_1t}\nonumber \\ &\qquad \qquad +C\delta_1^3\delta_3(\delta_3e^{-C\delta_3t})\nonumber \\ &\qquad \le\;C\delta_1\mathcal{G}_1^S + C(\delta_0+\varepsilon)^2\delta_1^3e^{-C\delta_1 t}+C\delta_0^4\bigl(\delta_1e^{-C\delta_1 t}+\delta_3e^{-C\delta_3t}\bigr). \label{eq:interaction-est} \end{align}\tag{110}\]

The same estimate holds for the \(3\)-shock.

Collecting 99 –110 and choosing \(\alpha,\beta\) sufficiently small, we conclude that \[\begin{align} |K_4| \le\;& C\delta_0^4\sum_{i\in\{1,3\}}\delta_i|\dot{X}_i|^2 + C\delta_0\sum_{i\in\{1,3\}}\mathcal{G}_i^S + C\delta_0(\varepsilon+\delta_0)^2\bigl(\delta_1e^{-C\delta_1 t}+\delta_3e^{-C\delta_3 t}\bigr) \nonumber\\ &+\frac{C\delta_C}{(1+t)^{\frac{5}{4}}} +\frac{C\delta_C^2}{1+t}\int_{\mathbb{R}}e^{-2c_0x^2/(1+t)}|(\phi,\zeta)|^2\,dx+\frac{1}{200}\mathcal{D}_{\mathrm{mac}}(U)\nonumber\\ &+C(\delta_0+\varepsilon)\mathcal{D}_{\mathrm{mic}}(t)+ C\int\bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2+\bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx. \label{eq:K4-final} \end{align}\tag{111}\]

Control of \(K_1\). By definition, \[K_1 = -\iint a\frac{\zeta}{\theta}\left(\xi_1\frac{|\xi|^2}{2}\right) \widetilde{\Pi}_{1x}\,d\xi\,dx.\] Integrating by parts in \(x\), we obtain \[K_1 = \iint \left(a\frac{\zeta}{\theta}\right)_x \left(\xi_1\frac{|\xi|^2}{2}\right) \widetilde{\Pi}_{1}\,d\xi\,dx.\] Hence, by Young’s inequality,

\[\begin{align} |K_1| \le\;& \underbrace{\frac{\alpha_1}{2}\int_{\mathbb{R}}\left(a_x\frac{\zeta}{\theta}\right)^2\,dx}_{K_{11}} +\frac{\alpha_1^{-1}}{2}\int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1\frac{\left| \xi\right|^2}{2} \widetilde{\Pi}_{1}\,d\xi \right)^2dx \nonumber\\ &+ \underbrace{\frac{\alpha_2}{2}\int_{\mathbb{R}}\left(a\frac{\zeta_x}{\theta}\right)^2\,dx}_{K_{12}} +\frac{\alpha_2^{-1}}{2}\int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1\frac{\left| \xi\right|^2}{2} \widetilde{\Pi}_{1}\,d\xi \right)^2dx \nonumber\\ &+ \underbrace{\frac{\alpha_3}{2}\int_{\mathbb{R}}\left(a\frac{\zeta\overline{\theta}_x}{\theta^2}\right)^2\,dx}_{K_{13}} +\frac{\alpha_3^{-1}}{2}\int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1\frac{\left| \xi\right|^2}{2} \widetilde{\Pi}_{1}\,d\xi \right)^2dx \nonumber\\ &+ \underbrace{\frac{\alpha_4}{2}\int_{\mathbb{R}}\left(a\frac{\zeta\zeta_x}{\theta^2}\right)^2\,dx}_{K_{14}} +\frac{\alpha_4^{-1}}{2}\int_{\mathbb{R}} \left( \int_{\mathbb{R}^3}\xi_1\frac{\left| \xi\right|^2}{2} \widetilde{\Pi}_{1}\,d\xi \right)^2dx. \label{eq:K1-Young} \end{align}\tag{112}\]

The terms \(K_{11}\) and \(K_{13}\) are controlled by \(\mathcal{G}_i^S\), and the terms \(K_{12}\) and \(K_{14}\) are controlled by the diffusion term \(\mathcal{D}_{\mathrm{mac}}(U)\).

By the same manner as \(K_4\),

\[\begin{align} & |K_1|\le\; C\delta_0^4\sum_{i\in\{1,3\}}\delta_i|\dot{X}_i|^2 + C\delta_0\sum_{i\in\{1,3\}}\mathcal{G}_i^S + C\delta_0(\varepsilon+\delta_0)^2\bigl(\delta_1e^{-C\delta_1 t}+\delta_3e^{-C\delta_3 t}\bigr) \nonumber\\ & \qquad \quad+\frac{C\delta_C}{(1+t)^{\frac{5}{4}}} +\frac{C\delta_C^2}{1+t}\int_{\mathbb{R}}e^{-\frac{2c_0x^2}{1+t}}|(\phi,\zeta)|^2\,dx+\frac{1}{200}\mathcal{D}_{\mathrm{mac}}(U)\nonumber\\ & \qquad \quad +C(\delta_0+\varepsilon)\mathcal{D}_{\mathrm{mic}}(t)+ C\int\bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2+\bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx. \label{eq:K1-final} \end{align}\tag{113}\]

Control of \(K_2\). We write \[\begin{align} K_2 &= \iint a\frac{\zeta}{\theta} \bigg[ u_1\xi_1^2\Pi_{1x} -\sum_{i\in\{1,3\}}(u_1^{S_i})^{-X_i}\xi_1^2\partial_x(\Pi_1^{S_i})^{-X_i} \bigg]d\xi\,dx \nonumber\\ &= \iint a\frac{\zeta}{\theta}u_1\xi_1^2 \widetilde{\Pi}_{1x}\,d\xi\,dx \nonumber\\ &\quad + \sum_{i\in\{1,3\}} \iint a\frac{\zeta}{\theta}\bigl(u_1-(u_1^{S_i})^{-X_i}\bigr)\xi_1^2\partial_x\bigl((\Pi_1^{S_i})^{-X_i}\bigr)\,d\xi\,dx \nonumber\\ &=:J^{\text{mic}}_1+J^{\text{mic}}_2. \label{eq:K2-split} \end{align}\tag{114}\]

The term \(J_1^{\text{mic}}\) is treated exactly as \(K_4\). For \(J_2^{\text{mic}}\), using the exponential decay of \(\bigl((\Pi_1^{S_i})^{-X_i}\bigr)_x\) (See Lemma 4), we obtain \[\begin{align} |J_2^{\text{mic}}| \le\;& C\delta_1^2\mathcal{G}_1^S + C\delta_1^2(\delta_1e^{-C\delta_1 t}+\delta_3e^{-C\delta_3 t}) \nonumber\\ &+ C\delta_3^2\mathcal{G}_3^S + C\delta_3^2(\delta_1e^{-C\delta_1 t}+\delta_3e^{-C\delta_3 t}). \label{eq:J2-est} \end{align}\tag{115}\] Hence, \[\begin{align} |K_2| \le\;& C\delta_0^4\sum_{i\in\{1,3\}}\delta_i|\dot{X}_i|^2 + C\delta_0\sum_{i\in\{1,3\}}\mathcal{G}_i^S + \frac{1}{200}\mathcal{D}_{\mathrm{mac}}(U) + C(\delta_0+\varepsilon)\mathcal{D}_{\mathrm{mic}}(t) \nonumber\\ &+C\delta_0(\varepsilon+\delta_0)^2\bigl(\delta_1e^{-C\delta_1 t}+\delta_3e^{-C\delta_3 t}\bigr)+\frac{C\delta_C}{1+t}\int_{\mathbb{R}}e^{-\frac{2c_0x^2}{1+t}}|(\phi,\zeta)|^2\,dx\nonumber\\ &+\frac{C\delta_C^4}{(1+t)^{\frac{5}{4}}}+ C\int\bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2+\bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx. \label{eq:K2-final} \end{align}\tag{116}\]

Finally, the mixed transverse kinetic terms \(K_3\) and \(K_5\) are controlled the same way as \(K_4\), since \[\int_{\mathbb{R}^3}\xi_1\xi_j\bigl((\Pi_1^{S_1})^{-X_1}+(\Pi_1^{S_3})^{-X_3}\bigr)\,d\xi=0, \qquad j=2,3.\] Therefore, \(K_3\) and \(K_5\) have the same bound as \(K_4\). Summing the above estimates yields ?? . ◻

Proof. Using the exponential decay of \((\Pi_1^{S_i})^{-X_i}\) (See Lemma 4) and Young’s inequality, we obtain \[\begin{align} |K_6| &\le C \sum_{i\in\{1,3\}} \iint a\frac{\zeta^2}{\theta\bar\theta} \left| \xi_1\left(\frac{|\xi|^2}{2}-(u_1^{S_i})^{-X_i}\xi_1\right) \partial_x(\Pi_1^{S_i})^{-X_i} \right|\,d\xi\,dx\\ &\le C\sum_{i\in\{1,3\}} \int \left| \zeta\right|^2 \left\{\left| \int \xi_1\frac{\left| \xi\right|^2}{2}\partial_x\bigl(\Pi_1^{S_{i}}\bigr)^{-X_{i}} d\xi\right|+\left| \int \xi_1^2\partial_x\bigl(\Pi_1^{S_{i}}\bigr)^{-X_{i}} d\xi\right|\right\}dx\\ &\le C \sum_{i\in\{1,3\}} \delta_i^2 \int_{\mathbb{R}}|\partial_x(v^{S_i})^{-X_i}|\,\zeta^2\,dx. \end{align}\] The right-hand side is bounded by \[C\delta_0\sum_{i=1,3}\mathcal{G}_i^S + \frac{C\delta_C}{1+t}\int_{\mathbb{R}}e^{-\frac{2c_0x^2}{1+t}}|(\phi,\zeta)|^2\,dx + C\delta_0(\varepsilon+\delta_0)^2\bigl(\delta_1e^{-C\delta_1 t}+\delta_3e^{-C\delta_3 t}\bigr),\] which proves ?? . ◻

Proof. Lemma 19 already gives the stated bound for \(K_1\), \(K_2\),\(K_3\), \(K_4\), and \(K_5\). Lemma  20 provides the corresponding bounds for the remaining term \(K_6\). Adding these inequalities yields ?? . ◻

Proof. See Lemma 12, Lemma 18, Proposition 3 and Proposition 4. Combining ?? , ?? , and ?? and choosing \(\delta_0+\varepsilon\) sufficiently small, we absorb the lower-order shock terms into the principal coercive term \(\sum_i\mathcal{G}_i^S\), and the dissipation contributions into \(\mathcal{D}_{\mathrm{mac}}(U)\). This proves ?? . ◻

Proof. Recall that 86 ,

\[\begin{align} \label{eq:energy2-1} \frac{d}{dt}\int a\eta(U\mid\bar{U}) dx \le \sum_{i\in\{1,3\}}\dot{X}_i\mathcal{Y}_i(U)+\sum_{l=1}^6 \mathcal{B}_l + \mathcal{S}_1+\mathcal{S}_2+ \sum_{l=1}^6 K_l -\mathfrak G(U)-\mathcal{D}_{\mathrm{mac}}(U). \end{align}\tag{117}\]

For any \(\lambda>0\), rearrange 117 ,

\[\begin{align} \label{eq:energy2-re} &\frac{d}{dt}\int a\eta(U\mid\bar{U}) dx + \sum_{i\in\{1,3\}}\frac{\delta_i}{4\mathfrak m_i}\left| \dot{X}_i\right|^2 +\frac{1}{2}\mathcal{D}_{\mathrm{mac}}(U) +\lambda \sum_{i\in\{1,3\}}\mathcal{G}_i^S \nonumber\\ &\le \sum_{i\in\{1,3\}} \frac{C}{\delta_i}\sum_{j=4}^6 |\mathcal{Y}_{ij}|^2 + \sum_{l=1}^6 \mathcal{B}_l + \mathcal{S}_1 + \mathcal{S}_2 + \sum_{l=1}^6 K_l -\mathfrak G(U) - \frac{1}{2}\mathcal{D}_{\mathrm{mac}}(U) +\lambda \sum_{i\in\{1,3\}}\mathcal{G}_i^S. \end{align}\tag{118}\]

For \(t\le 1\), we obtain the following rough estimate using 95 , ?? and the bootstrap assumptions.

\[\begin{align} \label{eq:coarsebd} \begin{aligned} &\sum_{i=1,3}\left(\frac{C}{\delta_i}\sum_{j=4}^6\left| \mathcal{Y}_{ij}\right|^2\right)+\sum_{l=1}^6 \left| \mathcal{B}_l\right| + \left| \mathcal{S}_1\right| + \left| \mathcal{S}_2\right| + \sum_{l=1}^6 \left| K_l\right| \\ &\qquad - \frac{1}{2}\mathcal{D}_{\mathrm{mac}}(U) - \mathfrak G(U) + \lambda \sum_{i\in\{1,3\}}\mathcal{G}_i^S\\ &\qquad \le C_\lambda \delta_0 + C(\delta_0+\varepsilon) \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx + C\int\bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2+\bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx. \end{aligned} \end{align}\tag{119}\] Using 119 , we have ?? . ◻

Proof. Multiply 121 by \(\widetilde{G}_{\text{rem}}/M_\#\) and integrate over \(\mathbb{R}_x\times\mathbb{R}^3_\xi\). Since \(\widetilde{G}_{\text{rem}}\in\mathfrak Z_M^\perp\), the coercivity estimate for the linearized collision operator yields \[-\iint \frac{\widetilde{G}_{\text{rem}}L_M\widetilde{G}_{\text{rem}}}{M_\#}\,d\xi\,dx \ge \lambda_{\mathrm{mic}} \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx\] for some \(\lambda_{\mathrm{mic}}>0\).

We estimate the source terms one by one.

For the shift terms, Young’s inequality gives \[\begin{align} \iint \dot{X}_i (G^{S_i}_x)^{-X_i}\frac{\widetilde{G}_{\text{rem}}}{M_\#}\,d\xi\,dx \le\;& \frac{\lambda_{\mathrm{mic}}}{32} \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx + C\delta_i^5|\dot{X}_i|^2 . \label{eq:mic-shift-est} \end{align}\tag{123}\] Similarly, \[\begin{align} \iint & \left(\frac{u_1}{v}-\frac{(u_1^{S_i})^{-X_i}}{(v^{S_i})^{-X_i}}\right) \bigl((G^{S_i})^{-X_i}\bigr)_x\frac{\widetilde{G}_{\text{rem}}}{M_\#}\,d\xi\,dx \nonumber\\ & \, \le\;\frac{\lambda_{\mathrm{mic}}}{32}\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx +C\delta_i^2\int_{\mathbb{R}}|\partial_x\bigl((v^{S_i})^{-X_i}\bigr)|^2|(u_1-(u_1^{S_i})^{-X_i},\,v-(v^{S_i})^{-X_i})|^2\,dx . \label{eq:mic-mismatch-est} \end{align}\tag{124}\] Using the decomposition \[U-(U^{S_1})^{-X_1} = (U-\bar U)+(\bar U-(U^{S_1})^{-X_1}),\] the first part is controlled by \(\mathcal{G}_1^S\) of ?? , while the second part is estimated by Lemma 7. Thus the last integral is bounded by \[C\delta_0\mathcal{G}_i^S + C\delta_0^3\bigl(\delta_1e^{-C\delta_1 t}+\delta_3e^{-C\delta_3 t}\bigr) + C\delta_i^3e^{-C\delta_i t}\int_{\mathbb{R}}\eta(U\mid \bar U)\,dx .\]

Next, note that \[P_1(\xi_1h)=\xi_1h-\sum_{j=0}^4\langle \xi_1h,\chi_j\rangle\chi_j.\] Hence \[\begin{align} -\iint \frac{1}{v}P_1(\xi_1\widetilde{G}_x)\frac{\widetilde{G}_{\text{rem}}}{M_\#}\,d\xi\,dx \le\;& C\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx + \frac{\lambda_{\mathrm{mic}}}{32} \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx . \label{eq:mic-P1Gx-est} \end{align}\tag{125}\] Also, \[\begin{align} & -\iint \frac{3}{2v\theta} P_1\left[ \xi_1M\left(\xi\cdot\psi_x+\frac{|\xi-u|^2}{2\theta}\zeta_x\right) \right] \frac{\widetilde{G}_{\text{rem}}}{M_\#}\,d\xi\,dx \nonumber \\ & \qquad \le\; C\|(\psi_x,\zeta_x)\|_{L^2_x}^2 + \frac{\lambda_{\mathrm{mic}}}{64} \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx . \label{eq:mic-source-est} \end{align}\tag{126}\]

We next treat the profile-difference terms. A representative estimate is \[\begin{align} &\iint \left[ -\frac{3\bigl((u_1^{S_1})^{-X_1}\bigr)_x}{2v\theta}P_1(\xi_1^2M) + \frac{3\bigl((u_1^{S_1})^{-X_1}\bigr)_x}{2(v^{S_1})^{-X_1}(\theta^{S_1})^{-X_1}} P_1^{S_1}\xi_1^2(M^{S_1})^{-X_1} \right] \frac{\widetilde{G}_{\text{rem}}}{M_\#}\,d\xi\,dx \nonumber\\ &\qquad\le C\delta_1^2\mathcal{G}_1^S + C\delta_0^3\bigl(\delta_1e^{-C\delta_1 t}+\delta_3e^{-C\delta_3 t}\bigr) + \frac{\lambda_{\mathrm{mic}}}{64} \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx . \label{eq:mic-profile-est} \end{align}\tag{127}\] The same estimate holds for the \(3\)-shock contribution. Likewise, the terms \[-\sum_{i=1,3}\left[ \frac{1}{v}P_1\bigl(\xi_1\bigl((G^{S_i})^{-X_i}\bigr)_x\bigr) - \frac{1}{(v^{S_i})^{-X_i}}P_1^{S_i}\bigl(\xi_1\bigl((G^{S_i})^{-X_i} \bigr)_x\bigr) \right]\] and \[\sum_{i=1,3}(L_M-L_1^S)(G^{S_i})^{-X_i}\] are profile-difference terms of the same type and are estimated in the same way.

For the nonlinear term, Lemmas 44, 45 in Appendix give \[\begin{align} \iint \mathcal{N}(\widetilde{G},\widetilde{G})\frac{\widetilde{G}_{\text{rem}}}{M_\#}\,d\xi\,dx \le\;& C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx \nonumber\\ &+\frac{\lambda_{\mathrm{mic}}}{64}\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx +\frac{C\delta_C^4}{(1+t)^{\frac{5}{4}}} \label{eq:mic-Q-est} \end{align}\tag{128}\] Here we used \[\bigl\lVert\widetilde{G}_C\bigr\rVert_{\nu,M_\#}^2 \approx |(u_{1x}^C,\theta_x^C)|^2 .\] The mixed terms \(\mathcal{N}((G^{S_i})^{-X_i},\widetilde{G})\) and \(\mathcal{N}((G^{S_1})^{-X_1},(G^{S_3})^{-X_3})\) are controlled in the same way and contribute only lower-order shock terms and exponentially decaying remainders.

Finally, differentiating \(\widetilde{G}_C\) in time yields \[\iint \widetilde{G}_{Ct}\frac{\widetilde{G}_{\text{rem}}}{M_\#}\,d\xi\,dx \le C\delta_0\|(\phi_t,\psi_t,\zeta_t)\|_{L^2_x}^2 + \frac{\lambda_{\mathrm{mic}}}{64} \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx .\]

Collecting 123 –128 , choosing \(\delta_0+\varepsilon\) sufficiently small, and absorbing the microscopic coercive terms into the left-hand side, we obtain ?? . ◻

Proof. Since \(v,\theta,\bar v,\bar\theta\) remain uniformly away from zero and infinity under 44 , the standard convexity property of \[\Phi(z)=z-1-\ln z\] implies \[\Phi\!\left(\frac{v}{\bar v}\right)\sim \left|\frac{\phi}{\bar v}\right|^2, \qquad \Phi\!\left(\frac{\theta}{\bar\theta}\right)\sim \left|\frac{\zeta}{\bar\theta}\right|^2.\] Together with the definition of \(\eta(U\mid\bar U)\), this yields ?? . Since the weight \(a\) is uniformly positive and bounded for sufficiently small \(\delta_0\), ?? follows. ◻

Proof. The right-hand side of ?? is integrable away from time 0, and hence contribute only \(O(\delta_0^{1/2})\) after time integration over \([1,t]\). ◻

Proof. See Lemma 22, Proposition 5, and Lemma 24. Integral ?? over \([0,T]\). Collecting all of these, we have the following estimate.

There exist constants \(c_0>0\) and \(C>0\) such that \[\begin{align} &\mathcal{E}_0(t)+ c_0\int_0^t \left[ \sum_{i=1,3}\mathcal{G}_i^S(s) + \mathcal{D}_{\mathrm{mac}}(U)(s) + \sum_{i=1,3}\delta_i|\dot{X}_i(s)|^2\right] ds\nonumber\\ &\qquad +c_0 \int_0^t \int_{\mathbb{R}}\bigl\lVert\widetilde{G}_{\mathrm{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx ds\nonumber\\ &\le C\mathcal{E}_0(0)+ C\int_0^t \frac{\delta_C}{1+s}\int_{\mathbb{R}}e^{-\frac{2c_0|x|^2}{1+s}}|(\phi,\zeta)|^2\,dx\,ds+ C\delta_0\int_0^t \|(\phi_t,\psi_t,\zeta_t)\|_{L^2_x}^2\,ds\nonumber\\ &\qquad+C\int_0^t\int_{\mathbb{R}}\bigl\lVert\widetilde{G}_t\bigr\rVert^2_{\nu,M_\#}+\bigl\lVert\widetilde{G}_x\bigr\rVert^2_{\nu,M_\#}\,dx\,ds +C\delta_0^{1/2}. \label{lacpiyof} \end{align}\tag{131}\]

The estimate ?? follows immediately from 120 , 130 and [eq:zero-order-final]. ◻

Proof. We have \[\begin{align} &\left(\frac{2}{3}\mu(\overline{\theta})\phi_x^2-v\psi_1\phi_x\right)_t + v_t \psi_1 \phi_x + (v\psi_1\phi_t)_x-(v_x\psi_1+v\psi_{1x})\phi_t + \frac{2}{3}\frac{\theta}{v}\phi_x^2 \\ &+ \sum_{i=1,3}\dot{X}_i\phi_x\left\{-\frac{4}{3}\mu(\overline{\theta})\partial_{xx}\bigl(v^{S_{i}}\bigr)^{-X_{i}}+v\partial_x\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right\}\\ &=\frac{2}{3}v\phi_x\left(-\frac{\theta}{v^2}+\frac{\overline{\theta}}{\overline{v}^2}\right)\overline{v}_x+v\phi_xQ_1+v\phi_x \int \xi_1^2 \widetilde{\Pi}_{1x} \,d\xi \\ &+\frac{2}{3}\phi_x\zeta_x+\frac{2}{3}v\phi_x\overline{\theta}_x\left(\frac{1}{v}-\frac{1}{\overline{v}}\right)+\frac{2}{3}\mu'(\overline{\theta})\overline{\theta}_t\phi_x^2-\frac{4}{3}\mu(\overline{\theta})v\phi_x\overline{u}_{1xx}\left(\frac{1}{v}-\frac{1}{\overline{v}}\right)\\ &-\frac{4}{3}\phi_x\left(\mu(\theta)-\mu(\overline{\theta})\right)_xu_{1x}-\frac{4}{3}\phi_x\mu'(\overline{\theta})\overline{\theta}_x\left(\psi_{1x}-\frac{\phi}{\overline{v}}\overline{u}_{1x}\right)-\frac{4}{3}\mu(\overline{\theta})v\phi_x\overline{u}_{1x}\left(\frac{\overline{v}_x}{\overline{v}^2}-\frac{v_x}{v^2}\right)\\ &+\frac{4}{3}\phi_xv\left(\mu(\theta)-\mu(\overline{\theta})\right)\left(\frac{u_{1x}}{v}\right)_x+\frac{4}{3}\frac{\mu(\overline{\theta})}{v}\phi_x\psi_{1x}v_x. \end{align}\]

We single out the term involving \(\phi_t\), since it requires a direct use of the continuity equation. The remaining terms are estimated in the same way as the corresponding macroscopic error terms in the first-order spatial energy estimate.

\[\begin{align} -\int (v_x\psi_1+v\psi_{1x})\phi_t\,dx \le\;& \int \left| \phi_x\psi_1\phi_t\right| + \left| \bar v_x\psi_1\phi_t\right| + \left| \phi\psi_{1x}\phi_t\right| + \left| \bar v\psi_{1x}\phi_t\right|\,dx \\ \le\;& \Bigl(\frac{1}{128}+C(\varepsilon+\delta_0)\Bigr)\int \left| \phi_x\right|^2\,dx + C\|\phi_t\|_{L^2}^2 + C\|\psi_{1x}\|_{L^2}^2. \end{align}\] Here we use the continuity equation \[\phi_t=\psi_{1x}+\sum_{i=1,3}\dot{X}_i \partial_x(v^{S_i})^{-X_i},\] which yields \[\|\phi_t\|_{L^2}^2 \le C\|\psi_{1x}\|_{L^2}^2 + C\sum_{i=1,3}|\dot{X}_i|^2\|\partial_x(v^{S_i})^{-X_i}\|_{L^2}^2 \le C\|\psi_{1x}\|_{L^2}^2 + C\sum_{i=1,3}\delta_i^3|\dot{X}_i|^2.\] Therefore, \[\begin{align} -\int (v_x\psi_1+v\psi_{1x})\phi_t\,dx \le\;& \Bigl(\frac{1}{128}+C(\varepsilon+\delta_0)\Bigr)\int \left| \phi_x\right|^2\,dx + C\|\psi_{1x}\|_{L^2}^2 + C\sum_{i=1,3}\delta_i^3|\dot{X}_i|^2 . \end{align}\]

The remaining terms on the right-hand side are lower-order macroscopic error terms or microscopic coupling terms. They are estimated exactly as in the first-order spatial energy estimate, using the previously established bounds for \(Q_1\), the profile derivatives of \(\bar U\), and the microscopic moments involving \(\widetilde{\Pi}_1\). Collecting all these estimates, we obtain the desired inequality. ◻

Proof. Differentiating the first momentum equation in the perturbed system with respect to \(x\), we obtain \[\begin{align} \begin{aligned} &\psi_{1tx}+\left(p-\overline{p}\right)_{xx}+\left(\bar{p}-p^{S_{1},-X_{1}}-p^C-p^{S_{3},-X_{3}}\right)_{xx}-\sum_{i=1,3}\dot{X}_i \partial_{xx}\bigl(u_1^{S_{i}}\bigr)^{-X_{i}} \\ &= -Q_{1x}^C-\frac{4}{3}\left(\frac{\mu\left(\theta^C\right)u_{1x}^C}{v^C}\right)_{xx}-\int \xi_1^2 \widetilde{G}_{xx} \,d\xi . \end{aligned} \end{align}\] Multiplying this equation by \(-\phi_{xx}\) and integrating over \(x\in\mathbb{R}\), we start from \[-\int \phi_{xx}\psi_{1tx}\,dx.\] By integration by parts in time, \[\begin{align} -\int \phi_{xx}\psi_{1tx}\,dx &= -\frac{d}{dt}\int \phi_{xx}\psi_{1x}\,dx +\int \phi_{xxt}\psi_{1x}\,dx \\ &= -\frac{d}{dt}\int \phi_{xx}\psi_{1x}\,dx -\int \phi_{xt}\psi_{1xx}\,dx . \end{align}\] Hence, \[\begin{align} -\int \phi_{xx}\psi_{1tx}\,dx \le -\frac{d}{dt}\int \phi_{xx}\psi_{1x}\,dx +\frac{1}{128}\|\phi_{xt}\|_{L^2}^2 + C\|\psi_{1xx}\|_{L^2}^2. \end{align}\] Now differentiate the continuity equation \[\phi_t-\psi_{1x}-\sum_{i=1,3}\dot{X}_i \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}=0\] with respect to \(x\). Then \[\phi_{xt} = \psi_{1xx} + \sum_{i=1,3}\dot{X}_i \partial_{xx}\bigl(v^{S_{i}}\bigr)^{-X_{i}},\] so that \[\begin{align} \|\phi_{xt}\|_{L^2}^2 &\le C\|\psi_{1xx}\|_{L^2}^2 + C\sum_{i=1,3} |\dot{X}_i|^2 \|\partial_{xx}\bigl(v^{S_{i}}\bigr)^{-X_{i}}\|_{L^2}^2 \\ &\le C\|\psi_{1xx}\|_{L^2}^2 + C\sum_{i=1,3}\delta_i^5 |\dot{X}_i|^2 \\ &\le C\|\psi_{1xx}\|_{L^2}^2 + C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i |\dot{X}_i|^2 . \end{align}\] Therefore, \[\begin{align} \label{eq:phixt-control} -\int \phi_{xx}\psi_{1tx}\,dx \le -\frac{d}{dt}\int \phi_{xx}\psi_{1x}\,dx + C\|\psi_{1xx}\|_{L^2}^2 + C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i |\dot{X}_i|^2 . \end{align}\tag{132}\]

Next, note that \[\begin{align} \begin{aligned} \left(p-\overline{p}\right)_{xx} = -\frac{p}{v}\phi_{xx} +\frac{2}{3v}\zeta_{xx} -\frac{1}{v}\left(p-\overline{p}\right)\overline{v}_{xx} -\frac{\phi}{v}\overline{p}_{xx} -\frac{2v_x}{v}\left(p-\overline{p}\right)_x -\frac{2\overline{p}_x}{v}\phi_x . \end{aligned} \end{align}\] The leading term gives the desired coercivity: \[-\int \phi_{xx}\left(-\frac{p}{v}\phi_{xx}\right)\,dx = \int \frac{p}{v}\phi_{xx}^2\,dx .\] The remaining terms are lower-order error terms. For example, \[\begin{align} -\int \frac{2}{3v}\phi_{xx}\zeta_{xx}\,dx &\le \frac{1}{128}\int \frac{p}{v}\phi_{xx}^2\,dx + C\|\zeta_{xx}\|_{L^2}^2, \\ \int \frac{1}{v}\left(p-\overline{p}\right)\overline{v}_{xx}\phi_{xx}\,dx &\le \frac{1}{128}\int \frac{p}{v}\phi_{xx}^2\,dx + C\int \left| \overline{v}_{xx}\right|^2\left| (\phi,\zeta)\right|^2\,dx, \\ \int \frac{\phi\phi_{xx}}{v}\overline{p}_{xx}\,dx &\le \frac{1}{128}\int \frac{p}{v}\phi_{xx}^2\,dx + C\int \left| \overline{p}_{xx}\right|^2\left| \phi\right|^2\,dx, \\ \int \frac{2v_x\phi_{xx}}{v}\left(p-\overline{p}\right)_x\,dx &\le \frac{1}{128}\int \frac{p}{v}\phi_{xx}^2\,dx + C(\delta_0+\varepsilon)^2\|(\phi_x,\zeta_x)\|_{L^2}^2 \\ &\qquad + C(\delta_0+\varepsilon)^2\int \left| (\overline{\theta}_x,\overline{v}_x)\right|^2\left| (\phi,\zeta)\right|^2\,dx, \\ \int \frac{2\overline{p}_x}{v}\phi_{xx}\phi_x\,dx &\le \frac{1}{128}\int \frac{p}{v}\phi_{xx}^2\,dx + C(\delta_0+\varepsilon)^2\|\phi_x\|_{L^2}^2 . \end{align}\]

We next estimate the profile-error and microscopic terms: \[\begin{align} -\int \phi_{xx} \left(\overline{p}-p^{S_{1},-X_{1}}-p^C-p^{S_{3},-X_{3}}\right)_{xx}\,dx &\le \frac{1}{128}\int \frac{p}{v}\phi_{xx}^2\,dx + C \|Q_{1x}\|_{L^2}^2, \\ \sum_{i=1,3} \int \dot{X}_i \partial_{xx}\bigl(u_1^{S_{i}}\bigr)^{-X_{i}} \phi_{xx}\,dx &\le \frac{1}{128}\int \frac{p}{v}\phi_{xx}^2\,dx + C\sum_{i=1,3} \delta_i^3 |\dot{X}_i|^2, \\ \int \phi_{xx} \left(Q_{1x}^C + \frac{4}{3}\left(\frac{\mu\left(\theta^C\right)u^C_{1x}}{v^C}\right)_{xx}\right)\,dx &\le \frac{1}{128}\int \frac{p}{v}\phi_{xx}^2\,dx + C \|Q_{1x}^C\|_{L^2}^2, \\ \int \phi_{xx} \int \xi_1^2 \widetilde{G}_{xx}\,d\xi\,dx &\le \frac{1}{128}\int \frac{p}{v}\phi_{xx}^2\,dx \\ &\qquad + C \int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2 dx . \end{align}\]

Combining these bounds with 132 , we obtain \[\begin{align} \begin{aligned} & -\frac{d}{dt}\int \phi_{xx}\psi_{1x}\,dx +\int \frac{p}{2v}\phi_{xx}^2\,dx \\ &\quad \le C \|(\psi_{1xx},\zeta_{xx})\|_{L^2}^2 + C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i |\dot{X}_i|^2 + C \int \left| (\overline{v}_{xx},\overline{p}_{xx})\right|^2\left| (\phi,\zeta)\right|^2\,dx + C \|Q_{1x}\|_{L^2}^2 \\ &\qquad + C \|Q_{1x}^C\|_{L^2}^2 + C(\delta_0+\varepsilon)^2 \|(\phi_x,\zeta_x)\|_{L^2}^2 + C(\delta_0+\varepsilon)^2 \int \left| (\overline{\theta}_x,\overline{v}_x)\right|^2\left| (\phi,\zeta)\right|^2\,dx\\ &\qquad + C \int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2 dx . \end{aligned} \end{align}\]

Finally, the terms involving \(Q_{1x}\), \(Q_{1x}^C\), \((\overline{v}_{xx},\overline{p}_{xx})\), and \((\overline{\theta}_x,\overline{v}_x)\) are estimated exactly as in the previous macroscopic energy bounds, using the profile decay estimates for the composite wave. In particular, \[\int \left| (\overline{v}_{xx},\overline{p}_{xx})\right|^2\left| (\phi,\zeta)\right|^2\,dx + \int \left| (\overline{\theta}_x,\overline{v}_x)\right|^2\left| (\phi,\zeta)\right|^2\,dx\] is bounded by \[C\delta_0 (\mathcal{G}_1^S+\mathcal{G}_3^S) + C\delta_C^2 \frac{1}{1+t}\int e^{-\frac{2c_0|x|^2}{1+t}}\left| (\phi,\zeta)\right|^2\,dx + C\delta_0^3(\delta_1e^{-C\delta_1 t}+\delta_Ce^{-Ct}+\delta_3e^{-C\delta_3 t}),\] while \[\|Q_{1x}\|_{L^2}^2+\|Q_{1x}^C\|_{L^2}^2\] is bounded by \[C\delta_0^3(\delta_1e^{-C\delta_1 t}+\delta_3e^{-C\delta_3 t})+C\frac{\delta_C^4}{(1+t)^{5/4}} + C(\varepsilon+\delta_0)\delta_3^2 e^{-C\delta_3t} + C(\varepsilon+\delta_0)\delta_1^2 e^{-C\delta_1t}.\] This proves ?? . ◻

Proof. Differentiating the perturbed system with respect to \(x\), we obtain \[\begin{align} \begin{aligned} &\phi_{xt}-\psi_{1xx}-\sum_{i=1,3} \dot{X}_i \partial_{xx}\bigl(v^{S_{i}}\bigr)^{-X_{i}} = 0,\\ &\psi_{1tx} + \left(p-p^{S_1,-X_1}-p^C-p^{S_3,-X_3}\right)_{xx}-\sum_{i=1,3} \dot{X}_i \partial_{xx}\bigl(u_1^{S_{i}}\bigr)^{-X_{i}} \\ &\qquad \quad= -Q_{1x}^C-\frac{4}{3}\left(\frac{\mu\left(\theta^C\right)u_{1x}^C}{v^C}\right)_{xx}-\int \xi_1^2 \widetilde{G}_{xx} d\xi,\\ &\psi_{itx}=-\int \xi_1\xi_i \widetilde{G}_{xx} d\xi,\quad (i=2,3),\\ &\zeta_{tx} +\left(pu_{1x}-p^{S_1,-X_1}\partial_x\bigl(u_1^{S_{1}}\bigr)^{-X_{1}}-p^Cu_{1x}^C-p^{S_3,-X_3}\partial_x\bigl(u_1^{S_{3}}\bigr)^{-X_{3}}\right)_x-\sum_{i=1,3}\dot{X}_i\partial_{xx}\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}\\ &\qquad = -\frac{1}{2}\int \xi_1 \left| \xi\right|^2 \widetilde{G}_{xx} d\xi +u_{1x}\int \xi_1^2 \widetilde{G}_x d\xi +u_1\int \xi_1^2 \widetilde{G}_{xx} d\xi \\ &\qquad\quad +\sum_{i=1,3} \left(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right)_x\int \xi_1^2 \partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}} d\xi\\ &\qquad\quad +\sum_{i=1,3} \left(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right)\int \xi_1^2 \partial_{xx}\bigl(G^{S_{i}}\bigr)^{-X_{i}} d\xi\\ &\qquad\quad +\sum_{k=2}^3 u_{kx} \int \xi_1\xi_k \widetilde{G}_x d\xi+\sum_{k=2}^3 u_k \int \xi_1\xi_k \widetilde{G}_{xx} d\xi \\ &\qquad\quad-\left(\frac{\alpha_{\rm{th}}\left(\theta^C\right)\theta_x^C}{v^C}\right)_{xx}-\left\{\frac{4}{3}\mu\left(\theta^C\right)\frac{\left(u_{1x}^C\right)^2}{v^C}+Q_2^C\right\}_x . \end{aligned} \end{align}\]

We estimate \(\phi_{xt}\), \(\psi_{xt}\), and \(\zeta_{xt}\) separately.

Step 1: estimate of \(\phi_{xt}\). Multiplying the first equation by \(\phi_{xt}\) and integrating over \(x\), we get \[\begin{align} \|\phi_{xt}\|_{L^2}^2 &\le \frac{1}{128}\|\phi_{xt}\|_{L^2}^2 + C\|\psi_{1xx}\|_{L^2}^2 + \frac{1}{128}\|\phi_{xt}\|_{L^2}^2 + C\sum_{i=1,3}\delta_i^5|\dot{X}_i|^2. \end{align}\] Hence, \[\begin{align} \label{eq:phixt-est} \|\phi_{xt}\|_{L^2}^2 \le C\|\psi_{1xx}\|_{L^2}^2 + C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i|\dot{X}_i|^2. \end{align}\tag{133}\]

Step 2: estimate of \(\psi_{1tx}\). Multiplying the differentiated first momentum equation by \(\psi_{1tx}\) and integrating over \(x\), we obtain \[\begin{align} \|\psi_{1tx}\|_{L^2}^2 \le\;& \frac{1}{128}\|\psi_{1tx}\|_{L^2}^2 + C\|(p-\bar p)_{xx}\|_{L^2}^2 + C\|Q_{1x}\|_{L^2}^2 + C\sum_{i=1,3}\delta_i^3|\dot{X}_i|^2 \\ &+ C\|Q_{1x}^C\|_{L^2}^2 + C\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2\,dx . \end{align}\] Since \[(p-\bar p)_{xx} = -\frac{p}{v}\phi_{xx} +\frac{2}{3v}\zeta_{xx} -\frac{1}{v}(p-\bar p)\bar v_{xx} -\frac{\phi}{v}\bar p_{xx} -\frac{2v_x}{v}(p-\bar p)_x -\frac{2\bar p_x}{v}\phi_x,\] we have \[\begin{align} \|(p-\bar p)_{xx}\|_{L^2}^2 \le\;& C\|(\phi_{xx},\zeta_{xx})\|_{L^2}^2 + C(\delta_0+\varepsilon)^2\|(\phi_x,\zeta_x)\|_{L^2}^2 \\ &+ C\int \left| (\bar v_{xx},\bar p_{xx})\right|^2\left| (\phi,\zeta)\right|^2\,dx + C(\delta_0+\varepsilon)^2\int \left| (\bar v_x,\bar\theta_x)\right|^2\left| (\phi,\zeta)\right|^2\,dx . \end{align}\] Therefore, \[\begin{align} \label{eq:psi1tx-est} \begin{aligned} \|\psi_{1tx}\|_{L^2}^2 \le\;& C\|(\phi_{xx},\zeta_{xx})\|_{L^2}^2 + C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i|\dot{X}_i|^2 + C(\delta_0+\varepsilon)^2\|(\phi_x,\zeta_x)\|_{L^2}^2 \\ &+ C\int \left| (\bar v_{xx},\bar p_{xx})\right|^2\left| (\phi,\zeta)\right|^2\,dx + C(\delta_0+\varepsilon)^2\int \left| (\bar v_x,\bar\theta_x)\right|^2\left| (\phi,\zeta)\right|^2\,dx \\ &+ C\|Q_{1x}\|_{L^2}^2 + C\|Q_{1x}^C\|_{L^2}^2 + C\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2\, dx . \end{aligned} \end{align}\tag{134}\]

Step 3: estimate of \(\psi_{itx}\), \(i=2,3\). Multiplying \[\psi_{itx}=-\int \xi_1\xi_i \widetilde{G}_{xx}\,d\xi\] by \(\psi_{itx}\) and integrating in \(x\), we get \[\label{eq:psiitx-est} \|\psi_{itx}\|_{L^2}^2 \le C\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2 \, dx, \qquad i=2,3.\tag{135}\]

Step 4: estimate of \(\zeta_{tx}\). Multiplying the differentiated energy equation by \(\zeta_{tx}\) and integrating over \(x\), we obtain \[\begin{align} & \|\zeta_{tx}\|_{L^2}^2 \\ & \, \le C\Bigl\|\left(pu_{1x}-p^{S_1,-X_1}\partial_x\bigl(u_1^{S_{1}}\bigr)^{-X_{1}}-p^Cu_{1x}^C-p^{S_3,-X_3}\partial_x\bigl(u_1^{S_{3}}\bigr)^{-X_{3}}\right)_x\Bigr\|_{L^2}^2 \\ &\quad + C\sum_{i=1,3}\delta_i^3|\dot{X}_i|^2 + C\|Q_{2x}\|_{L^2}^2 + C\|Q_{2x}^C\|_{L^2}^2 + C\|\theta_{xxx}^C\|_{L^2}^2 \\ &\quad + C\sum_{i=1,3}\int \left| \bigl(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\bigr)_x\right|^2 \left| \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\right|^2\,dx\\ &\quad + C(\delta_0+\varepsilon)\sum_{i=1,3}\int \left| u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right|^2 \left| \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\right|^2\,dx \\ &\quad + C\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2\, dx + C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\, dx. \end{align}\] Now \[\begin{align} &\left| \left(pu_{1x}-\bar p\,\bar u_{1x}\right)_x\right|\\ &\quad \le C \left| \psi_{1xx}\right| + C \left| (p-\bar p)_x\right|\left| (\bar u_{1x},\psi_{1x})\right| + C \left| \bar p_x\right|\left| \psi_{1x}\right| + C\left| (\phi,\zeta)\right|\left| \bar u_{1xx}\right|, \end{align}\] and therefore \[\begin{align} &\Bigl\|\left(pu_{1x}-\bar p\,\bar u_{1x}\right)_x\Bigr\|_{L^2}^2\\ &\quad \le C\|\psi_{1xx}\|_{L^2}^2 + C(\delta_0+\varepsilon)^2\|(\zeta_x,\psi_x,\phi_x)\|_{L^2}^2 \\ &\qquad + C(\delta_0+\varepsilon)^2\int \left| (\bar\theta_x,\bar v_x)\right|^2\left| (\phi,\zeta)\right|^2\,dx + C\int \left| \bar u_{1xx}\right|^2\left| (\phi,\zeta)\right|^2\,dx . \end{align}\] Hence \[\begin{align} \label{eq:zetatx-est} \begin{aligned} & \|\zeta_{tx}\|_{L^2}^2 \\ & \quad \le C\|\psi_{1xx}\|_{L^2}^2 + C(\delta_0+\varepsilon)^2\|(\zeta_x,\psi_x,\phi_x)\|_{L^2}^2 + C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i|\dot{X}_i|^2 \\ &\qquad + C\int \left| (\bar\theta_x,\bar v_x)\right|^2\left| (\phi,\zeta)\right|^2\,dx + C\int \left| \bar u_{1xx}\right|^2\left| (\phi,\zeta)\right|^2\,dx \\ &\qquad + C\|Q_{2x}\|_{L^2}^2 + C\|Q_{2x}^C\|_{L^2}^2 + C\|\theta_{xxx}^C\|_{L^2}^2 \\ &\qquad + C\sum_{i=1,3}\int \left| \partial_x\bigl(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\bigr)\right|^2 \left| \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\right|^2\,dx \\ &\qquad + C(\delta_0+\varepsilon)\sum_{i=1,3}\int \left| u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right|^2 \left| \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\right|^2\,dx \\ &\qquad + C\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2 \, dx + C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \, dx. \end{aligned} \end{align}\tag{136}\]

The profile terms in 134 and 136 are estimated exactly as in the previous macroscopic estimates. In particular, \[\int \left| (\bar v_{xx},\bar p_{xx},\bar u_{1xx})\right|^2\left| (\phi,\zeta)\right|^2\,dx + \int \left| (\bar\theta_x,\bar v_x)\right|^2\left| (\phi,\zeta)\right|^2\,dx\] is bounded by \[C\delta_0 (\mathcal{G}_1^S+\mathcal{G}_3^S) + C\delta_C^2 \frac{1}{1+t}\int e^{-\frac{2c_0|x|^2}{1+t}}\left| (\phi,\psi,\zeta)\right|^2\,dx + C\delta_0^3(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}),\] while \[\|Q_{1x}\|_{L^2}^2+\|Q_{1x}^C\|_{L^2}^2+\|Q_{2x}\|_{L^2}^2+\|Q_{2x}^C\|_{L^2}^2+\|\theta_{xxx}^C\|_{L^2}^2\] is bounded by \[C\delta_0^3(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}) + C\frac{\delta_C^4}{(1+t)^{\frac{5}{4}}} + C(\varepsilon+\delta_0)\delta_3^2 e^{-C\delta_3t} + C(\varepsilon+\delta_0)\delta_1^2 e^{-C\delta_1t}.\] Finally, the shock-interaction terms are controlled by the already established wave-interaction bounds, giving \[\begin{align} & \sum_{i=1,3}\int \left| \bigl(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\bigr)_x\right|^2 \left| \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\right|^2\,dx + (\delta_0+\varepsilon)\int \left| u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right|^2 \left| \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\right|^2\,dx\\ & \le C\delta_0(\mathcal{G}_1^S+\mathcal{G}_3^S) + C\delta_0^3(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}). \end{align}\]

Collecting 133 , 134 , 135 , and 136 , we obtain ?? . ◻

Proof. For convenience, we omit the shift \(X_1\). By Young’s inequality, for any \(\widetilde{\alpha}>0\), \[\begin{align} \iint_{\mathbb{R}\times\mathbb{R}^3} \bigl|v-v^{S_1}\bigr| \left|H\frac{G}{M_\#}\right|\,d\xi\,dx \le\;& \widetilde{\alpha} \int_{\mathbb{R}} |v-v^{S_1}|^2\bigl\lVert H\bigr\rVert_{M_\#}^2\,dx \\ &+ \frac{1}{\widetilde{\alpha}} \int_{\mathbb{R}} \bigl\lVert G\bigr\rVert_{M_\#}^2\,dx . \end{align}\] Using the assumption on \(H\), we obtain \[\begin{align} \iint_{\mathbb{R}\times\mathbb{R}^3} \bigl|v-v^{S_1}\bigr| \left|H\frac{G}{M_\#}\right|\,d\xi\,dx \le\;& C\widetilde{\alpha}\,\delta_1^{2k} \int_{\mathbb{R}} |v-v^{S_1}|^2\,|(v^{S_1})_x|^2\,dx \\ &+ \frac{1}{\widetilde{\alpha}} \int_{\mathbb{R}} \bigl\lVert G\bigr\rVert_{M_\#}^2\,dx . \end{align}\]

We now decompose \[v-v^{S_1} = \phi +(v^C-v_*)+(v^{S_3}-v^*),\] and therefore \[\begin{align} \int_{\mathbb{R}}|v-v^{S_1}|^2 |(v^{S_1})_x|^2\,dx \le\;& C\int_{\mathbb{R}}\phi^2 |(v^{S_1})_x|^2\,dx \\ &+ C\int_{\mathbb{R}}|v^C-v_*|^2 |(v^{S_1})_x|^2\,dx \\ &+ C\int_{\mathbb{R}}|v^{S_3}-v^*|^2 |(v^{S_1})_x|^2\,dx . \end{align}\]

For the perturbation term, since \(|(v^{S_1})_x|\lesssim \delta_1^2\) and \[\varphi_1+\varphi_3=1,\] we have \[\begin{align} \int_{\mathbb{R}}\phi^2 |(v^{S_1})_x|^2\,dx &\le C\delta_1^2\int_{\mathbb{R}}\phi^2 |(v^{S_1})_x|\,dx \\ &\le C\delta_1^2\int_{\mathbb{R}}|\varphi_1\phi|^2 |(v^{S_1})_x|\,dx + C\delta_1^2\int_{\mathbb{R}}|\varphi_3\phi|^2 |(v^{S_1})_x|\,dx . \end{align}\] By the definition of \(\mathcal{G}_1^S\) in ?? and Lemma 6, \[\begin{align} \int_{\mathbb{R}}\phi^2 |(v^{S_1})_x|^2\,dx \le C\delta_1^2\mathcal{G}_1^S + C\delta_1^4 e^{-C\delta_1 t}\int_{\mathbb{R}}\eta(U|\bar U)\,dx . \end{align}\]

Next, the interaction terms between the \(1\)-shock and the contact wave / \(3\)-shock are controlled by the previously established wave-interaction estimate (Lemma 7): \[\begin{align} &\int_{\mathbb{R}}|v^C-v_*|^2 |(v^{S_1})_x|^2\,dx + \int_{\mathbb{R}}|v^{S_3}-v^*|^2 |(v^{S_1})_x|^2\,dx\\ &\qquad \qquad \qquad \qquad \le C\delta_1^{3}\bigl(\delta_3^2e^{-C\delta_3 t}+(\delta_C^2+\delta_3^2)e^{-C\delta_1 t}\bigr). \end{align}\]

Combining the above bounds, we conclude that \[\begin{align} & \iint_{\mathbb{R}\times\mathbb{R}^3} \bigl|v-v^{S_1}\bigr| \left|H\frac{G}{M_\#}\right|\,d\xi\,dx \\ & \quad \le C\widetilde{\alpha}\delta_1^{2k+2}\mathcal{G}_1^S + C\widetilde{\alpha}\delta_1^{2k+4}e^{-C\delta_1 t}\int_{\mathbb{R}}\eta(U|\bar U)\,dx \\ &\qquad + C\widetilde{\alpha}\delta_1^{2k+3}\bigl(\delta_3^2e^{-C\delta_3 t}+(\delta_C^2+\delta_3^2)e^{-C\delta_1t}\bigr) + \frac{1}{\widetilde{\alpha}} \int_{\mathbb{R}} \bigl\lVert G\bigr\rVert_{M_\#}^2\,dx . \end{align}\] This proves the lemma. ◻

Proof. For convenience, we omit the shift \(X_1\). By Young’s inequality, for any \(\widetilde{\alpha}>0\), \[\begin{align} \iint_{\mathbb{R}\times\mathbb{R}^3} \bigl|\partial^\alpha_x (v-v^{S_1})\bigr| \left|H\frac{G}{M_\#}\right|\,d\xi\,dx \le\;& \widetilde{\alpha} \int_{\mathbb{R}} |\partial^\alpha_x (v-v^{S_1})|^2\bigl\lVert H\bigr\rVert_{M_\#}^2\,dx \\ &+ \frac{1}{\widetilde{\alpha}} \int_{\mathbb{R}} \bigl\lVert G\bigr\rVert_{M_\#}^2\,dx . \end{align}\] Using the assumption on \(H\), we infer \[\begin{align} \iint_{\mathbb{R}\times\mathbb{R}^3} \bigl|\partial^\alpha_x (v-v^{S_1})\bigr| \left|H\frac{G}{M_\#}\right|\,d\xi\,dx \le\;& C\widetilde{\alpha}\,\delta_1^{2k} \int_{\mathbb{R}} |\partial^\alpha_x (v-v^{S_1})|^2\,|(v^{S_1})_x|^2\,dx \\ &+ \frac{1}{\widetilde{\alpha}} \int_{\mathbb{R}} \bigl\lVert G\bigr\rVert_{M_\#}^2\,dx . \end{align}\]

We now decompose \[\partial_x^\alpha (v-v^{S_1}) = \partial_x^\alpha\phi + \partial_x^\alpha(v^C-v_*) + \partial_x^\alpha(v^{S_3}-v^*),\] and therefore \[\begin{align} \int_{\mathbb{R}} |\partial_x^\alpha (v-v^{S_1})|^2\,|(v^{S_1})_x|^2\,dx \le\;& C\int_{\mathbb{R}}|\partial_x^\alpha\phi|^2\,|(v^{S_1})_x|^2\,dx \\ &+ C\int_{\mathbb{R}}|\partial_x^\alpha(v^C-v_*)|^2\,|(v^{S_1})_x|^2\,dx \\ &+ C\int_{\mathbb{R}}|\partial_x^\alpha(v^{S_3}-v^*)|^2\,|(v^{S_1})_x|^2\,dx . \end{align}\]

For the perturbation term, since \(\|(v^{S_1})_x\|_{L^\infty_x}\le C\delta_1^2\), we obtain \[\begin{align} \int_{\mathbb{R}}|\partial_x^\alpha\phi|^2\,|(v^{S_1})_x|^2\,dx \le C\delta_1^2\|\partial_x^\alpha\phi\|_{L^2_x}^2. \end{align}\]

For the interaction terms involving the viscous contact wave and the \(3\)-shock profile, using the previously established derivative interaction bounds (Lemma 8), we derive that \[\begin{align} \int_{\mathbb{R}} |\partial_x^\alpha (v-v^{S_1})|^2\,|(v^{S_1})_x|^2\,dx \le\; C\delta_1^2\|\partial_x^\alpha\phi\|_{L^2_x}^2 + C\delta_1^{3} \bigl(\delta_3^{2(|\alpha|+1)}e^{-C\delta_3 t} +\delta_C^2e^{-C\delta_1t}\bigr). \end{align}\]

Substituting this estimate into the previous bound, we conclude that \[\begin{align} \iint_{\mathbb{R}\times\mathbb{R}^3} \bigl|\partial_x^\alpha (v-v^{S_1})\bigr| \left|H\frac{G}{M_\#}\right|\,d\xi\,dx \le\;& C\delta_1^{2k+2}\|\partial^\alpha\phi\|_{L^2_x}^2 \\ &+ C\delta_1^{2k+3} \bigl(\delta_3^{2(|\alpha|+1)}e^{-C\delta_3 t} +\delta_C^{2}e^{-C\delta_1t}\bigr) \\ &+ \frac{1}{\widetilde{\alpha}} \int_{\mathbb{R}} \bigl\lVert G\bigr\rVert_{M_\#}^2\,dx . \end{align}\] This proves the lemma. ◻

Proof. For convenience, we omit the shift \(X_1\). We divide the proof into the cases \(|\beta|=0\) and \(|\beta|=1\).

Case 1: \(|\beta|=0\). Since \[P_1(\xi_1H) = \xi_1H-\sum_{j=0}^4\langle \xi_1H,\chi_j\rangle_M\chi_j,\] we write \[\begin{align} &\iint \bigl|\partial_x^\alpha(v-v^{S_1})\bigr| \, \bigl|P_1(\xi_1H)\bigr| \, \Bigl|\frac{G}{M_\#}\Bigr|\,d\xi\,dx \\ &\le \iint \bigl|\partial_x^\alpha(v-v^{S_1})\bigr| \, \bigl|\xi_1H\bigr| \, \Bigl|\frac{G}{M_\#}\Bigr|\,d\xi\,dx \\ &\qquad + C\iint \bigl|\partial_x^\alpha(v-v^{S_1})\bigr| \,\bigl|\langle \xi_1H,\chi_j\rangle_M\bigr| \,\bigl|M\bigr| \,\Bigl|\frac{G}{M_\#}\Bigr|\,d\xi\,dx =:I_1+I_2 . \end{align}\]

For \(I_1\), Holder’s inequality in \(\xi\) gives \[\begin{align} I_1 \le& \int \bigl|\partial_x^\alpha(v-v^{S_1})\bigr| \, \bigl\lVert H\bigr\rVert_{\nu,M_\#} \, \bigl\lVert G\bigr\rVert_{\nu,M_\#} \, dx . \end{align}\] Using the assumption with \(j=0\), we obtain \[\begin{align} I_1 \le& C\delta_1^k \int \bigl|\partial_x^\alpha(v-v^{S_1})\bigr| \, \bigl|(v^{S_1})_x\bigr| \, \bigl\lVert G\bigr\rVert_{\nu,M_\#} \, dx \\ \le& C\delta_1^{k-1} \int \bigl|\partial_x^\alpha(v-v^{S_1})\bigr|^2 \, \bigl|(v^{S_1})_x\bigr|^2\,dx + C\delta_1 \int \bigl\lVert G\bigr\rVert_{\nu,M_\#}^2 \,dx . \end{align}\] If \(|\alpha|=0\), we use the zeroth-order interaction estimate (Lemma 7); if \(|\alpha|=1\), we use the derivative interaction estimate (Lemma 8). In this case, \[\begin{align} I_1 \le& C\delta_1 \int \bigl\lVert G\bigr\rVert_{\nu,M_\#}^2\,dx \\ &+ C\delta_1^{k+1} \bigl( \delta_1\delta_3^{4}e^{-C\delta_3 t} +\delta_C^2\delta_1e^{-C\delta_1t} \bigr) + C\delta_1^{k+1} \|\partial_x\phi\|_{L^2_x}^2 . \end{align}\] The term \(I_2\) is treated in exactly the same way, since \[|\langle \xi_1H,\chi_j\rangle_M| \le C\left( \int \frac{(1+|\xi|)|H|^2}{M_\#}\,d\xi \right)^{1/2}, \qquad \left|\frac{M}{M_\#}\right|\le C.\] This proves the estimate for \(|\beta|=0\). Also, we can compute this for \(|\alpha|=0\). That is,

\[\begin{align} \label{eq:kc3-1} &\iint_{\mathbb{R}\times\mathbb{R}^3} \bigl|v-(v^{S_1})^{-X_1}\bigr| \, \bigl|\partial_x^\beta P_1(\xi_1H)\bigr| \, \Bigl|\frac{G}{M_\#}\Bigr|\,d\xi\,dx \nonumber \\ &\le C\delta_1 \int_{\mathbb{R}} \bigl\lVert G\bigr\rVert_{\nu,M_\#}^2\,dx + C\delta_1^{k+1} \bigl( \delta_3^{4}\delta_1e^{-C\delta_3 t} +\delta_0^2\delta_1e^{-C\delta_1 t} \bigr) \nonumber \\ &\quad +C\delta_1^{k+1}\mathcal{G}_1^S + C\delta_1^{k+3} e^{-C\delta_1 t}\int_{\mathbb{R}}\eta(U|\bar U)\,dx . \end{align}\tag{137}\]

Case 2: \(|\beta|=1\). Differentiating the projection gives \[\partial_x P_1(\xi_1H) = \xi_1H_x - \sum_{j=0}^4 \Bigl( \partial_x\langle \xi_1H,\chi_j\rangle_M\,\chi_j + \langle \xi_1H,\chi_j\rangle_M\,\partial_x\chi_j \Bigr).\] Hence \[\begin{align} \left|\partial_x P_1(\xi_1H)\right| \le& |\xi_1H_x| + C|(v_x,u_x,\theta_x)|\,|M| \, \bigl\lVert H\bigr\rVert_{\nu,M_\#} + C|M|\,\bigl\lVert H_x\bigr\rVert_{\nu,M_\#}. \end{align}\] Accordingly, \[\begin{align} &\iint \bigl|\partial_x^\alpha(v-v^{S_1})\bigr| \, \bigl|\partial_xP_1(\xi_1H)\bigr| \, \Bigl|\frac{G}{M_\#}\Bigr|\,d\xi\,dx \\ &\le \iint \bigl|\partial_x^\alpha(v-v^{S_1})\bigr| \,\bigl|\xi_1H_x\bigr| \, \Bigl|\frac{G}{M_\#}\Bigr|\,d\xi\,dx \\ &\qquad + C\iint \bigl|(v_x,u_x,\theta_x)\bigr|\, \bigl|\partial_x^\alpha(v-v^{S_1})\bigr| \, \bigl\lVert H\bigr\rVert_{\nu,M_\#} \, \Bigl|\frac{MG}{M_\#}\Bigr|\,d\xi\,dx \\ &\qquad + C\iint \bigl|\partial_x^\alpha(v-v^{S_1})\bigr| \, \bigl\lVert H_x\bigr\rVert_{\nu,M_\#} \, \Bigl|\frac{MG}{M_\#}\Bigr|\,d\xi\,dx =:J_1+J_2+J_3 . \end{align}\]

Using the assumptions with \(j=0,1\), together with the a priori bound \[\|(v_x,u_x,\theta_x)\|_{L^\infty_x}\le C(\delta_0+\varepsilon),\] we estimate each \(J_\ell\) exactly as above and obtain \[\begin{align} J_\ell \le& C(\delta_0+\varepsilon) \int \bigl\lVert G\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &+ C(\delta_0+\varepsilon)\delta_1^{k} \bigl( \delta_1\delta_3^{|\alpha|+2}e^{-C\delta_3 t} +\delta_0^2\delta_1^{|\alpha|+1}e^{-C\delta_1 t} \bigr) \\ &\qquad + C(\delta_0+\varepsilon)\delta_1^{k+2} \|\partial^\alpha\phi\|_{L^2_x}^2 , \qquad \ell=1,2,3. \end{align}\] Summing these bounds gives the desired estimate for \(|\beta|=1\), and the proof is complete. ◻

Proof. Set \[\mathfrak R := -\frac{1}{v}P_1(\xi_1M_x) +\sum_{i=1,3}\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}P_1^{S_i}\Bigl(\xi_1\partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}} \Bigr) .\] We only prove the case \(|\alpha|=1\). We first decompose the Maxwellian part: \[\begin{align} &-\frac{1}{v}\xi_1 M_x + \sum_{i=1,3}\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}\xi_1 \partial_x \bigl(M^{S_{i}}\bigr)^{-X_{i}} \\ &\quad = -\frac{1}{v}\xi_1 \left(M-\bigl(M^{S_{1}}\bigr)^{-X_{1}}-\bigl(M^{S_{3}}\bigr)^{-X_{3}}\right)_x +\sum_{i=1,3}\left(\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}-\frac{1}{v}\right)\xi_1 \partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}}, \end{align}\] and similarly for the projected part, \[\begin{align} &-\frac{1}{v}\langle \xi_1 M_x, M\rangle_M\chi + \sum_{i=1,3} \frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}\bigl\langle \xi_1 \partial_x \bigl(M^{S_{i}}\bigr)^{-X_{i}}, \bigl(\chi^{S_{i}}\bigr)^{-X_{i}}\bigr\rangle_{S_i}\bigl(\chi^{S_{i}}\bigr)^{-X_{i}}\\ &\qquad = -\frac{1}{v}\bigl\langle \xi_1 \left(M-\bigl(M^{S_{1}}\bigr)^{-X_{1}}-\bigl(M^{S_{3}}\bigr)^{-X_{3}}\right)_x , \chi \bigr\rangle_M \chi \\ &\qquad \quad + \sum_{i=1,3} \left(\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}-\frac{1}{v}\right)\bigl\langle \xi_1 \partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}}, \chi \bigr\rangle_M \chi\\ &\qquad \quad + \sum_{i=1,3} \frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}} \Biggl\{\bigl\langle \xi_1 \partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}}, \bigl(\chi^{S_{i}}\bigr)^{-X_{i}} \bigr\rangle_{S_i}\bigl(\chi^{S_{i}}\bigr)^{-X_{i}} \Biggr. \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \Biggl.-\bigl\langle \xi_1 \partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}}, \chi \bigr\rangle_M\chi\Biggr\}. \end{align}\] Accordingly, we write \[\mathfrak R=\sum_{j=1}^5 \mathcal{K}_j,\] where \[\begin{align} \mathcal{K}_1 :=& \iint \left\{-\frac{1}{v}\xi_1\left(M-\bigl(M^{S_{1}}\bigr)^{-X_{1}}-\bigl(M^{S_{3}}\bigr)^{-X_{3}}\right)_x\right\}\frac{|G|}{M_\#} \,d\xi dx,\\ \mathcal{K}_{2i} :=& \iint \left(\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}-\frac{1}{v}\right)\xi_1 \partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}} \frac{|G|}{M_\#} \,d\xi dx,\qquad (i=1,3),\\ \mathcal{K}_{3} :=& -\iint \frac{1}{v} \Bigl\langle \xi_1\left(M-\bigl(M^{S_{1}}\bigr)^{-X_{1}}-\bigl(M^{S_{3}}\bigr)^{-X_{3}}\right)_x ,\chi \Bigr\rangle_M \chi \frac{|G|}{M_\#} \,d\xi dx,\\ \mathcal{K}_{4i} :=& \iint \left(\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}-\frac{1}{v}\right)\Bigl\langle \xi_1 \partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}}, \chi \Bigr\rangle_M \chi \frac{|G|}{M_\#} \,d\xi dx,\qquad (i=1,3),\\ \mathcal{K}_{5i} :=& \iint \frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}} \Bigl\{\Bigl\langle \xi_1\partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}},\bigl(\chi^{S_{i}}\bigr)^{-X_{i}} \Bigr\rangle_{S_i}\bigl(\chi^{S_{i}}\bigr)^{-X_{i}} \\ & \qquad \qquad \qquad \qquad -\Bigl\langle \xi_1\partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}},\chi \Bigr\rangle_M \chi\Bigr\} \frac{|G|}{M_\#} \,d\xi dx,\qquad (i=1,3), \end{align}\] and \(\mathcal{K}_j:=\mathcal{K}_{j1}+\mathcal{K}_{j3}\) for \(j=2,4,5\). Using Young’s inequality and the smooth dependence of \(M\) and \(\chi\) on the macroscopic variables, it suffices to estimate the differentiated forms of the representative terms \[\mathcal{H}_1,\;\mathcal{H}_2,\;\mathcal{H}_{3i},\;\mathcal{H}_{4i},\;\mathcal{H}_{5i},\] defined by \[\begin{align} \mathcal{H}_1 := & \iint \Bigl|P_1\xi_1\partial_{xx} \Bigl(M-\bigl(M^{S_{1}}\bigr)^{-X_{1}}-\bigl(M^{S_{3}}\bigr)^{-X_{3}}\Bigr) \Bigr|\,\Bigl|\frac{G}{M_\#} \Bigr| \,d\xi\, dx, \\ \mathcal{H}_2 := & \iint |v_x| \, \Bigl|\xi_1\Bigl(M-\bigl(M^{S_{1}}\bigr)^{-X_{1}}-\bigl(M^{S_{3}}\bigr)^{-X_{3}}\Bigr)_x\Bigr|\, \Bigl|\frac{G}{M_\#}\Bigr| \,d\xi \, dx, \\ \mathcal{H}_{3i} := & \iint \Bigl|v-\bigl(v^{S_{i}}\bigr)^{-X_{i}}\Bigr|\, \Bigl|\xi_1\partial_{xx}\bigl(M^{S_{i}}\bigr)^{-X_{i}}\Bigr| \, \Bigl|\frac{G}{M_\#}\Bigr| \,d\xi \, dx,\qquad (i=1,3), \\ \mathcal{H}_{4i} := & \iint \Bigl|\Bigl(v-\bigl(v^{S_{i}}\bigr)^{-X_{i}}\Bigr)_x\Bigr|\,\Bigl|\xi_1\partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}}\Bigr|\,\Bigl|\frac{G}{M_\#}\Bigr| \,d\xi \, dx,\qquad (i=1,3), \\ \mathcal{H}_{5i} := & \iint |v_x| \, \Bigl|v-\bigl(v^{S_{i}}\bigr)^{-X_{i}}\Bigr|\, \Bigl|\xi_1\partial_x\bigl(M^{S_{i}}\bigr)^{-X_{i}}\Bigr| \, \Bigl|\frac{G}{M_\#}\Bigr| \,d\xi \, dx,\qquad (i=1,3). \end{align}\] \(\mathcal{H}_1\) comes from \(\mathcal{K}_1+\mathcal{K}_3\). We only spell out the estimate of \(\mathcal{H}_2\), since the others are handled in the same way by Lemmas 30–32, the derivative wave-interaction estimate, and the standard decay bounds for the composite wave. By the cutoff decomposition, \[\begin{align} &\Bigl|\Bigl(M-\bigl(M^{S_{1}}\bigr)^{-X_{1}}-\bigl(M^{S_{3}}\bigr)^{-X_{3}}\Bigr)_x\Bigr| \\ &\quad \leq \varphi_1\Bigl|\Bigl(M-\bigl(M^{S_{1}}\bigr)^{-X_{1}}\Bigr)_x\Bigr| +\varphi_3\Bigl|\Bigl(M-\bigl(M^{S_{3}}\bigr)^{-X_{3}}\Bigr)_x\Bigr| +C\varphi_3 \Bigl|\bigl(M^{S_{1}}\bigr)^{-X_{1}}\,\partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}\Bigr| \\ &\qquad +C \varphi_1 \Bigl|\bigl(M^{S_{3}}\bigr)^{-X_{3}} \, \partial_x\bigl(v^{S_{3}}\bigr)^{-X_{3}}\Bigr|, \end{align}\] and therefore \[\begin{align} \mathcal{H}_2 \leq\;& \frac{1}{\widetilde{\alpha}}\iint \frac{|G|^2}{M_\#} \,d\xi dx + C\widetilde{\alpha}\sum_{i=1,3} \Biggl\{ \int \varphi_i^2|v_x|^2 \, \Bigl|\Bigl(v-\bigl(v^{S_{i}}\bigr)^{-X_{i}}\Bigr)_x\Bigr|^2 \,dx \Biggr.\\ &\qquad \qquad \qquad \qquad \qquad \qquad \quad \Biggl.+ \int (1-\varphi_i)^2 |v_x|^2\, \Bigl|\partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\Bigr|^2 \,dx \Biggr\}. \end{align}\] For \(i=1\), we decompose \[v_x-\partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}} = \phi_x+\bigl(v^C-v_*\bigr)_x+\bigl(\bigl(v^{S_{3}}\bigr)^{-X_{3}}-v^*\bigr)_x\] and obtain \[\begin{align} &\int \varphi_1^2 |v_x|^2 \, \Bigl|\Bigl(v-\bigl(v^{S_{1}}\bigr)^{-X_{1}}\Bigr)_x\Bigr|^2 dx \\ & \qquad \le C(\delta_0+\varepsilon)^2 \|\phi_x\|_{L^2}^2 + C\delta_0^3(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t})+ C\int |v_x^C|^4 \,dx \\ & \qquad \qquad + C(\varepsilon+\delta_0)^2\int \varphi_1^2 \, \Bigl|\partial_x\bigl(v^{S_{3}}\bigr)^{-X_{3}}\Bigr|^2 \,dx. \end{align}\] Since \[\int |v_x^C|^4 \,dx \le C\frac{\delta_C^4}{(1+t)^{3/2}}, \qquad \int \varphi_1^2 \Bigl|\partial_x\bigl(v^{S_{3}}\bigr)^{-X_{3}}\Bigr|^2 \,dx \le C\delta_3^2 e^{-C\delta_3t},\] it follows that \[\begin{align} \mathcal{H}_2 \leq\;& \frac{1}{\widetilde{\alpha}}\int \bigl\lVert G\bigr\rVert_{M_\#}^2 \,dx + C\widetilde{\alpha}(\delta_0+\varepsilon)^2\|\phi_x\|_{L^2}^2 + C\widetilde{\alpha}\delta_0^3(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t})\\ &+ C\widetilde{\alpha}\frac{\delta_C^4}{(1+t)^{3/2}} + C\widetilde{\alpha}(\varepsilon+\delta_0)^2\delta_3^2 e^{-C\delta_3t}. \end{align}\] The term \(\mathcal{H}_1\) can be treated in the same way as \(\mathcal{H}_2\). The key point is that the microscopic projection eliminates the leading density contribution, so no \(\phi_{xx}\)-term appears in the estimate. The remaining terms \(\mathcal{H}_{3i}\), \(\mathcal{H}_{4i}\), and \(\mathcal{H}_{5i}\), together with the differentiated projector-mismatch terms arising from \(\mathcal{K}_3\), \(\mathcal{K}_{4i}\), and \(\mathcal{K}_{5i}\), are estimated by the same combination of Young’s inequality, the smooth dependence of \(M\) and \(\chi\) on the macroscopic variables, Lemmas 30–32, and the derivative interaction bounds. Summing all these contributions gives the stated inequality. ◻

Proof. Since \(P(\xi)\) is a fixed polynomial and \(M_\#\) is a global Maxwellian, we have \[\left( \int_{\mathbb{R}^3_\xi} P(\xi)\,(L_M^{-1}g)_x\,d\xi \right)^2 \le C \int_{\mathbb{R}^3_\xi} \frac{|(L_M^{-1}g)_x|^2}{M_\#}\,d\xi .\] Next, by differentiating the identity \(L_M(L_M^{-1}g)=g\), we obtain \[(L_M^{-1}g)_x = L_M^{-1}(g_x) - L_M^{-1}\Bigl( \mathcal{N}(M_x,L_M^{-1}g)+\mathcal{N}(L_M^{-1}g,M_x) \Bigr).\] Therefore, \[\begin{align} \int_{\mathbb{R}^3_\xi} \frac{|(L_M^{-1}g)_x|^2}{M_\#}\,d\xi \le\;& C \int_{\mathbb{R}^3_\xi} \frac{|L_M^{-1}g_x|^2}{M_\#}\,d\xi \\ &+ C \int_{\mathbb{R}^3_\xi} \frac{\bigl|L_M^{-1}(\mathcal{N}(M_x,L_M^{-1}g)+\mathcal{N}(L_M^{-1}g,M_x))\bigr|^2}{M_\#}\,d\xi . \end{align}\] By the standard weighted estimate for \(L_M^{-1}\), \[\int_{\mathbb{R}^3_\xi} \frac{|L_M^{-1}h|^2}{M_\#}\,d\xi \le \frac{C}{\lambda_{\mathrm{mic}}^2} \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|h|^2}{M_\#}\,d\xi ,\] we obtain \[\begin{align} \int_{\mathbb{R}^3_\xi} \frac{|(L_M^{-1}g)_x|^2}{M_\#}\,d\xi \le\;& \frac{C}{\lambda_{\mathrm{mic}}^2} \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|g_x|^2}{M_\#}\,d\xi \\ &+ \frac{C}{\lambda_{\mathrm{mic}}^2} \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|\mathcal{N}(M_x,L_M^{-1}g)|^2}{M_\#}\,d\xi \\ &+ \frac{C}{\lambda_{\mathrm{mic}}^2} \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|\mathcal{N}(L_M^{-1}g,M_x)|^2}{M_\#}\,d\xi . \end{align}\] Using the standard bilinear estimate for the collision operator, \[\int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|\mathcal{N}(f,h)|^2}{M_\#}\,d\xi \le C \left( \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)|f|^2}{M_\#}\,d\xi \right) \left( \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|h|^2}{M_\#}\,d\xi \right),\] with \((f,h)=(M_x,L_M^{-1}g)\) and \((L_M^{-1}g,M_x)\), we deduce \[\begin{align} \int_{\mathbb{R}^3_\xi} \frac{|(L_M^{-1}g)_x|^2}{M_\#}\,d\xi \le\;& \frac{C}{\lambda_{\mathrm{mic}}^2} \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|g_x|^2}{M_\#}\,d\xi \\ &+ \frac{C}{\lambda_{\mathrm{mic}}^2} \left( \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)|M_x|^2}{M_\#}\,d\xi \right) \left( \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|L_M^{-1}g|^2}{M_\#}\,d\xi \right). \end{align}\] Applying once again the weighted estimate for \(L_M^{-1}\), \[\int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|L_M^{-1}g|^2}{M_\#}\,d\xi \le \frac{C}{\lambda_{\mathrm{mic}}^2} \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|g|^2}{M_\#}\,d\xi ,\] we conclude that \[\begin{align} \int_{\mathbb{R}^3_\xi} \frac{|(L_M^{-1}g)_x|^2}{M_\#}\,d\xi \le\;& \frac{C}{\lambda_{\mathrm{mic}}^2} \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|g_x|^2}{M_\#}\,d\xi \\ &+ \frac{C}{\lambda_{\mathrm{mic}}^4} \left( \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)|M_x|^2}{M_\#}\,d\xi \right) \left( \int_{\mathbb{R}^3_\xi} \frac{(1+|\xi|)^{-1}|g|^2}{M_\#}\,d\xi \right). \end{align}\] Combining this with the first inequality proves ?? . ◻

Proof. Recall that \[\widetilde{\Pi}_1 := \Pi_1-\bigl((\Pi_1^{S_{1}})^{-X_{1}}\bigr)-\bigl((\Pi_1^{S_{3}})^{-X_{3}}\bigr).\] By Young’s inequality in \(\xi\), \[\begin{align} \label{eq:kc6-reduce} \int_{\mathbb{R}} \left( \int_{\mathbb{R}^3_\xi} P(\xi)\,\widetilde{\Pi}_{1x}\,d\xi \right)^2 dx \le C \iint_{\mathbb{R}\times\mathbb{R}^3} \frac{|\widetilde{\Pi}_{1x}|^2}{M_\#}\,d\xi\,dx . \end{align}\tag{138}\] Recall the decomposition \[\begin{align} \widetilde{\Pi}_1 =&\; L_M^{-1}P_1\Biggl[\widetilde{G}_t-\frac{u_1}{v}\widetilde{G}_x+\frac{1}{v}P_1\Bigl(\xi_1\widetilde{G}_x\Bigr)-\mathcal{N}\Bigl(\widetilde{G},\widetilde{G}\Bigr)\Biggr] \\ &\; -L_M^{-1}\Biggl[\mathcal{N}\Bigl(\widetilde{G},\bigl(G^{S_{1}}\bigr)^{-X_{1}}+\bigl(G^{S_{3}}\bigr)^{-X_{3}}\Bigr)+\mathcal{N}\Bigl(\bigl(G^{S_{1}}\bigr)^{-X_{1}}+\bigl(G^{S_{3}}\bigr)^{-X_{3}},\widetilde{G}\Bigr)\Biggr] \\ &\; +L_M^{-1}\Biggl[\mathcal{N}\Bigl(\bigl(G^{S_{1}}\bigr)^{-X_{1}},\bigl(G^{S_{3}}\bigr)^{-X_{3}}\Bigr)+\mathcal{N}\Bigl(\bigl(G^{S_{3}}\bigr)^{-X_{3}},\bigl(G^{S_{1}}\bigr)^{-X_{1}}\Bigr)\Biggr] \\ &\; -\dot{X}_1\bigl(L_{1}^S\bigr)^{-1}\Bigl(\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}}\Bigr) -\dot{X}_3\bigl(L_{3}^S\bigr)^{-1}\Bigl(\partial_x\bigl((G^{S_{3}})^{-X_{3}}\bigr)\Bigr) +J, \end{align}\] where \[\begin{align} J :=&\; \sum_{i\in\{1,3\}} \Bigl(L_M^{-1}P_1-\bigl(L_{i}^S\bigr)^{-1}P_1^{S_i}\Bigr) \Bigl[\partial_t\bigl(G^{S_{i}}\bigr)^{-X_{i}}-\mathcal{N}\Bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}},\bigl(G^{S_{i}}\bigr)^{-X_{i}}\Bigr)\Bigr] \\ &+ \sum_{i\in\{1,3\}} \Biggl[\frac{\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}\bigl(L_{i}^S\bigr)^{-1}P_1^{S_i}-\frac{u_1}{v}L_M^{-1}P_1\Biggr] \Bigl(\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}\Bigr) \\ &+ \sum_{i\in\{1,3\}} \Biggl[\frac{1}{v}L_M^{-1}P_1\xi_1-\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}\bigl(L_{i}^S\bigr)^{-1}P_1^{S_i}\xi_1\Biggr] \Bigl(\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}\Bigr), \end{align}\] and \(L_{i}^{S}:=L_{\bigl(M^{S_{i}}\bigr)^{-X_{i}}}\). Accordingly, write \[\widetilde{\Pi}_1=\mathcal{P}^{(1)}+\mathcal{P}^{(2)}+\mathcal{P}^{(3)}+\mathcal{P}^{(4)}+\mathcal{P}^{(5)},\] where the five pieces correspond to the five displayed lines above. We estimate each contribution to \(\iint |\mathcal{P}_x|^2/M_\#\). Step 1: the main microscopic terms \(\mathcal{P}^{(1)}\). Applying Lemma 34 to \[g=\widetilde{G}_t-\frac{u_1}{v}\widetilde{G}_x+\frac{1}{v}P_1(\xi_1\widetilde{G}_x)-\mathcal{N}(\widetilde{G},\widetilde{G}),\] we obtain \[\begin{align} \label{eq:kc6-P1} \iint \frac{|(\mathcal{P}^{(1)})_x|^2}{M_\#}\,d\xi\,dx \le\;& C \iint \frac{(1+|\xi|)^{-1}|g_x|^2}{M_\#}\,d\xi\,dx \\ &+ C(\delta_0+\varepsilon) \iint \frac{(1+|\xi|)^{-1}|g|^2}{M_\#}\,d\xi\,dx . \nonumber \end{align}\tag{139}\] Using the product rule and the a priori smallness of \((v_x,u_x,\theta_x)\), we have \[\begin{align} |g_x| \le\;& |\widetilde{G}_{tx}| + C(\delta_0+\varepsilon)|\widetilde{G}_x| + C|\widetilde{G}_{xx}| + C(\delta_0+\varepsilon)|\widetilde{G}_t| \\ &+ C\,|\mathcal{N}(\widetilde{G},\widetilde{G})_x|, \end{align}\] and similarly for \(g\). By the standard bilinear collision estimate and the bootstrap smallness, \[\iint \frac{(1+|\xi|)^{-1}|\mathcal{N}(\widetilde{G},\widetilde{G})_x|^2}{M_\#}\,d\xi\,dx \le C(\delta_0+\varepsilon) \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2+\bigl\lVert\widetilde{G}_{\text{x}}\bigr\rVert_{\nu,M_\#}^2\,dx .\] Therefore, \[\begin{align} \label{eq:kc6-P1-final} \iint \frac{|(\mathcal{P}^{(1)})_x|^2}{M_\#}\,d\xi\,dx \le\;& C \int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2\,dx \\ &+ C(\delta_0+\varepsilon) \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_{t}\bigr\rVert_{\nu,M_\#}^2\,dx . \nonumber \end{align}\tag{140}\] Step 2: the mixed collision terms \(\mathcal{P}^{(2)}\). Using again Lemma 34, together with the standard bilinear estimate for \(\mathcal{N}\bigl(\widetilde{G},\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)\) and \(\mathcal{N}\bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}},\widetilde{G}\bigr)\), we obtain \[\begin{align} \label{eq:kc6-P2-final} \iint \frac{|(\mathcal{P}^{(2)})_x|^2}{M_\#}\,d\xi\,dx \le\;& C(\delta_0+\varepsilon) \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{\nu,M_\#}^2\,dx \\ &+ C(\delta_0+\varepsilon)^2\delta_0^3\left(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\right) . \nonumber \end{align}\tag{141}\] Step 3: the shock–shock interaction terms \(\mathcal{P}^{(3)}\). Since \(\bigl(G^{S_{1}}\bigr)^{-X_{1}}\) and \(\bigl(G^{S_{3}}\bigr)^{-X_{3}}\) are exponentially localized and spatially separated, \[\begin{align} \label{eq:kc6-P3-final} \iint \frac{|(\mathcal{P}^{(3)})_x|^2}{M_\#}\,d\xi\,dx \le C(\delta_0+\varepsilon)^2\delta_0^3 \left(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\right) . \end{align}\tag{142}\] Step 4: the shift terms \(\mathcal{P}^{(4)}\). By the explicit profile bounds, \[\iint \frac{\bigl|\partial_x\bigl(\bigl(L_{i}^S\bigr)^{-1}\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)\bigr|^2}{M_\#}\,d\xi\,dx \le C\delta_i^2, \qquad i=1,3.\] Hence \[\begin{align} \label{eq:kc6-P4-final} \iint \frac{|(\mathcal{P}^{(4)})_x|^2}{M_\#}\,d\xi\,dx \le C\sum_{i=1,3}\delta_i^2|\dot{X}_i|^2 \le C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i|\dot{X}_i|^2 . \end{align}\tag{143}\] Step 5: the coefficient-mismatch terms \(J=\mathcal{P}^{(5)}\). These are the most delicate terms. We only record representative estimates; the remaining variants are treated in the same way. First, for the difference of inverse linearized operators, \[\begin{align} \Bigl(L_M^{-1}-\bigl(L_{1}^S\bigr)^{-1} \Bigr) \bigl(\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}}\bigr) =\; & L_M^{-1}\Bigl( \mathcal{N}\bigl(\bigl(M^{S_{1}}\bigr)^{-X_{1}}-M,\bigl(L_{1}^S\bigr)^{-1}\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}}\bigr) \\ & + \mathcal{N}\bigl(\bigl(L_{1}^S\bigr)^{-1}\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}},\bigl(M^{S_{1}}\bigr)^{-X_{1}}-M\bigr) \Bigr), \end{align}\] and Lemma 34 gives \[\begin{align} & \iint \frac{\Bigl|\partial_x\Bigl\{\bigl(L^{-1}_M-\bigl(L_{1}^S\bigr)^{-1}\bigr)\bigl(\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}}\bigr)\Bigr\}\Bigr|^2}{M_\#}\,d\xi\,dx \\ & \quad \le C(\delta_0+\varepsilon)^2\int \bigl|\partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}\bigr|^2 \, \bigl|\bigl(v-\bigl(v^{S_{1}}\bigr)^{-X_{1}} \bigr)_x\bigr|^2\,dx \\ & \quad \qquad + C(\delta_0+\varepsilon)^2\int \bigl|\partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}\bigr|^2 \, \bigl|v-\bigl(v^{S_{1}}\bigr)^{-X_{1}}\bigr|^2\,dx . \end{align}\] By Lemmas 30–33, this is bounded by \[\begin{align} \label{eq:kc6-J1} C(\delta_0+\varepsilon)^2\delta_1\mathcal{G}_1^S + C(\delta_0+\varepsilon)^2\delta_0^3\left(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\right) + C(\delta_0+\varepsilon)^2\bigl\lVert(\phi_x,\psi_x,\zeta_x)\bigr\rVert_{L^2_x}^2 . \end{align}\tag{144}\] Next, for the coefficient mismatch \[\begin{align} & \frac{\bigl(u_1^{S_{1}}\bigr)^{-X_{1}}}{\bigl(v^{S_{1}}\bigr)^{-X_{1}}}\bigl(L_{1}^S\bigr)^{-1}-\frac{u_1}{v}L^{-1}_M \\ &\quad = \frac{\bigl(u_1^{S_{1}}\bigr)^{-X_{1}}-u_1}{\bigl(v^{S_{1}}\bigr)^{-X_{1}}}\bigl(L_{1}^{S}\bigr)^{-1} -\frac{u_1}{v\bigl(v^{S_{1}}\bigr)^{-X_{1}}}\bigl(\bigl(v^{S_{1}}\bigr)^{-X_{1}}-v\bigr)\bigl(L_{1}^S\bigr)^{-1} +\frac{u_1}{v}\bigl(\bigl(L^{S}_{1}\bigr)^{-1}-L^{-1}_M\bigr), \end{align}\] each term is treated exactly as above, giving \[\begin{align} \label{eq:kc6-J2} C(\delta_0+\varepsilon)^2\delta_1\mathcal{G}_1^S + C(\delta_0+\varepsilon)^2\delta_0^3\left(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\right) + C(\delta_0+\varepsilon)^2\bigl\lVert(\phi_x,\psi_x,\zeta_x)\bigr\rVert_{L^2_x}^2 . \end{align}\tag{145}\] Finally, for the projector mismatch terms, for example \[\begin{align} & \frac{1}{\bigl(v^{S_{1}}\bigr)^{-X_{1}}} L_M^{-1} \Biggl[\Bigl\langle \xi_1\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}},\chi_j-\bigl(\chi_j^{S_{1}}\bigr)^{-X_{1}} \Bigr\rangle_M \chi_j\Biggr], \\ & \frac{1}{\bigl(v^{S_{1}}\bigr)^{-X_{1}}} L_M^{-1} \Biggl[\Bigl\langle \xi_1\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}},\bigl(\chi_j^{S_{1}}\bigr)^{-X_{1}} \Bigr\rangle_{S_1}\Bigl(\chi_j-\bigl(\chi_j^{S_{1}}\bigr)^{-X_{1}}\Bigr)\Biggr], \end{align}\] we use the smooth dependence of \(\chi_j\) on the macroscopic variables and Lemma 34 to obtain \[\begin{align} \label{eq:kc6-J3} C(\delta_0+\varepsilon)^2\delta_1\mathcal{G}_1^S + C(\delta_0+\varepsilon)^2\delta_0^3\left(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\right) + C(\delta_0+\varepsilon)^2\bigl\lVert(\phi_x,\psi_x,\zeta_x)\bigr\rVert_{L^2_x}^2 . \end{align}\tag{146}\] The analogous \(i=3\) terms satisfy the same bounds. Therefore, \[\begin{align} \label{eq:kc6-P5-final} \iint \frac{|(\mathcal{P}^{(5)})_x|^2}{M_\#}\,d\xi\,dx \le\;& C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i \mathcal{G}_i^S + C(\delta_0+\varepsilon)\bigl\lVert(\phi_x,\psi_x,\zeta_x)\bigr\rVert_{L^2_x}^2 \\ &+ C(\delta_0+\varepsilon)^2\delta_0^3\left(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\right) + C \frac{\delta_C}{(1+t)^\frac{5}{4}}. \nonumber \end{align}\tag{147}\] Combining 138 , 140 , 141 , 142 , 143 , and 147 , we obtain ?? . ◻

Proof. Differentiate 37 with respect to \(x\), multiply the resulting equation by \(\widetilde{G}_x/M_\#\), and integrate over \((x,\xi)\in\mathbb{R}\times\mathbb{R}^3\). Then \[\begin{align} \frac{1}{2}\frac{d}{dt}\iint \frac{|\widetilde{G}_x|^2}{M_\#}\,d\xi\,dx = \mathcal{T}_1+\sum_{i=1,3}\mathcal{T}_{2i} +\mathcal{T}_3+\sum_{i=1,3}\mathcal{T}_{4i} +\sum_{i=1,3}\mathcal{T}_{5i} +\mathcal{T}_6+\mathcal{T}_7+\mathcal{T}_8+\mathcal{T}_9+\mathcal{T}_{10}, \end{align}\] where \[\begin{align} \mathcal{T}_1 &:= \iint (L_M\widetilde{G})_x\,\frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_{2i} &:= \iint \dot{X}_i\,\bigl(\partial_{xx}\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)\,\frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_3 &:= \iint \left(\frac{u_1}{v}\widetilde{G}_x-\frac{1}{v}P_1(\xi_1\widetilde{G}_x)\right)_x \frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_{4i} &:= \iint \Biggl[ \left(\frac{u_1}{v}-\frac{\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}\right)\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}} \Biggr]_x \frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_{5i} &:= -\iint \Biggl[ \frac{1}{v}P_1(\xi_1\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}) -\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}\bigl(P_1^{S_i}\bigr)(\xi_1\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}) \Biggr]_x \frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_6 &:= \iint \Biggl[ -\frac{1}{v}P_1(\xi_1M_x) +\frac{1}{\bigl(v^{S_{1}}\bigr)^{-X_{1}}}\bigl(P_1^{S_1}\bigr)(\xi_1\partial_x\bigl(M^{S_{1}}\bigr)^{-X_{1}})\Biggr.\\ &\qquad \qquad \Biggl. +\frac{1}{\bigl(v^{S_{3}}\bigr)^{-X_{3}}}\bigl(P_1^{S_3}\bigr)(\xi_1\partial_x\bigl(M^{S_{3}}\bigr)^{-X_{3}}) \Biggr]_x \frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_7 &:= \iint \bigl(\mathcal{N}(\widetilde{G},\widetilde{G})\bigr)_x \frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_8 &:= \sum_{i=1,3}\iint \Bigl[\mathcal{N}\bigl(\widetilde{G},\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)+\mathcal{N}\bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}},\widetilde{G}\bigr)\Bigr]_x \frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_9 &:= \iint \Bigl[\mathcal{N}\bigl(\bigl(G^{S_{1}}\bigr)^{-X_{1}},\bigl(G^{S_{3}}\bigr)^{-X_{3}}\bigr)+\mathcal{N}\bigl(\bigl(G^{S_{3}}\bigr)^{-X_{3}},\bigl(G^{S_{1}}\bigr)^{-X_{1}}\bigr)\Bigr]_x \frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_{10} &:= \sum_{i=1,3}\iint \Bigl[(L_M-L_{i}^S)\bigl(G^{S_{i}}\bigr)^{-X_{i}}\Bigr]_x \frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx. \end{align}\]

We estimate these terms one by one.

1. Estimate of \(\mathcal{T}_1\). Since \[L_M h=\mathcal{N}(M,h)+\mathcal{N}(h,M),\] we have \[\label{eq:GxE-LM-diff} (L_Mh)_x=L_Mh_x+\mathcal{N}(M_x,h)+\mathcal{N}(h,M_x).\tag{148}\] Hence \[\begin{align} \mathcal{T}_1 &= \iint L_M\widetilde{G}_x\,\frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx +\iint \Bigl(\mathcal{N}(M_x,\widetilde{G})+\mathcal{N}(\widetilde{G},M_x)\Bigr)\frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx . \end{align}\] By Lemma 45, more precisely ?? , \[\begin{align} \iint L_M\widetilde{G}_x\,\frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx \le -\lambda_{\mathrm{mic}} \iint \frac{(1+|\xi|)|\widetilde{G}_x|^2}{M_\#}\,d\xi\,dx . \end{align}\] For the remaining terms, Young’s inequality and the bilinear collision estimate ?? imply \[\begin{align} &\left| \iint \Bigl(\mathcal{N}(M_x,\widetilde{G})+\mathcal{N}(\widetilde{G},M_x)\Bigr)\frac{\widetilde{G}_x}{M_\#}\,d\xi\,dx \right| \\ &\le \frac{\lambda_{\mathrm{mic}}}{128}\iint \frac{(1+|\xi|)|\widetilde{G}_x|^2}{M_\#}\,d\xi\,dx +C\iint \frac{(1+|\xi|)^{-1}}{M_\#} \Bigl( |\mathcal{N}(M_x,\widetilde{G})|^2+|\mathcal{N}(\widetilde{G},M_x)|^2 \Bigr)\,d\xi\,dx . \end{align}\] Now we use the decomposition 43 , namely \[\widetilde{G}=\widetilde{G}_{\text{rem}}+\widetilde{G}_C,\] together with the explicit form of \(\widetilde{G}_C\), the smallness of \((v_x,u_x,\theta_x)\), and the standard bounds for the composite wave. Applying ?? once more, we obtain \[\begin{align} &\iint \frac{(1+|\xi|)^{-1}}{M_\#} \Bigl( |\mathcal{N}(M_x,\widetilde{G})|^2+|\mathcal{N}(\widetilde{G},M_x)|^2 \Bigr)\,d\xi\,dx \\ &\le C(\delta_0+\varepsilon)^2\|(\phi_x,\psi_x,\zeta_x)\|_{L^2_x}^2 +C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx \\ &\qquad +C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) +C\frac{\delta_C}{(1+t)^{3/2}} . \end{align}\] Therefore \[\begin{align} \label{eq:GxE-T1} \mathcal{T}_1 &\le -\frac{127}{128}\lambda_{\mathrm{mic}} \int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx \notag\\ &\quad +C(\delta_0+\varepsilon)^2\|(\phi_x,\psi_x,\zeta_x)\|_{L^2_x}^2 +C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx \notag\\ &\quad +C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) +C\frac{\delta_C}{(1+t)^{3/2}} . \end{align}\tag{149}\]

2. Estimate of \(\mathcal{T}_{2i}\). By Young’s inequality, \[\begin{align} |\mathcal{T}_{2i}| \le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx + C|\dot{X}_i|^2 \int \bigl\lVert\partial_{xx}\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr\rVert_{M_\#}^2\,dx . \end{align}\] By Lemma 4 with \(k=2\), together with translation invariance of the shifted profile, \[\int \bigl\lVert\partial_{xx}\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr\rVert_{M_\#}^2 \,dx \le C\delta_i^7 \le C(\delta_0+\varepsilon)\delta_i .\] Hence \[\label{eq:GxE-T2} |\mathcal{T}_{2i}| \le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx + C(\delta_0+\varepsilon)\delta_i|\dot{X}_i|^2 .\tag{150}\]

3. Estimate of \(\mathcal{T}_3\). Expanding the derivative gives \[\begin{align} |\mathcal{T}_3| &\le C\iint \frac{|\widetilde{G}_x||\widetilde{G}_{xx}|}{M_\#}\,d\xi\,dx + C\iint |(u_{1x},v_x)|\frac{|\widetilde{G}_x|^2}{M_\#}\,d\xi\,dx . \end{align}\] Using the a priori smallness of \((u_{1x},v_x)\) and Young’s inequality, we obtain \[\label{eq:GxE-T3} |\mathcal{T}_3| \le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx + C\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2\,dx .\tag{151}\]

4. Estimate of \(\mathcal{T}_{4i}\). After expanding the \(x\)-derivative and using Young’s inequality, we have \[\begin{align} |\mathcal{T}_{4i}| &\le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &\quad + C\iint \Bigl| \bigl(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}},\,v-\bigl(v^{S_{i}}\bigr)^{-X_{i}}\bigr) \Bigr|^2 \frac{|\partial_{xx}\bigl(G^{S_{i}}\bigr)^{-X_{i}}|^2}{M_\#}\,d\xi\,dx \\ &\quad + C\iint \Bigl| \bigl(u_{1x}-\partial_x\bigl(u_1^{S_{i}}\bigr)^{-X_{i}},\,v_x-\partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\bigr) \Bigr|^2 \frac{|\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}|^2}{M_\#}\,d\xi\,dx . \end{align}\] We now use Lemma 4, the shock localization estimate 237 , the interaction lemmas Lemma 7 and Lemma 8, the cutoff estimate Lemma 6, and the previously established macroscopic comparison lemmas Lemma 30–32. This yields \[\begin{align} \label{eq:GxE-T4} |\mathcal{T}_{4i}| &\le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx + C(\delta_0+\varepsilon)^2\mathcal{G}_i^S + C\delta_i^3e^{-C\delta_it}\int \eta(U|\overline{U})\,dx \notag\\ &\quad + C(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it} + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) + C\delta_i^2\|\phi_x\|_{L^2_x}^2 . \end{align}\tag{152}\]

5. Estimate of \(\mathcal{T}_{5i}\). Using the finite-dimensionality of \(P_1\) and the smooth dependence of the basis \(\{\chi_j\}_{j=0}^4\) on \((v,u,\theta)\), we write \[\bigl(P_1-\bigl(P_1^{S_i}\bigr)\bigr)f = \sum_{j=0}^4 \Bigl[ \langle f,\chi_j\rangle_M\chi_j - \bigl\langle f,\bigl(\chi_j^{S_{i}}\bigr)^{-X_{i}}\bigr\rangle_{S_i}\bigl(\chi_j^{S_{i}}\bigr)^{-X_{i}} \Bigr].\] Hence, after one application of Young’s inequality, \[\begin{align} |\mathcal{T}_{5i}| &\le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx \\ &\quad + C\int \Biggl[ \int \Bigl\{ \frac{1}{v}P_1(\xi_1\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}) -\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}P_1^{S_i}(\xi_1\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}) \Bigr\}_x \,d\xi \Biggr]^2 dx . \end{align}\] The differentiated coefficient/projector mismatch in the last line is exactly of the same type as the terms treated in Step 5 of the proof of Lemma 35; cf.@eq:eq:kc6-J1 , 145 , and 146 . Using those bounds, together with Lemma 4 for the shock profile moments, we obtain \[\begin{align} \label{eq:GxE-T5} |\mathcal{T}_{5i}| &\le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx + C(\delta_0+\varepsilon)^2\mathcal{G}_i^S + C\delta_i^3e^{-C\delta_it}\int \eta(U|\overline{U})\,dx \notag\\ &\quad + C(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it} + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) + C\delta_i^2\|\phi_x\|_{L^2_x}^2 . \end{align}\tag{153}\]

6. Estimate of \(\mathcal{T}_6\). This is exactly the differentiated source term treated in Lemma 33. Applying Lemma 33 with \(G=\widetilde{G}_x\) and choosing \(\widetilde{\alpha}>0\) fixed, we get \[\begin{align} \label{eq:GxE-T6} |\mathcal{T}_6| &\le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx + C(\delta_0+\varepsilon)^2\|(\phi_x,\psi_x,\zeta_x)\|_{L^2_x}^2 \notag\\ &\quad + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) + C\frac{\delta_C}{(1+t)^{3/2}} \notag\\ &\quad + C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t} + C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t} + C\|(\psi_{xx},\zeta_{xx})\|_{L^2_x}^2 \notag\\ &\quad + C(\delta_0+\varepsilon)^2(\mathcal{G}_1^S+\mathcal{G}_3^S). \end{align}\tag{154}\]

7. Estimate of \(\mathcal{T}_7\). Using \[(\mathcal{N}(f,g))_x=\mathcal{N}(f_x,g)+\mathcal{N}(f,g_x),\] we have \[(\mathcal{N}(\widetilde{G},\widetilde{G}))_x = \mathcal{N}(\widetilde{G}_x,\widetilde{G})+\mathcal{N}(\widetilde{G},\widetilde{G}_x).\] Therefore, by Young’s inequality and ?? , \[\begin{align} |\mathcal{T}_7| &\le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{\nu,M_\#}^2\,dx \\ &\quad + C\iint \frac{(1+|\xi|)^{-1}}{M_\#} \Bigl( |\mathcal{N}(\widetilde{G}_x,\widetilde{G})|^2 + |\mathcal{N}(\widetilde{G},\widetilde{G}_x)|^2 \Bigr)\,d\xi\,dx . \end{align}\] Using again 43 , ?? , and the a priori smallness, \[\begin{align} \label{eq:GxE-T7} &|\mathcal{T}_7| \le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)\int \bigl\lVert\tilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx\nonumber \\ &\qquad \qquad + C(\delta_0+\varepsilon)^2\|(\phi_x,\psi_x,\zeta_x)\|_{L^2_x}^2 . \end{align}\tag{155}\]

8. Estimate of \(\mathcal{T}_8\). Expanding the derivative and using ?? , \[\begin{align} |\mathcal{T}_8| &\le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &\quad + C\sum_{i=1,3}\iint \frac{(1+|\xi|)^{-1}}{M_\#} \Bigl( \bigl|\mathcal{N}\bigl(\widetilde{G}_x,\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)\bigr|^2 + \bigl|\mathcal{N}\bigl(\widetilde{G},\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)\bigr|^2 \\ & + \bigl|\mathcal{N}\bigl(\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}},\widetilde{G}\bigr)\bigr|^2 + \bigl|\mathcal{N}\bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}},\widetilde{G}_x\bigr)\bigr|^2 \Bigr)\,d\xi\,dx . \end{align}\] By Lemma 4, 43 , the same mixed-collision bookkeeping used in Step 2 of Lemma 35, and in particular 141 , we obtain \[\begin{align} \label{eq:GxE-T8} |\mathcal{T}_8| &\le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx + C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx \notag\\ &\quad + C(\delta_0+\varepsilon)^2(\mathcal{G}_1^S+\mathcal{G}_3^S) + C(\varepsilon+\delta_0)^2 \bigl(\delta_1^2e^{-C\delta_1t}+\delta_3^2e^{-C\delta_3t}\bigr) \notag\\ &\quad + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) + C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int \eta(U|\overline{U})\,dx . \end{align}\tag{156}\]

9. Estimate of \(\mathcal{T}_9\). Similarly, \[\begin{align} |\mathcal{T}_9| & \le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx \\ &\quad + C\iint \frac{(1+|\xi|)^{-1}}{M_\#} \Bigl( \bigl|\mathcal{N}\bigl(\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}},\bigl(G^{S_{3}}\bigr)^{-X_{3}}\bigr)\bigr|^2 + \bigl|\mathcal{N}\bigl(\bigl(G^{S_{1}}\bigr)^{-X_{1}},\partial_x\bigl(G^{S_{3}}\bigr)^{-X_{3}}\bigr)\bigr|^2 \\ & + \bigl|\mathcal{N}\bigl(\partial_x\bigl(G^{S_{3}}\bigr)^{-X_{3}},\bigl(G^{S_{1}}\bigr)^{-X_{1}}\bigr)\bigr|^2 + \bigl|\mathcal{N}\bigl(\bigl(G^{S_{3}}\bigr)^{-X_{3}},\partial_x\bigl(G^{S_{1}}\bigr)^{-X_{1}}\bigr)\bigr|^2 \Bigr)\,d\xi\,dx . \end{align}\] By Lemma 4, the exponential localization and separation of the two shifted shock layers, and exactly the same shock–shock interaction estimate as in Step 3 of Lemma 35, cf.@eq:eq:kc6-P3-final , we infer \[\label{eq:GxE-T9} |\mathcal{T}_9| \le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t} + C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t}.\tag{157}\]

10. Estimate of \(\mathcal{T}_{10}\). Since \[L_M h=\mathcal{N}(M,h)+\mathcal{N}(h,M),\] we have \[\begin{align} &(L_M-L_{i}^S)\bigl(G^{S_{i}}\bigr)^{-X_{i}} \\ &\qquad = \mathcal{N}\bigl(M-\bigl(M^{S_{i}}\bigr)^{-X_{i}},\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr) +\mathcal{N}\bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}},M-\bigl(M^{S_{i}}\bigr)^{-X_{i}}\bigr), \end{align}\] and hence \[\begin{align} & \Bigl((L_M-L_{i}^S)\bigl(G^{S_{i}}\bigr)^{-X_{i}}\Bigr)_x\\ &\qquad = \mathcal{N}\bigl(\bigl(M-\bigl(M^{S_{i}}\bigr)^{-X_{i}}\bigr)_x,\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr) +\mathcal{N}\bigl(M-\bigl(M^{S_{i}}\bigr)^{-X_{i}},\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr) \\ &\qquad \quad +\mathcal{N}\bigl(\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}},M-\bigl(M^{S_{i}}\bigr)^{-X_{i}}\bigr) +\mathcal{N}\bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}},(M-\bigl(M^{S_{i}}\bigr)^{-X_{i}})_x\bigr). \end{align}\] Therefore, by Young’s inequality and ?? , \[\begin{align} |\mathcal{T}_{10}| &\le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &\quad + C\sum_{i=1,3}\iint \frac{(1+|\xi|)^{-1}}{M_\#} \Bigl( \bigl|\mathcal{N}\bigl((M-\bigl(M^{S_{i}}\bigr)^{-X_{i}})_x,\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)\bigr|^2 \\ &\qquad + \bigl|\mathcal{N}\bigl(M-\bigl(M^{S_{i}}\bigr)^{-X_{i}},\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)\bigr|^2 \\ &\qquad + \bigl|\mathcal{N}\bigl(\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}},M-\bigl(M^{S_{i}}\bigr)^{-X_{i}}\bigr)\bigr|^2 \\ &\qquad + \bigl|\mathcal{N}\bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}},(M-\bigl(M^{S_{i}}\bigr)^{-X_{i}})_x\bigr)\bigr|^2 \Bigr)\,d\xi\,dx . \end{align}\] Now \((M-\bigl(M^{S_{i}}\bigr)^{-X_{i}})\) and \((M-\bigl(M^{S_{i}}\bigr)^{-X_{i}})_x\) are controlled by \[\begin{align} &\bigl(v-\bigl(v^{S_{i}}\bigr)^{-X_{i}},u-\bigl(u^{S_{i}}\bigr)^{-X_{i}},\theta-\bigl(\theta ^{S_{i}}\bigr)^{-X_{i}}\bigr),\\ &\qquad \bigl(v_x-\partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}},u_x-\partial_x\bigl(u^{S_{i}}\bigr)^{-X_{i}},\theta_x-\partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}\bigr), \end{align}\] respectively. Using Lemma 4, Lemma 6, Lemma 7, Lemma 8, and the macroscopic comparison lemmas Lemma 30–33, we conclude that \[\begin{align} \label{eq:GxE-T10} |\mathcal{T}_{10}| &\le \frac{\lambda_{\mathrm{mic}}}{128}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)^2\|(\phi_x,\psi_x,\zeta_x)\|_{L^2_x}^2 \notag\\ &\quad + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) + C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t} + C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t} \notag\\ &\quad + C(\delta_0+\varepsilon)^2(\mathcal{G}_1^S+\mathcal{G}_3^S) + C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int \eta(U|\overline{U})\,dx . \end{align}\tag{158}\]

Finally, summing 149 , 150 , 151 , 152 , 153 , 154 , 155 , 156 , 157 , and 158 , and absorbing all small fractions of \[\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx\] into the left-hand side, we obtain \[\begin{align} &\frac{d}{dt}\iint \frac{|\widetilde{G}_x|^2}{M_\#}\,d\xi\,dx +\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{M_\#}^2 \,dx \\ &\le C(\delta_0+\varepsilon)^2\|(\phi_x,\psi_x,\zeta_x)\|_{L^2_x}^2 +C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_Ce^{-Ct}+\delta_3e^{-C\delta_3t}\bigr) +C\frac{\delta_C}{(1+t)^{3/2}} \\ &\quad +C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t} +C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t} +C\|(\psi_{xx},\zeta_{xx})\|_{L^2_x}^2 \\ &\quad +C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i|\dot{X}_i|^2 +C(\delta_0+\varepsilon)^2(\mathcal{G}_1^S+\mathcal{G}_3^S) +C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &\quad +C\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2 \,dx +C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int \eta(U\mid\bar U)\,dx . \end{align}\] This proves the lemma. ◻

Proof. By Lemma 4, together with the equivalence between \(M_\#\) and the fixed global Maxwellian near the shock profile, we have \[\label{eq:ddotX-profile} \int_{\mathbb{R}}\bigl\lVert\partial_x \bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr\rVert_{M_\#}^2 \,dx \le C\delta_i^5.\tag{159}\] Hence it suffices to estimate \(\delta_i^3|\ddot X_i|^2\).

Set \[\mathfrak Y_i(t,x) := a(t,x) \Bigl( \partial_x\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\,\psi_1 + \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\,\frac{\overline{p}}{\overline{v}}\phi + \frac{\partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}}{\overline{\theta}}\zeta \Bigr)(t,x).\] By the definition of the dynamical shifts 40 , \[\dot{X}_i(t) = -\frac{M_i}{\delta_i}\int_{\mathbb{R}}\mathfrak Y_i(t,x)\,dx,\] and therefore \[\ddot X_i(t) = -\frac{M_i}{\delta_i}\int_{\mathbb{R}}\partial_t\mathfrak Y_i(t,x)\,dx .\]

Observe that \[\begin{align} \left| \dot{X}_i\right| \leq C\delta_i^{-1} \bigl\lVert(\phi,\psi_1,\zeta)\bigr\rVert_{L^{\infty}} \int \bigl|\partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\bigr| dx \leq C\varepsilon \end{align}\] by the bootstrap assumption. Since \(M_i=O(1)\), it follows that \[\label{eq:ddotX-reduce} \delta_i^3|\ddot X_i|^2 \le C\delta_i\left(\int_{\mathbb{R}}\partial_t\mathfrak Y_i\,dx\right)^2.\tag{160}\]

We decompose \[\partial_t\mathfrak Y_i = a_t\,\mathcal{Z}_i + a\,\partial_t\mathcal{Z}_i, \qquad \mathcal{Z}_i := \partial_x\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\,\psi_1 + \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}_x\,\frac{\overline{p}}{\overline{v}}\phi + \frac{\partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}}{\overline{\theta}}\zeta.\] Accordingly, write \[\left(\int_{\mathbb{R}}\partial_t\mathfrak Y_i\,dx\right)^2 \le C(I_{i,1}+I_{i,2}+I_{i,3}),\] where \[\begin{align} I_{i,1} &:= \Biggl( \int_{\mathbb{R}} a \Bigl( \partial_x\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\,\psi_{1t} + \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\,\frac{\overline{p}}{\overline{v}}\phi_t + \frac{\partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}}{\overline{\theta}}\zeta_t \Bigr)\,dx \Biggr)^2,\\ I_{i,2} &:= \left( \int_{\mathbb{R}} a_t\,\mathcal{Z}_i\,dx \right)^2,\\ I_{i,3} &:= \Biggl( \int_{\mathbb{R}} a \Bigl( (\partial_x\bigl(u_1^{S_{i}}\bigr)^{-X_{i}})_t\,\psi_1 + \Bigl(\partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}}\frac{\overline{p}}{\overline{v}}\Bigr)_t\phi + \Bigl(\frac{\partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}}}{\overline{\theta}}\Bigr)_t\zeta \Bigr)\,dx \Biggr)^2. \end{align}\]

For \(I_{i,1}\), using Lemma 3, Lemma 5, and the bound \[\int_{\mathbb{R}}|\partial_x(v^{S_i})^{-X_i}|\,dx \le C\delta_i,\] we obtain by Young’s inequality \[\begin{align} I_{i,1} &\le \bigl\lVert\partial_x \bigl(v^{S_{i}}\bigr)^{-X_{i}}\bigr\rVert^2_{L_x^2}\, \bigl\lVert(\phi_t,\psi_{1t},\zeta_t)\bigr\rVert_{L^2_x}^2\nonumber\\ &\le C\delta_i^3\|(\phi_t,\psi_{1t},\zeta_t)\|_{L^2_x}^2 . \label{eq:ddotX-I1} \end{align}\tag{161}\]

For \(I_{i,2}\), we differentiate the weight 39 . Since \[a_t = -(\sigma_1+\dot{X}_1)\,\partial_x(a_1)^{-X_1} -(\sigma_3+\dot{X}_3)\,\partial_x(a_3)^{-X_3},\] and \(a_i'=\delta_i^{-1/2}(v^{S_i})'\), Lemma 3 implies \[|a_t| \le C\delta_1^{-1/2}|(v^{S_1})_x^{-X_1}| + C\delta_3^{-1/2}|(v^{S_3})_x^{-X_3}|.\] Therefore, by Young’s inequality, ?? , 237 , Lemma 7, and the exponential separation of the two shocks, \[\begin{align} I_{i,2} &\le C\Bigl(\int \delta_1^{-1/2}|\partial_x \bigl(v^{S_{1}}\bigr)^{-X_{1}}|\,|\partial_x \bigl(v^{S_{i}}\bigr)^{-X_{i}}| \,|(\phi,\psi_1,\zeta)| dx \Bigr)^2 \nonumber \\ & \qquad+ C\Bigl(\int \delta_3^{-1/2}|\partial_x \bigl(v^{S_{3}}\bigr)^{-X_{3}}|\,|\partial_x \bigl(v^{S_{i}}\bigr)^{-X_{i}}| \,|(\phi,\psi_1,\zeta)| dx \Bigr)^2 \nonumber\\ &\le C\Bigl(\delta_1^{-1}\int |\partial_x \bigl(v^{S_{1}}\bigr)^{-X_{1}}|^2 dx\Bigr)\Bigl(\int |\partial_x \bigl(v^{S_{i}}\bigr)^{-X_{i}}|^2\, |(\phi,\psi_1,\zeta)|^2 dx \Bigr) \nonumber \\ &\qquad + C\Bigl(\delta_3^{-1}\int |\partial_x \bigl(v^{S_{3}}\bigr)^{-X_{3}}|^2 dx\Bigr)\Bigl(\int |\partial_x \bigl(v^{S_{i}}\bigr)^{-X_{i}}|^2\, |(\phi,\psi_1,\zeta)|^2 dx \Bigr)\nonumber \\ &\le C(\delta_1^2+\delta_3^2) \,\mathcal{G}_i^S + C(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it} + C\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\overline{U})\,dx . \label{eq:ddotX-I2} \end{align}\tag{162}\]

For \(I_{i,3}\), we use the shifted-profile identities \[\partial_t \bigl(h^{S_{i}}\bigr)^{-X_{i}} = -(\sigma_i+\dot{X}_i)\partial_x \bigl(h^{S_{i}}\bigr)^{-X_{i}},\] valid for \(h=v^{S_i},u_1^{S_i},\theta^{S_i},G^{S_i}\), together with Lemma 3, Lemma 5, ?? , and ?? . Since each time derivative of a shifted shock profile contributes one additional \(x\)-derivative, we get \[\begin{align} I_{i,3} \le\;& C\delta_i\,\mathcal{G}_i^S + C(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it} + C\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\overline{U})\,dx . \label{eq:ddotX-I3} \end{align}\tag{163}\]

Combining 160 , 161 , 162 , and 163 , we arrive at \[\begin{align} \delta_i^3|\ddot X_i|^2 \le & \; C\delta_i\|(\phi_t,\psi_{1t},\zeta_t)\|_{L^2_x}^2 + C\delta_i\mathcal{G}_i^S + C(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it} \nonumber \\ &\qquad + C\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\overline{U})\,dx . \label{eq:ddotX-pre} \end{align}\tag{164}\] Now we invoke Lemma 25, namely ?? , and use \(\delta_i\le C(\delta_0+\varepsilon)\). This gives \[\begin{align} \delta_i^3|\ddot X_i|^2 \le\;& C(\delta_0+\varepsilon)\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 + C(\delta_0+\varepsilon)(\mathcal{G}_1^S+\mathcal{G}_3^S) \\ &+ C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i|\dot{X}_i|^2 + C(\delta_0+\varepsilon) \int_{\mathbb{R}}\bigl\lVert\widetilde{G}_{x}\bigr\rVert_{\nu,M_\#}^2\,dx \\ &+ C\delta_C^2\frac{1}{1+t} \int_{\mathbb{R}}e^{-\frac{2c_0|x|^2}{1+t}}|(\phi,\psi,\zeta)|^2\,dx + C\frac{\delta_C}{(1+t)^{3/2}} \\ &+ C(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it} + C\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\overline{U})\,dx . \end{align}\] Finally, combining this with 159 proves ?? . ◻

Proof. Differentiating 37 with respect to \(t\), multiplying the resulting equation by \(\widetilde{G}_t/M_\#\), and integrating over \((x,\xi)\in\mathbb{R}\times\mathbb{R}^3\), we obtain

\[\begin{align} \frac{1}{2}\frac{d}{dt}\iint \frac{|\widetilde{G}_t|^2}{M_\#}\,d\xi\,dx = \mathcal{T}_1+\sum_{i=1,3}\mathcal{T}_{2i} +\mathcal{T}_3+\sum_{i=1,3}\mathcal{T}_{4i} +\sum_{i=1,3}\mathcal{T}_{5i} +\mathcal{T}_6+\mathcal{T}_7+\mathcal{T}_8+\mathcal{T}_9+\mathcal{T}_{10}, \end{align}\]

where

\[\begin{align} \mathcal{T}_1 &:= \iint (L_M\widetilde{G})_t\frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_{2i} &:= \iint \dot{X}_i\,\partial_{xt}\bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr) \frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx + \iint \ddot X_i\,\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}\frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_3 &:= \iint \Bigl( \frac{u_1}{v}\widetilde{G}_x-\frac{1}{v}P_1(\xi_1\widetilde{G}_x)\Bigr)_t \frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_{4i} &:= \iint \Biggl[ \Bigl( \frac{u_1}{v}-\frac{\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}} \Bigr)\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}} \Biggr]_t \frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_{5i} &:= -\iint \Biggl[\frac{1}{v}P_1(\xi_1\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}})-\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}P_1^{S_i}(\xi_1\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}})\Biggr]_t\frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_6&:= \iint\Biggl[-\frac{1}{v}P_1(\xi_1M_x)+\frac{1}{\bigl(v^{S_{1}}\bigr)^{-X_{1}}}P_1^{S_1}(\xi_1\partial_x\bigl(M^{S_{1}}\bigr)^{-X_{1}})\Biggr.\\ &\qquad \qquad \Biggl.+\frac{1}{\bigl(v^{S_{3}}\bigr)^{-X_{3}}}P_1^{S_3}(\xi_1\partial_x\bigl(M^{S_{3}}\bigr)^{-X_{3}})\Biggr]_t\frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_7&:= \iint \bigl(\mathcal{N}(\widetilde{G},\widetilde{G})\bigr)_t \frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_8&:= \sum_{i=1,3}\iint\Bigl(\mathcal{N}\bigl(\widetilde{G},\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)+\mathcal{N}\bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}},\widetilde{G}\bigr)\Bigr)_t\frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_9&:= \iint \Bigl[ \mathcal{N}\bigl(\bigl(G^{S_{1}}\bigr)^{-X_{1}},\bigl(G^{S_{3}}\bigr)^{-X_{3}}\bigr)+\mathcal{N}\bigl(\bigl(G^{S_{3}}\bigr)^{-X_{3}},\bigl(G^{S_{1}}\bigr)^{-X_{1}}\bigr) \Bigr]_t \frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx,\\ \mathcal{T}_{10} &:= \sum_{i=1,3}\iint \Bigl( (L_M-L_{i}^S)\bigl(G^{S_{i}}\bigr)^{-X_{i}} \Bigr)_t \frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx. \end{align}\]

Estimate of \(\mathcal{T}_1\). Since \[L_M h=\mathcal{N}(M,h)+\mathcal{N}(h,M),\] we have \[(L_Mh)_t=L_Mh_t+\mathcal{N}(M_t,h)+\mathcal{N}(h,M_t).\] Therefore, \[\begin{align} \mathcal{T}_1 &= \iint L_M\widetilde{G}_t\frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx + \iint \Bigl(\mathcal{N}(M_t,\widetilde{G})+\mathcal{N}(\widetilde{G},M_t)\Bigr)\frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx. \end{align}\] By the coercivity estimate ?? , \[\iint L_M\widetilde{G}_t\frac{\widetilde{G}_t}{M_\#}\,d\xi\,dx \le -\lambda_{\mathrm{mic}} \iint \frac{(1+|\xi|)|\widetilde{G}_t|^2}{M_\#}\,d\xi\,dx.\] Using Young’s inequality, ?? , the decomposition 43 , and Lemma 25, we obtain \[\begin{align} \label{eq:GtE-T1} \begin{aligned} \mathcal{T}_1 \le\;& -\frac{127}{128}\lambda_{\mathrm{mic}} \int \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)^2\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 \\ &+ C\delta_C^2\frac{1}{1+t} \int_{\mathbb{R}}e^{-\frac{2c_0|x|^2}{1+t}}|(\phi,\psi,\zeta)|^2\,dx + C\frac{\delta_C}{(1+t)^{3/2}} \\ &+ C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr). \end{aligned} \end{align}\tag{165}\]

Estimate of \(\mathcal{T}_{2i}\). Since \[\partial_t\bigl(G^{S_{i}}\bigr)^{-X_{i}}=-(\sigma_i+\dot{X}_i)\partial_x\bigl(G^{S_{i}}\bigr)^{-X_{i}}, \qquad \partial_{xt}\bigl(G^{S_{i}}\bigr)^{-X_{i}}=-(\sigma_i+\dot{X}_i)\partial_{xx}\bigl(G^{S_{i}}\bigr)^{-X_{i}},\] Lemma 4 yields \[\iint \frac{|\bigl((G^{S_{i}})^{-X_{i}}\bigr)_{xt}|^2}{M_\#}\,d\xi\,dx \le C\delta_i^7 \le C(\delta_0+\varepsilon)\delta_i .\] Therefore, by Young’s inequality and Lemma 37, \[\begin{align} \label{eq:GtE-T2} \begin{aligned} |\mathcal{T}_{2i}| \le\;& \frac{\lambda_{\mathrm{mic}}}{64} \int \bigl\lVert\widetilde{G}_{t}\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)\delta_i|\dot{X}_i|^2 \\ &+ C(\delta_0+\varepsilon)^2\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 + C(\delta_0+\varepsilon)^2(\mathcal{G}_1^S+\mathcal{G}_3^S) \\ &+ C(\delta_0+\varepsilon)\int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &+ C\delta_C^2\frac{1}{1+t} \int_{\mathbb{R}}e^{-\frac{2c_0|x|^2}{1+t}}|(\phi,\psi,\zeta)|^2\,dx + C\frac{\delta_C}{(1+t)^{3/2}} \\ &+ C(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it} + C\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx . \end{aligned} \end{align}\tag{166}\]

Estimate of \(\mathcal{T}_3\). Expanding the time derivative gives terms of the form \[\frac{u_{1t}}{v}\widetilde{G}_x,\qquad \frac{u_1}{v}\widetilde{G}_{xt},\qquad \frac{v_t}{v^2}P_1(\xi_1\widetilde{G}_x),\qquad \frac{1}{v}(P_1)_t(\xi_1\widetilde{G}_x),\qquad \frac{1}{v}P_1(\xi_1\widetilde{G}_{xt}).\] Using Young’s inequality, the boundedness of \(P_1\), and Lemma 25, we get \[\begin{align} \label{eq:GtE-T3} \begin{aligned} |\mathcal{T}_3| \le\;& \frac{\lambda_{\mathrm{mic}}}{128} \int \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 \,dx + C\int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)^2\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 \\ &+ C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &+ C\delta_C^2\frac{1}{1+t} \int_{\mathbb{R}}e^{-\frac{2c_0|x|^2}{1+t}}|(\phi,\psi,\zeta)|^2\,dx . \end{aligned} \end{align}\tag{167}\]

Estimate of \(\mathcal{T}_{4i}\), \(\mathcal{T}_{5i}\), \(\mathcal{T}_6\), and \(\mathcal{T}_{10}\). These are the time-differentiated counterparts of the corresponding terms in the proof of Lemma 36. When the derivative falls on a macroscopic coefficient, we use Lemma 25; when it falls on the projected polynomial moments, we use Lemmas 33–35; and when it falls on a shifted shock profile, we use the identities \[\partial_t \bigl(h^{S_{i}}\bigr)^{-X_{i}} = -(\sigma_i+\dot{X}_i)\partial_x \bigl(h^{S_{i}}\bigr)^{-X_{i}}, \qquad h=v^{S_i},u_1^{S_i},\theta^{S_i},G^{S_i},\] together with Lemmas 3–5, Lemma 6, Lemma 7, and Lemma 8. Proceeding exactly as in Lemma 36, we obtain \[\begin{align} \label{eq:GtE-T45610} \begin{aligned} &\sum_{i=1,3}\bigl(|\mathcal{T}_{4i}|+|\mathcal{T}_{5i}|\bigr)+|\mathcal{T}_6|+|\mathcal{T}_{10}| \\ &\le \frac{\lambda_{\mathrm{mic}}}{16} \int \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)^2\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 \\ &\quad + C\|(\psi_{xt},\zeta_{xt})\|_{L^2_x}^2 + C(\delta_0+\varepsilon)^2(\mathcal{G}_1^S+\mathcal{G}_3^S) + C(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i|\dot{X}_i|^2 \\ &\quad + C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &\quad + C\delta_C^2\frac{1}{1+t} \int_{\mathbb{R}}e^{-\frac{2c_0|x|^2}{1+t}}|(\phi,\psi,\zeta)|^2\,dx + C\frac{\delta_C}{(1+t)^{3/2}} \\ &\quad + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) + C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t} + C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t} \\ &\quad + C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid \bar U)\,dx . \end{aligned} \end{align}\tag{168}\]

Estimate of \(\mathcal{T}_7\), \(\mathcal{T}_8\), and \(\mathcal{T}_9\). Using \[(\mathcal{N}(f,g))_t=\mathcal{N}(f_t,g)+\mathcal{N}(f,g_t),\] together with ?? , Lemma 4, and the same bookkeeping as in 141 –142 , we infer \[\begin{align} \begin{aligned} |\mathcal{T}_7|+|\mathcal{T}_8| \le\;& \frac{\lambda_{\mathrm{mic}}}{64} \iint \frac{(1+|\xi|)|\widetilde{G}_t|^2}{M_\#}\,d\xi\,dx \\ &+ C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &+ C(\delta_0+\varepsilon)^2(\mathcal{G}_1^S+\mathcal{G}_3^S) + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) \\ &+ C(\varepsilon+\delta_0)^2\bigl(\delta_1^2e^{-C\delta_1t}+\delta_3^2e^{-C\delta_3t}\bigr) + C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx , \end{aligned} \end{align}\] and \[\label{eq:GtE-T9} |\mathcal{T}_9| \le \frac{\lambda_{\mathrm{mic}}}{128} \int \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t} + C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t}.\tag{169}\]

Finally, summing 165 , 166 , 167 , 168 , and 169 , and absorbing all small fractions of \[\int \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 \,dx\] into the left-hand side, we obtain ?? . This completes the proof. ◻

Proof. We start from the perturbed equation for \(\widetilde{f}\): \[\begin{align} \begin{aligned} &v\widetilde{f}_t-u_1\widetilde{f}_x+\xi_1\widetilde{f}_x -\sum_{i=1,3}v\dot{X}_i\partial_x\bigl(F^{S_{i}}\bigr)^{-X_{i}} +\sum_{i=1,3} v\Bigl( \frac{\xi_1-u_1}{v}-\frac{\xi_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}} \Bigr)\partial_x\bigl(F^{S_{i}}\bigr)^{-X_{i}} \\ &= vL_M\widetilde{G} + v\mathcal{N}(\widetilde{G},\widetilde{G}) + v\sum_{i=1,3}\Bigl((L_M-L_{i}^S)\bigl(G^{S_{i}}\bigr)^{-X_{i}}\Bigr) \\ &\quad + v\sum_{i=1,3} \Bigl( \mathcal{N}\bigl(\widetilde{G},\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr) + \mathcal{N}\bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}},\widetilde{G}\bigr) \Bigr) \\ &\quad+ v\Bigl( \mathcal{N}\bigl(\bigl(G^{S_{1}}\bigr)^{-X_{1}},\bigl(G^{S_{3}}\bigr)^{-X_{3}}\bigr) + \mathcal{N}\bigl(\bigl(G^{S_{3}}\bigr)^{-X_{3}},\bigl(G^{S_{1}}\bigr)^{-X_{1}}\bigr) \Bigr). \end{aligned} \label{eq:ftx-start} \end{align}\tag{171}\] Apply \(\partial_t\partial_x\) to 171 , multiply by \(\widetilde{f}_{tx}/M_\#\), and integrate over \((x,\xi)\in\mathbb{R}\times\mathbb{R}^3\). We write the resulting terms as \[\mathcal{I}_1+\mathcal{I}_2+\sum_{i=1,3}\mathcal{I}_{3i} +\sum_{i=1,3}\mathcal{I}_{4i} +\mathcal{I}_5+\sum_{i=1,3}\mathcal{I}_{6i} +\sum_{i=1,3}\mathcal{I}_{7i} +\mathcal{I}_8+\mathcal{I}_9,\] corresponding respectively to \[(v\widetilde{f}_t)_{tx},\quad (u_1\widetilde{f}_x)_{tx},\quad \Bigl(v\dot{X}_i\partial_x\bigl(F^{S_{i}}\bigr)^{-X_{i}}\Bigr)_{tx},\quad \Biggl( v\Bigl(\frac{\xi_1-u_1}{v}-\frac{\xi_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}\Bigr)\partial_x\bigl(F^{S_{i}}\bigr)^{-X_{i}} \Biggr)_{tx},\] \[(v\mathcal{N}(\widetilde{G},\widetilde{G}))_{tx},\quad \bigl(v(L_M-L_{i}^S)\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)_{tx},\quad \Bigl( v\bigl(\mathcal{N}\bigl(\widetilde{G},\bigl(G^{S_{i}}\bigr)^{-X_{i}}\bigr)+\mathcal{N}\bigl(\bigl(G^{S_{i}}\bigr)^{-X_{i}},\widetilde{G}\bigr)\bigr) \Bigr)_{tx},\] \[(vL_M\widetilde{G})_{tx},\qquad \Bigl( v\bigl(\mathcal{N}\bigl(\bigl(G^{S_{1}}\bigr)^{-X_{1}},\bigl(G^{S_{3}}\bigr)^{-X_{3}}\bigr)+\mathcal{N}\bigl(\bigl(G^{S_{3}}\bigr)^{-X_{3}},\bigl(G^{S_{1}}\bigr)^{-X_{1}}\bigr)\bigr) \Bigr)_{tx}.\]

Step 1: the transport terms \(\mathcal{I}_1+\mathcal{I}_2\). Expanding derivatives, we have \[\begin{align} \mathcal{I}_1 &= \iint \frac{(v\widetilde{f}_t)_{tx}\widetilde{f}_{tx}}{M_\#}\,d\xi\,dx \\ &= \iint \frac{v_{tx}\widetilde{f}_t\widetilde{f}_{tx}}{M_\#}\,d\xi\,dx + \iint \frac{(v_t\widetilde{f}_{xx}+v_x\widetilde{f}_{tt})\widetilde{f}_{tx}}{M_\#}\,d\xi\,dx + \iint \frac{v\widetilde{f}_{ttx}\widetilde{f}_{tx}}{M_\#}\,d\xi\,dx. \end{align}\] The last term gives \[\iint \frac{v\widetilde{f}_{ttx}\widetilde{f}_{tx}}{M_\#}\,d\xi\,dx = \frac{d}{dt}\iint \frac{v|\widetilde{f}_{tx}|^2}{2M_\#}\,d\xi\,dx - \iint \frac{v_t|\widetilde{f}_{tx}|^2}{2M_\#}\,d\xi\,dx.\] Since \(v\) is uniformly comparable to a positive constant, the energy \(\iint v|\widetilde{f}_{tx}|^2/M_\#\) is equivalent to \(\iint |\widetilde{f}_{tx}|^2/M_\#\). Next, using \[\widetilde{f}_{tx}=\widetilde{G}_{tx}+\mathcal{M}_{tx}, \qquad \widetilde{f}_t=\widetilde{G}_t+\mathcal{M}_t,\] together with 170 , Lemma 25, and Young’s inequality, we get \[\begin{align} \begin{aligned} |\mathcal{I}_1| \le\;& \frac{d}{dt}\iint \frac{v|\widetilde{f}_{tx}|^2}{2M_\#}\,d\xi\,dx + C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2\,dx \\ &+ C(\delta_0+\varepsilon)\|(\phi_{xt},\psi_{xt},\zeta_{xt})\|_{L^2_x}^2 + C(\delta_0+\varepsilon)\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 \\ &+ C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{t}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &+ C(\delta_0+\varepsilon)(\mathcal{G}_1^S+\mathcal{G}_3^S) + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) \\ &+ C\frac{\delta_C}{(1+t)^{3/2}} + C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t} + C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t} \\ &+ C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\overline{U})\,dx + C\delta_C^2\frac{1}{1+t}\int_{\mathbb{R}}e^{-\frac{2c_0|x|^2}{1+t}}|(\phi,\psi,\zeta)|^2\,dx . \end{aligned} \label{eq:ftx-I1} \end{align}\tag{172}\]

Similarly, \[\mathcal{I}_2 = -\iint \frac{(u_1\widetilde{f}_x)_{tx}\widetilde{f}_{tx}}{M_\#}\,d\xi\,dx .\] Expanding derivatives and integrating by parts in \(x\) in the top-order term \(\iint u_1\widetilde{f}_{txx}\widetilde{f}_{tx}/M_\#\), we obtain \[\begin{align} \begin{aligned} |\mathcal{I}_2| \le\;& C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2\,dx + C(\delta_0+\varepsilon)\|(\phi_{xx},\psi_{xx},\zeta_{xx})\|_{L^2_x}^2 \\ &+ C(\delta_0+\varepsilon)\|(\phi_{xt},\psi_{xt},\zeta_{xt})\|_{L^2_x}^2 + C(\delta_0+\varepsilon)\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 \\ &+ C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 +\bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx \\ &+ C(\delta_0+\varepsilon)(\mathcal{G}_1^S+\mathcal{G}_3^S) + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) \\ &+ C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t} + C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t} + C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx . \end{aligned} \label{eq:ftx-I2} \end{align}\tag{173}\]

Step 2: the shift term \(\mathcal{I}_{3i}\). Since \[\partial_t\bigl(F^{S_{i}}\bigr)^{-X_{i}} = -(\sigma_i+\dot{X}_i)\partial_x \bigl(F^{S_{i}}\bigr)^{-X_{i}}, \qquad \partial_{tx}\bigl(F^{S_{i}}\bigr)^{-X_{i}} = -(\sigma_i+\dot{X}_i)\partial_{xx}\bigl(F^{S_{i}}\bigr)^{-X_{i}},\] we may expand \[\begin{align} (v\dot{X}_i\partial_x\bigl(F^{S_{i}}\bigr)^{-X_{i}})_{tx} &= v_{tx}\dot{X}_i\partial_x\bigl(F^{S_{i}}\bigr)^{-X_{i}} + v_t\dot{X}_i\partial_{xx}\bigl(F^{S_{i}}\bigr)^{-X_{i}} + v_x\ddot X_i\partial_x\bigl(F^{S_{i}}\bigr)^{-X_{i}} \\ &\quad + v_x\dot{X}_i\partial_{tx}\bigl(F^{S_{i}}\bigr)^{-X_{i}} + v\ddot X_i\partial_{xx}\bigl(F^{S_{i}}\bigr)^{-X_{i}} + v\dot{X}_i\partial_{txx}\bigl(F^{S_{i}}\bigr)^{-X_{i}}\\ &=:\sum_{l=1}^5 \mathcal{I}_{3i}^l \end{align}\]

For each term, we have the following estimates.

\[\begin{align} &|\mathcal{I}_{3i}^1| \le C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{f}_{tx}\bigr\rVert_{M_\#}^2 dx +C(\delta_0+\varepsilon)\delta_i|\dot{X}_i|^2 \\ &|\mathcal{I}_{3i}^3| + |\mathcal{I}_{3i}^5| \le C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{f}_{tx}\bigr\rVert_{M_\#}^2 dx + C\delta_i^3 |\ddot{X}_i|^2\\ &|\mathcal{I}_{3i}^2|+|\mathcal{I}_{3i}^4| +|\mathcal{I}_{3i}^6| \le C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{f}_{tx}\bigr\rVert_{M_\#}^2 dx + C(\delta_0+\varepsilon)\delta_i|\dot{X}_i|^2 \end{align}\]

By Young’s inequality, 164 , 173 , Lemma 4, and Lemma 37, we obtain \[\begin{align} \begin{aligned} |\mathcal{I}_{3i}| \le\;& C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)\delta_i|\dot{X}_i|^2 \\ &+ C(\delta_0+\varepsilon)(\mathcal{G}_1^S+\mathcal{G}_3^S) + C(\delta_0+\varepsilon)\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 \\ &+ C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &+ C\frac{\delta_C}{(1+t)^{3/2}} + C\delta_C^2\frac{1}{1+t}\int_{\mathbb{R}}e^{-\frac{2c_0|x|^2}{1+t}}|(\phi,\psi,\zeta)|^2\,dx \\ &+ C(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it} + C\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx . \end{aligned} \label{eq:ftx-I3} \end{align}\tag{174}\]

Step 3: the coefficient-mismatch term \(\mathcal{I}_{4i}\). Write \[\begin{align} &\frac{\xi_1-u_1}{v}-\frac{\xi_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}\\ &\quad = -\frac{\xi_1}{v\bigl(v^{S_{i}}\bigr)^{-X_{i}}}(v-\bigl(v^{S_{i}}\bigr)^{-X_{i}}) +\frac{u_1}{v\bigl(v^{S_{i}}\bigr)^{-X_{i}}}(v-\bigl(v^{S_{i}}\bigr)^{-X_{i}}) -\frac{1}{\bigl(v^{S_{i}}\bigr)^{-X_{i}}}(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}). \end{align}\] After applying \(\partial_{tx}\), every term is a linear combination of products of:

• a first- or second-order macroscopic derivative of \(v,u_1\),

• a factor \(v-\bigl(v^{S_{i}}\bigr)^{-X_{i}}\) or \(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\),

• and a profile derivative \(\partial_t^{\gamma_0}\partial_x^{\gamma_1} \bigl(F^{S_{i}}\bigr)^{-X_{i}}\) with \(|\gamma|=|\gamma_0|+|\gamma_1|\le 3\).

Therefore, using Young’s inequality, Lemma 4, Lemma 6, Lemma 7, Lemma 8, and the same weighted shock-interaction bounds as in Lemma 36, we get \[\begin{align} \begin{aligned} |\mathcal{I}_{4i}| \le\;& C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)\mathcal{G}_i^S \\ &+ C(\delta_0+\varepsilon)\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 + C(\delta_0+\varepsilon)\|(\phi_{xx},\psi_{xx},\zeta_{xx})\|_{L^2_x}^2 \\ &+ C(\delta_0+\varepsilon)\|(\phi_{xt},\psi_{xt},\zeta_{xt})\|_{L^2_x}^2 + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) \\ &+ C(\varepsilon+\delta_0)^2\delta_i^2e^{-C\delta_it} + C\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx . \end{aligned} \label{eq:ftx-I4} \end{align}\tag{175}\]

Step 4: the quadratic nonlinear term \(\mathcal{I}_5\). Expanding by Leibniz’ rule, \[(v\mathcal{N}(\widetilde{G},\widetilde{G}))_{tx} = v_{tx}\mathcal{N}(\widetilde{G},\widetilde{G}) + v_t\bigl(\mathcal{N}(\widetilde{G}_x,\widetilde{G})+\mathcal{N}(\widetilde{G},\widetilde{G}_x)\bigr)\] \[\qquad + v_x\bigl(\mathcal{N}(\widetilde{G}_t,\widetilde{G})+\mathcal{N}(\widetilde{G},\widetilde{G}_t)\bigr) + v\bigl(\mathcal{N}(\widetilde{G}_{tx},\widetilde{G})+\mathcal{N}(\widetilde{G}_t,\widetilde{G}_x) +\mathcal{N}(\widetilde{G}_x,\widetilde{G}_t)+\mathcal{N}(\widetilde{G},\widetilde{G}_{tx})\bigr).\] Using ?? , the one-dimensional Sobolev bound \[\left\| \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \right\|_{L^\infty_x} \le C\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{\nu,M_\#}^2\,dx,\] and Young’s inequality, we infer \[\begin{align} \begin{aligned} |\mathcal{I}_5| \le\;& C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &+ C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 \\ &+ C(\delta_0+\varepsilon)\|(\phi_{xx},\psi_{xx},\zeta_{xx})\|_{L^2_x}^2 + C(\delta_0+\varepsilon)\|(\phi_{xt},\psi_{xt},\zeta_{xt})\|_{L^2_x}^2 \\ &+ C(\delta_0+\varepsilon)(\mathcal{G}_1^S+\mathcal{G}_3^S) + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) \\ &+ C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t} + C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t} + C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx . \end{aligned} \label{eq:ftx-I5} \end{align}\tag{176}\]

Step 5: the mixed shock terms \(\mathcal{I}_{6i}\), \(\mathcal{I}_{7i}\), and \(\mathcal{I}_9\). These terms are handled exactly as above by combining:

• the bilinear collision estimate ?? ,

• the profile bounds from Lemma 4,

• the interaction lemmas Lemma 6, Lemma 7, and Lemma 8,

• and the already established lower-order bounds in Lemma 36 and Lemma 38.

In particular, differentiating \[(L_M-L_{i}^S)\bigl(G^{S_{i}}\bigr)^{-X_{i}}\] in \((t,x)\) produces only coefficient mismatches of first and second order, hence these terms are of the same type as in Step 4 and the proof of Lemma 36. Consequently, \[\begin{align} \begin{aligned} &\sum_{i=1,3}\bigl(|\mathcal{I}_{6i}|+|\mathcal{I}_{7i}|\bigr)+|\mathcal{I}_9| \\ &\le C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2\,dx + C(\delta_0+\varepsilon)(\mathcal{G}_1^S+\mathcal{G}_3^S) \\ &\quad + C(\delta_0+\varepsilon)\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 + C(\delta_0+\varepsilon)\|(\phi_{xx},\psi_{xx},\zeta_{xx})\|_{L^2_x}^2 \\ &\quad + C(\delta_0+\varepsilon)\|(\phi_{xt},\psi_{xt},\zeta_{xt})\|_{L^2_x}^2 + C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^ + \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &\quad + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) + C\frac{\delta_C}{(1+t)^{3/2}} \\ &\quad + C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t} + C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t} + C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx . \end{aligned} \label{eq:ftx-I679} \end{align}\tag{177}\]

Step 6: the linear collision term \(\mathcal{I}_8\). This is where the coercive term appears. By differentiating \(vL_M\widetilde{G}\), we write \[(vL_M\widetilde{G})_{tx} = vL_M\widetilde{G}_{tx} +\mathcal{R}_{tx},\] where \(\mathcal{R}_{tx}\) is a sum of lower-order terms involving \(v_t,v_x,v_{tx}\), \(M_t,M_x,M_{tx}\), and \(\widetilde{G},\widetilde{G}_t,\widetilde{G}_x\). Hence \[\mathcal{I}_8 = \iint \frac{vL_M\widetilde{G}_{tx}\,\widetilde{f}_{tx}}{M_\#}\,d\xi\,dx + \iint \frac{\mathcal{R}_{tx}\,\widetilde{f}_{tx}}{M_\#}\,d\xi\,dx .\] Using \(\widetilde{f}_{tx}=\widetilde{G}_{tx}+\mathcal{M}_{tx}\), we split the first term: \[\begin{align} \iint \frac{vL_M\widetilde{G}_{tx}\,\widetilde{f}_{tx}}{M_\#}\,d\xi\,dx = \iint \frac{v\widetilde{G}_{tx}L_M\widetilde{G}_{tx}}{M_\#}\,d\xi\,dx + \iint \frac{vL_M\widetilde{G}_{tx}\,\mathcal{M}_{tx}}{M_\#}\,d\xi\,dx . \end{align}\] By ?? , \[\iint \frac{v\widetilde{G}_{tx}L_M\widetilde{G}_{tx}}{M_\#}\,d\xi\,dx \le -\lambda_{\mathrm{mic}} \int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2 \,dx .\] The cross term with \(\mathcal{M}_{tx}\) is controlled by 170 , Lemma 25, and Young’s inequality; the remainder \(\mathcal{R}_{tx}\) is estimated by ?? exactly as in Steps 1, 4, and 5. Thus \[\begin{align} \begin{aligned} \mathcal{I}_8 \le\;& -\frac{127}{128}\lambda_{\mathrm{mic}} \int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 \\ &+ C(\delta_0+\varepsilon)\|(\phi_{xt},\psi_{xt},\zeta_{xt})\|_{L^2_x}^2 + C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 +\bigl\lVert\widetilde{G}_{t}\bigr\rVert_{\nu,M_\#}^2+\bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2\,dx \\ &+ C(\delta_0+\varepsilon)(\mathcal{G}_1^S+\mathcal{G}_3^S) + C\delta_0^3\bigl(\delta_1e^{-C\delta_1t}+\delta_3e^{-C\delta_3t}\bigr) \\ &+ C\frac{\delta_C}{(1+t)^{3/2}} + C(\varepsilon+\delta_0)^2\delta_1^2e^{-C\delta_1t} + C(\varepsilon+\delta_0)^2\delta_3^2e^{-C\delta_3t} \\ &+ C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx + C\delta_C^2\frac{1}{1+t}\int_{\mathbb{R}}e^{-\frac{2c_0|x|^2}{1+t}}|(\phi,\psi,\zeta)|^2\,dx . \end{aligned} \label{eq:ftx-I8} \end{align}\tag{178}\]

Finally, summing 172 , 173 , 174 , 175 , 176 , 177 , and 178 , and absorbing the small multiples of \[\int \bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{\nu,M_\#}^2 \,dx\] into the left-hand side, we obtain ?? . ◻

Proof sketch. Apply \(\partial_x^2\) to 171 , multiply by \(\widetilde{f}_{xx}/M_\#\), and integrate over \((x,\xi)\in\mathbb{R}\times\mathbb{R}^3\). The resulting terms are treated by the same decomposition as in the proof of Lemma 39, with the following simplifications.

(1) There is no \(\ddot X_i\)-term. Indeed, \[\partial_x^2(v\dot{X}_i\partial_x\bigl(F^{S_{i}}\bigr)^{-X_{i}}) = v_{xx}\dot{X}_i\partial_x\bigl(F^{S_{i}}\bigr)^{-X_{i}} + 2v_x\dot{X}_i\partial_{xx}\bigl(F^{S_{i}}\bigr)^{-X_{i}} + v\dot{X}_i\partial^3_x\bigl(F^{S_{i}}\bigr)^{-X_{i}},\] so only \(\dot{X}_i\) appears. Hence the shift contribution is easier than in the \(\widetilde{f}_{tx}\)-estimate and is controlled directly by Lemma 4 and Young’s inequality.

(2) The coercive term again comes from \((vL_M\widetilde{G})_{xx}\). Writing \[(vL_M\widetilde{G})_{xx}=vL_M\widetilde{G}_{xx}+\mathcal{R}_{xx},\] we have \[\iint \frac{vL_M\widetilde{G}_{xx}\,\widetilde{f}_{xx}}{M_\#}\,d\xi\,dx = \iint \frac{v\widetilde{G}_{xx}L_M\widetilde{G}_{xx}}{M_\#}\,d\xi\,dx + \iint \frac{vL_M\widetilde{G}_{xx}\,\mathcal{M}_{xx}}{M_\#}\,d\xi\,dx .\] The first term is estimated by ?? ; the second by 170 , exactly as in Step 6 of Lemma 39.

(3) The only terms which are genuinely different from the \(\widetilde{f}_{tx}\)-case are those where two spatial derivatives hit the transport coefficients. A representative example is \[\iint \frac{v_{xx}\widetilde{f}_t\,\widetilde{f}_{xx}}{M_\#}\,d\xi\,dx .\] Using \(\widetilde{f}_t=\widetilde{G}_t+\mathcal{M}_t\), Young’s inequality, 170 , and Lemma 25, we obtain \[\begin{align} & \left| \iint \frac{v_{xx}\widetilde{f}_t\,\widetilde{f}_{xx}}{M_\#}\,d\xi\,dx \right| \\ &\quad \le C(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2+\bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 \,dx + C(\delta_0+\varepsilon)\|(\phi_{xx},\psi_{xx},\zeta_{xx})\|_{L^2_x}^2 \\ &\qquad + C(\delta_0+\varepsilon)\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 + C\delta_C^2\frac{1}{1+t}\int_{\mathbb{R}}e^{-\frac{2c_0|x|^2}{1+t}}|(\phi,\psi,\zeta)|^2\,dx \\ &\qquad + C\frac{\delta_C}{(1+t)^{3/2}} + C(\delta_0+\varepsilon)(\mathcal{G}_1^S+\mathcal{G}_3^S) + C\sum_{i=1,3}\delta_i^3e^{-C\delta_it}\int_{\mathbb{R}}\eta(U\mid\bar U)\,dx . \end{align}\] The term with \(u_{1xx}\widetilde{f}_x\widetilde{f}_{xx}\) is handled in the same way.

(4) All remaining terms are identical in structure to those treated in the proof of Lemma 39, with \(t\)-derivatives replaced by \(x\)-derivatives. Hence they are estimated by the same ingredients: Lemma 4, Lemma 6, Lemma 7, Lemma 8, Lemma 36, Lemma 38, Lemma 25, and ?? .

Collecting all contributions and absorbing the small multiples of \[\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2 \,dx\] into the left-hand side, we obtain ?? . ◻

Proof. This follows immediately by adding ?? and ?? . ◻

Proof. We combine the previously established high-order differential estimates: the second-order macroscopic estimate, Lemma 36, Lemma 38, Lemma 39, and Lemma 40. Thus there exist positive constants \(C_1,\dots,C_5\) such that, for any positive parameters \(\lambda_1,\lambda_2,\lambda_3,\lambda_4\), one has \[\begin{align} \begin{aligned} &\frac{d}{dt} \Bigg\{ \|(\phi_x,\psi_x,\zeta_x)\|_{L^2_x}^2 +\lambda_1\int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{M_\#}^2 + \bigl\lVert\widetilde{G}_{t}\bigr\rVert_{M_\#}^2 \,dx +\lambda_2\int_{\mathbb{R}} \bigl\lVert\widetilde{f}_{xx}\bigr\rVert_{M_\#}^2 + \bigl\lVert\widetilde{f}_{xt}\bigr\rVert_{M_\#}^2\,dx \Bigg\} \\ &\quad +\int_{\mathbb{R}} \bigl( \lambda_4 \phi_{xx}^2+\psi_{1xx}^2+\psi_{2xx}^2+\psi_{3xx}^2+\zeta_{xx}^2 \bigr)\,dx +\lambda_3\|(\phi_{xt},\psi_{xt},\zeta_{xt})\|_{L^2_x}^2 \\ &\quad +\frac{\lambda_1}{2}\int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 \,dx +\frac{\lambda_2}{2}\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_{xt}\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &\le C_{\mathrm{high}}(\delta_0+\varepsilon)\sum_{i=1,3}\delta_i|\dot{X}_i|^2 + C_{\mathrm{high}}(\delta_0+\varepsilon)\sum_{i=1,3}\mathcal{G}_i^S + C_{\mathrm{high}}(\delta_0+\varepsilon)\sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2 \\ &\quad + C_{\mathrm{high}}(\delta_0+\varepsilon)\int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx + C_{\mathrm{high}}\delta_C^2 \frac{1}{1+t}\int_{\mathbb{R}}e^{-\frac{2c|x|^2}{1+t}}|(\phi,\psi,\zeta)|^2\,dx \\ &\quad + C_{\mathrm{mic}1}\int \bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{\nu,M_\#}^2 \,dx + C_{\mathrm{mic}2}\int \bigl\lVert\widetilde{G}_{xt}\bigr\rVert_{\nu,M_\#}^2 \,dx \\ &\quad + C_{\mathrm{mic}3}\int \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{\nu,M_\#}^2 \,dx + C_{\mathrm{mic}4}\int \bigl\lVert\widetilde{G}_{t}\bigr\rVert_{\nu,M_\#}^2 \,dx + \lambda_4\frac{d}{dt}\int_{\mathbb{R}}\phi_{xx}\psi_{1x}\,dx , \end{aligned} \label{eq:high-order-precombine} \end{align}\tag{181}\] where \[C_{\mathrm{high}}:=C_1+\lambda_1C_2+\lambda_2C_3+\lambda_3C_4+\lambda_4C_5,\] and similarly \(C_{\mathrm{mic}1},C_{\mathrm{mic}2},C_{\mathrm{mic}3},C_{\mathrm{mic}4}\) denote the corresponding linear combinations of \(C_1,\dots,C_5\).

Choose \(\lambda_1,\lambda_2,\lambda_3,\lambda_4\) successively so that \[\begin{align} C_{\mathrm{mic}1}<\frac{\lambda_2}{4}, \qquad C_{\mathrm{mic}2}<\frac{\lambda_2}{4}, \qquad C_{\mathrm{mic}3}<\frac{\lambda_1}{8}, \qquad C_{\mathrm{mic}4}<\frac{\lambda_1}{8}. \label{eq:gamma-choice-1} \end{align}\tag{182}\] Then the four kinetic terms on the right-hand side of 181 are absorbed into the left-hand side.

Next, move the derivative term to the left and define \[\begin{align} &\mathcal{E}_h(t) := \|(\phi_x,\psi_x,\zeta_x)(t)\|_{L^2_x}^2 +\lambda_1\int \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{M_\#}^2 + \bigl\lVert\widetilde{G}_{t}\bigr\rVert_{M_\#}^2 \,dx \nonumber\\ &\quad +\lambda_2\int \bigl\lVert\widetilde{f}_{xx}\bigr\rVert_{M_\#}^2 + \bigl\lVert\widetilde{f}_{xt}\bigr\rVert_{M_\#}^2 \,dx -\lambda_4\int_{\mathbb{R}}\phi_{xx}\psi_{1x}\,dx . \label{eq:high-order-modified-energy} \end{align}\tag{183}\]

By Young’s inequality, for any \(\kappa_1>0\), \[\begin{align} \begin{aligned} \lambda_4\left|\int_{\mathbb{R}}\phi_{xx}\psi_{1x}\,dx\right| \le& \kappa_1\|\phi_{xx}\|_{L^2_x}^2 + C_{\kappa_1}\lambda_4^2\|\psi_{1x}\|_{L^2_x}^2 \\ \le&\kappa_1\iint \frac{\left| \widetilde{f}_{xx}\right|^2}{M_\#} d\xi dx + C\kappa_1\varepsilon^2\bigl\lVert(\phi_x,\psi_x,\zeta_x)\bigr\rVert_{L^2}^2 + C\kappa_1\delta + C_{\kappa_1} \lambda_4^2 \bigl\lVert\psi_{1x}\bigr\rVert_{L^2}^2. \end{aligned} \label{eq:young-cross-high} \end{align}\tag{184}\] Choose \(\kappa_1>0\) sufficiently small so that the first term on the right-hand side of 184 is absorbed into the \(\lambda_4\phi_{xx}^2\)-term already present on the left-hand side of 181 . Since \(\psi_{1x}\) is lower order, the second term is controlled by the low-order energy. Hence, for \(\lambda_4\) sufficiently small, the modified energy \(\mathcal{E}_h(t)\) is equivalent to \[\|(\phi_x,\psi_x,\zeta_x)(t)\|_{L^2_x}^2 + \mathcal{K}_{\mathrm{high}}\] up to a harmless multiple of \(\|(\phi_x,\psi_x,\zeta_x)(t)\|_{L^2_x}^2\).

Integrating in time over \([0,T]\), using 182 , and absorbing the cross term contribution yields ?? . ◻

Proof. Set \[\omega_G(t,x):=\frac{1}{\sqrt{1+t}}e^{-\frac{\beta |x|^2}{1+t}}, \quad h(t,x):=\int_{-\infty}^x \omega_G(t,y)\,dy, \quad H(t,x):=\int_{-\infty}^x \omega_G(t,y)^2\,dy .\] Then \[h_x=\omega_G,\qquad H_x=\omega_G^2,\qquad \omega_{Gt}=\frac{1}{4\beta}\omega_{Gxx},\qquad h_t=\frac{1}{4\beta}h_{xx},\] and \(h\), \(H\) are uniformly bounded on \([0,T]\times\mathbb{R}\).

We apply weighted multipliers directly to the macroscopic perturbation system 35 –36 . More precisely, we multiply the three momentum equations by \(\psi_i h^2\) \((i=1,2,3)\), and we combine the continuity equation 35 \(_1\) with the temperature equation 36 \(_2\) against \[H\Bigl(\frac{2}{3}\zeta-\bar p\,\phi\Bigr).\] After summing the resulting identities and integrating by parts in \(x\), the terms in which one derivative falls on \(H\) produce \(H_x=\omega_G^2\), hence yield the localized coercive quantity \[\int_{\mathbb{R}}\omega_G(t,x)^2\,|(\phi,\psi,\zeta)(t,x)|^2\,dx .\]

More precisely, one obtains \[\begin{align} \frac{d}{dt}\mathcal{J}_G(t) + c\int_{\mathbb{R}}\omega_G(t,x)^2\,|(\phi,\psi,\zeta)(t,x)|^2\,dx \le \mathcal{R}_G(t)+\mathcal{K}_G(t), \label{eq:WG-identity} \end{align}\tag{185}\] where \(\mathcal{J}_G(t)\) is a weighted interaction functional satisfying \[|\mathcal{J}_G(t)| \le C\|(\phi,\psi,\zeta)(t)\|_{L^2_x}^2 , \label{eq:WG-J-bound}\tag{186}\] \(\mathcal{R}_G(t)\) contains only macroscopic remainder terms, and \(\mathcal{K}_G(t)\) contains the microscopic moments coming from the \(\Pi_1\)-terms in 35 –36 .

We first estimate the macroscopic part. By construction, \(\mathcal{R}_G(t)\) is made of

(i) terms in which derivatives hit the weights \(h\) and \(H\);

(ii) the macroscopic error terms \(Q_1\), \(Q_2\), the profile interaction terms, and the modulation terms already treated in the zeroth-order analysis;

(iii) lower-order quadratic terms involving \((\phi_x,\psi_x,\zeta_x)\).

Since \(h\) and \(H\) are bounded, \(H_x=\omega_G^2\), and \[|\omega_{Gt}|+|h_t| \le C\left(\frac{1}{1+t}+\frac{|x|^2}{(1+t)^2}\right)\omega_G \le \frac{C}{1+t}\omega_G,\] all contributions of type (i) are bounded by \[C\sup_{t\in[0,T]}\|(\phi,\psi,\zeta)(t)\|_{L^2_x}^2 + C\int_0^T \|(\phi_x,\psi_x,\zeta_x)\|_{L^2_x}^2\,dt .\] For the terms of type (ii), we invoke the macroscopic zeroth-order structure already developed above: by 95 , the full collection of macroscopic error terms and lower-order shift contributions is controlled by \[C\sum_{i=1,3}\delta_i |\dot{X}_i|^2 + C\sum_{i=1,3}\mathcal{G}_i^S + C\delta_C\frac{1}{1+t}\int_{\mathbb{R}}e^{-2c|x|^2/(1+t)}|(\phi,\psi,\zeta)|^2\,dx + C\|(\phi_x,\psi_x,\zeta_x)\|_{L^2_x}^2 ,\] up to a harmless \(\lambda \mathcal{D}_{\mathrm{mac}}(U)\)-term, which we absorb by taking \(\lambda>0\) sufficiently small. The remaining pure profile terms are estimated by the Gaussian decay of the viscous contact wave and the exponential localization of the shock profiles, hence their time integrals are bounded by \(C\delta^{1/2}\) and the microscopic parts. More detail, we compute each Gaussian estiamtes as followed.

\[\begin{align} \label{eq:Gaussian1} & \left(\bar p \phi + \zeta \right)_t - \sum_{i\in\{1,3\}} \dot{X}_i \left( \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}} \bar p + \partial\bigl(\theta^{S_{i}}\bigr)^{-X_{i}} \right) + \bigl(p-\bar p\bigr)\psi_x \nonumber \\ & \quad = \bar p_t \phi - \bar u_x (p-\bar p)+\left(\alpha_{\rm {th}}(\theta)\frac{\theta_x}{v}-\alpha_{\rm{th}}(\bar \theta)\frac{\bar \theta_x}{\bar v}\right)_x \nonumber \\ & \qquad + \left(\mu(\theta)\frac{(u_x)^2}{v}-\mu(\bar \theta)\frac{(\bar u_x)^2}{\bar v}\right) - Q_2 + \frac{\mu(\theta)(\psi_{2x}^2 + \psi_{3x}^2)}{v} + K_{\theta} \end{align}\tag{187}\]

where

\[\begin{align} K_{\theta} :=& \sum_{j=2}^3 \psi_j \int \xi_1 \xi_j \Pi_{1x} d\xi - \int \xi_1 \frac{\left| \xi\right|^2}{2} \widetilde{\Pi}_{1x} d\xi \\ & \quad + u_1\int \xi_1^2 \Pi_{1x} d\xi - \sum_{i\in\{1,3\}} \bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\int \xi_1^2 \partial_x\bigl(\Pi_1^{S_{i}}\bigr)^{-X_{i}} d\xi. \end{align}\]

Take \(\mathcal{A}_G:=\bar p \phi + \zeta\) and observe that

\[\begin{align} \label{eq:Gaussian2} &\left[\mathcal{A}_G^2 \frac{h^2}{2}\right]_t - \frac{1}{4\beta}\mathcal{A}_G^2 h \omega_{Gx} - \sum_{i\in\{1,3\}} \dot{X}_i \left( \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}} \bar p + \partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}} \right) \mathcal{A}_G h^2 + \bigl(p-\bar p\bigr)\psi_x \mathcal{A}_G h^2 \nonumber \\ & = \left(\alpha_{\rm{th}}(\theta)\frac{\theta_x}{v}-\alpha_{\rm{th}}(\bar \theta)\frac{\bar \theta_x}{\bar v}\right)_x \mathcal{A}_G h^2 + \left[ \bar p_t \phi - \bar u_x (p-\bar p) + \left(\mu(\theta)\frac{(u_x)^2}{v}-\mu(\bar \theta)\frac{(\bar u_x)^2}{\bar v}\right) \nonumber\right. \\ & \left. \quad - Q_2 + \frac{\mu(\theta)(\psi_{2x}^2 + \psi_{3x}^2)}{v} + K_{\theta} \right] \mathcal{A}_G h^2 \end{align}\tag{188}\]

Applying the energy estimate to 188 ,

\[\begin{align} \label{eq:Gaussian3} &\frac{1}{4\beta} \int_0^T \int \mathcal{A}_G^2 \omega_G^2 dx dt = \int \mathcal{A}_G(0,x)^2 \frac{h_0^2}{2} dx \nonumber \\ &\quad - \int \mathcal{A}_G^2 \frac{h^2}{2} dx + \sum_{i\in\{1,3\}} \int_0^T \dot{X}_i \int \left( \partial_x \bigl(v^{S_{i}}\bigr)^{-X_{i}} \bar p + \partial_x \bigl(\theta^{S_{i}}\bigr)^{-X_{i}} \right) \mathcal{A}_G h^2 dx dt \nonumber \\ &\quad -\int_0^T\int \frac{1}{4\beta}(\mathcal{A}_G^2)_x h\omega_G dx dt -\int_0^T \int (p-\bar p)\psi_x \mathcal{A}_Gh^2 dx dt \nonumber\\ &\quad +\int_0^T \int \left[\bar p_t \phi -\bar u_x(p-\bar p) - Q_2 + \left(\mu(\theta)\frac{(u_x)^2}{v}-\mu(\bar \theta)\frac{(\bar u_x)^2}{\bar v}\right) \right] \mathcal{A}_Gh^2 dx dt \nonumber \\ &\quad -\int_0^T \int \left(\alpha_{\rm{th}}(\theta)\frac{\theta_x}{v}-\alpha_{\rm{th}}(\bar \theta)\frac{\bar \theta_x}{\bar v}\right) (\mathcal{A}_G h^2)_x dx dt +\int_0^T \int \left[\frac{\mu(\theta)(\psi_{2x}^2+\psi_{3x}^2)}{v}\right] \mathcal{A}_G h^2 dx dt \nonumber \\ &\quad +\int_0^T \int K_{\theta} \mathcal{A}_G h^2 dx dt =: \sum_{i=1}^9 J_i. \end{align}\tag{189}\]

See (D.16) in [37]. Then, \(J_i \, i=1,\ldots,7\) can be controlled as below.

\[\begin{align} \label{eq:Gaussian4} &\sum_{i=1}^7 J_i \le C \sup_{t\in[0,T]} \bigl\lVert(\phi,\zeta)\bigr\rVert_{L^2}^2 \nonumber\\ &\quad + C \sum_{i\in\{1,3\}} \delta_i \int_0^T \left| \dot{X}_i\right|^2 dt + (\nu+C\delta_0) \int_0^T \int \omega_G^2 \left| (\phi,\zeta)\right|^2 dx dt \nonumber\\ &\quad + C_{\nu} \int_0^T \bigl\lVert(\phi_x,\psi_x,\zeta_x)\bigr\rVert_{L^2}^2 dt + C\sum_{i\in\{1,3\}}\int_0^T \mathcal{G}_i^S dt +C\delta_0^{\frac{1}{3}}. \end{align}\tag{190}\]

Since \(h\) is bounded, \(J_8\le C \int_0^T \bigl\lVert\psi_x\bigr\rVert^2_{L^2} dt\). The remain part \(J_9\) is purely kinetic term.

\[\begin{align} \label{eq:Gaussian5} &J_9 \le \int_0^T \iint (\zeta+\bar p \phi) \left[\sum_{j=2}^3 \psi_k \xi_1 \xi_j \Pi_{1x} \right] d\xi dx dt \nonumber \\ &\quad -\int_0^T \iint (\zeta+ \bar p \phi) \left[\xi_1 \frac{\left| \xi\right|^2}{2} \widetilde{\Pi}_{1x} d\xi\right] d\xi dx dt \nonumber \\ &\quad +\int_0^T \iint (\zeta+\bar p \phi)\left(u_1 \xi_1^2 \Pi_{1x} - \sum_{i\in\{1,3\}} \bigl(u_1^{S_{i}}\bigr)^{-X_{i}} \xi_1^2 \partial_x\bigl(\Pi_1^{S_{i}}\bigr)^{-X_{i}} \right) d\xi dx dt\\ &\qquad \quad =:J_{91}+J_{92}+J_{93}. \end{align}\tag{191}\]

By the similar argument of Lemma 19, we have the following bounds.

\[\begin{align} \label{eq:Gaussian6} &J_9 \le C(\delta_0+\varepsilon) \sum_{i\in\{1,3\}} \delta_i \int_0^T \left| \dot{X}_i\right|^2 dt + C\delta_0 \int_0^T \int \omega_G^2 \left| (\phi,\psi,\zeta)\right|^2 dx dt \nonumber\\ &\qquad + C \int_0^T \bigl\lVert(\phi_x,\psi_x,\zeta_x)\bigr\rVert_{L^2}^2 dt + C\delta_0 \sum_{i\in\{1,3\}}\int_0^T \mathcal{G}_i^S dt \nonumber\\ &\qquad + C(\delta_0+\varepsilon)\int_0^T \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\, dx \,dt \nonumber\\ &\qquad + C\int_0^T \int \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \, dx\, dt +C\delta_0^{\frac{1}{3}}. \end{align}\tag{192}\]

Also, we can apply to \(H(\frac{2}{3}\zeta-\bar \phi)\) (\(\mathcal{B}_G:=\frac{2}{3}\zeta-\bar \phi\)) by the similar argument, we have the following estimate.

\[\begin{align} \label{eq:Gaussian7} & \int_0^T \int \omega_G^2 \left[\frac{1}{2v}\mathcal{B}_G^2 + \frac{4\bar p |\psi_1|^2}{3}\right] dx dt = \int H\psi_1\mathcal{B}_G|_{t=0}^{t=T} dx \nonumber \\ & \quad + \int_0^T \int v_x \frac{H}{2v^2}\mathcal{B}_G^2 dx dt - \int_0^T \int \psi\mathcal{B}_G H_t dx dt \nonumber \\ & \quad + \sum_{i\in\{1,3\}}\int_0^T \dot{X}_i \int H \left[\bar p \psi \partial_x\bigl(v^{S_{i}}\bigr)^{-X_{i}} -\psi \partial_x\bigl(\theta^{S_{i}}\bigr)^{-X_{i}} - \mathcal{B}_G \partial_x\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right] dx dt \nonumber\\ & \quad -\int_0^T \int \bar p_x H \frac{4}{3}\psi_1^2 dx dt +\int_0^T \int \frac{2}{3} H \psi\left[(p-\bar p)\psi_x+\bar u_x(p-\bar p)\right] dx dt \nonumber\\ & \quad +\int_0^T \int H\psi \phi \bar p_t dx dt + \int_0^T \int \frac{2}{3}H\psi_1 \left(\mu(\theta)\frac{u_{1x}^2}{v}-\mu(\bar{\theta})\frac{(\bar u_{1x})^2}{\bar v}\right) dx dt \nonumber\\ & \quad +\int_0^T \int \left(H\mathcal{B}_G\right)_x\left(\mu(\theta)\frac{u_x}{v}-\mu(\overline{\theta})\frac{\bar u_x}{\bar v}\right) dx dt \nonumber \\ & \quad+ \int_0^T \int \frac{2}{3} (H\psi)_x\left(\alpha_{\rm{th}}(\theta)\frac{\theta_x}{v}-\alpha_{\rm{th}}(\bar \theta)\frac{\bar \theta_x}{\bar v}\right) dx dt + \int_0^T \int \left[\frac{2}{3}HQ_2 + H\mathcal{B}_GQ_1\right] dx dt \nonumber\\ &\quad + \int_0^T \int \left[\frac{2}{3} H K_\theta + H\mathcal{B}_G K_u\right] dx dt \end{align}\tag{193}\] where \[\begin{align} K_u:= -\int \xi_1^2 \widetilde{\Pi}_{1x} d\xi. \end{align}\]

By using the result of [37] and Lemma 19, we have

\[\begin{align} \label{eq:Gaussian8} & \int_0^T \int \omega_G^2 \left[\frac{1}{2v}\mathcal{B}_G^2 + \frac{4\bar p |\psi_1|^2}{3}\right] dx dt \le C \sup_{t\in[0,T]} \bigl\lVert(\phi,\psi,\zeta)\bigr\rVert_{L^2}^2 \nonumber\\ &\quad + C \sum_{i\in\{1,3\}} \delta_i \int_0^T \left| \dot{X}_i\right|^2 dt + C \delta_0 \int_0^T \int \omega_G^2 \left| (\phi,\psi,\zeta)\right|^2 dx dt \nonumber\\ &\quad + C \int_0^T \bigl\lVert(\phi_x,\psi_x,\zeta_x)\bigr\rVert_{L^2}^2 dt + C(\delta_0+\varepsilon)\sum_{i\in\{1,3\}}\int_0^T \mathcal{G}_i^S dt \nonumber\\ &\qquad + C(\delta_0+\varepsilon)\int_0^T \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\, dx\, dt \nonumber\\ &\qquad + C\int_0^T \int \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx\, dt +C\delta_0^{\frac{1}{3}}. \end{align}\tag{194}\]

Finally, consider the \(\psi_{i},\, i=2,3\) cases. Multiply to 36 by \(\psi_ih^2\). \[\begin{align} \label{eq:Gaussian9} h^2\psi_i\psi_{it} = h^2\psi_i\left(\frac{\mu(\theta)\psi_{ix}}{v}\right)_x - h^2 \int \psi_i \xi_1 \xi_i \Pi_{1x} d\xi. \end{align}\tag{195}\]

Note that \[\begin{align} (\psi_i^2 \frac{h^2}{2})_t =& h^2\psi_i\psi_{it} + \frac{1}{4\beta}\omega_{Gx} \psi_i^2\\ =& h^2\psi_i\psi_{it} + \left[\frac{1}{4\beta}\psi_i^2 h \omega_G\right]_x -\frac{1}{4\beta}\psi_i^2 \omega_G^2 -\frac{1}{4\beta} (\psi_i^2)_x h \omega_G. \end{align}\]

Therefore, \[\begin{align} \label{eq:Gaussian10} &\frac{1}{4\beta}\int_0^T\int \psi_i^2 \omega_G^2 dx dt = \int \psi_i(0,x)^2 \frac{h_0^2}{2} dx \nonumber\\ &\qquad -\int \psi_i^2 \frac{h^2}{2}dx -\int_0^T \int \frac{1}{4\beta} (\psi_i^2)_x h\omega_G dx dt \nonumber \\ &\qquad - \int_0^T (\psi_i^2 h)_x \frac{\mu(\theta)\psi_{ix}}{v} dx dt - \int_0^T h^2 \int K_{gi} dx dt \end{align}\tag{196}\] where \[\begin{align} K_{gi}:= -\int \psi_i\xi_1 \xi_i \Pi_{1x} d\xi. \end{align}\]

By using Lemma 19, we have the following estimate.

\[\begin{align} \label{eq:Gaussian11} &\int_0^T\int \psi_i^2 \omega_G^2 dx dt \le C \sup_{t\in[0,T]} \bigl\lVert(\phi,\psi,\zeta)\bigr\rVert_{L^2}^2 \nonumber\\ &\qquad + C \sum_{i\in\{1,3\}} \delta_i \int_0^T \left| \dot{X}_i\right|^2 dt + C \delta_0 \int_0^T \int W^2 \left| (\phi,\psi,\zeta)\right|^2 dx dt \nonumber\\ &\qquad + C \int_0^T \bigl\lVert(\phi_x,\psi_x,\zeta_x)\bigr\rVert_{L^2}^2 dt + C(\delta_0+\varepsilon)\sum_{i\in\{1,3\}}\int_0^T \mathcal{G}_i^S dt \nonumber\\ &\qquad + C(\delta_0+\varepsilon)\int_0^T \int \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx \,dt \nonumber\\ &\qquad + C\int_0^T \int \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2+\bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 d\xi dx dt +C\delta_0^{\frac{1}{3}}. \end{align}\tag{197}\]

Collecting 190 , 192 , 194 , and 197 , Lemma 41 was proved. ◻

Proof of Proposition 1. We combine the zeroth-order estimate from Section 6, the differentiated estimates from Section 7, and the Gaussian-weighted estimate of Lemma 41.

For \(t\in[0,T]\), let \(\mathfrak{R}(t)^2\) denote the full left-hand side of ?? , with \(T\) replaced by \(t\). Since ?? does not explicitly contain the first-order time derivatives, we also introduce the auxiliary quantity \[\mathcal{E}_t(t)^2 := \int_0^t \|(\phi_t,\psi_t,\zeta_t)(s)\|_{L^2_x}^2\,ds .\] It is enough to prove \[\mathfrak{R}(t)^2+\mathfrak T(t)^2 \le C\bigl(\mathcal{E}(0)^2+\delta_0^{1/2}\bigr), \qquad 0\le t\le T.\]

We also set \[\mathcal{W}_C(t) := \int_0^t \frac{1}{1+s}\int_{\mathbb{R}} e^{-\frac{2c|x|^2}{1+s}}|(\phi,\psi,\zeta)(s,x)|^2\,dx\,ds ,\] and \[\mathcal{K}(t) := \int_0^t\int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx\,ds + \mathcal{K}_{\mathrm{high}}(t).\]

Step 1: Zeroth-order control. By Lemma 23, more precisely by ?? , the weighted relative entropy is equivalent to the \(L^2_x\)-norm of the macroscopic perturbation. In addition, 130 identifies the zeroth-order energy with \[\|(\phi,\psi,\zeta)(t)\|_{L^2_x}^2 + \int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{M_\#}^2\,dx .\] Moreover, by the definition of the dissipation \(\mathcal{D}_{\mathrm{mac}}(U)\) in 86 , together with the bootstrap bound \(\mathcal{E}(T)^2\le \varepsilon^2\) in 44 , one has \[\mathcal{D}_{\mathrm{mac}}(U)(t)\ge c\,\|(\psi_x,\zeta_x)(t)\|_{L^2_x}^2 .\] Therefore Proposition 6, namely ?? , yields \[\begin{align} \begin{aligned} &\sup_{0\le s\le t} \left\{ \|(\phi,\psi,\zeta)(s)\|_{L^2_x}^2 + \int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{M_\#}^2\,dx \right\} \\ &\quad + \int_0^t \sum_{i=1,3}\delta_i |\dot{X}_i(s)|^2\,ds + \int_0^t \sum_{i=1,3}\mathcal{G}_i^S(s)\,ds + \int_0^t \|(\psi_x,\zeta_x)(s)\|_{L^2_x}^2\,ds \\ &\quad + \int_0^t\int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx\,ds \\ &\le C\bigl(\mathcal{E}(0)^2+\delta_0^{1/2}\bigr) + C\delta_C\mathcal{W}_C(t) + C\int_0^t\int_{\mathbb{R}} \bigl\lVert\widetilde{G}_t\bigr\rVert_{\nu,M_\#}^2 + \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 \,dx\,ds . \end{aligned} \label{eq:priest-step1} \end{align}\tag{198}\]

Step 2: The low-order differentiated estimate. We next integrate ?? over \([0,t]\). The energy functional there contains the cross term \(\int_{\mathbb{R}} v\psi_1\phi_x\,dx\), but by the bootstrap bound \(\mathcal{E}(T)^2\le \varepsilon^2\) in 44 and Young’s inequality, \[\left|\int_{\mathbb{R}} v\psi_1\phi_x\,dx\right| \le \frac{1}{4}\|\phi_x\|_{L^2_x}^2 + C\|\psi_1\|_{L^2_x}^2 ,\] and the \(L^2_x\)-norm of \(\psi_1\) is already controlled by 198 . Using also the bound for \(\int_0^t \|(\psi_x,\zeta_x)(s)\|_{L^2_x}^2\,ds\) furnished by 198 , we obtain \[\begin{align} \begin{aligned} &\sup_{0\le s\le t} \left\{ \|\phi_x(s)\|_{L^2_x}^2 + \int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{M_\#}^2 \,dx \right\} + \mathcal{E}_t(t)^2 \\ &\quad + \int_0^t\int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2\,dx\,ds \\ &\le C\bigl(\mathcal{E}(0)^2+\delta_0^{1/2}\bigr) + C\varepsilon^2\int_0^t \|\psi_{1xx}(s)\|_{L^2_x}^2\,ds + C\delta_C \mathcal{W}_C(t) + C\mathcal{K}_{\mathrm{high}}(t). \end{aligned} \label{eq:priest-step2} \end{align}\tag{199}\]

Step 3: High-order closure. We now invoke Proposition 7, namely ?? . This estimate provides the remaining first-order spatial derivatives \((\psi_x,\zeta_x)\), the second-order macroscopic terms \[(\phi_{xx},\psi_{xx},\zeta_{xx},\phi_{xt},\psi_{xt},\zeta_{xt}),\] and the differentiated microscopic quantities \[\widetilde{G}_x,\qquad \widetilde{G}_t,\qquad \widetilde{G}_{xx},\qquad \widetilde{G}_{xt}.\] Its right-hand side depends only on lower-order quantities, namely \[\int_0^t \sum_{i=1,3}\delta_i |\dot{X}_i|^2\,ds,\qquad \int_0^t \sum_{i=1,3}\mathcal{G}_i^S\,ds,\qquad \int_0^t \sum_{|\beta|=1}\|\partial^\beta(\phi,\psi,\zeta)\|_{L^2_x}^2\,ds,\] and \[\int_0^t\int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{\text{rem}}\bigr\rVert_{\nu,M_\#}^2 \,dx\,ds ,\] which are precisely the quantities controlled by 198 and 199 . Therefore \[\begin{align} \begin{aligned} &\sup_{0\le s\le t} \left\{ \|(\phi_x,\psi_x,\zeta_x)(s)\|_{L^2_x}^2 + \int_{\mathbb{R}} \bigl\lVert\widetilde{G}_{x}\bigr\rVert_{M_\#}^2+\bigl\lVert\widetilde{G}_{t}\bigr\rVert_{M_\#}^2+\bigl\lVert\widetilde{G}_{xx}\bigr\rVert_{M_\#}^2+\bigl\lVert\widetilde{G}_{tx}\bigr\rVert_{M_\#}^2 \,dx \right\} \\ &\quad + \int_0^t \|(\phi_{xx},\psi_{xx},\zeta_{xx},\phi_{xt},\psi_{xt},\zeta_{xt})(s)\|_{L^2_x}^2\,ds \\ &\quad + \int_0^t\mathcal{K}_{\mathrm{high}}(s)\,ds \\ &\le C\bigl(\mathcal{E}(0)^2+\delta_0^{1/2}\bigr) + C\delta_C\mathcal{W}_C(t) + C(\delta_0+\varepsilon)\bigl(\mathfrak{R}(t)^2+\mathcal{E}_t(t)^2\bigr). \end{aligned} \label{eq:priest-step3} \end{align}\tag{200}\]

Step 4: Control of the Gaussian-weighted term. We now apply Lemma 41 on the interval \([0,t]\). Since the right-hand side of ?? contains only the lower-order quantities already controlled by 198 , 199 , and 200 , we obtain \[\delta_C\mathcal{W}_C(t) \le C\delta_C\bigl(\mathcal{E}(0)^2+\delta_0^{1/3}\bigr) + C\delta_C(\delta_0+\varepsilon)\bigl(\mathcal{E}(t)^2+\mathfrak T(t)^2\bigr). \label{eq:priest-step4}\tag{201}\]

Step 5: Final absorption. Combining 198 , 199 , 200 , and 201 , we find \[\mathfrak{R}(t)^2+\mathcal{E}_t(t)^2 \le C\bigl(\mathcal{E}(0)^2+\delta_0^{1/2}\bigr) + C\varepsilon^2\int_0^t \|\psi_{1xx}(s)\|_{L^2_x}^2\,ds + C(\delta_0+\varepsilon)\bigl(\mathfrak{R}(t)^2+\mathcal{E}_t(t)^2\bigr).\] Since \(\int_0^t \|\psi_{1xx}(s)\|_{L^2_x}^2\,ds\) is already one of the components of \(\mathfrak{R}(t)^2\), it follows that \[\mathfrak{R}(t)^2+\mathcal{E}_t(t)^2 \le C\bigl(\mathcal{E}(0)^2+\delta_0^{1/2}\bigr) + C(\delta_0+\varepsilon)\bigl(\mathfrak{R}(t)^2+\mathcal{E}_t(t)^2\bigr).\] Choosing \(\delta_0>0\) and \(\varepsilon>0\) sufficiently small, we absorb the last term into the left-hand side and conclude that \[\mathfrak{R}(t)^2+\mathcal{E}_t(t)^2 \le C\bigl(\mathcal{E}(0)^2+\delta_0^{1/2}\bigr), \qquad 0\le t\le T.\] In particular, \[\mathfrak{R}(t)^2 \le C\bigl(\mathcal{E}(0)^2+\delta_0^{1/2}\bigr), \qquad 0\le t\le T.\] Taking \(t=T\), and recalling the definition of \(\mathfrak{R}(T)^2\), we obtain exactly ?? . This proves Proposition 1. ◻

Proof. For a single 3-shock, we set \(\delta_1=\delta_C=0\). Then the interaction estimates, as well as the error terms generated by the cutoff functions \(\varphi_i\) (See 41 ) introduced to separate the two shocks in the a-priori estimates ?? , disappear. In particular, the term \(\delta_0^{1/2}\) on the right hand side of the composite-wave a priori estimate ?? from the wave interactions, and therefore it is absent in the single-shock case. In addition, the zeroth perturbed microscopic variable \(\widetilde{G}_{\mathrm{rem}}=\widetilde{G} = G-\bigl(G^S\bigr)^{-X}\) since there is no contact discontinuity. All the remaining estimates follow directly from the estimates established in the previous section. ◻

Proof. The estimate ?? is precisely the exponential convergence of the travelling shock profile to its end Maxwellians.

To prove the pointwise estimate ?? , fix \((\tau,y)\in[0,T]\times\mathbb{R}\), and set \[z:=\frac{y-s\tau-X^\kappa(\tau)}{\kappa}.\] Then \[\bar f^\kappa(\tau,y,\xi)=F^S(z,\xi).\]

If \(z\le0\), then by the definition of \(M_{E,\kappa}\), \[M_{E,\kappa}(\tau,y,\xi)=M_-(\xi),\] and therefore, using ?? with the minus sign, \[\left\| \bar f^\kappa(\tau,y,\cdot)-M^{E,\kappa}(\tau,y,\cdot) \right\|_{M_\#} = \|F^S(z,\cdot)-M_-\|_{M_\#} \le Ce^{-c|z|}.\]

If \(z>0\), then \[M_{E,\kappa}(\tau,y,\xi)=M_+(\xi),\] and ?? with the plus sign gives \[\left\| \bar f^\kappa(\tau,y,\cdot)-M_{E,\kappa}(\tau,y,\cdot) \right\|_{M_\#} = \|F^S(z,\cdot)-M_+\|_{M_\#} \le Ce^{-c|z|}.\]

Since \[|z|=\frac{|y-s\tau-X^\kappa(\tau)|}{\kappa},\] the two cases combine to yield ?? .

Finally, if \[|y-s\tau-X^\kappa(\tau)|\ge h,\] then ?? implies \[\left\| \bar f^\kappa(\tau,y,\cdot)-M_{E,\kappa}(\tau,y,\cdot) \right\|_{M_\#} \le Ce^{-ch/\kappa}.\] Taking the supremum over all such \((\tau,y)\) gives ?? . ◻

Proof of Theorem 3. For the single-shock problem, define \[\bar U^\kappa(\tau,y) := U^S\!\left(\frac{y-s\tau-X^\kappa(\tau)}{\kappa}\right), \qquad \bar M^\kappa(\tau,y,\xi) := M[\bar U^\kappa(\tau,y)](\xi),\] and \[\bar f^\kappa(\tau,y,\xi) := F^S\!\left(\frac{y-s\tau-X^\kappa(\tau)}{\kappa},\xi\right).\] By the decomposition \(F^S=M[U^S]+G^S\), we have \[\bar f^\kappa(\tau,y,\xi) = \bar M^\kappa(\tau,y,\xi) + G^S\!\left(\frac{y-s\tau-X^\kappa(\tau)}{\kappa},\xi\right).\]

Let \[U^\kappa=(v^\kappa,u^\kappa,\theta^\kappa), \qquad (\phi^\kappa,\psi^\kappa,\zeta^\kappa) := U^\kappa-\bar U^\kappa .\] Hence \[f^\kappa-\bar f^\kappa = \bigl(M[U^\kappa]-\bar M^\kappa\bigr) + \widetilde{G}^\kappa . \label{eq:single-shock-away-pointwise-decomp}\tag{206}\]

By Proposition 8, for every fixed \(T>0\), \[\mathcal{E}^\kappa(T)^2\le C\,\mathcal{E}^\kappa(0)^2 .\] Recalling the definition of \(\mathcal{E}^\kappa(T)\), and using \(\widetilde{G}^\kappa=\widetilde{G}_1^\kappa\), we infer that \[\sup_{0\le\tau\le T} \|(\phi^\kappa,\psi^\kappa,\zeta^\kappa)(\tau)\|_{H^1_y} \le C\,\mathcal{E}^\kappa(0), \label{eq:single-shock-away-H1-fluid}\tag{207}\] and \[\sup_{0\le\tau\le T} \|\widetilde{G}_1^\kappa(\tau,\cdot,\cdot)\|_{H^1_y(L^2_\xi(M_\#))} \le C\,\mathcal{E}^\kappa(0). \label{eq:single-shock-away-H1-micro}\tag{208}\]

Applying Lemma 46 with \(H=\mathbb{R}^3\) to \((\phi^\kappa,\psi^\kappa,\zeta^\kappa)\), and with \(H=L^2_\xi(M_\#^{-1})\) to \(\widetilde{G}^\kappa\), we obtain \[\sup_{0\le\tau\le T}\sup_{y\in\mathbb{R}} |U^\kappa(\tau,y)-\bar U^\kappa(\tau,y)| \le C\,\mathcal{E}^\kappa(0), \label{eq:single-shock-away-Linfty-fluid}\tag{209}\] and \[\sup_{0\le\tau\le T}\sup_{y\in\mathbb{R}} \|\widetilde{G}^\kappa(\tau,y,\cdot)\|_{M_\#} \le C\,\mathcal{E}^\kappa(0). \label{eq:single-shock-away-Linfty-micro}\tag{210}\]

Next we compare the Maxwellian parts. Since the single shock profile \(U^S\) connects the fixed end states \(U_-\) and \(U_+\), its range is contained in a compact set \[K_S\Subset \mathbb{R}_+\times\mathbb{R}\times\mathbb{R}_+ .\] By 209 , and after possibly shrinking the bootstrap constant \(\kappa_0\), the range of \(U^\kappa\) on \([0,T]\times\mathbb{R}\) is contained in a fixed compact set \[K\Subset \mathbb{R}_+\times\mathbb{R}\times\mathbb{R}_+\] depending only on \(K_S\) and \(\kappa_0\). Therefore Lemma 47 applies uniformly on \(K\), and yields \[\sup_{0\le\tau\le T}\sup_{y\in\mathbb{R}} \|M[U^\kappa(\tau,y)]-\bar M^\kappa(\tau,y)\|_{M_\#} \le C\,\mathcal{E}^\kappa(0). \label{eq:single-shock-away-Maxwellian}\tag{211}\]

Combining 206 , 210 , and 211 , we obtain the uniform estimate \[\|f^\kappa(\tau,y,\cdot)-\bar f^\kappa(\tau,y,\cdot)\|_{M_\#} \le C\,\mathcal{E}^\kappa(0), \qquad (\tau,y)\in[0,T]\times\mathbb{R}. \label{eq:single-shock-away-pointwise-step1}\tag{212}\]

On the other hand, Lemma 42 gives the pointwise exponential tail estimate \[\|\bar f^\kappa(\tau,y,\cdot)-M_{E,\kappa}(\tau,y,\cdot)\|_{M_\#} \le C e^{-c|y-s\tau-X^\kappa(\tau)|/\kappa}, \qquad (\tau,y)\in[0,T]\times\mathbb{R}. \label{eq:single-shock-away-pointwise-step2}\tag{213}\]

Therefore, by the triangle inequality,

\[\begin{align} & \left\| f^\kappa(\tau,y,\cdot)-M_{E,\kappa}(\tau,y,\cdot) \right\|_{M_\#} \\ &\quad \le \left\| f^\kappa(\tau,y,\cdot)-\bar f^\kappa(\tau,y,\cdot) \right\|_{M_\#} + \left\| \bar f^\kappa(\tau,y,\cdot)-M_{E,\kappa}(\tau,y,\cdot) \right\|_{M_\#} \\ &\quad \le C\,\mathcal{E}^\kappa(0) + C e^{-c|y-s\tau-X^\kappa(\tau)|/\kappa}\\ &\quad \le C\, \kappa \mathcal{E}(0) + C e^{-c|y-s\tau-X^\kappa(\tau)|/\kappa} \end{align}\]

which proves ?? . ◻

Proof of Theorem 3. For the single-shock problem, define \[\bar U^\kappa(\tau,y) := U^S\!\left(\frac{y-s\tau-X^\kappa(\tau)}{\kappa}\right), \qquad \bar M^\kappa(\tau,y,\xi) := M[\bar U^\kappa(\tau,y)](\xi),\] and \[\bar f^\kappa(\tau,y,\xi) := F^S\!\left(\frac{y-s\tau-X^\kappa(\tau)}{\kappa},\xi\right).\] Then \[\bar f^\kappa(\tau,y,\xi) = \bar M^\kappa(\tau,y,\xi) + G^S\!\left(\frac{y-s\tau-X^\kappa(\tau)}{\kappa},\xi\right).\]

Let \[U^\kappa=(v^\kappa,u^\kappa,\theta^\kappa), \qquad (\phi^\kappa,\psi^\kappa,\zeta^\kappa) := U^\kappa-\bar U^\kappa .\] Since there is no contact component in the present setting, \[\widetilde{G}_0^\kappa\equiv0, \qquad \widetilde{G}^\kappa=\widetilde{G}_1^\kappa,\] and therefore \[f^\kappa-\bar f^\kappa = \bigl(M[U^\kappa]-\bar M^\kappa\bigr) + \widetilde{G}_1^\kappa . \label{eq:single-shock-away-pointwise-decomp1}\tag{214}\]

By Proposition 8, for every fixed \(T>0\), \[\mathcal{E}^\kappa(T)^2\le C\,\mathcal{E}^\kappa(0)^2.\] Hence \[\sup_{0\le\tau\le T} \|(\phi^\kappa,\psi^\kappa,\zeta^\kappa)(\tau)\|_{H^1_y} \le C\,\mathcal{E}^\kappa(0),\] and \[\sup_{0\le\tau\le T} \|\widetilde{G}_1^\kappa(\tau,\cdot,\cdot)\|_{H^1_y(L^2_\xi(M_\#^{-1}))} \le C\,\mathcal{E}^\kappa(0).\] By Lemma 46, it follows that \[\sup_{0\le\tau\le T}\sup_{y\in\mathbb{R}} |U^\kappa(\tau,y)-\bar U^\kappa(\tau,y)| \le C\,\mathcal{E}^\kappa(0),\] and \[\sup_{0\le\tau\le T}\sup_{y\in\mathbb{R}} \|\widetilde{G}_1^\kappa(\tau,y,\cdot)\|_{L^2_\xi(M_\#^{-1})} \le C\,\mathcal{E}^\kappa(0).\] By the local Lipschitz continuity of the Maxwellian map on compact positive sets, \[\sup_{0\le\tau\le T}\sup_{y\in\mathbb{R}} \|M[U^\kappa(\tau,y)]-\bar M^\kappa(\tau,y)\|_{L^2_\xi(M_\#^{-1})} \le C\,\mathcal{E}^\kappa(0).\] Combining these bounds with 214 , we obtain the uniform estimate \[\|f^\kappa(\tau,y,\cdot)-\bar f^\kappa(\tau,y,\cdot)\|_{L^2_\xi(M_\#^{-1})} \le C\,\mathcal{E}^\kappa(0), \qquad (\tau,y)\in[0,T]\times\mathbb{R}. \label{eq:single-shock-away-pointwise-step1-1}\tag{215}\]

Next, Lemma 42 gives the pointwise exponential tail estimate \[\|\bar f^\kappa(\tau,y,\cdot)-M^{E,\kappa}(\tau,y,\cdot)\|_{L^2_\xi(M_\#^{-1})} \le C e^{-c|y-s\tau-X^\kappa(\tau)|/\kappa}, \qquad (\tau,y)\in[0,T]\times\mathbb{R}. \label{eq:single-shock-away-pointwise-step2-1}\tag{216}\]

Therefore, by the triangle inequality, \[\begin{align} \left\| f^\kappa(\tau,y,\cdot)-M^{E,\kappa}(\tau,y,\cdot) \right\|_{L^2_\xi(M_\#^{-1})} &\le \left\| f^\kappa(\tau,y,\cdot)-\bar f^\kappa(\tau,y,\cdot) \right\|_{L^2_\xi(M_\#^{-1})} \\ &\quad + \left\| \bar f^\kappa(\tau,y,\cdot)-M^{E,\kappa}(\tau,y,\cdot) \right\|_{L^2_\xi(M_\#^{-1})} \\ &\le C\,\mathcal{E}^\kappa(0) + C e^{-c|y-s\tau-X^\kappa(\tau)|/\kappa}, \end{align}\] which proves ?? .

Finally, if \((\tau,y)\in\Omega^\kappa_{h,T}\), then \[|y-s\tau-X^\kappa(\tau)|\ge h,\] so ?? immediately yields [eq:single-shock-away-modulated]. ◻

Proof of Theorem 3. For the single-shock problem, define \[\bar U^\kappa(\tau,y) := U^S\!\left(\frac{y-s\tau-X^\kappa(\tau)}{\kappa}\right), \qquad \bar M^\kappa(\tau,y,\xi) := M[\bar U^\kappa(\tau,y)](\xi),\] and \[\bar f^\kappa(\tau,y,\xi) := F^S\!\left(\frac{y-s\tau-X^\kappa(\tau)}{\kappa},\xi\right).\] Then \[\bar f^\kappa(\tau,y,\xi) = \bar M^\kappa(\tau,y,\xi) + G^S\!\left(\frac{y-s\tau-X^\kappa(\tau)}{\kappa},\xi\right).\]

Let \[U^\kappa=(v^\kappa,u^\kappa,\theta^\kappa)\] denote the macroscopic field associated with \(f^\kappa\), and set \[(\phi^\kappa,\psi^\kappa,\zeta^\kappa) := U^\kappa-\bar U^\kappa .\] Since there is no contact component in the present setting, we have \[\widetilde{G}_0^\kappa\equiv0, \qquad \widetilde{G}^\kappa=\widetilde{G}_1^\kappa .\] Hence \[f^\kappa-\bar f^\kappa = \bigl(M[U^\kappa]-\bar M^\kappa\bigr) + \widetilde{G}_1^\kappa . \label{eq:single-shock-away-decomp}\tag{217}\]

By Proposition 8, for every fixed \(T>0\), \[\mathcal{E}^\kappa(T)^2\le C\,\mathcal{E}^\kappa(0)^2 .\] Recalling the definition of \(\mathcal{E}^\kappa(T)\), and using \(\widetilde{G}^\kappa=\widetilde{G}_1^\kappa\), we infer that \[\sup_{0\le\tau\le T} \|(\phi^\kappa,\psi^\kappa,\zeta^\kappa)(\tau)\|_{H^1_y} \le C\,\mathcal{E}^\kappa(0), \label{eq:single-shock-away-H1-fluid1}\tag{218}\] and \[\sup_{0\le\tau\le T} \|\widetilde{G}_1^\kappa(\tau,\cdot,\cdot)\|_{H^1_y(L^2_\xi(M_\#^{-1}))} \le C\,\mathcal{E}^\kappa(0). \label{eq:single-shock-away-H1-micro1}\tag{219}\]

Applying Lemma 46 with \(H=\mathbb{R}^3\) to \((\phi^\kappa,\psi^\kappa,\zeta^\kappa)\), and with \(H=L^2_\xi(M_\#^{-1})\) to \(\widetilde{G}_1^\kappa\), we obtain \[\sup_{0\le\tau\le T}\sup_{y\in\mathbb{R}} |U^\kappa(\tau,y)-\bar U^\kappa(\tau,y)| \le C\,\mathcal{E}^\kappa(0), \label{eq:single-shock-away-Linfty-fluid1}\tag{220}\] and \[\sup_{0\le\tau\le T}\sup_{y\in\mathbb{R}} \|\widetilde{G}_1^\kappa(\tau,y,\cdot)\|_{L^2_\xi(M_\#^{-1})} \le C\,\mathcal{E}^\kappa(0). \label{eq:single-shock-away-Linfty-micro1}\tag{221}\]

Next we compare the Maxwellian parts. Since the single shock profile \(U^S\) connects the fixed end states \(U_-\) and \(U_+\), its range is contained in a compact set \[K_S\Subset \mathbb{R}_+\times\mathbb{R}\times\mathbb{R}_+ .\] By 220 , and after possibly shrinking the bootstrap constant \(\kappa_0\), the range of \(U^\kappa\) on \([0,T]\times\mathbb{R}\) is contained in a fixed compact set \[K\Subset \mathbb{R}_+\times\mathbb{R}\times\mathbb{R}_+\] depending only on \(K_S\) and \(\kappa_0\). Therefore Lemma 47 applies uniformly on \(K\), and yields \[\sup_{0\le\tau\le T}\sup_{y\in\mathbb{R}} \|M[U^\kappa(\tau,y)]-\bar M^\kappa(\tau,y)\|_{L^2_\xi(M_\#^{-1})} \le C\,\mathcal{E}^\kappa(0). \label{eq:single-shock-away-Maxwellian1}\tag{222}\]

Combining 217 , 221 , and 222 , we obtain \[\sup_{0\le\tau\le T}\sup_{y\in\mathbb{R}} \|f^\kappa(\tau,y,\cdot)-\bar f^\kappa(\tau,y,\cdot)\|_{L^2_\xi(M_\#^{-1})} \le C\,\mathcal{E}^\kappa(0). \label{eq:single-shock-away-step1}\tag{223}\]

On the other hand, Lemma 42 yields \[\sup_{(\tau,y)\in\Omega^\kappa_{h,T}} \|\bar f^\kappa(\tau,y,\cdot)-M^{E,\kappa}(\tau,y,\cdot)\|_{L^2_\xi(M_\#^{-1})} \le Ce^{-ch/\kappa}. \label{eq:single-shock-away-step2}\tag{224}\]

Therefore, by the triangle inequality, \[\|f^\kappa-M^{E,\kappa}\|_{L^2_\xi(M_\#^{-1})} \le \|f^\kappa-\bar f^\kappa\|_{L^2_\xi(M_\#^{-1})} + \|\bar f^\kappa-M^{E,\kappa}\|_{L^2_\xi(M_\#^{-1})}.\] Taking the supremum over \((\tau,y)\in\Omega^\kappa_{h,T}\), and using 223 and 224 , yields [eq:single-shock-away-modulated]. Since \(\mathcal{E}^\kappa(0)\to0\) and \(e^{-ch/\kappa}\to0\) for every fixed \(h>0\), the convergence statement follows. ◻

Proof. The proof is the same as that in [65]. ◻

Proof of Corollary 3. By Proposition 9, together with the same local existence and continuation argument used in the proof of Theorem 2, there exists a unique global normalized solution in the moving frame, together with a shift \(X_3=X_3(t)\).

Let \(f\) denote this normalized solution, and define the rescaled physical solution and shift by 51 and 52 . Equivalently, \[f^\kappa(\tau,y,\xi)=f\!\left(\frac{\tau}{\kappa},\frac{y}{\kappa},\xi\right), \qquad X_3^\kappa(\tau)=\kappa X_3\!\left(\frac{\tau}{\kappa}\right).\] Then \(f^\kappa\) exists uniquely on \([0,T]\) for every \(T>0\), since \(f\) exists globally on \([0,\infty)\).

We first compare \(f^\kappa\) with the rescaled composite approximate wave \(\bar f^{\,\kappa}\). By Proposition 9, after the same scaling argument as in the proof of Theorem 2, we obtain the perturbation estimate \[\begin{align} \int_0^T\int_{\mathbb{R}} \bigl\lVert f^\kappa-\bar f^{\kappa}\bigr\rVert_{M_\#}^2 \,dy\,d\tau \le C\kappa\bigl((\varepsilon_1+\delta_{\mathrm{res},0})^2+\delta_{\mathrm{res},0}^{1/2}\bigr)T . \label{eq:main-step-1-rare-cor} \end{align}\tag{227}\]

Next we prove compactness of the rescaled shift family. By 52 , \[\frac{d}{d\tau}X_3^\kappa(\tau) = \dot{X}_3\!\left(\frac{\tau}{\kappa}\right)\] for a.e. \(\tau\in(0,T)\), and therefore \[\operatorname{TV}(X_3^\kappa;[0,T]) = \int_0^T \left|\frac{d}{d\tau}X_3^\kappa(\tau)\right|\,d\tau = \kappa \int_0^{T/\kappa} |\dot{X}_3(t)|\,dt.\] By Cauchy–Schwarz and Proposition 9, \[\operatorname{TV}(X_3^\kappa;[0,T]) \le \kappa^{1/2}T^{1/2} \left(\int_0^{T/\kappa} |\dot{X}_3(t)|^2\,dt\right)^{1/2} \le C_T \delta_S^{-1/2}.\] Hence \(\{X_3^\kappa\}_{\kappa>0}\) is uniformly bounded in \(BV(0,T)\), and thus, after extraction of a subsequence if necessary, there exists \(X_3^0\in BV([0,T])\) such that \[X_3^\kappa \to X_3^0 \qquad\text{in }L^1(0,T).\]

We now compare the rescaled composite wave with the shifted Euler Maxwellian profile. Combining the smooth rarefaction approximation estimate from Lemma 43 with the corresponding contact-wave estimate, we obtain \[\begin{align} \int_0^T\int_{\mathbb{R}} \bigl\lVert\bar f^{\,\kappa}-M_{X_3^\kappa}[U^E]\bigr\rVert_{M_\#}^2\,dy\,d\tau \le C(\delta_R+\delta_C)\kappa^{1/2}T^{3/2}+C\delta_S\kappa T . \label{eq:main-step-2-rare-cor} \end{align}\tag{228}\]

Finally, by the mean value theorem and the exponential decay of the Euler shock profile, for each \(\tau\in[0,T]\), \[\begin{align} \int_{\mathbb{R}} \bigl\lVert M_{X_3^\kappa}[U^E]-M_{X_3^0}[U^E]\bigr\rVert_{M_\#}^2 \,dy \le C\delta_{\mathrm{res},0}^2|X_3^\kappa(\tau)-X_3^0(\tau)|. \end{align}\] Integrating in \(\tau\) yields \[\begin{align} \int_0^T\int_{\mathbb{R}} \bigl\lVert M_{X_3^\kappa}[U^E]-M_{X_3^0}[U^E]\bigr\rVert_{M_\#}^2 \,dy\,d\tau \le C\delta_{\mathrm{res},0}^2\|X_3^\kappa-X_3^0\|_{L^1(0,T)}. \label{eq:main-step-3-rare-cor} \end{align}\tag{229}\]

Combining 227 , 228 , and 229 , and using that \(M_\#\) is uniformly bounded from above and below by positive constants, we conclude that \[\begin{align} &\int_0^T \int_{\mathbb{R}} \bigl\lVert f^\kappa-M_{X_3^0}[U^E]\bigr\rVert_{M_\#}^2 \,dy\,d\tau\\ &\qquad \le C\kappa\bigl((\varepsilon_1+\delta_{\mathrm{res},0})^2+\delta_{\mathrm{res},0}^{1/2}\bigr)T + C(\delta_R+\delta_C)\kappa^{1/2}T^{3/2} + C\delta_{\mathrm{res},0}^2\|X_3^\kappa-X_3^0\|_{L^1(0,T)}. \end{align}\] This is exactly ?? . ◻

Proof. It is enough to prove ?? for \(g\in C_c^\infty(\mathbb{R}_y;H)\), the general case following by density. For such \(g\), \[\frac{d}{dy}\|g(y)\|_H^2 = 2\langle g_y(y),g(y)\rangle_H .\] Hence, for every \(y\in\mathbb{R}\), \[\|g(y)\|_H^2 = 2\int_{-\infty}^y \langle g_z(z),g(z)\rangle_H\,dz \le 2\int_{\mathbb{R}}\|g_z(z)\|_H\,\|g(z)\|_H\,dz .\] By Cauchy–Schwarz, \[\|g(y)\|_H^2 \le 2\|g_y\|_{L^2_y(H)}\|g\|_{L^2_y(H)} .\] Taking the supremum in \(y\) proves ?? . ◻

Proof. Let \(U,V\in K\), and define \(U_\vartheta:=(1-\vartheta)V+\vartheta U\), \(\vartheta\in[0,1]\). Enlarging \(K\) slightly if necessary, we may assume that the whole segment \(\{U_\vartheta:\vartheta\in[0,1]\}\) lies in a compact set \(K'\Subset \mathbb{R}_+\times\mathbb{R}\times\mathbb{R}_+\). By the fundamental theorem of calculus, \[M[U]-M[V] = \int_0^1 DM[U_\vartheta](U-V)\,d\vartheta .\] Hence \[\|M[U]-M[V]\|_{M_\#} \le |U-V| \sup_{W\in K'} \|DM[W]\|_{\mathcal{L}(\mathbb{R}^3,L^2_\xi(M_\#))}.\] It therefore suffices to bound \(DM[W]\) uniformly on \(K'\). Each partial derivative \(\partial_{W_j}M[W]\) is of the form \(P_j(\xi;W)M[W](\xi)\), where \(P_j\) is a polynomial of degree at most \(2\) in \(\xi\), with coefficients depending continuously on \(W\). Since \(K'\) is compact and bounded away from vacuum and zero temperature, there exist positive constants \(C,c\) such that \[\frac{M[W](\xi)^2}{M_\#(\xi)} \le Ce^{-c|\xi|^2}, \qquad W\in K',\;\xi\in\mathbb{R}^3.\] Therefore \[\sup_{W\in K'} \sum_j \|\partial_{W_j}M[W]\|_{M_\#} <\infty,\] which implies ?? . ◻

Proof. We do not reproduce the proof of the corresponding estimates for \(3\)-shock wave here, since it is essentially the same as the proof given in [38]. In particular, the estimates for the \(3\)-shock profile will be used in the form established in [20]. Our purpose in the above discussion is only to make explicit the scalar parametrization and the auxiliary estimates from [38], which are invoked in the proof of [20]. We first recall the scalr parametrization of a Boltzmann shock profile. By (295) in [20], there exists a scalar function \(\eta=\eta(y)\) such that \[\label{eq:scalar-param-zeta} \left\{ \begin{align} \eta_y &= a_0(\eta)\bigl(\eta-\eta_-\bigr)\bigl(\eta-\eta_+\bigr), \\ \eta(-\infty) &= \eta_-, \qquad \eta(+\infty) = \eta_+, \end{align} \right.\tag{231}\] where \(\eta_->0>\eta_+\), \(|\eta_- - \eta_+| = O(\delta_3)\), and \(a_0\) is a smooth function satisfying \(a_0(\eta)\ge c>0\). The elementary consequences of 231 , including the monotonicity, exponential decay, the higher derivative estimates of \(\eta\), are obtained in [20]. In particular, we shall use \[\label{eq:zeta-basic-estimates} \eta_y<0, \qquad |\eta-\eta_\pm| \lesssim \delta_3 e^{-c \delta_3 |y|} \quad \text{as } y\to \pm \infty,\tag{232}\] and \[\label{eq:zeta-derivative-estimates} |\eta_y| \lesssim \delta_3^2 e^{-c\delta_3 |y|}, \qquad |\partial_y^k \eta| \lesssim \delta_3^{k-1}|\eta_y|, \qquad k\ge 2.\tag{233}\] by (295) in [20] and Lemma A.1 in [38]. Under this parametrization, the macroscopic part of the shock profile satisfies \[\label{eq:macro-zeta-comparison} \bigl|v_y^{S_3}\bigr| \sim \bigl|u_{1y}^{S_3}\bigr| \sim \bigl| \theta_y^{S_3}\bigr| \sim |\eta_y|.\tag{234}\] Moreover, by (289) and (291) in [20], the microscopic component can be represented in the form \[G^{S_3}(y,\xi) = \sqrt{M_\#} \, \Gamma(\chi(\eta(y)),\xi),\] where \(M_\#\) is a fixed global Maxwellian, \(\Gamma\) is smooth and \(\chi=\chi(\eta)\) is determined by \[\label{eq:chi-equation} \chi(\eta) = b(\eta,\chi(\eta)) (\eta- \eta_-) (\eta-\eta_+),\tag{235}\] where \(b\) is smooth and uniformly bounded above and below by postivie constants. Consequently, \[\label{eq:G-chi-relation} \Bigl( \int_{\mathbb{R}^3} \frac{(1+|\xi|)\bigl|G^{S_3}(y,\xi)\bigr|^2}{M_\#}\, d\xi \Bigr)^{\frac{1}{2}} = |\chi(\eta(y))|\, c_0 (\chi(\eta(y))),\tag{236}\] where \(c_0\) is smooth and satisfies \(c_0\ge c>0\).

By referring to [38], we obtain the corresponding estimate for the \(3\)-shock. We prove the \(1\)-shock estimate by using the reflection symmetry from the corresponding \(3\)-shock estimate. Let \[\begin{align} U^{S_1} = (v^{S_1},u^{S_1},\theta^{S_1}) \end{align}\] be the \(1\)-shock profile connecting \(U_-=(v_-,u_-,\theta_-)\) to \(U_*=(v_*,u_*,\theta_*)\). Define the reflected profiles by \[\begin{align} \widetilde{v}^{S_3}(z) := v^{S_1}(-z), \quad \widetilde{u}^{S_3}(z) := -u^{S_1}(-z), \quad \widetilde{\theta}^{S_3}(z) := \theta^{S_1}(-z). \end{align}\] Then \(\widetilde{U}^{S_3}\) is a \(3\)-shock profile connecting \(\widetilde{U}^*=(v_*,-u_*,\theta_*)\) to \(\widetilde{U}_+=(v_-,-u_-,\theta_-)\). In particular, we denote \(\widetilde{v}^*=v_*\), \(\widetilde{v}_+=v_-\), \(\widetilde{\theta}^*= \theta_*\), \(\widetilde{p}^*=p_*\) and \(\widetilde{p}_+=p_-\). Note that \(\widetilde{p}^{S_3}(z) = p^{S_1}(-z)\) since \(p=p(v,\theta)\). Moreover the shock strength is preserved under reflection, namely, \(\widetilde{\delta}_3=\delta_1\). Applying the \(3\)-shock estimate to the reflected profile gives \[\begin{align} &\Biggl|\frac{\widetilde{p}^{S_3}-\widetilde{p}_+}{\widetilde{v}^{S_3}-\widetilde{v}_+}- \frac{\widetilde{p}^{S_3}-\widetilde{p}^*}{\widetilde{v}^{S_3}-\widetilde{v}^*}-\frac{5\widetilde{p}^*}{9\bigl(\widetilde{v}^*\bigr)^2}\Bigl(\frac{10\mu(\widetilde{\theta}^*)-9\alpha_{\rm th}(\widetilde{\theta}^*)}{10\mu(\widetilde{\theta}^*)+3\alpha_{\rm th}(\widetilde{\theta}^*)}+3\Bigr)\Biggr| \le C\widetilde{\delta}_3^2. \end{align}\] Using the above identifications, this becomes \[\begin{align} &\Biggl|\frac{p^{S_1}-p_-}{v^{S_1}-v_-}- \frac{p^{S_1}-p_*}{v^{S_1}-v_*}-\frac{5p_*}{9\bigl(v_*\bigr)^2}\Bigl(\frac{10\mu(\theta_*)-9\alpha_{\rm th}(\theta_*)}{10\mu(\theta_*)+3\alpha_{\rm th}(\theta_*)}+3\Bigr)\Biggr| \le C\delta_1^2. \end{align}\] This proves the desired \(1\)-shock estimate. The remaining estimates for \(1\)-shock can be derived in a similar manner. ◻

Proof. The proof follows a similar argument to that in [34]. We provide the details for completeness. Let \[x_1(t):=\sigma_1 t+X_1(t), \qquad x_3(t):=\sigma_3 t+X_3(t).\] By the exponential localization of each viscous shock profile, there exist positive constants \(c\) and \(C\) such that \[\label{eq:shock-tail-proof} \bigl|\partial_x(v^{S_i})^{-X_i}(t,x)\bigr| \le C\delta_i^2 e^{-c\delta_i|x-x_i(t)|}, \qquad i=1,3.\tag{237}\] Moreover, in the present \(1\)-shock/\(3\)-shock configuration, the two shifted shock locations remain separated linearly in time. More precisely, after possibly decreasing \(\delta_0>0\), we may assume that there exists a constant \(c_1,c_3>0\) such that \[\label{eq:shock-separation-proof} x_1(t)\le -c_1 t, \qquad x_3(t)\ge c_3 t, \qquad t>0.\tag{238}\]

We first prove the estimate for \(\varphi_3 |(v^{S_1})_x^{-X_1}|\). By the definition of \(\varphi_1\) and \(\varphi_3\), if \(\varphi_3(t,x)\neq0\), then \[x\ge \frac{x_1(t)}{2}.\] Since \(x_1(t)<0\) by 238 , it follows that \[x-x_1(t) \ge -\frac{x_1(t)}{2} \ge \frac{c_1}{2}\,t>0.\] Hence, by 237 , \[\varphi_3(t,x)\bigl|\partial_x(v^{S_1})^{-X_1}(t,x)\bigr| \le C\delta_1^2 e^{-c\delta_1|x-x_1(t)|} \le C\delta_1^2 e^{-C\delta_1 t}.\] This proves the first estimate in ?? .

Similarly, if \(\varphi_1(t,x)\neq0\), then \[x\le \frac{x_3(t)}{2}.\] Since \(x_3(t)>0\) by 238 , we obtain \[x_3(t)-x \ge \frac{x_3(t)}{2} \ge \frac{c_3}{2}\,t>0.\] Using again 237 , we get \[\varphi_1(t,x)\bigl|\partial_x(v^{S_3})^{-X_3}(t,x)\bigr| \le C\delta_3^2 e^{-C\delta_3 t},\] which proves the second pointwise estimate.

We next prove the integral estimates. By the pointwise bound just proved, \[\int_{\mathbb{R}}\varphi_3(t,x)\bigl|\partial_x(v^{S_1})^{-X_1}(t,x)\bigr|\,dx \le C\delta_1^2 \int_{\{x\ge x_1(t)/2\}} e^{-c\delta_1|x-x_1(t)|}\,dx.\] Since \(x\ge x_1(t)/2\) implies \(|x-x_1(t)|\ge |x_1(t)|/2\ge c_1 t/2\), we have \[\int_{\{x\ge x_1(t)/2\}} e^{-c\delta_1|x-x_1(t)|}\,dx \le C\int_{c_1 t/2}^{\infty} e^{-c\delta_1 z}\,dz \le C\delta_1^{-1}e^{-C\delta_1 t}.\] Therefore, \[\int_{\mathbb{R}}\varphi_3(t,x)\bigl|\partial_x(v^{S_1})^{-X_1}(t,x)\bigr|\,dx \le C\delta_1 e^{-C\delta_1 t}.\] The estimate \[\int_{\mathbb{R}}\varphi_1(t,x)\bigl|\partial_x(v^{S_3})^{-X_3}(t,x)\bigr|\,dx \le C\delta_3 e^{-C\delta_3 t}\] is proved in the same way.

This completes the proof. ◻

Proof. The proof follows a similar argument to that in [37]. We provide the details for completeness. We only prove ?? and ?? , since ?? –?? follow in exactly the same way.

Set \[\xi_1(t):=\sigma_1 t+X_1(t), \qquad \xi_3(t):=\sigma_3 t+X_3(t).\] By the standard pointwise estimates for the viscous shock profiles and the viscous contact wave obtained earlier, we have \[\begin{align} \tag{239} |\partial_x(v^{S_1})^{-X_1}(t,x)| &\le C\delta_1^2 e^{-c\delta_1|x-\xi_1(t)|}, \quad x-\xi_1(t)\in \mathbb{R} \\ \tag{240} |(v^{S_3}-v^*,\theta^{S_3}-\theta^*)^{-X_3}(t,x)| &\le C\delta_3 e^{-c\delta_3|x-\xi_3(t)|}, \quad x-\xi_3(t)\leq0, \\ \tag{241} |(v^C-v_*,\theta^C-\theta_*)(t,x)| &\le C\delta_C(1+t)^{-1/2}e^{-c_0x^2/(1+t)}, \quad x\leq 0. \end{align}\] Moreover, since the shock speeds are distinct and the shifts are bounded in the regime of interest, there exists a positive constant \(c_1\) such that \[\label{eq:proof-int-sep} |\xi_3(t)-\xi_1(t)|\ge c_1(1+t), \qquad |\xi_1(t)|+|\xi_3(t)|\ge c_1(1+t).\tag{242}\]

We first prove ?? . We divide into two regions \(A_1:=\{x-\sigma_1t\geq -\frac{\sigma_1}{2}t\}\) and \(A_2:=\mathbb{R}-A_1\). Since \[\begin{align} & x-\xi_1(t) \geq -\frac{\sigma_1}{2}t + C\varepsilon t\geq -\frac{\sigma_1t}{4}>0 \quad \text{for}\, x\in A_1,\\ & x\leq \frac{\sigma_1}{2}t <0,\quad \text{for} \, x\in A_2, \end{align}\] we have the following observations. \[\begin{align} \label{eq:keysep1} \left| \partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}\right|\leq C \delta_1^2 e^{-C\delta_1\left| x-\xi_1(t)\right|} \leq C \delta_1^2 e^{-\frac{C\delta_1}{2}\left| x-\xi_1(t)\right|}e^{-C\delta_1t}. \end{align}\tag{243}\]

By 239 , 241 and 243 , \[\begin{align} &\left\| |\partial_x(v^{S_1})^{-X_1}| \,|(v^C-v_*,\theta^C-\theta_*)| \right\|_{L^2_x}^2 \\ &\le \int_{A_1}|\partial_x(v^{S_1})^{-X_1}|^2 \,|(v^C-v_*,\theta^C-\theta_*)|^2 dx + \int_{A_2}|\partial_x(v^{S_1})^{-X_1}|^2 \,|(v^C-v_*,\theta^C-\theta_*)|^2 dx\\ &\le C\delta_C^2 \int_{A_1} \delta_1^4 e^{-2c\delta_1\left| x-\xi_1(t)\right|} dx + \frac{C}{1+t}\int_{A_2} \delta_1^4\delta_C^2 e^{-2c\delta_1\left| x-\xi_1(t)\right|}e^{-2c_0x^2/(1+t)} dx \\ &\le C\delta_C^2 \delta_1^3 e^{-C\delta_1t} + C\delta_C^2\delta_1^3e^{-Ct} \end{align}\] Therefore \[\left\| |\partial_x(v^{S_1})^{-X_1}| \,|(v^C-v_*,\theta^C-\theta_*)| \right\|_{L^2_x}^2 \le C\delta_1^3\delta_C^2 e^{-c\delta_1 t},\] which gives ?? .

Next, we prove ?? . Separate \(\mathbb{R}\) into \[\Omega_1 := \{x:x-\sigma_3t\leq -\frac{\sigma_3t}{2}\}, \qquad \Omega_2 := \mathbb{R}\setminus \Omega_1.\] Since \[\begin{align} &x-\xi_3(t)=x-\sigma_3t-X_3(t)\leq -\frac{\sigma_3t}{2}+C\varepsilon t\leq -\frac{\sigma_3t}{4}<0 \quad \text{for}\, x\in \Omega_1,\\ &x-\xi_1(t)=x-\sigma_1t-X_1(t)\geq \frac{\sigma_3t}{2}-\sigma_1t-C\varepsilon t\geq \frac{\sigma_3 t}{4}>0 \quad \text{for}\, x\in \Omega_2, \end{align}\] we have the following estimate. \[\begin{align} &\bigl\lVert\left| \partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}\right|\left| \left(\bigl(v^{S_{3}}\bigr)^{-X_{3}}-v^*,\bigl(\theta^{S_{3}}\bigr)^{-X_{3}}-\theta^*\right)\right|\bigr\rVert^2\\ &\le \int_{\Omega_1}\left| \partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}\right|^2\left| \left(\bigl(v^{S_{3}}\bigr)^{-X_{3}}-v^*,\bigl(\theta^{S_{3}}\bigr)^{-X_{3}}-\theta^*\right)\right|^2 dx\\ &\qquad+ \int_{\Omega_2} \left| \partial_x\bigl(v^{S_{1}}\bigr)^{-X_{1}}\right|^2\left| \left(\bigl(v^{S_{3}}\bigr)^{-X_{3}}-v^*,\bigl(\theta^{S_{3}}\bigr)^{-X_{3}}-\theta^*\right)\right|^2 dx \\ &\le C\delta_1^4\delta_3^2\int_{\Omega_1} e^{-2c\delta_1\left| x-\xi_1(t)\right|}e^{-2c\delta_3\left| x-\xi_3(t)\right|} dx + C\delta_1^4\delta_3^2\int_{\Omega_2} e^{-2c\delta_1\left| x-\xi_1(t)\right|} dx \\ &\le C\delta_1^3\delta_3^2e^{-c\delta_3t}+C\delta_1^3\delta_3^2e^{-c\delta_1t} \end{align}\] Thus \[\left\| |\partial_x(v^{S_1})^{-X_1}| \,|(v^{S_3}-v^*,\theta^{S_3}-\theta^*)^{-X_3}| \right\|_{L^2_x}^2 \le C\delta_1^3\delta_3^2 \bigl(e^{-c\delta_3 t}+e^{-c\delta_1 t}\bigr),\] and after taking square roots, \[\left\| |\partial_x(v^{S_1})^{-X_1}| \,|(v^{S_3}-v^*,\theta^{S_3}-\theta^*)^{-X_3}| \right\|_{L^2_x} \le C\delta_1^{3/2}\delta_3 \bigl(e^{-c\delta_3 t}+e^{-c\delta_1 t}\bigr).\] This proves ?? . ◻

Proof. The solutions are satisfied the following equations: \[\begin{align} \begin{aligned} &v_t-u_{1x}=0\\ &u_{1t}+p_x=-\int \xi_1^2 G_x d\xi\\ &u_{it}=-\int \xi_1\xi_i G_x d\xi,\quad (i=2,3)\\ &\theta_t+pu_{1x}=u_1\int \xi_1^2 G_x d\xi+\sum_{i=2}^3 u_i \int \xi_1\xi_i G_x d\xi-\frac{1}{2}\int \xi_1\left| \xi\right|^2 G_x d\xi. \end{aligned} \end{align}\]

For the approxiate solutions, we have the following systems: \[\begin{align} \begin{aligned} &v_t^C-u_{1x}^C=0\\ &u_{1t}^C+p_x^C=\frac{4}{3}\left(\frac{\mu\left(\theta^C\right)u_{1x}^C}{v^C}\right)_x+Q_1^C\\ &u_i^C = 0\quad (i=2,3)\\ &\theta_t^C + p^Cu_{1x}^C = \left(\frac{\alpha_{\rm{th}}\left(\theta^C\right)\theta_x^C}{v^C}\right)_x+\frac{4}{3}\mu\left(\theta^C\right)\frac{\left(u_{1x}^C\right)^2}{v^C}+Q_2^C \end{aligned} \end{align}\]

and

\[\begin{align} \begin{aligned} &\bigl((v^{S_{i}})^{-X_{i}}\bigr)_t+\dot{X}_i\bigl((v^{S_{i}})^{-X_{i}}\bigr)_x-\bigl((u_1^{S_{i}})^{-X_{i}}\bigr)_x=0\\ &\bigl((u_1^{S_{i}})^{-X_{i}}\bigr)_t+\dot{X}_i\bigl((u_1^{S_{i}})^{-X_{i}}\bigr)_x+\bigl((p^{S_{i}})^{-X_{i}}\bigr)_x=-\int\xi_1^2 \bigl((G^{S_{i}})^{-X_{i}}\bigr)_x d\xi\\ &\bigl(u_j^{S_{i}}\bigr)^{-X_{i}}\equiv 0 \quad(j=2,3)\\ &\bigl((\theta^{S_{i}})^{-X_{i}}\bigr)_t+\dot{X}_i\bigl((\theta^{S_{i}})^{-X_{i}}\bigr)_x+\bigl(p^{S_{i}}\bigr)^{-X_{i}}\bigl((u_1^{S_{i}})^{-X_{i}}\bigr)_x\\ &\qquad =\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\int \xi_1^2 \bigl((G^{S_{i}})^{-X_{i}}\bigr)_x d\xi -\frac{1}{2}\int \xi_1\left| \xi\right|^2 \bigl((G^{S_{i}})^{-X_{i}}\bigr)_xd\xi. \end{aligned} \end{align}\] Therefore, the perturbed systems are written by the micro part \(\widetilde{G}\):

\[\begin{align} \begin{aligned} &\phi_t-\psi_{1x}-\sum_{i=1,3} \dot{X}_i \bigl((v^{S_{i}})^{-X_{i}}\bigr)_x = 0\\ &\psi_{1t} + \left(p-p^{S_{1},-X_{1}}-p^C-p^{S_{3},-X_{3}}\right)_x-\sum_{i=1,3} \dot{X}_i \bigl((u_1^{S_{i}})^{-X_{i}}\bigr)_x \\ & \qquad = -Q_1^C-\frac{4}{3}\left(\frac{\mu\left(\theta^C\right)u_{1x}^C}{v^C}\right)_x-\int \xi_1^2 \widetilde{G}_x d\xi\\ &\psi_{it}=-\int \xi_1\xi_i \widetilde{G}_x d\xi,\quad (i=2,3)\\ &\zeta_t +\left(pu_{1x}-p^{S_{1},-X_{1}}\bigl((u_1^{S_{1}})^{-X_{1}}\bigr)_x-p^Cu_{1x}^C-p^{S_{3},-X_{3}}\bigl((u_1^{S_{3}})^{-X_{3}}\bigr)_x\right)-\sum_{i=1,3}\dot{X}_i\bigl((\theta^{S_{i}})^{-X_{i}}\bigr)_x\\ &\qquad \qquad = -\frac{1}{2}\int \xi_1 \left| \xi\right|^2 \widetilde{G}_x d\xi +u_1\int \xi_1^2 \widetilde{G}_x d\xi \\ &\qquad \qquad +\sum_{i=1,3} \left(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right)\int \xi_1^2 \bigl((G^{S_{i}})^{-X_{i}}\bigr)_x d\xi+\sum_{j=2}^3 u_j \int \xi_1\xi_j \widetilde{G}_x d\xi\\ &\qquad \qquad -\left(\frac{\alpha_{\rm{th}}\left(\theta^C\right)\theta_x^C}{v^C}\right)_x-\frac{4}{3}\mu\left(\theta^C\right)\frac{\left(u_{1x}^C\right)^2}{v^C}-Q_2^C. \end{aligned} \end{align}\]

We can easily get the following estimate for \(\phi_t\).

\[\begin{align} \begin{aligned} \bigl\lVert\phi_t\bigr\rVert_{L^2}^2 \leq C \int \left| \psi_{1x}\right|^2 dx + C\sum_{i=1,3} \left| \dot{X}_i\right|^2 \delta_i^2 \int \left| \bigl((v^{S_{i}})^{-X_{i}}\bigr)_x\right| dx \leq C \int \left| \psi_{1x}\right|^2 dx + C\sum_{i=1,3} \delta_i^3 \left| \dot{X}_i\right|^2 \end{aligned} \end{align}\]

For the estimate \(\psi_{1t}\), observe that \[\begin{align} \begin{aligned} \left(p-\overline{p}\right)_x =& \left(\frac{\zeta}{v}-\frac{\overline{\theta}\phi}{v\overline{v}}\right)_x \\ \leq & C \left| \left(\zeta_x,\zeta\overline{v}_x,\zeta\phi_x,\phi_x,\overline{\theta}_x\phi,\overline{v}_x\phi,\phi_x\phi\right)\right|\\ \leq & C\left| \left(\zeta_x,\phi_x\right)\right|+C\left| \left(\overline{\theta}_x,\overline{v}_x\right)\right|\left| \left(\zeta,\phi\right)\right|. \end{aligned} \end{align}\] By using this, we can compute as belowed: \[\begin{align} \begin{aligned} & \int \psi_{1t}\left(p-p^{S_{1},-X_{1}}-p^C-p^{S_{3},-X_{3}}\right)_x dx \\ & \quad \leq \frac{1}{64}\bigl\lVert\psi_{1t}\bigr\rVert_{L^2}^2 + C\int \left| Q_1\right|^2 dx +C \int \left| \left(p-\overline{p}\right)_x\right|^2 dx \\ & \quad \leq \frac{1}{64}\bigl\lVert\psi_{1t}\bigr\rVert_{L^2}^2 + C\int \left| Q_1\right|^2 dx + C\int \left| \left(\phi_x,\zeta_x\right)\right|^2 dx \\ & \qquad + C \int \left| \left(\overline{\theta}_x,\overline{v}_x\right)\right|^2 \left| \left(\zeta,\phi\right)\right|^2 dx . \end{aligned} \end{align}\]

By using Young’s inequality, we have

\[\begin{align} \begin{aligned} \int \psi_{1t}\frac{4}{3} \left(\frac{\mu\left(\theta^C\right)u_{1x}^C}{v^C}\right)_x dx \leq & C \int \left| \psi_{1t}\right| \left| \left(u_{1xx}^C,\theta_x^Cu_{1x}^C,u_{1x}^Cv_x^C\right)\right| dx \\ \leq & \frac{1}{64}\bigl\lVert\psi_{1t}\bigr\rVert_{L^2}^2 + C \int \left| Q_1^C\right|^2 dx \end{aligned} \end{align}\] and \[\begin{align} \begin{aligned} \int \psi_{1t} \int \xi_1^2 \widetilde{G}_x d\xi dx \leq & \frac{1}{64}\bigl\lVert\psi_{1t}\bigr\rVert_{L^2}^2 +C \int \left\{\int \xi_1^2 \widetilde{G}_x d\xi\right\}^2 dx\\ \leq & \frac{1}{64}\bigl\lVert\psi_{1t}\bigr\rVert_{L^2}^2 + C \int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 dx. \end{aligned} \end{align}\]

Therefore, we have the following estimate, \[\begin{align} \begin{aligned} \bigl\lVert\psi_{1t}\bigr\rVert_{L^2}^2\leq &C \sum_{i=1,3}\delta_i^3 \left| \dot{X}_i\right|+ C\int \left| \left(\phi_x,\zeta_x\right)\right|^2 dx+C\int \left| Q_1\right|^2 dx+ C \int \left| \left(\overline{\theta}_x,\overline{v}_x\right)\right|^2 \left| \left(\zeta,\phi\right)\right|^2 dx \\ &+C \int \left| Q_1^C\right|^2 dx + C \int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 dx . \end{aligned} \end{align}\]

For the estimate of \(\psi_{it}\), we have

\[\begin{align} \begin{aligned} \int \left| \psi_{it}\right|^2 dx = \bigl\lVert\psi_{it}\bigr\rVert_{L^2}^2, \end{aligned} \end{align}\] and \[\begin{align} \begin{aligned} \int \psi_{it} \int \xi_1\xi_i \widetilde{G}_x d\xi dx \leq& \frac{1}{64}\bigl\lVert\psi_{it}\bigr\rVert_{L^2}^2 + C \int \left\{\int \xi_1\xi_i \widetilde{G}_x d\xi\right\}^2 dx\\ \leq & \frac{1}{64}\bigl\lVert\psi_{it}\bigr\rVert_{L^2}^2 + C \int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 dx. \end{aligned} \end{align}\] Therefore, \[\begin{align} \begin{aligned} \bigl\lVert\psi_{it}\bigr\rVert_{L^2}^2 \leq C \int \bigl\lVert\widetilde{G}_x\bigr\rVert_{\nu,M_\#}^2 dx. \end{aligned} \end{align}\]

To estimate \(\zeta_t\), we can observe \[\begin{align} \begin{aligned} & \int \zeta_t\left(pu_{1x}-p^{S_{1},-X_{1}}\bigl((u_1^{S_{1}})^{-X_{1}}\bigr)_x-p^Cu_{1x}^C-p^{S_{3},-X_{3}}\bigl((u_1^{S_{3}})^{-X_{3}}\bigr)_x\right) dx\\ & \leq \frac{1}{64} \bigl\lVert\zeta_t\bigr\rVert_{L^2}^2 + C \int \left| Q_2\right|^2 dx + \int \left| \left(pu_{1x}-\overline{p}\overline{u}_{1x}\right)\right|^2 dx\\ & \leq \frac{1}{64} \bigl\lVert\zeta_t\bigr\rVert_{L^2}^2 + C \int \left| Q_2\right|^2 dx + C \int \left| \psi_{1x}\right|^2 dx + C \int \left| \overline{u}_{1x}\right|^2\left| \left(\phi,\zeta\right)\right|^2 dx. \end{aligned} \end{align}\] Furthermore, \(\bigl\lVert\theta_{xx}^C\bigr\rVert_{L^2}^2\) is globally time integrable since its time decay is \((1+t)^{-3/2}\) we have the following inequality,

\[\begin{align} \begin{aligned} & \int \zeta_t \left(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right)\int \xi_1^2 \bigl((G^{S_{i}})^{-X_{i}}\bigr)_x d\xi dx \\ & \qquad \leq \frac{1}{64} \bigl\lVert\zeta_t\bigr\rVert_{L^2}^2 + C\int \left(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right)^2 \left\{\int \xi_1^2 \bigl((G^{S_{i}})^{-X_{i}}\bigr)_x d\xi\right\}^2 dx\\ & \qquad \leq \frac{1}{64} \bigl\lVert\zeta_t\bigr\rVert_{L^2}^2 + C\int \left(u_1-\bigl(u_1^{S_{i}}\bigr)^{-X_{i}}\right)^2 \left| \bigl((v^{S_{i}})^{-X_{i}}\bigr)_x\right|^2 dx. \end{aligned} \end{align}\]

The other terms are computed by the similar method of the estimate of \(\psi_{1t}\). ◻

Proof. Using the same method as [38], it holds that \[\begin{align} &\left(\frac{\overline{p}\theta}{2v}\phi_x^2+\frac{3\overline{p}v}{4}\psi_{1x}^2+\sum_{k=2}^3\frac{\psi_{kx}^2}{2}+\frac{\zeta_x^2}{2}\right)_t+2\overline{p}\psi_{1xx}^2\mu\left(\overline{\theta}\right)+\sum_{k=2}^3\frac{\mu(\theta)}{v}\psi_{kxx}^2+\frac{\alpha_{\rm{th}}\left(\overline{\theta}\right)}{v}\zeta_{xx}^2\\ &=-\sum_{i=1,3}\dot{X}_i\left[\phi_{xx}\bigl((v^{S_{i}})^{-X_{i}}\bigr)_x\frac{\overline{p}\theta}{v}+\psi_{1xx}\bigl((u_1^{S_{i}})^{-X_{i}}\bigr)_x\frac{3\overline{p}v}{2}+\zeta_{xx}\bigl((\theta^{S_{i}})^{-X_{i}}\bigr)_x\right]+\frac{\phi_x^2}{2}\left(\frac{\overline{p}\theta}{v}\right)_t\\ &+\frac{\psi_{1x}^2}{2}\left(\frac{3\overline{p}v}{2}\right)_t-\left(\frac{\overline{p}\theta}{v}\right)_x\phi_x\phi_t+\left(\frac{\overline{p}\theta}{v}\right)_x\phi_x\psi_{1x}-\left(\frac{3\overline{p}v}{2}\right)_x\psi_{1x}\psi_{1t}+\psi_{1xx}\overline{p}v\overline{\theta}_x\left(\frac{1}{v}-\frac{1}{\overline{v}}\right)\\ &-\psi_{1xx}\overline{p}v\overline{v}_x\left(\frac{\theta}{v^2}-\frac{\overline{\theta}}{\overline{v}^2}\right)-\psi_{1x}\overline{p}_x\zeta_x+\zeta_{xx}u_{1x}\left(p-\overline{p}\right)-2\overline{p}v\psi_{1xx}\left(\left(\mu(\theta)-\mu\left(\overline{\theta}\right)\right)\frac{u_{1x}}{v}\right)_x\\ &-2\overline{p}v\psi_{1xx}\left[\left(\mu\left(\overline{\theta}\right)\right)_x\left(\frac{u_{1x}}{v}-\frac{\overline{u}_{1x}}{\overline{v}}\right)-\mu\left(\overline{\theta}\right)\frac{v_x}{v^2}\psi_{1x}+\mu\left(\overline{\theta}\right)\left\{\overline{u}_{1x}\left(\frac{1}{v}-\frac{1}{\overline{v}}\right)\right\}_x\right]\\ &-\zeta_{xx}\left(\left(\alpha_{\rm{th}}(\theta)-\alpha_{\rm{th}}\left(\overline{\theta}\right)\right)\frac{\theta_x}{v}\right)_x+\sum_{k=2}^3\frac{\psi_{kx}^2}{2}\left(\frac{\mu(\theta)}{v}\right)_{xx}-\zeta_{xx}\sum_{k=2}^3\frac{\mu(\theta)}{v}\psi_{kx}^2+\zeta_{xx}Q_2\\ &+\psi_{1xx}\frac{3\overline{p}v}{2}Q_1 -\zeta_{xx}\left[\left(\alpha_{\rm{th}}\left(\overline{\theta}\right)\right)_x\left(\frac{\theta_{x}}{v}-\frac{\overline{\theta}_{x}}{\overline{v}}\right)-\alpha_{\rm{th}}\left(\overline{\theta}\right)\frac{v_x}{v^2}\zeta_{x}+\alpha_{\rm{th}}\left(\overline{\theta}\right)\left\{\overline{\theta}_{x}\left(\frac{1}{v}-\frac{1}{\overline{v}}\right)\right\}_x\right]\\ &-\frac{4}{3}\zeta_{xx}\left(\frac{\mu(\theta)u_{1x}^2}{v}-\frac{\mu\left(\overline{\theta}\right)\overline{u}_{1x}^2}{\overline{v}}\right)+\psi_{1xx}\frac{3\overline{p}v}{2}\int \xi_1^2 \widetilde{\Pi}_{1x} d\xi +\sum_{k=2}^3 \psi_{kxx}\int \xi_1\xi_i \Pi_{1x}d\xi\\ & + \zeta_{xx}\int \xi_1 \frac{\left| \xi\right|^2}{2}\widetilde{\Pi}_{1x} d\xi - \zeta_{xx}\sum_{k=2}^3 \psi_k\int \xi_1\xi_i \Pi_{1x}d\xi\\ &-\zeta_{xx}\left[u_1\int \xi_1^2 \Pi_{1x}d\xi-\bigl(u_1^{S_{1}}\bigr)^{-X_{1}}\int\xi_1^2 \bigl((\Pi_1^{S_{1}})^{-X_{1}}\bigr)_xd\xi -\bigl(u_1^{S_{3}}\bigr)^{-X_{3}}\int \xi_1^2 \bigl((\Pi_1^{S_{3}})^{-X_{3}}\bigr)_xd\xi\right]\\ &+ (\mathcal{R}^{(1)})_x. \end{align}\] Here \(\mathcal{R}^{(1)}\) denotes a collection of lower-order flux terms produced by integration by parts and differentiation of the coefficients. Since the estimate is carried out over \(\mathbb{R}_x\), these divergence terms do not contribute after integration in \(x\).

Collecting the above identity, estimating the fluid error terms as in the zeroth-order energy argument, and controlling the microscopic terms by the auxiliary estimates from the high-order microscopic analysis, we obtain ?? . ◻