&+\frac{\rho_{7}\rho_{1}}{2C_{f}}\Big{(}\sum_{p\neq 1,p\neq
7}\gamma_{1p}^{1}\frac{w_{1}}{\rho_{p}}+\sum_{p\neq 1,q\neq 1\atop p\neq
q}\gamma_{pq}^{1}\frac{w_{q}}{\rho_{p}}\Big{)}(\partial_{y_{1}}v_{p}-w_{p}\partial_{y_{1}}\rho_{p})\\\
&+\frac{\rho_{7}\rho_{1}}{2C_{f}}\sum_{p\neq 1,q\neq 1\atop p\neq
q}\gamma_{pq}^{1}\frac{w_{p}}{\rho_{q}}(\partial_{y_{1}}v_{q}-w_{q}\partial_{y_{1}}\rho_{q})+\frac{\rho_{1}}{2C_{f}}\gamma_{17}^{1}w_{1}\partial_{y_{1}}v_{7}\\\
=&B_{21}^{\eta}\tau_{1}^{(7)}+B_{22}^{\eta}\pi_{1}^{(7)}+B_{23}^{\eta},\end{split}$
(5.17)

where one can check as in (5.16) $B_{21}^{\eta},B_{22}^{\eta},B_{23}^{\eta}$
are also uniformly bounded.

Next, we check the initial data of $\tau_{1}^{(7)}$ and $\pi_{m}^{(7)}$. Since
$\rho_{1}(z_{1},0)=1,$ for fixed $\eta$, it follows from (5.7) that

$\begin{split}\tau_{1}^{(7)}(z_{1},0)=&\partial_{z_{1}}\rho_{1}(z_{1},0)-\frac{\rho_{1}(z_{1},0)}{2C_{f}}\partial_{s_{1}}\rho_{1}(z_{1},0)\\\
=&-\frac{1}{2C_{f}}\sum_{k}c_{1k}^{1}w_{k}^{(\eta)}(z_{1},0)=O(W_{0}^{(\eta)})<+\infty.\end{split}$

And with $v_{1}(z_{1},0)=w_{1}^{(\eta)}(z_{1},0)$, we similarly deduce

$\begin{split}\pi_{1}^{(7)}(z_{1},0)=&\partial_{z_{1}}v_{1}(z_{1},0)-\frac{\rho_{1}(z_{1},0)}{2C_{f}}\partial_{s_{1}}v_{1}(z_{1},0)\\\
=&\partial_{z_{1}}w_{1}^{(\eta)}(z_{1},0)-\frac{1}{2C_{f}}\sum_{q\neq 1,q\neq
p}\gamma_{pq}^{1}w_{p}(z_{1},0)w_{q}(z_{1},0)\rho_{1}(z_{1},0)\\\
=&O(\partial_{z_{1}}w_{1}^{(\eta)}(z_{1},0)+[W_{0}^{(\eta)}]^{2})<+\infty.\end{split}$

Applying Grönwall’s inequality to (5.16) and (5.17), for $t\leq T_{\eta}^{*}$
we deduce that $\tau_{1}^{(7)}:=\partial_{y_{1}}\rho_{1}$ is bounded by a
constant depending on $\eta$.

Back to (5.9), we hence prove

$\partial_{z_{1}}\rho_{1}=\partial_{y_{1}}\rho_{1}+O(v_{1}+\sum_{m\neq
1}w_{m}\rho_{1}).$

With the bounds for $J(t)$, $S(t)$ and $V(t)$ in Section 4.2, consequently, we
arrive at the conclusion that $\partial_{z_{1}}\rho_{1}$ is bounded by
$C_{\eta}$, which is a constant depending on $\eta$. Note that the dependence
on $\eta$ causes no trouble since the blow-up is derived for each fixed $\eta$
in Subsection 5.3. ∎

###  5.3. Ill-posedness mechanism

We prove our main result Theorem 1.3 in this section. For our proof, one can
see that the ill-posedness is driven by the shock formation.

###### Proof of Theorem 1.3.

First, the ill-posedness builds on an instantaneous shock formation. More
precisely, for each $\eta$ the solution yields a shock at a finite time
$T_{\eta}^{*}$ and $T_{\eta}^{*}\to 0$ as $\eta\to 0$. With initial data (5.1)
and (5.4), employing the estimates in Subsection 4.1 and Subsection 4.2, we
know that a shock forms in $\mathcal{R}_{1}$ at time $T_{\eta}^{*}$ with

$\frac{1}{(1+\varepsilon)^{3}|c_{11}^{1}(0)|W_{0}^{(\eta)}}\leq
T_{\eta}^{*}\leq\frac{1}{(1-\varepsilon)^{3}|c_{11}^{1}(0)|W_{0}^{(\eta)}}.$
(5.18)

By (5.1), we see that $w_{1}^{(\eta)}(z,0)=\theta\cdot O(|\ln z|^{\alpha})$
with $0<\alpha<\frac{1}{2}$. As $\eta\to 0$, the support of the initial data
shrinks to the origin and hence $|w_{1}^{(\eta)}(z,0)|\to\infty$. It then
follows from the definition of $W_{0}^{(\eta)}$ in (4.4) and the above
estimate (5.18) that $T_{\eta}^{*}\to 0$.

Furthermore, we will show the $H^{1}$-norm of solutions to (1.1) blows up at
$t=T_{\eta}^{*}$. In particular, we give a lower bound estimate of the
$H^{2}$-norm in the following spatial region

$\Omega_{T_{\eta}^{*}}=\\{(x,x_{2},x_{3},T_{\eta}^{*}):x=X_{1}(z,T_{\eta}^{*}),\
(x-\lambda_{1}T_{\eta}^{*}-\frac{3\eta}{2})^{2}+x_{2}^{2}+x_{3}^{2}\leq\frac{\eta^{2}}{4},\
\text{for}\ \eta\leq z\leq 2\eta\\}.$ (5.19)

To calculate
$\int_{\Omega_{T_{\eta}^{*}}}|w_{1}(x,x_{2},x_{3},T_{\eta}^{*})|^{2}dxdx_{2}dx_{3}$,
we first compute the area element $\int dx_{2}dx_{3}$. By the definition of
characteristics, for a characteristic line $\mathcal{C}_{1}$ starting from
$z$, its terminal at time $T_{\eta}^{*}$ is $x=z+\lambda_{1}T_{\eta}^{*}$.
Thus, one can verify that the area of $\Omega_{T_{\eta}^{*}}$ is

$\begin{split}\int_{x_{2}^{2}+x_{3}^{2}\leq\frac{\eta^{2}}{4}-(x-\lambda_{1}T_{\eta}^{*}-\frac{3\eta}{2})^{2}}dx_{2}dx_{3}=&\pi\Big{[}\frac{\eta^{2}}{4}-(z-\frac{3\eta}{2})^{2}\Big{]}=\pi(z-\eta)(2\eta-z).\end{split}$
(5.20)

Employing characteristic coordinates, we obtain

$\|w_{1}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}^{2}=\pi\int_{\frac{\eta}{2}}^{\eta}\big{|}\frac{v_{1}}{\rho_{1}}(z,T_{\eta}^{*})\big{|}^{2}\rho_{1}(z,T_{\eta}^{*})(z-\eta)(2\eta-z)dz.$
(5.21)

Recall the estimate of $v_{1}$ in (4.71)

$v_{1}(z,t)=w_{1}^{(\eta)}(z,0)+O(\eta[W_{0}^{(\eta)}]^{2}).$ (5.22)

We integrate in a subinterval $(z_{0},z_{0}^{*}]\subseteq[\eta,2\eta]$ where
$|w_{1}^{(\eta)}(z,0)|>\frac{1}{2}W_{0}^{(\eta)}$ for $z\in(z_{0},z_{0}^{*}]$.
Then in this subinterval, it follows from (5.22) that $|v_{1}(z,t)|$ admits a
lower bound $|v_{1}(z,t)|\geq\frac{1}{4}W_{0}^{(\eta)}$. Applying this in
(5.21) and using the fact that $\rho_{1}(z_{0},T_{\eta}^{*})=0$, we get

$\begin{split}\|w_{1}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}^{2}&\geq
C\int_{z_{0}}^{z_{0}^{*}}\big{|}\frac{v_{1}}{\rho_{1}}(z,T_{\eta}^{*})\big{|}^{2}\rho_{1}(z,T_{\eta}^{*})(z-\eta)(2\eta-z)dz\\\
&\geq C(W_{0}^{(\eta)})^{2}(z_{0}-\eta)(2\eta-
z_{0}^{*})\underbrace{\int_{z_{0}}^{z_{0}^{*}}\frac{1}{\rho_{1}(z,T_{\eta}^{*})-\rho_{1}(z_{0},T_{\eta}^{*})}dz}_{\text{by
$\rho_{1}(z_{0},T_{\eta}^{*})=0$}}\\\ &\geq
C(W_{0}^{(\eta)})^{2}(z_{0}-\eta)(2\eta-
z_{0}^{*})\underbrace{\int_{z_{0}}^{z_{0}^{*}}\frac{1}{(z-z_{0})\sup|\partial_{z}\rho_{1}|}dz}_{\text{by
mean value theorem}}\\\ &\geq
C_{\eta}\int_{z_{0}}^{z_{0}^{*}}\frac{1}{z-z_{0}}dz=+\infty.\end{split}$
(5.23)

This shows the blow-up of the $L^{2}$-norm of $w_{1}$. In (5.23), we crucially
use the fact that $|\partial_{z}\rho_{1}|$ is bounded from Proposition 5.1.

The last step is going back to the original physical unknowns $\Phi$ and show
the blow-up of $\|\Phi\|_{H^{1}}$. Due to the boundedness of
$\\{\sup_{(x,T_{\eta}^{*})\in\mathcal{R}_{i}}w_{i}(x,T_{\eta}^{*})\\}_{i=2,\cdots,7}$
in (4.65), (4.82) and (4.90), it holds

$\Big{\\{}\sup_{(x,T_{\eta}^{*})\in\mathcal{R}_{i}}w_{i}(x,T_{\eta}^{*})\Big{\\}}_{i=2,\cdots,7}\leq
CW_{0}^{(\eta)},$ (5.24)

together with the fact that the first components of $r_{2},\ r_{4}$ and
$r_{6}$ are zero, we deduce

$\begin{split}\|\partial_{x}u_{1}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}=&\|\sum_{k=1}^{7}w_{k}r_{k}^{1}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}\geq
C\Big{(}\|w_{1}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}-\sum_{k=3,5,7}\|w_{k}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}\Big{)}\\\
\geq&C\Big{(}\|w_{1}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}-3\eta^{\frac{3}{2}}W_{0}^{(\eta)}\Big{)}\geq+\infty.\end{split}$

Here $r_{k}^{i}$ denotes the $i^{\text{th}}$ component of the right
eigenvector $r_{k}$. Hence the blow-up of
$\|w_{1}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}$ implies that
$\|u_{1}\|_{H^{1}(\Omega_{T_{\eta}^{*}})}$ is also infinite. Similarly, we can
show $\|\partial_{x}\varrho\|_{L^{2}(\Omega_{T_{\eta}^{*}})}=+\infty.$

$\begin{split}\|\partial_{x}\varrho\|_{L^{2}(\Omega_{T_{\eta}^{*}})}=&\|\sum_{k=1}^{7}w_{k}r_{k}^{4}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}\geq
C\Big{(}\|w_{1}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}-\sum_{k=3,4,5,7}\|w_{k}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}\Big{)}\\\
\geq&C\Big{(}\|w_{1}\|_{L^{2}(\Omega_{T_{\eta}^{*}})}-4\eta^{\frac{3}{2}}W_{0}^{(\eta)}\Big{)}\geq+\infty.\end{split}$

This finishes the proof of Theorem 1.3. ∎

##  6\. Case of $H_{1}=0$

In this section, we explain how to deal with the case when $H_{1}=0$. We
analyze the detailed subtle structures of the system (1.4) under this
scenario. And we explain the steps to prove shock formation and ill-posedness
when $H_{1}=0$.

With $H_{1}=0$, $\Phi$ also satisfies (2.3):

$\partial_{t}\Phi+A(\Phi)\partial_{x}\Phi=0,$

but with a different coefficient matrix, which is

$A(\Phi)=\begin{pmatrix}u_{1}&0&0&c^{2}/\varrho&\mu_{0}H_{2}/\varrho&\mu_{0}H_{3}/\varrho&\partial_{S}p/\varrho\\\
0&u_{1}&0&0&0&0&0\\\ 0&0&u_{1}&0&0&0&0\\\ \varrho&0&0&u_{1}&0&0&0\\\
H_{2}&0&0&0&u_{1}&0&0\\\ H_{3}&0&0&0&0&u_{1}&0\\\
0&0&0&0&0&0&u_{1}\end{pmatrix}.$ (6.1)

The eigenvalues of (6.1) are

$\begin{split}&\lambda_{1}=u_{1}+C_{f},\quad\lambda_{2}=\lambda_{3}=\lambda_{4}=\lambda_{5}=\lambda_{6}=u_{1},\quad\lambda_{7}=u_{1}-C_{f}.\end{split}$
(6.2)

Here, $C_{f}$ denotes the fast wave speed. It is defined as

$C_{f}=\Big{\\{}\frac{\mu_{0}}{\varrho}(H_{2}^{2}+H_{3}^{2})+c^{2}\Big{\\}}^{1/2}.$

And the slow and Alfvén wave speeds $C_{s},C_{a}$ vanish. The eigenvalues for
$A(\Phi)$ in (6.1) satisfy

$\begin{split}\lambda_{7}<\lambda_{6}=\lambda_{5}=\lambda_{4}=\lambda_{3}=\lambda_{2}<\lambda_{1}.\end{split}$
(6.3)

There exist seven linearly independent right eigenvectors:

$\begin{split}&r_{1}=\left(\begin{array}[]{cc}-1\\\ 0\\\ 0\\\
-\varrho/C_{f}\\\ -H_{2}/C_{f}\\\ -H_{3}/C_{f}\\\
0\end{array}\right),r_{2}=\left(\begin{array}[]{cc}0\\\ 1\\\ 0\\\ 0\\\ 0\\\
0\\\ 0\end{array}\right),r_{3}=\left(\begin{array}[]{cc}0\\\ 0\\\ 1\\\ 0\\\
0\\\ 0\\\ 0\end{array}\right),r_{4}=\left(\begin{array}[]{cc}0\\\ 0\\\ 0\\\
-\partial_{S}p/c^{2}\\\ 0\\\ 0\\\ 1\end{array}\right),\\\
&r_{5}=\left(\begin{array}[]{cc}0\\\ 0\\\ 0\\\ -\mu_{0}H_{2}/c^{2}\\\ 1\\\
0\\\ 0\end{array}\right),r_{6}=\left(\begin{array}[]{cc}0\\\ 0\\\ 0\\\
-\mu_{0}H_{3}/c^{2}\\\ 0\\\ 1\\\
0\end{array}\right),r_{7}=\left(\begin{array}[]{cc}-1\\\ 0\\\ 0\\\
\varrho/C_{f}\\\ H_{2}/C_{f}\\\ H_{3}/C_{f}\\\
0\end{array}\right).\end{split}$ (6.4)

In view of (6.2)-(6.4), when $H_{1}=0$, our system is a non-strictly
hyperbolic system with a quintuple eigenvalue
$\lambda_{2}=\lambda_{3}=\lambda_{4}=\lambda_{5}=\lambda_{6}=u_{1}$. The dual
left eigenvectors are

$\begin{split}&l_{1}=\Big{(}-\frac{1}{2},0,0,-\frac{c^{2}}{2\varrho
C_{f}},-\frac{\mu_{0}H_{2}}{2\varrho C_{f}},-\frac{\mu_{0}H_{3}}{2\varrho
C_{f}},-\frac{\partial_{S}p}{2\varrho C_{f}}\Big{)},\\\
&l_{2}=(0,1,0,0,0,0,0),\quad l_{3}=(0,0,1,0,0,0,0),\quad
l_{4}=(0,0,0,0,0,0,1),\\\ &l_{5}=\Big{(}0,0,0,-\frac{c^{2}H_{2}}{\varrho
C_{f}^{2}},\frac{\mu_{0}H_{3}^{2}+\varrho c^{2}}{\varrho
C_{f}^{2}},-\frac{\mu_{0}H_{2}H_{3}}{\varrho
C_{f}^{2}},-\frac{H_{2}\partial_{S}p}{\varrho C_{f}^{2}}\Big{)},\\\
&l_{6}=\Big{(}0,0,0,-\frac{c^{2}H_{3}}{\varrho
C_{f}^{2}},-\frac{\mu_{0}H_{2}H_{3}}{\varrho
C_{f}^{2}},\frac{\mu_{0}H_{2}^{2}+\varrho c^{2}}{\varrho
C_{f}^{2}},-\frac{H_{3}\partial_{S}p}{\varrho C_{f}^{2}}\Big{)},\\\
&l_{7}=\Big{(}-\frac{1}{2},0,0,\frac{c^{2}}{2\varrho
C_{f}},\frac{\mu_{0}H_{2}}{2\varrho C_{f}},\frac{\mu_{0}H_{3}}{2\varrho
C_{f}},\frac{\partial_{S}p}{2\varrho C_{f}}\Big{)}.\end{split}$ (6.5)

Since

$\begin{split}&\nabla_{\Phi}\lambda_{1}=\Big{(}1,0,0,\frac{(\gamma-1)\varrho
c^{2}-\mu_{0}(H_{2}^{2}+H_{3}^{2})}{2\varrho^{2}C_{f}},\frac{\mu_{0}H_{2}}{C_{f}\varrho},\frac{\mu_{0}H_{3}}{C_{f}\varrho},\frac{c^{2}}{2C_{f}}\Big{)},\\\
&\nabla_{\Phi}\lambda_{2,3,4,5,6}=(1,0,0,0,0,0,0),\\\
&\nabla_{\Phi}\lambda_{7}=\Big{(}1,0,0,-\frac{(\gamma-1)\varrho
c^{2}-\mu_{0}(H_{2}^{2}+H_{3}^{2})}{2\varrho^{2}C_{f}},-\frac{\mu_{0}H_{2}}{C_{f}\varrho},-\frac{\mu_{0}H_{3}}{C_{f}\varrho},-\frac{c^{2}}{2C_{f}}\Big{)},\\\
\end{split}$ (6.6)

by (2.28) we have $c_{im}^{i}=0$ for $i,m=2,3,4,5,6.$ Via a direct
calculation, it holds that $c_{11}^{1}(\Phi)=-\big{(}\frac{1}{2}+\frac{\gamma
c^{2}}{2C_{f}^{2}}+\frac{\mu_{0}(H_{2}^{2}+H_{3}^{2})}{\varrho
C_{f}^{2}}\big{)}<0$. Thus, the first characteristic is genuinely nonlinear.
For other coefficients, noting that $l_{i},r_{i},\nabla_{\Phi}r_{i}$ are all
regular, one can easily verify that all the coefficients $\gamma^{i}_{im}$,
$\gamma^{i}_{km}$ are of $O(1)$. The structure of the decomposed equations for
$H_{1}=0$ then coincides with (1.6)-(1.7). Since
$\lambda_{2}=\lambda_{3}=\lambda_{4}=\lambda_{5}=\lambda_{6}$, the
corresponding characteristic strips
$\mathcal{R}_{2},\mathcal{R}_{3},\mathcal{R}_{4},\mathcal{R}_{5},\mathcal{R}_{6}$
are also identical. We consider characteristics that propagate in and out of
the following three characteristic strips:
$\\{\mathcal{R}_{1},\mathcal{R}_{2},\mathcal{R}_{7}\\}$. These three strips
will be completely separated when $t>t_{0}^{(\eta)}$. See the following
picture.

$2\eta$$-2\eta$$t=0$$t=t_{0}^{(\eta)}$$\mathcal{R}_{1}$$\mathcal{R}_{2}$$\mathcal{R}_{7}$
Figure 6. Separation of three characteristic strips.

The initial data are given in (5.1)-(5.4) and satisfy the following conditions

$\begin{split}&W_{0}^{(\eta)}=\sup_{z_{1}}(w_{1}^{(\eta)}(z_{1},0))=O(\theta),\quad\max_{i=2,3,4,5,6,7}\sup_{z_{i}}|w_{i}^{(\eta)}(z_{i},0)|\leq
C\eta(W_{0}^{(\eta)})^{2}.\end{split}$ (6.7)

The main quantities to be estimated are the same as those in (4.6)-(4.10)
except slight changes in the following two quantities:

$\displaystyle
V_{\bar{2}}(t):=\max\sup_{(x^{\prime},t^{\prime})\notin\mathcal{R}_{\bar{2}},\atop
0\leq t^{\prime}\leq
t}\\{|w_{2}(x^{\prime},t^{\prime})|,|w_{3}(x^{\prime},t^{\prime})|,|w_{4}(x^{\prime},t^{\prime})|,|w_{5}(x^{\prime},t^{\prime})|,|w_{6}(x^{\prime},t^{\prime})|\\},$
$\displaystyle\check{W}(t):=\sup_{(x^{\prime},t^{\prime})\in\mathcal{R}_{\bar{2}},\atop
0\leq t^{\prime}\leq t}\\{w_{2},w_{3},w_{4},w_{5},w_{6}\\}.$

In the same fashion as in Section 4, for $t\in[0,T_{\eta}^{*})$ with
$T_{\eta}^{*}\sim\frac{1}{W_{0}^{(\eta)}}$, we obtain the following estimates:

$\displaystyle S(t)=$ $\displaystyle O(1),\quad J(t)=O(W_{0}^{(\eta)}),\quad
V(t)=O\big{(}\eta(W_{0}^{(\eta)})^{2}\big{)},\quad\check{W}(t)=O\big{(}\eta(W_{0}^{(\eta)})^{2}\big{)}.$

Finally, following the same argument as in Section 5, one can deduce that the
$H^{2}$-norm of the solution to (6.1) blows up. In conclusion, Theorem 1.2-1.3
also hold when $H_{1}=0$.

##  7\. Proof of $H^{2}$ ill-posedness for 3D compressible Euler equations
(Theorem 1.4)

In this section, we study the case with the absence of magnetic field, i.e.,
$H=0$. The MHD system (1.1) then reduces to the 3D compressible Euler
equations (1.14)

$\left\\{\begin{split}&\partial_{t}\varrho+\nabla\cdot(\varrho u)=0,\\\
&\varrho\\{\partial_{t}+(u\cdot\nabla)\\}u+\nabla p=0,\\\
&\partial_{t}S+(u\cdot\nabla)S=0.\end{split}\right.$

To establish ill-posedness via Lindblad-type approach, we study the above
system under planar symmetry

$\left\\{\begin{split}&\varrho\partial_{t}u_{1}+\varrho
u_{1}\partial_{x}u_{1}+c^{2}\partial_{x}\varrho+\partial_{S}p\partial_{x}S=0,\\\
&\varrho\partial_{t}u_{2}+\varrho u_{1}\partial_{x}u_{2}=0,\\\
&\varrho\partial_{t}u_{3}+\varrho u_{1}\partial_{x}u_{3}=0,\\\
&\partial_{t}\varrho+u_{1}\partial_{x}\varrho+\varrho\partial_{x}u_{1}=0,\\\
&\partial_{t}S+u_{1}\partial_{x}S=0.\end{split}\right.$ (7.1)

Recall the definition of vorticity $\omega:=\nabla\times u$. Under plane
symmetry, the vorticity reads
$\omega=(0,-\partial_{x}u_{3},\partial_{x}u_{2})$. The system (7.1) therefore
allows non-trivial vorticity and entropy.

In a similar fashion, we can also construct examples of shock formation and
$H^{2}$ ill-posedness for (7.1). In particular, according to the low-
regularity local well-posedness result in [46] by Wang (See also [13] by
Disconzi-Luo-Mazzone-Speck and [52] by Zhang-Andersson), we will see that the
$H^{2}$ ill-posedness is sharp.

To see the connection, we first review studies on the quasilinear wave
equation. The sharp local well-posedness results were obtained by Tataru-Smith
[42] and Wang [45]. And the sharp ill-posedness, as mentioned before, was
constructed by Lindblad [31]. For Einstein vacuum equations, the sharp local
well-posedness was established by Klainerman-Rodnianski-Szeftel [26].

For the 3D compressible Euler equations, the classical local well-posedness
result holds if the initial data for $(u,\varrho)$ are in $H^{s}$ with
$s>\frac{5}{2}$. In [46], Wang proved the low-regularity local well-posedness
requiring the initial data of velocity $u$, density $\varrho$ and vorticity
$\omega$ satisfying $(u,\varrho,\omega)\in H^{s}\times H^{s}\times
H^{s^{\prime}}$ with $2<s^{\prime}<s$. Interested readers are also referred to
Disconzi-Luo-Mazzone-Speck [13] and Zhang-Andersson [52]. In [46], it is also
conjectured that the sharp local well-posedness results would be for initial
data satisfying $(u,\varrho,\omega)\in H^{s}\times H^{s}\times
H^{s-\frac{1}{2}}$ with $s>2$. For the 3D incompressible case, the
corresponding sharp low-regularity ill-posedness was proved by Bourgain-Li [4]
for initial datum satisfying $(u,\omega)\in H^{\frac{5}{2}}\times
H^{\frac{3}{2}}$ and the ill-posedness is driven by the evolution of $\omega$.
Considering the 3D incompressible case as a special scenario for the 3D
compressible Euler equations, Bourgain-Li provided a sharp result with respect
to the regularity of vorticity. Our Theorem 1.4 provides a sharp companion
with respect to the regularity of the fluid velocity $u$ and density
$\varrho$. In particular, our initial data leading to the ill-posedness
satisfy $(u,\varrho)\in H^{2}\times H^{2}$ and $\omega\in C^{\infty}\subseteq
H^{\frac{3}{2}}$. See Figure 7.

$s$: regularity for the velocity$s^{\prime}$: regularity for the
vorticity2$\frac{5}{2}$$\frac{3}{2}$2LWP proved by Wang [46](See also
Disconzi-Luo-Mazzone-Speck [13], Zhang-Andersson [52])LWP by the Conjecture in
[46]Bourgain-Li’s ill-posedness [4]An-Chen-Yin’s ill-posedness Figure 7. Low
regularity local well-posedness (LWP) and ill-posedness for 3D Euler
equations.

With $H=0$, denoting $\tilde{\Phi}:=(u_{1},u_{2},u_{3},\varrho-1,S)^{T}(x,t)$,
the 3D compressible Euler system (7.1) reads

$\partial_{t}\tilde{\Phi}+A(\tilde{\Phi})\partial_{x}\tilde{\Phi}=0$ (7.2)

with

$A(\tilde{\Phi})=\begin{pmatrix}u_{1}&0&0&c^{2}/\varrho&\partial_{S}p/\varrho\\\
0&u_{1}&0&0&0\\\ 0&0&u_{1}&0&0\\\ \varrho&0&0&u_{1}&0\\\
0&0&0&0&u_{1}\end{pmatrix}.$ (7.3)

The eigenvalues of (7.3) are

$\lambda_{1}=u_{1}+c,\quad\lambda_{2}=\lambda_{3}=\lambda_{4}=u_{1},\quad\lambda_{5}=u_{1}-c$
(7.4)

with $c=\sqrt{\partial_{\varrho}p}$ being the sound speed. And we have a
triple eigenvalue $\lambda_{2}=\lambda_{3}=\lambda_{4}=u_{1}$. The five
linearly independent right eigenvectors are

$\begin{split}&r_{1}=\left(\begin{array}[]{cc}-1\\\ 0\\\ 0\\\ -\varrho/c\\\
0\end{array}\right),r_{2}=\left(\begin{array}[]{cc}0\\\ 1\\\ 0\\\ 0\\\
0\end{array}\right),r_{3}=\left(\begin{array}[]{cc}0\\\ 0\\\ 1\\\ 0\\\
0\end{array}\right),r_{4}=\left(\begin{array}[]{cc}0\\\ 0\\\ 0\\\
-\partial_{S}p/c^{2}\\\
1\end{array}\right),r_{5}=\left(\begin{array}[]{cc}-1\\\ 0\\\ 0\\\
\varrho/c\\\ 0\end{array}\right).\end{split}$ (7.5)

The dual left eigenvectors are

$\begin{split}&l_{1}=(-\frac{1}{2},0,0,-\frac{c}{2\varrho},-\frac{\partial_{S}p}{2\varrho
c}),\quad l_{2}=(0,1,0,0,0),\quad l_{3}=(0,0,1,0,0),\\\
&l_{4}=(0,0,0,0,1),\quad
l_{5}=(-\frac{1}{2},0,0,\frac{c}{2\varrho},\frac{\partial_{S}p}{2\varrho
c}).\end{split}$ (7.6)

With these setup, one can then prove the shock formation and the $H^{2}$ ill-
posedness as in Section 6. In particular, with the initial datum of $w_{1}$
taken as in (5.2), the $H^{1}$-norms of the velocity $u_{1}$ and density
$\varrho$ will blow up at the shock formation time.

Moreover, the vorticity satisfies

$\partial_{x}u_{2}=\sum_{k=1}^{5}w_{k}r_{k}^{2}=w_{2},\quad\partial_{x}u_{3}=\sum_{k=1}^{5}w_{k}r_{k}^{3}=w_{3}.$

Therefore, with the following initial data for $w_{i}$ (similar to (5.4)):

$w_{i}^{(\eta)}(z,0)=\theta^{2}\int_{\mathbb{R}}\zeta_{\frac{\eta}{10}}(y)\chi(z-y)dy,\quad\text{with}\quad
i=2,3,4,5,$

we have that the initial datum of $\omega$ is smooth and hence belongs to
$H^{\frac{3}{2}}(\mathbb{R}^{3})$.

##  Appendix A Proof of Proposition 3.1

In Proposition 3.1, we analyze and bound the behaviours of $c^{i}_{im}$,
$\gamma^{4}_{4m}$ and $\gamma^{4}_{km}$. In this appendix, we complete the
proof of Proposition 3.1 by proving that the coefficients $\gamma^{i}_{im}$,
$\gamma^{i}_{km}$ with $i\neq 4$ satisfy (3.8) when $H_{1}\neq 0$. This is
achieved via a thorough calculation of all the coefficients. As one will see,
the formula of certain coefficients can be very complicated. In spite of these
complexities, we show that all these coefficients are of $O(1)$. This is owing
to the careful design of right eigenvectors in (2.9) which eliminates all
potentially singular terms, e.g.
${(C_{a}^{2}-C_{s}^{2})}^{-1},{H_{2}}^{-1},{H_{3}}^{-1}$.

We begin with calculating the coefficients $\gamma^{1}_{km}$ as below.

$\displaystyle\gamma^{1}_{12}=$
$\displaystyle(\lambda_{1}-\lambda_{2})\bigg{\\{}\frac{-C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}-\frac{C_{a}^{2}-3C_{s}^{2}}{2C_{f}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{f}-H_{2}\partial_{H_{3}}C_{f})\bigg{\\}},$
$\displaystyle\gamma^{1}_{13}=$
$\displaystyle-(\lambda_{1}-\lambda_{3})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}$
$\displaystyle-
H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}$
$\displaystyle-\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}-\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}-\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}$
$\displaystyle-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}+\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{1}_{14}=$
$\displaystyle-(\lambda_{1}-\lambda_{4})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma}\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}+\frac{2C_{a}^{2}}{C_{f}^{3}}\partial_{S}C_{f}\bigg{)}$
$\displaystyle+\frac{C_{a}^{2}-C_{s}^{2}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{f}}\partial_{\rho}C_{f}-\frac{\partial_{S}C_{f}}{C_{f}}+\frac{1}{\gamma}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{1}_{15}=$
$\displaystyle-(\lambda_{1}-\lambda_{5})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}$
$\displaystyle-
H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}$
$\displaystyle-\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}-\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}-\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}C_{f}\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}+\frac{1}{C_{f}}-\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}-\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}$
$\displaystyle-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}+\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{1}_{16}=$
$\displaystyle(\lambda_{1}-\lambda_{6})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{a}^{2}-3C_{s}^{2}}{2C_{f}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{f}-H_{2}\partial_{H_{3}}C_{f})\bigg{\\}},$
$\displaystyle\gamma^{1}_{17}=$
$\displaystyle-\frac{(\lambda_{1}-\lambda_{7})(C_{a}^{2}-C_{s}^{2})}{C_{f}^{2}-C_{s}^{2}}\bigg{[}\frac{\rho(C_{a}^{2}-C_{f}^{2})}{C_{f}^{3}}\partial_{\rho}C_{f}+\frac{C_{a}^{2}}{C_{f}^{2}}-\frac{H_{2}}{C_{f}}\partial_{H_{2}}C_{f}-\frac{H_{3}}{C_{f}}\partial_{H_{3}}C_{f}\bigg{]},$
$\displaystyle\gamma^{1}_{23}=$ $\displaystyle 0,\ \gamma^{1}_{24}=0,\
\gamma^{1}_{25}=0,\
\gamma^{1}_{26}=\frac{(\lambda_{2}-\lambda_{6})C_{f}(C_{f}+C_{a})(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\
\gamma^{1}_{27}=0,$ $\displaystyle\gamma^{1}_{32}=$
$\displaystyle-(\lambda_{3}-\lambda_{2})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}-\frac{C_{f}^{2}}{C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{1}_{34}=$
$\displaystyle-(\lambda_{3}-\lambda_{4})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\rho\partial_{\rho}C_{s}-\partial_{S}C_{s}+\frac{C_{s}}{\gamma}\bigg{)}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{2C_{a}^{2}}{C_{s}^{3}}\partial_{S}C_{s}+\frac{1}{\gamma}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{1}_{35}=$
$\displaystyle-(\lambda_{3}-\lambda_{5})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2\rho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{2}\partial_{\rho}\bigg{(}\rho\big{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\big{)}\bigg{)}^{2}+\frac{2\rho
C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2\rho
C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}+\frac{C_{a}^{2}}{C_{s}^{3}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}+\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{1}_{36}=$
$\displaystyle(\lambda_{3}-\lambda_{6})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}-\frac{C_{f}^{2}}{C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{1}_{37}=$
$\displaystyle-(\lambda_{3}-\lambda_{7})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}-\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{1}_{42}=$ $\displaystyle 0,\
\gamma^{1}_{43}=-\frac{(\lambda_{4}-\lambda_{3})C_{f}^{2}C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)},\
\gamma^{1}_{45}=-\frac{(\lambda_{4}-\lambda_{3})C_{f}^{2}C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)},\
\gamma^{1}_{46}=0,$ $\displaystyle\gamma^{1}_{47}=$
$\displaystyle\frac{(\lambda_{4}-\lambda_{7})C_{f}^{2}C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)},$
$\displaystyle\gamma^{1}_{52}=$
$\displaystyle-(\lambda_{5}-\lambda_{2})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{C_{f}^{2}}{C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{1}_{53}=$
$\displaystyle-(\lambda_{5}-\lambda_{3})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2\rho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{2}\partial_{\rho}\bigg{(}\rho\big{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\big{)}\bigg{)}^{2}+\frac{2\rho
C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2\rho
C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle-\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}+\frac{C_{a}^{2}}{C_{s}^{3}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}+\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{1}_{54}=$
$\displaystyle-(\lambda_{5}-\lambda_{4})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\rho\partial_{\rho}C_{s}-\partial_{S}C_{s}+\frac{C_{s}}{\gamma}\bigg{)}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{2C_{a}^{2}}{C_{s}^{3}}\partial_{S}C_{s}+\frac{1}{\gamma}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{1}_{56}=$
$\displaystyle(\lambda_{5}-\lambda_{6})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{C_{f}^{2}}{C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{1}_{57}=$
$\displaystyle-(\lambda_{5}-\lambda_{7})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}+\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{1}_{62}=$
$\displaystyle\frac{(\lambda_{6}-\lambda_{2})C_{f}(C_{f}-C_{a})(C_{a}^{2}-C_{s})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\
\gamma^{1}_{63}=0,\ \gamma^{1}_{64}=0,\ \gamma^{1}_{65}=0,\
\gamma^{1}_{67}=0,$ $\displaystyle\gamma^{1}_{72}=$
$\displaystyle-(\lambda_{7}-\lambda_{2})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{a}^{2}+C_{s}^{2}}{2C_{f}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{f}-H_{2}\partial_{H_{3}}C_{f})\bigg{\\}},$
$\displaystyle\gamma^{1}_{73}=$
$\displaystyle-(\lambda_{7}-\lambda_{3})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}\bigg{]}+\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{1}_{74}=$
$\displaystyle-(\lambda_{7}-\lambda_{4})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{\gamma}\partial_{\rho}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{2C_{a}^{2}}{C_{f}^{3}}\partial_{S}C_{f}\bigg{]}$
$\displaystyle+\frac{C_{a}^{2}-C_{s}^{2}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{f}}\partial_{\rho}C_{f}-\frac{\partial_{S}C_{f}}{C_{f}}+\frac{1}{\gamma}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{1}_{75}=$
$\displaystyle(\lambda_{7}-\lambda_{5})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}\bigg{]}+\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{1}_{76}=$
$\displaystyle(\lambda_{7}-\lambda_{6})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{a}^{2}+C_{s}^{2}}{2C_{f}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{f}-H_{2}\partial_{H_{3}}C_{f})\bigg{\\}}.$

As one can see, the coefficients $\gamma^{1}_{km}$ have very complex forms.
Nevertheless, thanks to the proper right eigenvectors, there is no singular
factors like ${(C_{a}^{2}-C_{s}^{2})}^{-1},{H_{2}}^{-1},{H_{3}}^{-1}$. Hence,
every term in the formula is a simple function of $\Phi$ (or its derivatives),
which is clearly of $O(1)$. Moreover, the potentially singular factor
$\frac{1}{H_{2}^{2}+H_{3}^{2}}$ in the left eigenvectors, as we explained in
Proposition 3.1, can always be cancelled by a numerator which is either
$H_{2}^{2}+H_{3}^{2}$ or $0$. Then, the combination of these $O(1)$ terms is
also regular.

For $i=2$, we obtain

$\displaystyle\gamma^{2}_{21}=$
$\displaystyle\frac{\lambda_{2}-\lambda_{1}}{4}\bigg{(}\frac{C_{a}}{C_{f}}-1\bigg{)}^{2},\
\gamma^{2}_{23}=\frac{(\lambda_{2}-\lambda_{3})}{4}\bigg{(}\frac{C_{a}}{C_{s}}-1\bigg{)}^{2},\
\gamma^{2}_{24}=-\frac{\lambda_{2}-\lambda_{4}}{4\gamma},$
$\displaystyle\gamma^{2}_{25}=$
$\displaystyle-\frac{(\lambda_{2}-\lambda_{5})}{4}\bigg{(}\frac{C_{a}}{C_{s}}+1\bigg{)}^{2},\
\gamma^{2}_{26}=0,\
\gamma^{2}_{27}=-\frac{(\lambda_{2}-\lambda_{7})}{4}\bigg{(}\frac{C_{a}}{C_{f}}+1\bigg{)}^{2},$
$\displaystyle\gamma^{2}_{13}=$ $\displaystyle 0,\ \gamma^{2}_{14}=0,\
\gamma^{2}_{15}=0,\
\gamma^{2}_{16}=\frac{(\lambda_{1}-\lambda_{6})(C_{a}+C_{f})}{2C_{f}},\
\gamma^{2}_{17}=0,$ $\displaystyle\gamma^{2}_{31}=$ $\displaystyle 0,\
\gamma^{2}_{34}=0,\ \gamma^{2}_{35}=0,\
\gamma^{2}_{36}=-\frac{(\lambda_{3}-\lambda_{6})(C_{a}+C_{s})}{2C_{s}},\
\gamma^{2}_{37}=0,$ $\displaystyle\gamma^{2}_{41}=$ $\displaystyle 0,\
\gamma^{2}_{43}=0,\ \gamma^{2}_{45}=0,\ \gamma^{2}_{46}=0,\
\gamma^{2}_{47}=0,$ $\displaystyle\gamma^{2}_{51}=$ $\displaystyle 0,\
\gamma^{2}_{53}=0,\ \gamma^{2}_{54}=0,\
\gamma^{2}_{56}=-\frac{(\lambda_{5}-\lambda_{6})(C_{a}-C_{s})}{2C_{s}},\
\gamma^{2}_{57}=0,$ $\displaystyle\gamma^{2}_{61}=$
$\displaystyle\frac{\lambda_{6}-\lambda_{1}}{4}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}+1\bigg{)},\
\gamma^{2}_{63}=-\frac{(\lambda_{6}-\lambda_{3})(C_{a}^{2}-C_{s}^{2})}{4C_{s}^{2}},\
\gamma^{2}_{64}=-\frac{\lambda_{6}-\lambda_{4}}{4\gamma},$
$\displaystyle\gamma^{2}_{65}=$
$\displaystyle\frac{(\lambda_{6}-\lambda_{5})(C_{a}^{2}-C_{s}^{2})}{4C_{s}^{2}},\
\gamma^{2}_{67}=-\frac{\lambda_{6}-\lambda_{7}}{4}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}+1\bigg{)},$
$\displaystyle\gamma^{2}_{71}=$ $\displaystyle 0,\ \gamma^{2}_{73}=0,\
\gamma^{2}_{74}=0,\ \gamma^{2}_{75}=0,\
\gamma^{2}_{76}=\frac{(\lambda_{7}-\lambda_{6})(C_{f}-C_{a})}{2C_{f}}.$

One can easily see that $\gamma^{2}_{km}$ are all uniformly bounded.

Next, we calculate $\gamma^{3}_{km}$.

$\displaystyle\gamma^{3}_{31}=$
$\displaystyle-(\lambda_{3}-\lambda_{1})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}$
$\displaystyle-
H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}$
$\displaystyle-\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}-\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}-\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}$
$\displaystyle-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}+\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{3}_{32}=$
$\displaystyle(\lambda_{3}-\lambda_{2})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{3C_{f}^{2}-C_{a}^{2}}{2C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{3}_{34}=$
$\displaystyle-(\lambda_{3}-\lambda_{4})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}-\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma}\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}+\frac{2C_{a}^{2}}{C_{s}^{3}}\partial_{S}C_{s}\bigg{)}$
$\displaystyle-\frac{C_{a}^{2}-C_{f}^{2}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{s}}\partial_{\rho}C_{s}-\frac{\partial_{S}C_{s}}{C_{s}}+\frac{1}{\gamma}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{3}_{35}=$
$\displaystyle\frac{(\lambda_{3}-\lambda_{5})(C_{a}^{2}-C_{f}^{2})}{C_{f}^{2}-C_{s}^{2}}\bigg{[}\frac{\rho(C_{a}^{2}-C_{s}^{2})}{C_{s}^{3}}\partial_{\rho}C_{s}+\frac{C_{a}^{2}}{C_{s}^{2}}-\frac{H_{2}}{C_{s}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}}\partial_{H_{3}}C_{s}\bigg{]},$
$\displaystyle\gamma^{3}_{36}=$
$\displaystyle-(\lambda_{3}-\lambda_{6})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{3C_{f}^{2}-C_{a}^{2}}{2C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{3}_{37}=$
$\displaystyle-(\lambda_{3}-\lambda_{7})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}$
$\displaystyle+H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}$
$\displaystyle-\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}-\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}-\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}$
$\displaystyle+\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}+\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}-\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}-\frac{1}{C_{s}}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{3}_{12}=$
$\displaystyle-(\lambda_{1}-\lambda_{2})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\bigg{[}\frac{C_{s}^{2}}{(C_{f}^{2}-C_{s}^{2})C_{f}}-\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2C_{f}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{]}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{3}_{14}=$
$\displaystyle(\lambda_{1}-\lambda_{4})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}+\frac{2C_{a}^{2}}{C_{f}^{3}}\partial_{S}C_{f}+\frac{1}{\gamma}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{f}^{2}}\partial_{\rho}C_{f}-\frac{\partial_{S}C_{f}}{C_{f}^{2}}+\frac{1}{\gamma
C_{f}}\bigg{)}\bigg{\\}},$ $\displaystyle\gamma^{3}_{15}=$
$\displaystyle-(\lambda_{1}-\lambda_{5})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle-\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}\bigg{]}-\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{3}_{16}=$
$\displaystyle(\lambda_{1}-\lambda_{6})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\bigg{[}\frac{C_{s}^{2}}{(C_{f}^{2}-C_{s}^{2})C_{f}}-\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2C_{f}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{]}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{3}_{17}=$
$\displaystyle-(\lambda_{1}-\lambda_{7})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2\rho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{2}\partial_{\rho}\bigg{(}\rho\big{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\big{)}\bigg{)}^{2}+\frac{2\rho
C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2\rho
C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle-\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{C_{a}^{2}}{C_{f}^{3}}-\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}-\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}\bigg{]}-\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{3}_{21}=$ $\displaystyle 0,\ \gamma^{3}_{24}=0,\
\gamma^{3}_{25}=0,\
\gamma^{3}_{26}=-\frac{C_{s}(\lambda_{2}-\lambda_{6})(C_{a}+C_{s})(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\
\gamma^{3}_{27}=0,$ $\displaystyle\gamma^{3}_{41}=$
$\displaystyle\frac{(\lambda_{4}-\lambda_{1})(C_{a}^{2}-C_{f}^{2})C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\ \gamma^{3}_{42}=0,\
\gamma^{3}_{45}=-\frac{(\lambda_{4}-\lambda_{5})C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\ \gamma^{3}_{46}=0,$
$\displaystyle\gamma^{3}_{47}=$
$\displaystyle-\frac{(\lambda_{4}-\lambda_{7})(C_{a}^{2}-C_{f}^{2})C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},$ $\displaystyle\gamma^{3}_{51}=$
$\displaystyle-(\lambda_{5}-\lambda_{1})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}-\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{3}_{52}=$
$\displaystyle(\lambda_{5}-\lambda_{2})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}-\frac{C_{a}^{2}+C_{f}^{2}}{2C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{3}_{54}=$
$\displaystyle(\lambda_{5}-\lambda_{4})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}-\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{\gamma}\partial_{\rho}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}+\frac{2C_{a}^{2}}{C_{s}^{3}}\partial_{S}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{a}^{2}-C_{f}^{2}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{s}}\partial_{\rho}C_{s}-\frac{\partial_{S}C_{s}}{C_{s}}+\frac{1}{\gamma}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{3}_{56}=$
$\displaystyle(\lambda_{5}-\lambda_{6})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{C_{a}^{2}+C_{f}^{2}}{2C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{3}_{57}=$
$\displaystyle(\lambda_{5}-\lambda_{7})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}-\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{3}_{61}=$ $\displaystyle 0,\
\gamma^{3}_{62}=-\frac{C_{s}(\lambda_{6}-\lambda_{2})(C_{s}-C_{a})(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\
\gamma^{3}_{64}=0,\ \gamma^{3}_{65}=0,\ \gamma^{3}_{67}=0,$
$\displaystyle\gamma^{3}_{71}=$
$\displaystyle-(\lambda_{7}-\lambda_{1})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2\rho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{2}\partial_{\rho}\bigg{(}\rho\big{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\big{)}\bigg{)}^{2}+\frac{2\rho
C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2\rho
C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{C_{a}^{2}}{C_{f}^{3}}-\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}-\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}\bigg{]}-\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{3}_{72}=$
$\displaystyle(\lambda_{7}-\lambda_{2})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\bigg{[}\frac{C_{s}^{2}}{(C_{f}^{2}-C_{s}^{2})C_{f}}+\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2C_{f}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{]}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{3}_{74}=$
$\displaystyle(\lambda_{7}-\lambda_{4})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}+\frac{2C_{a}^{2}}{C_{f}^{3}}\partial_{S}C_{f}+\frac{1}{\gamma}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{f}^{2}}\partial_{\rho}C_{f}-\frac{\partial_{S}C_{f}}{C_{f}^{2}}+\frac{1}{\gamma
C_{f}}\bigg{)}\bigg{\\}},$ $\displaystyle\gamma^{3}_{75}=$
$\displaystyle-(\lambda_{7}-\lambda_{5})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle-\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}\bigg{]}+\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{3}_{76}=$
$\displaystyle(\lambda_{7}-\lambda_{6})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle-\bigg{[}\frac{C_{s}^{2}}{(C_{f}^{2}-C_{s}^{2})C_{f}}+\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2C_{f}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{]}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}}.$

Note that the coefficients $\gamma^{3}_{km}$ take similar forms to that in
$\gamma^{1}_{km}$. In the same fashion, one can verify that $\gamma^{3}_{km}$
are all regular.

Next, for $i>4$, the remaining coefficients can be estimated in a similar
manner based on the above calculations. This can be verified easily by
noticing the following two facts: first, the potentially singular
$\frac{1}{H_{2}^{2}+H_{3}^{2}}$ term appears only in the $2^{\text{nd}}$,
$3^{\text{rd}}$, $5^{\text{th}}$, $6^{\text{th}}$ components of $l_{i}$ and
$l_{i}^{1,4,7}$ are all regular. Since all $\nabla_{\Phi}r_{k}\cdot r_{m}$ are
of order $1$, it holds that $l_{i}^{j}\cdot(\nabla_{\Phi}r_{k}\cdot
r_{m})_{j}=O(1)$ with $j=1,4,7$. Then, we only need to check

$\sum\limits_{j=2,3}l_{i}^{j}\cdot(\nabla_{\Phi}r_{k}\cdot
r_{m})_{j},\quad\sum\limits_{j=5,6}l_{i}^{j}\cdot(\nabla_{\Phi}r_{k}\cdot
r_{m})_{j}.$

Second, the components of $l_{i}$ satisfy the following symmetry properties

$l_{i}^{1,2,3}=l_{8-i}^{1,2,3},\quad l_{i}^{4,5,6,7}=-l_{8-i}^{4,5,6,7}.$

Therefore, the rest coefficients $\gamma^{i}_{km}$ with $i>4$ can be regarded
as counterparts of the known ones and thus can be easily verified to be of
$O(1)$. Now we also list all the other coefficients with for $i>4$:

$\displaystyle\gamma^{5}_{51}=$
$\displaystyle-(\lambda_{5}-\lambda_{1})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}$
$\displaystyle+H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}$
$\displaystyle-\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}-\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}-\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle-\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}$
$\displaystyle+\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}+\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}-\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}-\frac{1}{C_{s}}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{5}_{52}=$
$\displaystyle(\lambda_{5}-\lambda_{2})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{3C_{f}^{2}-C_{a}^{2}}{2C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{5}_{53}=$
$\displaystyle\frac{(\lambda_{3}-\lambda_{5})(C_{a}^{2}-C_{f}^{2})}{C_{f}^{2}-C_{s}^{2}}\bigg{[}\frac{\rho(C_{a}^{2}-C_{s}^{2})}{C_{s}^{3}}\partial_{\rho}C_{s}+\frac{C_{a}^{2}}{C_{s}^{2}}-\frac{H_{2}}{C_{s}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}}\partial_{H_{3}}C_{s}\bigg{]},$
$\displaystyle\gamma^{5}_{54}=$
$\displaystyle-(\lambda_{5}-\lambda_{4})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}-\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}+\frac{2C_{a}^{2}}{C_{s}^{3}}\partial_{S}C_{s}\bigg{]}$
$\displaystyle-\frac{C_{a}^{2}-C_{f}^{2}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{s}}\partial_{\rho}C_{s}-\frac{\partial_{S}C_{s}}{C_{s}}+\frac{1}{\gamma}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{5}_{56}=$
$\displaystyle-(\lambda_{5}-\lambda_{6})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{3C_{f}^{2}-C_{a}^{2}}{2C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{5}_{57}=$
$\displaystyle(\lambda_{5}-\lambda_{7})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}$
$\displaystyle-
H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}$
$\displaystyle-\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}-\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}-\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}]$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}$
$\displaystyle-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}+\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{5}_{12}=$
$\displaystyle-(\lambda_{1}-\lambda_{2})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle-\bigg{[}\frac{C_{s}^{2}}{(C_{f}^{2}-C_{s}^{2})C_{f}}+\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2C_{f}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{]}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{5}_{13}=$
$\displaystyle(\lambda_{1}-\lambda_{3})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle-\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}\bigg{]}+\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{5}_{14}=$
$\displaystyle(\lambda_{1}-\lambda_{4})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}+\frac{2C_{a}^{2}}{C_{f}^{3}}\partial_{S}C_{f}+\frac{1}{\gamma}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{f}^{2}}\partial_{\rho}C_{f}-\frac{\partial_{S}C_{f}}{C_{f}^{2}}+\frac{1}{\gamma
C_{f}}\bigg{)}\bigg{\\}},$ $\displaystyle\gamma^{5}_{16}=$
$\displaystyle(\lambda_{1}-\lambda_{6})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle-\bigg{[}\frac{C_{s}^{2}}{(C_{f}^{2}-C_{s}^{2})C_{f}}+\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2C_{f}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{]}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{5}_{17}=$
$\displaystyle-(\lambda_{1}-\lambda_{7})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2\rho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{2}\partial_{\rho}\bigg{(}\rho\big{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\big{)}\bigg{)}^{2}+\frac{2\rho
C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2\rho
C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle-\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{C_{a}^{2}}{C_{f}^{3}}-\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}-\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}\bigg{]}+\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{5}_{21}=$ $\displaystyle 0,\ \gamma^{5}_{23}=0,\
\gamma^{5}_{24}=0,\
\gamma^{5}_{26}=-\frac{C_{s}(\lambda_{2}-\lambda_{6})(C_{a}-C_{s})(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\
\gamma^{5}_{27}=0,$ $\displaystyle\gamma^{5}_{31}=$
$\displaystyle-(\lambda_{3}-\lambda_{1})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}-\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{5}_{32}=$
$\displaystyle(\lambda_{3}-\lambda_{2})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}-\frac{C_{a}^{2}+C_{f}^{2}}{2C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{5}_{34}=$
$\displaystyle-(\lambda_{3}-\lambda_{4})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}-\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{\gamma}\partial_{\rho}\rho(\frac{C_{a}^{2}}{C_{s}^{2}}-1)+\frac{2C_{a}^{2}}{C_{s}^{3}}\partial_{S}C_{s}\bigg{]}$
$\displaystyle-\frac{C_{a}^{2}-C_{f}^{2}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{s}}\partial_{\rho}C_{s}-\frac{\partial_{S}C_{s}}{C_{s}}+\frac{1}{\gamma}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{5}_{36}=$
$\displaystyle-(\lambda_{3}-\lambda_{6})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}-\frac{C_{a}^{2}+C_{f}^{2}}{2C_{s}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{5}_{37}=$
$\displaystyle(\lambda_{3}-\lambda_{7})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}-\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{5}_{41}=$
$\displaystyle-\frac{(\lambda_{4}-\lambda_{1})(C_{a}^{2}-C_{f}^{2})C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\ \gamma^{5}_{42}=0,\
\gamma^{5}_{43}=-\frac{(\lambda_{4}-\lambda_{3})C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\ \gamma^{5}_{46}=0,$
$\displaystyle\gamma^{5}_{47}=$
$\displaystyle\frac{(\lambda_{4}-\lambda_{7})(C_{a}^{2}-C_{f}^{2})C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},$ $\displaystyle\gamma^{5}_{61}=$
$\displaystyle 0,\
\gamma^{5}_{62}=\frac{C_{s}(\lambda_{6}-\lambda_{2})(C_{s}+C_{a})(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\
\gamma^{5}_{63}=0,\ \gamma^{5}_{64}=0,\ \gamma^{5}_{67}=0,$
$\displaystyle\gamma^{5}_{71}=$
$\displaystyle-(\lambda_{7}-\lambda_{1})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2\rho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{2}\partial_{\rho}\bigg{(}\rho\big{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\big{)}\bigg{)}^{2}+\frac{2\rho
C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2\rho
C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{C_{a}^{2}}{C_{f}^{3}}-\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}-\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}\bigg{]}+\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{5}_{72}=$
$\displaystyle-(\lambda_{7}-\lambda_{2})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\bigg{[}\frac{C_{s}^{2}}{(C_{f}^{2}-C_{s}^{2})C_{f}}-\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2C_{f}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{]}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{5}_{73}=$
$\displaystyle(\lambda_{7}-\lambda_{3})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle-\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}\bigg{]}-\frac{C_{s}^{2}(C_{a}^{2}-C_{f}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{5}_{74}=$
$\displaystyle(\lambda_{7}-\lambda_{4})\bigg{\\{}\frac{C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}+\frac{2C_{a}^{2}}{C_{f}^{3}}\partial_{S}C_{f}+\frac{1}{\gamma}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{s}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{f}^{2}}\partial_{\rho}C_{f}-\frac{\partial_{S}C_{f}}{C_{f}^{2}}+\frac{1}{\gamma
C_{f}}\bigg{)}\bigg{\\}},$ $\displaystyle\gamma^{5}_{76}=$
$\displaystyle(\lambda_{7}-\lambda_{6})\bigg{\\{}\frac{-C_{s}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\bigg{[}\frac{C_{s}^{2}}{(C_{f}^{2}-C_{s}^{2})C_{f}}-\frac{C_{s}(C_{a}^{2}-C_{f}^{2})}{2C_{f}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{]}(H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s})\bigg{\\}},$
$\displaystyle\gamma^{6}_{61}=$
$\displaystyle\frac{(\lambda_{6}-\lambda_{1})}{4}\bigg{(}\frac{C_{a}}{C_{f}}+1\bigg{)}^{2},\
\gamma^{6}_{62}=0,\
\gamma^{6}_{63}=\frac{(\lambda_{6}-\lambda_{3})}{4}\bigg{(}\frac{C_{a}}{C_{s}}+1\bigg{)}^{2},$
$\displaystyle\gamma^{6}_{64}=$
$\displaystyle-\frac{\lambda_{6}-\lambda_{4}}{4\gamma},\
\gamma^{6}_{65}=-\frac{(\lambda_{6}-\lambda_{5})}{4}\bigg{(}\frac{C_{a}}{C_{s}}-1\bigg{)}^{2},\
\gamma^{6}_{67}=-\frac{\lambda_{6}-\lambda_{7}}{4}\bigg{(}\frac{C_{a}}{C_{f}}-1\bigg{)}^{2},$
$\displaystyle\gamma^{6}_{12}=$
$\displaystyle-\frac{(\lambda_{1}-\lambda_{2})(C_{f}-C_{a})}{2C_{f}},\
\gamma^{6}_{13}=0,\ \gamma^{6}_{14}=0,\ \gamma^{6}_{15}=0,\
\gamma^{6}_{17}=0,$ $\displaystyle\gamma^{6}_{21}=$
$\displaystyle\frac{\lambda_{2}-\lambda_{1}}{4}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}+1\bigg{)},\
\gamma^{6}_{23}=-\frac{(\lambda_{2}-\lambda_{3})(C_{a}^{2}-C_{s}^{2})}{4C_{s}^{2}},\
\gamma^{6}_{24}=-\frac{\lambda_{2}-\lambda_{4}}{4\gamma},$
$\displaystyle\gamma^{6}_{25}=$
$\displaystyle\frac{(\lambda_{2}-\lambda_{5})(C_{a}^{2}-C_{s}^{2})}{4C_{s}^{2}},\
\gamma^{6}_{27}=-\frac{\lambda_{2}-\lambda_{7}}{4}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}+1\bigg{)},$
$\displaystyle\gamma^{6}_{31}=$ $\displaystyle 0,\
\gamma^{6}_{32}=\frac{(\lambda_{3}-\lambda_{2})(C_{a}-C_{s})}{2C_{s}},\
\gamma^{6}_{34}=0,\ \gamma^{6}_{35}=0,\ \gamma^{6}_{37}=0,$
$\displaystyle\gamma^{6}_{41}=$ $\displaystyle 0,\ \gamma^{6}_{42}=0,\
\gamma^{6}_{43}=0,\ \gamma^{6}_{45}=0,\ \gamma^{6}_{47}=0,$
$\displaystyle\gamma^{6}_{51}=$ $\displaystyle 0,\
\gamma^{6}_{52}=\frac{(\lambda_{5}-\lambda_{2})(C_{a}+C_{s})}{2C_{s}},\
\gamma^{6}_{53}=0,\ \gamma^{6}_{54}=0,\ \gamma^{6}_{57}=0,$
$\displaystyle\gamma^{6}_{71}=$ $\displaystyle 0,\
\gamma^{6}_{72}=-\frac{(\lambda_{7}-\lambda_{2})(C_{a}+C_{f})}{2C_{f}},\
\gamma^{6}_{73}=0,\ \gamma^{6}_{74}=0,\ \gamma^{6}_{75}=0,$
$\displaystyle\gamma^{7}_{71}=$
$\displaystyle-\frac{(\lambda_{1}-\lambda_{7})(C_{a}^{2}-C_{s}^{2})}{C_{f}^{2}-C_{s}^{2}}\bigg{[}\frac{\rho(C_{a}^{2}-C_{f}^{2})}{C_{f}^{3}}\partial_{\rho}C_{f}+\frac{C_{a}^{2}}{C_{f}^{2}}-\frac{H_{2}}{C_{f}}\partial_{H_{2}}C_{f}-\frac{H_{3}}{C_{f}}\partial_{H_{3}}C_{f}\bigg{]},$
$\displaystyle\gamma^{7}_{72}=$
$\displaystyle-(\lambda_{7}-\lambda_{2})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{a}^{2}-3C_{s}^{2}}{2C_{f}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{f}-H_{2}\partial_{H_{3}}C_{f})\bigg{\\}},$
$\displaystyle\gamma^{7}_{73}=$
$\displaystyle(\lambda_{7}-\lambda_{3})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}$
$\displaystyle-
H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}$
$\displaystyle-\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}-\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}-\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}C_{f}(\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}})\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}+\frac{1}{C_{f}}-\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}-\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}$
$\displaystyle-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}+\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{7}_{74}=$
$\displaystyle-(\lambda_{7}-\lambda_{4})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma}\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}+\frac{2C_{a}^{2}}{C_{f}^{3}}\partial_{S}C_{f}\bigg{)}$
$\displaystyle+\frac{C_{a}^{2}-C_{s}^{2}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{f}}\partial_{\rho}C_{f}-\frac{\partial_{S}C_{f}}{C_{f}}+\frac{1}{\gamma}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{7}_{75}=$
$\displaystyle(\lambda_{7}-\lambda_{5})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}$
$\displaystyle-
H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}+\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}$
$\displaystyle-\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}-\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}-\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}$
$\displaystyle-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}+\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{7}_{76}=$
$\displaystyle(\lambda_{7}-\lambda_{6})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{a}^{2}-3C_{s}^{2}}{2C_{f}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{f}-H_{2}\partial_{H_{3}}C_{f})\bigg{\\}},$
$\displaystyle\gamma^{7}_{12}=$
$\displaystyle-(\lambda_{1}-\lambda_{2})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{a}^{2}+C_{s}^{2}}{2C_{f}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{f}-H_{2}\partial_{H_{3}}C_{f})\bigg{\\}},$
$\displaystyle\gamma^{7}_{13}=$
$\displaystyle-(\lambda_{1}-\lambda_{3})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}\bigg{]}+\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{7}_{14}=$
$\displaystyle-(\lambda_{1}-\lambda_{4})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{\gamma}\partial_{\rho}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{2C_{a}^{2}}{C_{f}^{3}}\partial_{S}C_{f}\bigg{]}$
$\displaystyle+\frac{C_{a}^{2}-C_{s}^{2}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{\rho}{\gamma
C_{f}}\partial_{\rho}C_{f}-\frac{\partial_{S}C_{f}}{C_{f}}+\frac{1}{\gamma}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{7}_{15}=$
$\displaystyle(\lambda_{1}-\lambda_{5})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}+H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{f}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{f}^{3}}\partial_{H_{2}}C_{f}+\frac{2C_{a}^{2}H_{3}}{C_{f}^{3}}\partial_{H_{3}}C_{f}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{H_{2}}{C_{f}^{2}}\partial_{H_{2}}C_{f}+\frac{H_{3}}{C_{f}^{2}}\partial_{H_{3}}C_{f}-\bigg{(}\frac{1}{C_{f}}+\frac{\rho\partial_{\rho}C_{f}}{C_{f}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}-\frac{1}{C_{f}}\bigg{]}+\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{7}_{16}=$
$\displaystyle(\lambda_{1}-\lambda_{6})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{f}}-C_{f}\bigg{)}\bigg{]}+\frac{C_{a}^{2}+C_{s}^{2}}{2C_{f}(C_{f}^{2}-C_{s}^{2})}(H_{3}\partial_{H_{2}}C_{f}-H_{2}\partial_{H_{3}}C_{f})\bigg{\\}},$
$\displaystyle\gamma^{7}_{21}=$ $\displaystyle 0,\ \gamma^{7}_{23}=0,\
\gamma^{7}_{24}=0,\ \gamma^{7}_{25}=0,\
\gamma^{7}_{26}=\frac{(\lambda_{2}-\lambda_{6})C_{f}(C_{a}-C_{f})(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},$
$\displaystyle\gamma^{7}_{31}=$
$\displaystyle(\lambda_{3}-\lambda_{1})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}+\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{7}_{32}=$
$\displaystyle-(\lambda_{3}-\lambda_{2})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{C_{f}^{2}}{C_{s}(C_{f}^{2}-C_{s}^{2})}\bigg{(}H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s}\bigg{)}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{7}_{34}=$
$\displaystyle-(\lambda_{3}-\lambda_{4})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\rho\partial_{\rho}C_{s}-\partial_{S}C_{s}+\frac{C_{s}}{\gamma}\bigg{)}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{2C_{a}^{2}}{C_{s}^{3}}\partial_{S}C_{s}+\frac{1}{\gamma}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{7}_{35}=$
$\displaystyle-(\lambda_{3}-\lambda_{5})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2\rho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{2}\partial_{\rho}\bigg{(}\rho(\frac{C_{a}^{2}}{C_{s}^{2}}-1)\bigg{)}^{2}+\frac{2\rho
C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2\rho
C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}+\frac{C_{a}^{2}}{C_{s}^{3}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}-\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{7}_{36}=$
$\displaystyle(\lambda_{3}-\lambda_{6})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{C_{f}^{2}}{C_{s}(C_{f}^{2}-C_{s}^{2})}\bigg{(}H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s}\bigg{)}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{7}_{41}=$
$\displaystyle\frac{(\lambda_{4}-\lambda_{1})C_{f}^{2}C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)},\
\gamma^{7}_{42}=0,\
\gamma^{7}_{43}=\frac{(\lambda_{4}-\lambda_{3})C_{f}^{2}C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)},$
$\displaystyle\gamma^{7}_{45}=$
$\displaystyle\frac{(\lambda_{4}-\lambda_{3})C_{f}^{2}C_{s}^{2}}{2\gamma
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)},\
\gamma^{7}_{46}=0,$ $\displaystyle\gamma^{7}_{51}=$
$\displaystyle(\lambda_{5}-\lambda_{1})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}+\frac{2C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\bigg{(}\frac{1}{C_{s}}+\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{)}\bigg{(}\frac{C_{a}^{2}}{C_{f}^{2}}-1\bigg{)}+\frac{1}{C_{s}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}-\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{7}_{52}=$
$\displaystyle-(\lambda_{5}-\lambda_{2})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}-\frac{C_{f}^{2}}{C_{s}(C_{f}^{2}-C_{s}^{2})}\bigg{(}H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s}\bigg{)}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{7}_{53}=$
$\displaystyle-(\lambda_{5}-\lambda_{3})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\rho\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{3}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}$
$\displaystyle-\frac{C_{f}^{2}C_{s}^{2}}{2\rho
C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{1}{2}\partial_{\rho}\bigg{(}\rho\big{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\big{)}\bigg{)}^{2}+\frac{2\rho
C_{a}^{2}H_{2}}{C_{s}^{3}}\partial_{H_{2}}C_{s}+\frac{2\rho
C_{a}^{2}H_{3}}{C_{s}^{3}}\partial_{H_{3}}C_{s}\bigg{]}$
$\displaystyle-\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho\partial_{\rho}C_{s}}{C_{s}^{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1\bigg{)}+\frac{C_{a}^{2}}{C_{s}^{3}}-\frac{H_{2}}{C_{s}^{2}}\partial_{H_{2}}C_{s}-\frac{H_{3}}{C_{s}^{2}}\partial_{H_{3}}C_{s}\bigg{]}-\frac{C_{f}^{2}(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{\\}},$
$\displaystyle\gamma^{7}_{54}=$
$\displaystyle-(\lambda_{5}-\lambda_{4})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{\rho}{\gamma}\partial_{\rho}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-\partial_{S}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}\rho\partial_{\rho}C_{s}-\partial_{S}C_{s}+\frac{C_{s}}{\gamma}\bigg{)}$
$\displaystyle+\frac{C_{f}^{2}C_{s}^{2}}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{[}\frac{2C_{a}^{2}}{C_{s}^{3}}\partial_{S}C_{s}+\frac{1}{\gamma}\bigg{(}\frac{C_{a}^{2}}{C_{s}^{2}}-1+\rho\partial_{\rho}\frac{C_{a}^{2}}{C_{s}^{2}}\bigg{)}\bigg{]}\bigg{\\}},$
$\displaystyle\gamma^{7}_{56}=$
$\displaystyle(\lambda_{5}-\lambda_{6})\bigg{\\{}\frac{C_{f}}{2(C_{f}^{2}-C_{s}^{2})}\bigg{[}H_{3}\partial_{H_{2}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}-H_{2}\partial_{H_{3}}\bigg{(}\frac{C_{a}^{2}}{C_{s}}-C_{s}\bigg{)}\bigg{]}-\frac{C_{f}^{2}}{C_{s}(C_{f}^{2}-C_{s}^{2})}\bigg{(}H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s}\bigg{)}$
$\displaystyle+\frac{C_{f}(C_{a}^{2}-C_{s}^{2})}{2C_{s}^{2}(C_{f}^{2}-C_{s}^{2})}\bigg{(}H_{3}\partial_{H_{2}}C_{s}-H_{2}\partial_{H_{3}}C_{s}\bigg{)}\bigg{\\}},$
$\displaystyle\gamma^{7}_{61}=$ $\displaystyle 0,\
\gamma^{7}_{62}=-\frac{(\lambda_{6}-\lambda_{2})C_{f}(C_{f}+C_{a})(C_{a}^{2}-C_{s}^{2})}{2C_{a}^{2}(C_{f}^{2}-C_{s}^{2})},\
\gamma^{7}_{63}=0,\ \gamma^{7}_{64}=0,\ \gamma^{7}_{65}=0.$

With all above calculations, we now conclude that all the coefficients
$c^{i}_{im}$ and $\gamma^{i}_{km}$ in the decomposed system (1.4) are of order
$1$. This finishes the proof of Proposition 3.1.

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