the left-hand-side in the first variation formula reads

$\displaystyle\frac{d}{d\theta}A(\zeta+\theta\zeta_{1})\Bigg{|}_{\theta=0}$
$\displaystyle=\int_{\mathbb{S}^{2}}\left(-\rho\Delta_{g_{0}}\zeta+\frac{\rho\mathrm{Hess}_{g_{0}}(\zeta)(\nabla_{0}\zeta,\nabla_{0}\zeta)}{\rho^{2}+|\nabla_{0}\zeta|_{g_{0}}^{2}}+2\rho^{2}\right)\frac{\zeta_{1}}{\sqrt{\rho^{2}+|\nabla_{0}\zeta|_{g_{0}}^{2}}}d\mu_{0},$

Thus the mean curvature operator reads

(5.2)
$H(\zeta)=\frac{1}{\rho\sqrt{\rho^{2}+|\nabla_{0}\zeta|_{g_{0}}^{2}}}\left(\Delta_{g_{0}}\zeta-\frac{\mathrm{Hess}_{g_{0}}(\zeta)(\nabla_{0}\zeta,\nabla_{0}\zeta)}{\rho^{2}+|\nabla_{0}\zeta|_{g_{0}}^{2}}\right)+\frac{2}{\rho\sqrt{\rho^{2}+|\nabla_{0}\zeta|_{g_{0}}^{2}}}.$

We lift (5.2) to $\mathrm{SU}(2)$. Recalling that
$\sum_{j=1}^{3}X_{j}\zeta^{\sharp}\sigma[X_{j}]$, where the left-invariant
vector fields $X_{i}$ are as in (3.3), we compute the lift of the Hessian form
as

$\displaystyle\mathrm{Hess}_{G_{0}}(\zeta^{\sharp})(\nabla_{G_{0}}\zeta^{\sharp},\nabla_{G_{0}}\zeta^{\sharp})$
$\displaystyle=\sum_{i,j=1}^{3}\left(X_{i}X_{j}\zeta^{\sharp}-(\nabla_{X_{i}}X_{j})\zeta^{\sharp}\right)\cdot
X_{i}\zeta^{\sharp}\cdot X_{j}\zeta^{\sharp}$
$\displaystyle=\mathbf{Op}(\beta_{2}(x,l)^{2})\zeta^{\sharp}.$

Here we used that $\nabla_{X_{i}}X_{j}=[X_{i},X_{j}]/2$ (with respect to the
bi-invariant metric) on the Lie group $\mathrm{SU}(2)$. Recalling the symbol
$\lambda^{(1)}=\sqrt{\beta_{2}^{2}+\beta_{1}l(l+1)}/(\rho^{\sharp})^{2}$ in
(4.21), we find that the first term in (5.2) are lifted as

(5.3)
$\displaystyle\frac{\Delta_{G_{0}}\zeta^{\sharp}}{\rho^{\sharp}(x)\beta_{1}(x)^{1/2}}-\frac{\mathrm{Hess}_{G_{0}}(\zeta^{\sharp})(\nabla_{G_{0}}\zeta^{\sharp},\nabla_{G_{0}}\zeta^{\sharp})}{\rho^{\sharp}(x)\beta_{1}(x)^{3/2}}$
$\displaystyle=\mathbf{Op}\left(\frac{-l(l+1)}{\rho^{\sharp}(x)\beta_{1}(x)^{1/2}}-\frac{\beta_{2}^{2}(x,l)}{\rho^{\sharp}(x)\beta_{1}(x)^{3/2}}\right)\zeta^{\sharp}$
$\displaystyle=-\mathbf{Op}\big{(}h^{(2)}(x,l)\big{)}\zeta^{\sharp}.$

By Bony’s para-linearization theorem, (5.3) is then para-linearized as (with
the “mod” notation in 4.1 used)

$-T_{h^{(2)}}\zeta^{\sharp}+T_{\tilde{h}^{(1)}}\cdot\nabla_{G_{0}}\zeta^{\sharp}+T_{\tilde{h}_{0}}\zeta^{\sharp}\mod
H^{2s-3}_{x},$

where $\tilde{h}^{(1)},\tilde{h}_{0}$ are $C^{\infty}$ function of
$\big{(}\zeta^{\sharp},\nabla_{G_{0}}\zeta^{\sharp},\nabla_{G_{0}}^{2}\zeta^{\sharp}\big{)}$.
Finally, the last term in (5.2) is also lifted and para-linearized similarly.
This completes the proof. ∎

We are now at the place to para-linearize the second order terms in the
expression of $\partial_{t}\phi$, and derive the evolution equation of
$w^{\sharp}$.

######  Proposition 5.2.

Suppose $s>3$ and $\zeta\in H^{s+0.5}$, $\phi\in H^{s}$. The evolution of
$w^{\sharp}$ is given by

$\partial_{t}w^{\sharp}=-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}w^{\sharp}-T_{h}\zeta^{\sharp}+f_{2}(\zeta,\phi).$

The error term satisfies

$\|f_{2}(\zeta,\phi)\|_{H^{s}_{x}}\leq
K\big{(}\|\zeta\|_{H^{s+0.5}_{x}}\big{)}\|\phi\|_{H^{s}_{x}},$

where $K$ is an increasing function vanishing linearly at zero.

###### Proof.

We start by looking at the quadratic terms in the equation for
$\partial_{t}\phi$. By a direct computation using our coordinate system
$(x,y)$ introduced in last section, we argue just as in Lemma 3.26 of [ABZ11]
as follows. Still write $\rho=1+\zeta$. Then with

$F(\mu,a,b,c)=\frac{\left(\mu^{2}a+b\cdot
c\right)^{2}}{2\mu^{2}\big{(}\mu^{2}+|b|_{g_{0}}^{2}\big{)}},$

the quadratic part in the right-hand-side of the equation for $\phi$ in (EQ)
may be written as

(5.4)
$\displaystyle-\frac{|\nabla_{0}\phi|_{g_{0}}^{2}}{2\rho^{2}}+\frac{\left(\rho^{2}D[\zeta]\phi+\nabla_{0}\zeta\cdot\nabla_{0}\phi\right)^{2}}{2\rho^{2}\big{(}\rho^{2}+|\nabla_{0}\zeta|_{g_{0}}^{2}\big{)}}=-\frac{|\nabla_{0}\phi|_{g_{0}}^{2}}{2\rho^{2}}+F(\rho,D[\zeta]\phi,\nabla_{0}\zeta,\nabla_{0}\phi).$

Bony’s para-linearization theorem implies that

(5.5)
$\displaystyle\left(-\frac{|\nabla_{0}\phi|_{g_{0}}^{2}}{2\rho^{2}}\right)^{\sharp}$
$\displaystyle=-T_{(\rho^{\sharp})^{-2}\nabla_{G_{0}}\phi^{\sharp}}\cdot\nabla_{G_{0}}\phi^{\sharp}$
$\displaystyle=-T_{(\rho^{\sharp})^{-2}\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}\phi^{\sharp}-T_{(\rho^{\sharp})^{-2}\mathfrak{b}^{\sharp}\nabla_{G_{0}}\zeta^{\sharp}}\cdot\nabla_{G_{0}}\phi^{\sharp}\mod
H^{2s-3}_{x},$

and (here subscript for $F$ stands for partial differentiation)

$\big{(}F(\rho,D[\zeta]\phi,\nabla_{0}\zeta,\nabla_{0}\phi)\big{)}^{\sharp}=T_{F_{\mu}^{\sharp}}\rho^{\sharp}+T_{F_{a}^{\sharp}}\big{(}D[\zeta]\phi\big{)}^{\sharp}+T_{F_{b}^{\sharp}}\cdot\nabla_{G_{0}}\zeta^{\sharp}+T_{F_{c}^{\sharp}}\cdot\nabla_{G_{0}}\phi^{\sharp}\mod
H^{2s-3}_{x}.$

Since $\rho\in H^{s+0.5}_{x}$, we find $T_{F_{\mu}^{\sharp}}\rho^{\sharp}\in
H^{s+0.5}_{x}$ and thus can be disregarded. At
$(\mu,a,b,c)=(\rho,D[\zeta]\phi,\nabla_{0}\zeta,\nabla_{0}\phi)$, we compute,
recalling (4.22),

$\displaystyle F_{a}$ $\displaystyle=\frac{\mu^{2}a+b\cdot
c}{\mu^{2}+|b|_{g_{0}}^{2}}=\mathfrak{b},$ $\displaystyle F_{b}$
$\displaystyle=\frac{\mu^{2}a+b\cdot
c}{\mu^{2}\big{(}\mu^{2}+|b|_{g_{0}}^{2}\big{)}}\left(c-\frac{(\mu^{2}a+b\cdot
c)b}{\mu^{2}+|b|_{g_{0}}^{2}}\right)=\mathfrak{bv},$ $\displaystyle F_{c}$
$\displaystyle=\frac{\mu^{2}a+b\cdot
c}{\mu^{2}\big{(}\mu^{2}+|b|_{g_{0}}^{2}\big{)}}b=\frac{\mathfrak{b}\nabla_{0}\zeta}{\rho^{2}}.$

Thus

(5.6)
$\big{(}F(\rho,D[\zeta]\phi,\nabla_{0}\zeta,\nabla_{0}\phi)\big{)}^{\sharp}=T_{\mathfrak{b}^{\sharp}}\big{(}D[\zeta]\phi\big{)}^{\sharp}+T_{\mathfrak{b}^{\sharp}}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}\zeta^{\sharp}+T_{(\rho^{\sharp})^{-2}\mathfrak{b}^{\sharp}\nabla_{G_{0}}\phi^{\sharp}}\cdot\nabla_{G_{0}}\phi^{\sharp}\mod
H^{s}_{x}.$

Combining (5.5) and (5.6), after suitable cancellation and commutation
(causing acceptable error), we find that (5.4) lifts to $\mathrm{SU}(2)$ as

$-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}w^{\sharp}+T_{\mathfrak{b}^{\sharp}}\big{(}D[\zeta]\phi\big{)}^{\sharp}\mod
H_{x}^{s}.$

Consequently, noticing that
$\partial_{t}w^{\sharp}=\partial_{t}\phi^{\sharp}-T_{\partial_{t}\mathfrak{b}^{\sharp}}\phi^{\sharp}-T_{\mathfrak{b}^{\sharp}}\partial_{t}\zeta^{\sharp}$,
we find

$\partial_{t}w^{\sharp}=-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}w^{\sharp}-T_{h}\zeta^{\sharp}+T_{\partial_{t}\mathfrak{b}^{\sharp}}\phi^{\sharp}\mod
H_{x}^{s}.$

The error caused by para-linearization is quadratic in $\zeta$ and $\phi$,
since the para-linearization already includes the linear part of the nonlinear
expression. Finally, the regularity of
$T_{\partial_{t}\mathfrak{b}^{\sharp}}\phi^{\sharp}$ is obtained by applying
Proposition 4.1-4.2, identically as in Lemma 3.27. of [ABZ11]. This finishes
the proof. ∎

Just as in [ABZ11], we obtain the para-linearization of system (EQ), lifted to
$\mathrm{SU}(2)$:

(5.7) $\displaystyle\partial_{t}\zeta^{\sharp}$
$\displaystyle=T_{\lambda}w^{\sharp}-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}\zeta^{\sharp}+f_{1}(\zeta,\phi)$
$\displaystyle\partial_{t}w^{\sharp}$
$\displaystyle=-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}w^{\sharp}-T_{h}\zeta^{\sharp}+f_{2}(\zeta,\phi),$

where the errors $f_{1},f_{2}$ takes the explicit form

(5.8) $\displaystyle f_{1}(\zeta,\phi)$
$\displaystyle=\big{(}D[\zeta]\phi\big{)}^{\sharp}-T_{\lambda}\big{(}w^{\sharp}-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}\zeta^{\sharp}\big{)}$
$\displaystyle f_{2}(\zeta,\phi)$
$\displaystyle=\left(-\frac{|\nabla_{0}\phi|_{g_{0}}^{2}}{2\rho^{2}}+\frac{\left(\rho^{2}D[\zeta]\phi+\nabla_{0}\zeta\cdot\nabla_{0}\phi\right)^{2}}{2\rho^{2}\big{(}\rho^{2}+|\nabla_{0}\zeta|_{g_{0}}^{2}\big{)}}+H(\zeta)\right)^{\sharp}+T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}w^{\sharp}+T_{h}\zeta^{\sharp}-T_{\partial_{t}\mathfrak{b}^{\sharp}}w^{\sharp},$

and are affordable and in fact quadratic:

$\|f_{1}(\zeta,\phi)\|_{H_{x}^{s+0.5}}+\|f_{2}(\zeta,\phi)\|_{H_{x}^{s}}\lesssim\left(\|\zeta\|_{H_{x}^{s+0.5}}+\|\phi\|_{H_{x}^{s}}\right)^{2}.$

We already know that (5.7) is equivalent to (EQ) at least when
$\|\zeta(t)\|_{H^{s+0.5}_{x}}$ remains sufficiently small. Thus we have
successfully derived the para-linear form of (EQ).

###  5.2. Adjoint of Para-differential Operators

In order to symmetrize the system (5.7), we need to understand the adjoint
properties of the Dirichlet-Neumann operator and the mean curvature operator.
We start from the former. We still employ the notation $\rho=1+\zeta$, and
define the symbol $\lambda$ as in (4.21). The following proposition reflects
the self-adjointness of the Dirichlet-Neumann operator.

######  Proposition 5.3.

Still fix $s>3$ and $\zeta\in H^{s+0.5}$, $\phi\in H^{s}$. Then the symbol
$\lambda$ of the Dirichlet-Neumann operator satisfies

(5.9)
$(\rho^{\sharp})^{-2}\lambda^{\bullet;1,\mathscr{Q}}=\lambda\\#_{1;\mathscr{Q}}(\rho^{\sharp})^{-2}\mod\mathcal{A}^{-1}_{s-2.5}$

Here the operations $\\#_{r;\mathscr{Q}}$ and ${\bullet;r,\mathscr{Q}}$ are
defined in (3.10)-(3.11). Thus the para-differential operator $T_{\lambda}$
satisfies

$T_{(\rho^{\sharp})^{-2}}T_{\lambda}^{*}=T_{\lambda}T_{(\rho^{\sharp})^{-2}}\mod\mathbf{Op}\Sigma^{-0.5}_{<1/2}.$

###### Proof.

If we directly manipulate with (4.17) and (4.21), the algebra in the proof of
(5.9) would be extremely involved. Thus we proceed differently by recalling
the geometric interpretation of the Dirichlet-Neumann operator.

Assume for the moment that $\zeta$ is _smooth_. Recall that the linear
operator taking the Dirichlet boundary value $\phi$ to the Neumann boundary
value $\bar{\nabla}\Phi\cdot N(\iota)=\big{(}N_{0}\cdot
N(\iota)\big{)}D[\zeta]\phi$ on $M_{\zeta}$ is self-adjoint with respect to
the surface measure of $M_{\zeta}$; this is the starting point of our
argument. Recall that $N_{0}\cdot
N(\iota)=\rho/\sqrt{\rho^{2}+|\nabla_{0}\zeta|_{g_{0}}^{2}}$, while the
surface measure of $M_{\zeta}$ is
$d\mu(\iota)=\rho\sqrt{\rho^{2}+|\nabla_{0}\zeta|_{g_{0}}^{2}}d\mu_{0}$.
Consequently, we find, for smooth functions $\phi$, $\phi_{1}$ defined on
$\mathbb{S}^{2}$,

$\int_{\mathbb{S}^{2}}D[\zeta]\phi\cdot\phi_{1}\rho^{2}d\mu_{0}=\int_{\mathbb{S}^{2}}\phi\cdot
D[\zeta]\phi_{1}\cdot\rho^{2}d\mu_{0}.$

Lifting to $\mathrm{SU}(2)$, this is exactly

$\int_{\mathrm{SU}(2)}\mathfrak{D}\phi^{\sharp}\cdot\phi_{1}^{\sharp}\cdot(\rho^{\sharp})^{2}d\mu(G_{0})=\int_{\mathrm{SU}(2)}\phi^{\sharp}\cdot\mathfrak{D}\phi_{1}^{\sharp}\cdot(\rho^{\sharp})^{2}d\mu(G_{0}),$

where $\mathfrak{D}\phi^{\sharp}:=\big{(}D[\zeta]\phi\big{)}^{\sharp}$. As a
result, we have

(5.10)
$(\rho^{\sharp})^{-2}\mathfrak{D}^{*}=\mathfrak{D}\circ(\rho^{\sharp})^{-2}.$

On the other hand, if $\zeta$ is smooth, the para-linearization formula of the
Dirichlet-Neumann operator shows that in fact

$\mathfrak{D}=\mathbf{Op}(\lambda)+\mathbf{Op}\mathscr{S}^{-1}_{1,0}.$

Converting (5.10) to the symbolic side modulo symbols of order less than $-1$,
this gives

(5.11)
$(\rho^{\sharp})^{-2}\lambda^{\bullet;1,\mathscr{Q}}-\lambda\\#_{1;\mathscr{Q}}(\rho^{\sharp})^{-2}\in\mathscr{S}^{-1}_{1,0}.$

If we denote the left-hand-side of (5.11) as $Q(\zeta;x,l)$, then (5.11) of
course implies

(5.12)
$\displaystyle\big{\|}\mathbf{D}_{\mu}^{\alpha}Q(\zeta;x,l)\big{\|}_{C^{s-2.5}_{*}}$
$\displaystyle=O(l^{-1-|\alpha|}),\quad\text{for all multi-index }\alpha.$

This matrix norm estimate has been deduced for smooth $\zeta$. However, as we
can see from the definition of $\lambda$ and symbolic operation, the concrete
expression of entries of $Q(\zeta;x,l)$ only contains up to second order
derivative of $\zeta$. Thus we can approximate a given $\zeta\in H^{s+0.5}$ by
smooth functions and conclude that (5.12) continues to hold for $\zeta\in
H^{s+0.5}$. Thus for $\zeta\in H^{s+0.5}$,

$(\rho^{\sharp})^{-2}\lambda^{\bullet;1,\mathscr{Q}}-\lambda\\#_{1;\mathscr{Q}}(\rho^{\sharp})^{-2}$

is a symbol of order $-1$. This proves (5.9). ∎

We then proceed to the mean curvature.

######  Proposition 5.4.

Suppose $s>3$, $\zeta\in H^{s}$. Then with the symbol $h=h^{(2)}+h^{(1)}$ as
in Proposition 5.1, there holds a precise equality

(5.13)
$(\rho^{\sharp})^{-2}h^{\bullet;2,\mathscr{Q}}=h\\#_{2,\mathscr{Q}}(\rho^{\sharp})^{-2}.$

Thus

$T_{(\rho^{\sharp})^{-2}}T_{h}^{*}=T_{h}T_{(\rho^{\sharp})^{-2}}\mod\mathbf{Op}\Sigma^{0.5}_{<1/2}.$

###### Proof.

Of course we may directly deal with the expression of $h(x,l)$. But there is a
method that could tactfully avoid such lengthy computation. Consider a
variation of the surface $M_{\zeta}$:

$x\to\big{(}1+\zeta(x)+\theta_{1}\zeta_{1}(x)+\theta_{2}\zeta_{2}(x)\big{)}N_{0}.$

By second variation formula for area functional, the bilinear form

$\frac{\partial^{2}A}{\partial\theta_{1}\partial\theta_{2}}(\zeta+\theta_{1}\zeta_{1}+\theta_{2}\zeta_{2})\Bigg{|}_{\theta=0}$

is symmetric in $\zeta_{1},\zeta_{2}$. On the other hand, the second variation
obviously equals

$\int_{\mathbb{S}^{2}}H^{\prime}(\zeta)\zeta_{2}\cdot\zeta_{1}\rho^{2}d\mu_{0}+2\int_{\mathbb{S}^{2}}H(\zeta)\zeta_{1}\zeta_{2}\rho
d\mu_{0}.$

Thus the second order differential operator $\rho^{2}H^{\prime}(\zeta)$ is
self-adjoint with respect to $\mu_{0}$. Lifting to $\mathrm{SU}(2)$, it
follows that the second order differential operator
$(\rho^{\sharp})^{2}\big{(}H^{\prime}(\zeta)\big{)}^{\sharp}$ is self-adjoint
with respect to $\mu(G_{0})$. But we also notice that the symbol $h$ is
nothing but the symbol of the second order differential operator
$\big{(}H^{\prime}(\zeta)\big{)}^{\sharp}$. By the symbolic calculus theorem
of para-differential operators, this proves (5.13). ∎

###  5.3. Symmetrization

We are at the place to symmetrize the para-differential system (5.7). The
general procedure is identical to that in [ABZ11] but with different
computational details due to the non-flat geometry in our scenario.

The symmetrizer that we are looking for should take the form

$\left(\begin{matrix}T_{p}&\\\ &T_{q}\end{matrix}\right),$

where $p\in\mathcal{A}^{1/2}_{s-2.5}$ should be close to the symbol of
$|\nabla_{G_{0}}|^{0.5}$, and $q=q(x)$ is a scalar function. We require that
for some elliptic symbol $\gamma$ of order 1.5, there holds

(5.14) $\left(\begin{matrix}T_{p}&\\\
&T_{q}\end{matrix}\right)\left(\begin{matrix}&T_{\lambda}\\\
-T_{h}&\end{matrix}\right)=\left(\begin{matrix}&T_{\gamma}\\\
-T_{\gamma}&\end{matrix}\right)\left(\begin{matrix}T_{p}&\\\
&T_{q}\end{matrix}\right)\mod\left(\begin{matrix}&\mathbf{Op}\Sigma^{0}_{<1/2}\\\
\mathbf{Op}\Sigma^{0.5}_{<1/2}&\end{matrix}\right)$

The operator norms of the error term will be bounded linearly by
$\|\zeta\|_{H_{x}^{s+0.5}}$.

The elliptic symbol $\gamma$ will be the one that satisfy

(5.15)
$T_{\gamma}^{2}=\frac{T_{h}T_{\lambda}+T_{\lambda}^{*}T_{h}^{*}}{2}\mod\mathbf{Op}\Sigma^{0.5}_{<1/2}.$

This ensures that $T_{\gamma}$ is approximately self-adjoint. Our
symmetrization is finished by several steps.

Step 1: solve $\gamma$. We look for a solution
$\gamma=\gamma^{(1.5)}+\gamma^{(0.5)}\in\mathcal{A}^{1.5}_{s-1.5}+\mathcal{A}^{0.5}_{s-2.5}$
of (5.15). The highest order symbol $\gamma^{(1.5)}$ is just fixed as

$\gamma^{(1.5)}=\sqrt{\lambda^{(1)}h^{(2)}},$

which is Hermitian on each representation space $\mathcal{H}_{[}l]$. Note that
we used the commutativity of $h^{(2)}$ and $\lambda^{(1)}$, as shown in
Proposition 5.1. For the next order symbol $\gamma^{(0.5)}$, we just have to
fix it by looking at the next order symbols in the composition formula of
para-differential operators. That is, we require

$\gamma\\#_{1;\mathscr{Q}}\gamma=\frac{h\\#_{1,\mathscr{Q}}\lambda+\lambda^{\bullet;1,\mathscr{Q}}\\#_{1,\mathscr{Q}}h^{\bullet;1,\mathscr{Q}}}{2},$

or equivalently,

$\displaystyle\sqrt{\lambda^{(1)}h^{(2)}}\gamma^{(0.5)}+\gamma^{(0.5)}\sqrt{\lambda^{(1)}h^{(2)}}$
$\displaystyle+\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}\sqrt{\lambda^{(1)}h^{(2)}}\partial_{\mu}\sqrt{\lambda^{(1)}h^{(2)}}$
$\displaystyle=\frac{h\\#_{1,\mathscr{Q}}\lambda+\lambda^{\bullet;1,\mathscr{Q}}\\#_{1,\mathscr{Q}}h^{\bullet;1,\mathscr{Q}}}{2}-\lambda^{(1)}h^{(2)}.$

Since $[\sqrt{\lambda^{(1)}h^{(2)}},\gamma^{(0.5)}]$ is a symbol of order 1 by
Proposition 2.3, we can just replace $\gamma^{(0.5)}\gamma^{(1.5)}$ by
$\gamma^{(1.5)}\gamma^{(0.5)}$, and solve $\gamma^{(0.5)}$ as

$\gamma^{(0.5)}=\frac{(\lambda^{(1)}h^{(2)})^{-1/2}}{2}\left(\frac{h\\#_{1,\mathscr{Q}}\lambda+\lambda^{\bullet;1,\mathscr{Q}}\\#_{1,\mathscr{Q}}h^{\bullet;1,\mathscr{Q}}}{2}-\lambda^{(1)}h^{(2)}-\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}\sqrt{\lambda^{(1)}h^{(2)}}\partial_{\mu}\sqrt{\lambda^{(1)}h^{(2)}}\right).$

Step 2: solve $q$. We seek for a scalar function $q=q(x)$ defined on
$\mathrm{SU}(2)$ such that

(5.16)
$T_{q}T_{h}T_{\lambda}=\frac{T_{h}T_{\lambda}+T_{\lambda}^{*}T_{h}^{*}}{2}\cdot
T_{q}\mod\mathbf{Op}\Sigma^{1.5}_{<1/2}.$

This is the step that we should employ the almost self-adjoint properties of
$T_{\lambda}$ and $T_{h}$. Note that we have already proved, in Proposition
5.3-5.4, that

$\displaystyle T_{(\rho^{\sharp})^{-2}}T_{\lambda}^{*}$
$\displaystyle=T_{\lambda}T_{(\rho^{\sharp})^{-2}}$
$\displaystyle\mod\mathbf{Op}\Sigma^{-0.5}_{<1/2}$ $\displaystyle
T_{(\rho^{\sharp})^{-2}}T_{h}^{*}$
$\displaystyle=T_{h}T_{(\rho^{\sharp})^{-2}}$
$\displaystyle\mod\mathbf{Op}\Sigma^{0.5}_{<1/2}.$

So (5.16) becomes

$2T_{q}T_{h}T_{\lambda}=\left(T_{h}T_{\lambda}+T_{(\rho^{\sharp})^{2}\lambda}T_{h}T_{(\rho^{\sharp})^{-2}}\right)T_{q}\mod\mathbf{Op}\Sigma^{1.5}_{<1/2}.$

Here we used that $\rho\in H^{s+0.5}$, so
$T_{(\rho^{\sharp})^{2}}T_{h}=T_{(\rho^{\sharp})^{2}h}\mod\mathbf{Op}\Sigma^{3.5-s}_{<1/2}$.
Since only symbols of order $\geq 2$ is of our concern in the above equality,
we can simply use the composition formula of para-differential operators and
then neglect symbols of order $\leq 1$. Thus in order that (5.16) holds, it
suffices to require

(5.17) $\displaystyle
2q\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}h^{(2)}\partial_{\mu}\lambda^{(1)}$
$\displaystyle=q\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}h^{(2)}\partial_{\mu}\lambda^{(1)}$
$\displaystyle\quad+2\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}(\lambda^{(1)}h^{(2)})\partial_{\mu}q$
$\displaystyle\quad+q(\rho^{\sharp})^{2}\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}\lambda^{(1)}\partial_{\mu}\left(\frac{h^{(2)}}{(\rho^{\sharp})^{2}}\right)$
$\displaystyle\quad+q(\rho^{\sharp})^{2}\sum_{\mu\in\\{+,-,0\\}}\lambda^{(1)}\mathbf{D}_{\mu}h^{(2)}\partial_{\mu}(\rho^{\sharp})^{-2}\mod\mathcal{A}^{1}_{s-3}.$

Being tedious at a first glance, there are several key features of the above
equality that can be exploited. Recall that
$h^{(2)}=(\rho^{\sharp})^{3}\beta_{1}^{-3/2}(\lambda^{(1)})^{2}$, and from the
expression of $\lambda^{(1)}$, i.e. formula (4.21), we known that
$\lambda^{(1)}$ is a quasi-homogeneous symbol of order 1 in the sense of
Definition 2.12, where the vector field is just
$\nabla_{G_{0}}\zeta^{\sharp}/\beta_{1}(x)$. Proposition 2.9 thus ensures that
the commutators $[\lambda^{(1)},\partial_{\mu}\lambda^{(1)}]$ and
$[\lambda^{(1)},\mathbf{D}_{\mu}\lambda^{(1)}]$ are rough symbols of order 1
and 0 respectively. Consequently, we find

$\displaystyle\mathbf{D}_{\mu}h^{(2)}$
$\displaystyle=\frac{2(\rho^{\sharp})^{3}}{\beta_{1}^{3/2}}\mathbf{D}_{\mu}\lambda^{(1)}\cdot\lambda^{(1)}$
$\displaystyle\mod\mathcal{A}^{1}_{s-3},$
$\displaystyle\mathbf{D}_{\mu}(\lambda^{(1)}h^{(2)})$
$\displaystyle=\frac{3(\rho^{\sharp})^{3}}{\beta_{1}^{3/2}}\mathbf{D}_{\mu}\lambda^{(1)}\cdot(\lambda^{(1)})^{2}$
$\displaystyle\mod\mathcal{A}^{1}_{s-3},$
$\displaystyle\partial_{\mu}\left(\frac{h^{(2)}}{(\rho^{\sharp})^{2}}\right)$
$\displaystyle=(\lambda^{(1)})^{2}\partial_{\mu}\big{(}\rho^{\sharp}\beta_{1}^{-3/2}\big{)}+\frac{2\rho^{\sharp}\lambda^{(1)}\partial_{\mu}\lambda^{(1)}}{\beta_{1}^{3/2}}$
$\displaystyle\mod\mathcal{A}^{1}_{s-3},$

leading to cancellation between the left-hand-side and the first/thrid terms
in the right-hand-side of (5.17) modulo symbol of order $1$. Thus (5.17) is in
fact equivalent to

$\displaystyle\frac{6(\rho^{\sharp})^{3}}{\beta_{1}^{3/2}}\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}\lambda^{(1)}\cdot(\lambda^{(1)})^{2}\partial_{\mu}q$
$\displaystyle+2q\big{(}\rho^{\sharp}\big{)}^{2}\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}\lambda^{(1)}\cdot(\lambda^{(1)})^{2}\partial_{\mu}\big{(}\rho^{\sharp}\beta_{1}^{-3/2}\big{)}$
$\displaystyle+\frac{2q(\rho^{\sharp})^{5}}{\beta_{1}^{3/2}}\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}\lambda^{(1)}\cdot(\lambda^{(1)})^{2}\partial_{\mu}(\rho^{\sharp})^{-2}=0\mod\mathcal{A}^{1}_{s-3}.$

In order that this equality holds, it suffices to require

$\frac{6(\rho^{\sharp})^{3}}{\beta_{1}^{3/2}}\frac{\nabla_{G_{0}}q}{q}+2\big{(}\rho^{\sharp}\big{)}^{2}\nabla_{G_{0}}\big{(}\rho^{\sharp}\beta_{1}^{-3/2}\big{)}+\frac{2(\rho^{\sharp})^{5}}{\beta_{1}^{3/2}}\nabla_{G_{0}}(\rho^{\sharp})^{-2}=0,$

which can be solved as

(5.18)
$q={\beta_{1}^{3/2}}{(\rho^{\sharp})^{1/3}}=(1+\zeta^{\sharp})^{1/3}\sqrt{(1+\zeta^{\sharp})^{2}+|\nabla_{G_{0}}\zeta^{\sharp}|_{G_{0}}^{2}}.$

Step 3: solve $p$. Finally, the symbol $p=p^{(0.5)}+p^{(-0.5)}$ is solved
directly from the equation

$T_{p}T_{\lambda}=T_{\gamma}T_{q}\mod\mathbf{Op}\Sigma^{0}_{<1/2},$

or equivalently

$p\\#_{1;\mathscr{Q}}\lambda=\gamma\\#_{1;\mathscr{Q}}q\mod\mathbf{Op}\Sigma^{0}_{<1/2}.$

The principal symbol of $p$ is just

(5.19)
$p^{(0.5)}=\frac{(\rho^{\sharp})^{1.5}}{\beta_{1}^{3/4}}\sqrt{\lambda^{(1)}}.$

The sub-principal symbol $p^{(-0.5)}$ is then fixed as

$p^{(-0.5)}=(\lambda^{(1)})^{-1}\gamma^{(0.5)}q+(\lambda^{(1)})^{-1}\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}\gamma^{(1.5)}\partial_{\mu}q.$

Thus we have had all symbols $\gamma,p,q$ defined and matching (5.14). The
operator norm of the error term in (5.14) is bounded by
$\|\zeta\|_{H_{x}^{s+0.5}}$ linearly when $\zeta\simeq 0$, as a consequence of
Theorem 2.11-2.12.

Step 4: symmetrization. We now apply the symmetrizer
$\left(\begin{matrix}T_{p}&\\\ &T_{q}\end{matrix}\right)$ to the para-
differential sysmtem (5.7). We first notice that the expressions for
$\partial_{t}p$ and $\partial_{t}q$ actually do not explicitly contain time
derivative of $\zeta$ and $\phi$, since we assumed _in a priori_ that
$(\zeta,\phi)$ solves the autonomous partial differential equation (EQ). In
order not to cause unnecessary confusion, we denote the symbols

(5.20)
$a+b:=\partial_{t}p^{(0.5)}+\partial_{t}p^{(-0.5)}\in\mathcal{A}^{0.5}_{s-3}+\mathcal{A}^{-0.5}_{s-4},\quad
c:=\partial_{t}q\in\mathcal{A}^{0}_{s-2}.$

Here the regularity of $\partial_{t}p^{(-0.5)}$ is justified by noting that it
is of the type indicated in Proposition 2.10.

Hence we compute

(5.21) $\displaystyle\partial_{t}\left(\begin{matrix}T_{p}\zeta^{\sharp}\\\
T_{q}w^{\sharp}\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}&T_{\gamma}\\\
-T_{\gamma}&\end{matrix}\right)\left(\begin{matrix}T_{p}\zeta^{\sharp}\\\
T_{q}w^{\sharp}\end{matrix}\right)-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}\left(\begin{matrix}T_{p}\zeta^{\sharp}\\\
T_{q}w^{\sharp}\end{matrix}\right)$
$\displaystyle\quad+\left(\begin{matrix}(T_{p}T_{\lambda}-T_{\gamma}T_{q})w^{\sharp}\\\
(-T_{q}T_{h}+T_{\gamma}T_{p})\zeta^{\sharp}\end{matrix}\right)+\left(\begin{matrix}\left(T_{a+b}-\big{[}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}},T_{p}\big{]}\right)\zeta^{\sharp}\\\
\left(T_{c}-\big{[}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}},T_{q}\big{]}\right)w^{\sharp}\end{matrix}\right)+\left(\begin{matrix}T_{p}f_{1}(\zeta,\phi)\\\
T_{q}f_{2}(\zeta,\phi)\end{matrix}\right),$

where the mappings $f_{1},f_{2}$ are as in (5.8). We just define the sum of
last three terms as the mapping $f_{3}(\zeta,\phi)$.

Due to the para-differential reduction, we know that $f_{1},f_{2}$ are
controlled by
$\left(\|\zeta\|_{H_{x}^{s+0.5}}+\|\phi\|_{H_{x}^{s}}\right)^{2}$ as (5.8)
indicates. Due to (5.14), the operator norms of
$T_{p}T_{\lambda}-T_{\gamma}T_{q}$ and $-T_{q}T_{h}+T_{\gamma}T_{p}$ are
controlled linearly by $\|\zeta\|_{H^{s+0.5}_{x}}$ when $\zeta$ is small.

It remains to show that given $\alpha\in\mathbb{R}$,

(5.22)
$\big{[}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}},T_{p}\big{]}\quad\text{and}\quad
T_{a+b}$

both have operator norms in $\mathcal{L}(H^{\alpha+0.5},H^{\alpha})$
controlled linearly by $\|\zeta\|_{H^{s+0.5}_{x}}+\|\phi\|_{H_{x}^{s}}$ when
$(\zeta,\phi)\simeq 0$, while

(5.23)
$\big{[}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}},T_{q}\big{]}\quad\text{and}\quad
T_{c}\zeta^{\sharp}$

both have operator norms in $\mathcal{L}(H^{\alpha},H^{\alpha})$ controlled
linearly by $\|\zeta\|_{H^{s+0.5}_{x}}+\|\phi\|_{H_{x}^{s}}$ when
$(\zeta,\phi)\simeq 0$.

The claim for first operators in (5.22)-(5.23) follows from the fact that
$\big{[}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}},T_{p}\big{]}$ and
$\big{[}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}},T_{q}\big{]}$ are para-
differential operators of order 1/2 and 0 respectively, since commutator with
a vector field does not increase the order of the operator. The control of the
norms is a consequence of Corollary 2.11.1. AS for the claim for second
operators in (5.22)-(5.23), this is exactly the content of Lemma 4.10. of
[ABZ11]:

######  Lemma 5.1 (Equivalent form of Lemma 4.10., [ABZ11]).

The para-differential operators $T_{a}$ and $T_{b}$ are both of order $0.5$:
for all $\alpha\in\mathbb{R}$, there is an increasing function $K$ vanishing
linearly at 0, such that

$\big{\|}T_{a}\big{\|}_{\mathcal{L}(H^{\alpha},H^{\alpha-0.5})}+\big{\|}T_{b}\big{\|}_{\mathcal{L}(H^{\alpha},H^{\alpha-0.5})}\leq
K\big{(}\|\zeta\|_{H_{x}^{s+0.5}}\big{)}.$

The para-differential operator $T_{c}$ has order 0: for all
$\alpha\in\mathbb{R}$, there is an increasing function $K$ anishing linearly
at 0, such that

$\big{\|}T_{c}\big{\|}_{\mathcal{L}(H^{\alpha},H^{\alpha})}\leq
K\big{(}\|\zeta\|_{H_{x}^{s+0.5}}\big{)}.$

######  Remark 5.1.

The only technical difficulty with the proof of this Lemma is that the symbol
$b(x,l)$ is too rough in $x$. In fact, from the expression of $p^{(-0.5)}$,
one concludes that $p^{(-0.5)}$ has $H^{s-1.5}$ regularity in $x$.
Differentiating in $t$ reduces its regularity to $H^{s-3}\subset C_{*}^{s-4}$.
Thus $b(x,l)$ is a symbol of order $-0.5$ in $l$ and $C_{*}^{s-4}$ regularity
in $x$. By Proposition 2.7, $T_{b}$ is a para-differential operator of order
at most $0.5$.

To summarize, introducing the diagonal unknown
$\left(\begin{matrix}T_{p}\zeta^{\sharp}\\\
T_{q}w^{\sharp}\end{matrix}\right)$, the original system (EQ) is equivalent to

(5.24) $\partial_{t}\left(\begin{matrix}T_{p}\zeta^{\sharp}\\\
T_{q}w^{\sharp}\end{matrix}\right)=\left(\begin{matrix}&T_{\gamma}\\\
-T_{\gamma}&\end{matrix}\right)\left(\begin{matrix}T_{p}\zeta^{\sharp}\\\
T_{q}w^{\sharp}\end{matrix}\right)-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}u+f_{3}(\zeta,\phi),$

where the error $f_{3}(\zeta,\phi)$ collects the last three terms in (5.21),
and
$\|f_{3}(\zeta,\phi)\|_{H^{s}_{x}}\lesssim\left(\|\zeta\|_{H_{x}^{s+0.5}}+\|\phi\|_{H_{x}^{s}}\right)^{2}$.
This is exactly (1.13), We hence complete the proof of Theorem 1.2.

##  6\. Energy Estimate and Local Well-posedness

We finally complete the energy estimate for (1.13), and prove local well-
posedness of the original system (EQ). For simplicity, we sketch the proof of
local well-posedness for small amplitude solutions. Since the argument is
rather standard, and all the key estimates are already done in [ABZ11], we
will only present in detail the parts that are different from [ABZ11]; for
those that are identical to [ABZ11], we will directly cite the corresponding
Lemmas from [ABZ11] and show how they aid our proof.

###  6.1. Energy Estiamte

In [ABZ11], the authors used the $2s/3$’th order power of the principal symbol
$\gamma$ to construct a suitable energy functional. The reason is that the
Poisson bracket $\\{(\gamma^{(1.5)})^{2\alpha/3},\gamma\\}$ vanishes up to
order $\alpha-0.5$, hence $[T_{(\gamma^{(1.5)})^{2\alpha/3}},T_{\gamma}]$ is
in fact of order $\leq\alpha$. A similar result applies to the spherical
system (1.13) as well, although not as trivial as in the Euclidean case.

######  Proposition 6.1.

Given $\zeta\in H^{s+0.5}$, let $\gamma^{(1.5)}$ be the rough symbol as given
in Subsection 5.3. Then
$\big{[}T_{(\gamma^{(1.5)})^{2\alpha/3}},T_{\gamma}\big{]}$ is a para-
differential operator of order $\leq\alpha$, instead of just $\alpha+0.5$.
Quantitatively, its operator norm for $H^{s}\to H^{s}$ is controlled linearly
by $\|\zeta\|_{H_{x}^{s+0.5}}$ when $\zeta\simeq 0$.

###### Proof.

Recall from Subsection 5.3 that

$\gamma^{(1.5)}=\sqrt{\lambda^{(1)}h^{(2)}}=(\rho^{\sharp})^{1.5}\beta_{1}^{-3/4}(\lambda^{(1)})^{1.5},$

which may be abbreviated as

$\gamma^{(1.5)}(x,\xi)=g(x)\kappa(\xi)^{1.5}f\left(\frac{b(x,\xi)}{\kappa(\xi)}\right),$

where $g(x)$ is some scalar function, $b$ is the symbol of a vector field, all
depending on first order derivatives of $\zeta$; $\kappa$ is the symbol of
$|\nabla_{G_{0}}|$, and $f(z)=(1+z^{2})^{3/4}$. By our symbolic calculus
formula for para-differential operators, namely Theorem 2.11, the commutator
$\big{[}T_{(\gamma^{(1.5)})^{2\alpha/3}},T_{\gamma}\big{]}$ in fact equals the
para-differential operator corresponding to

(6.1)
$\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}(\gamma^{(1.5)})^{2\alpha/3}\partial_{\mu}\gamma^{(1.5)}-\sum_{\mu\in\\{+,-,0\\}}\mathbf{D}_{\mu}\gamma^{(1.5)}\partial_{\mu}(\gamma^{(1.5)})^{2\alpha/3},$

modulo an operator of class $\mathbf{Op}\Sigma^{\alpha}_{<1/2}$. Obviously
$\kappa(\xi)^{1.5}f\left(\frac{b(x,\xi)}{\kappa(\xi)^{2}}\right)$ is a quasi-
homogeneous symbol of order 1.5 in the sense of Definition 2.12. We can now
apply Proposition 2.8-2.9 as follows.

We first compute, by Proposition 2.8, that

$\mathbf{D}_{\mu}\kappa^{1.5}=\frac{3}{2}\kappa^{0.5}\mathbf{D}_{\mu}\kappa\mod\mathscr{S}^{-0.5}_{1,0},\quad\mathbf{D}_{\mu}\kappa^{2\alpha/3}=\frac{2\alpha}{3}\kappa^{2\alpha/3-1}\mathbf{D}_{\mu}\kappa\mod\mathscr{S}^{\alpha-2}_{1,0}.$

Thus by Proposition 2.9, together with the Leibniz property of
$\mathbf{D}_{\mu}$,

$\displaystyle\mathbf{D}_{\mu}(\gamma^{(1.5)})^{2\alpha/3}$
$\displaystyle=\alpha
g^{2\alpha/3}\kappa^{\alpha-1}f\left(\frac{b}{\kappa}\right)^{2\alpha/3}\mathbf{D}_{\mu}\kappa+\frac{2\alpha}{3}g^{2\alpha/3}\kappa^{\alpha-1}f\left(\frac{b}{\kappa}\right)^{2\alpha/3-1}f^{\prime}\left(\frac{b}{\kappa}\right)\mathbf{D}_{\mu}b$
$\displaystyle\quad+\frac{2\alpha}{3}g^{2\alpha/3}\kappa^{\alpha-2}f\left(\frac{b}{\kappa}\right)^{2\alpha/3-1}f^{\prime}\left(\frac{b}{\kappa}\right)\mathbf{D}_{\mu}\kappa
b$ $\displaystyle\mod\mathcal{A}^{\alpha-2}_{s-1.5}$
$\displaystyle=\frac{2\alpha}{3}(\gamma^{(1.5)})^{2\alpha/3-1}\mathbf{D}_{\mu}\gamma^{(1.5)}$
$\displaystyle\mod\mathcal{A}^{\alpha-2}_{s-1.5}.$

On the other hand, by Proposition 2.9,

$\displaystyle\partial_{\mu}(\gamma^{(1.5)})^{2\alpha/3}$
$\displaystyle=\frac{2\alpha}{3}g^{2\alpha/3-1}\kappa^{\alpha}\partial_{\mu}gf\left(\frac{b}{\kappa}\right)^{2\alpha/3}+\frac{2\alpha}{3}g^{2\alpha/3}\kappa^{\alpha}f\left(\frac{b}{\kappa}\right)^{2\alpha/3-1}f^{\prime}\left(\frac{b}{\kappa}\right)\partial_{\mu}b$
$\displaystyle\mod\mathcal{A}^{\alpha-2}_{s-2.5}$
$\displaystyle=\frac{2\alpha}{3}(\gamma^{(1.5)})^{2\alpha/3-1}\partial_{\mu}\gamma^{(1.5)}$
$\displaystyle\mod\mathcal{A}^{\alpha-2}_{s-2.5}.$

Thus (6.1) in fact equals $0\mod\mathcal{A}^{\alpha-1}_{s-2.5}$. Consequently
$\big{[}T_{(\gamma^{(1.5)})^{2\alpha/3}},T_{\gamma}\big{]}$ is a para-
differential operator of order $\leq\alpha$, not just $\alpha+0.5$. ∎

We can now state the energy estimate:

######  Proposition 6.2.

Let

$(\zeta,\phi)\in\big{(}C_{T}H^{s+0.5}_{x}\times
C_{T}H^{s}_{x}\big{)}\cap\big{(}C_{T}^{1}H^{s-1}_{x}\times
C_{T}^{1}H^{s-1.5}_{x}\big{)}$

be given, with norm in this space less than some small $R$. We define the good
unknown $w^{\sharp}$ as in (5.1), the symbols
$\lambda,h,\mathfrak{b}^{\sharp},\mathfrak{v}^{\sharp}$ as in Section 5.
Suppose $f\in C_{T}H^{s+0.5}_{x}\times C_{T}H^{s}_{x}$. Then the _linear_
Cauchy problem

(6.2) $\partial_{t}\left(\begin{matrix}\eta^{\sharp}\\\
v^{\sharp}\end{matrix}\right)=\left(\begin{matrix}-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}&T_{\lambda}v^{\sharp}\\\
-T_{h}&-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}\end{matrix}\right)\left(\begin{matrix}\eta^{\sharp}\\\
v^{\sharp}\end{matrix}\right)+f$

in the unknown $(\eta,v)$ admits a unique solution

$(\eta,v)\in\big{(}C_{T}H^{s+0.5}_{x}\times
C_{T}H^{s}_{x}\big{)}\cap\big{(}C_{T}^{1}H^{s-1}_{x}\times
C_{T}^{1}H^{s-1.5}_{x}\big{)},$

satisfying the following energy estimate for any real number $\alpha\leq s$:

$\displaystyle\|\eta(t)\|_{H^{\alpha+0.5}_{x}}+\|v(t)\|_{H^{\alpha}_{x}}\leq
C_{\alpha}e^{C_{\alpha}Rt}\left(\|\eta(0)\|_{H^{\alpha+0.5}_{x}}+\|v(0)\|_{H^{\alpha}_{x}}+\int_{0}^{t}\big{\|}f(\tau);{H^{\alpha+0.5}_{x}\times
H^{\alpha}_{x}}\big{\|}d\tau\right).$

###### Proof.

If we define symbols $p,q,\gamma,a,b,c$ corresponding to $(\zeta,\phi)$ as in
Subsection 5.3 (especially (5.20)), and set
$u=(T_{p}\zeta^{\sharp},T_{q}v^{\sharp})$, then the procedure in Subsection
5.3 transforms (6.2) into the following equivalent form:

(6.3) $\displaystyle\partial_{t}u$
$\displaystyle=\left(\begin{matrix}&T_{\gamma}\\\
-T_{\gamma}&\end{matrix}\right)u-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}u+\left(\begin{matrix}\left(T_{a+b}-\big{[}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}},T_{p}\big{]}\right)T_{p}^{-1}\\\
\left(T_{c}-\big{[}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}},T_{q}\big{]}\right)T_{q}^{-1}\end{matrix}\right)u$
$\displaystyle\quad+\left(\begin{matrix}&(T_{p}T_{\lambda}-T_{\gamma}T_{q})T_{q}^{-1}\\\
(-T_{q}T_{h}+T_{\gamma}T_{p})T_{p}^{-1}&\end{matrix}\right)u+\left(\begin{matrix}T_{p}&\\\
&T_{q}\end{matrix}\right)f.$

For any $\alpha\leq s$, we simply apply the operator
$\Gamma_{\alpha}:=T_{(\gamma^{(1.5)})^{2\alpha/3}}$ to (6.3). We first notice
that

$\big{\|}\Gamma_{\alpha}u\big{\|}_{L^{2}_{x}}\simeq\|u\|_{H^{\alpha}_{x}},$

and the implicit constants in this equivalence of norms depend only on
$\|\zeta\|_{H^{s+0.5}_{x}}$. In fact, $\gamma^{(1.5)}(x,l)^{2\alpha/3}$ is a
symbol of order $\alpha$ and involves only up to first order derivative of
$\zeta$. Furthermore, by the symbolic calculus formula,
$\Gamma_{\alpha}\cdot\Gamma_{-\alpha}=\mathrm{Id}+\mathbf{Op}\Sigma^{-1}_{<1/2}$,
so at least when $\|\zeta\|_{H^{s+0.5}_{x}}$ stays close to 0 (which is the
case by our assumption), the inverse of the operator $\Gamma_{\alpha}$ is a
para-differential operator of order $-\alpha$. This gives the equivalence of
norms. The estimate on the implicit constants is a direct consequence of
Theorem 2.11.

By Proposition 6.1, $\big{[}\Gamma_{\alpha},T_{\gamma}\big{]}$ is a para-
differential operator of order $\alpha$. The symbols in this commutator
involve only up to second order derivatives of $\zeta$. We next notice that
$\big{[}\Gamma_{\alpha},T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}\big{]}$
is also a para-differential operators of order $\leq\alpha$. This is because
commuting with a vector field does not increase the order of an operator. The
commutator of $\Gamma_{\alpha}$ with

$\left(T_{a+b}-\big{[}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}},T_{p}\big{]}\right)T_{p}^{-1},\quad\left(T_{c}-\big{[}T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}},T_{q}\big{]}\right)T_{q}^{-1}$

and

$(T_{p}T_{\lambda}-T_{\gamma}T_{q})T_{q}^{-1},\quad(-T_{q}T_{h}+T_{\gamma}T_{p})T_{p}^{-1}$

are all of order $\leq\alpha$, due to Lemma 5.1 and (5.14). Furthermore,
Proposition 2.9 ensures that

$\partial_{t}\Gamma_{\alpha}=\frac{2\alpha}{3}T_{(\gamma^{(1.5)})^{2\alpha/3-1}\partial_{t}\gamma^{(1.5)}}+\text{lower
order para-differential operator}.$

Since $\gamma^{(1.5)}$ involves first order derivative of $\zeta$, while by
the assumption we have $\partial_{t}\zeta\in C_{T}^{0}H^{s-1}_{x}$, it follows
that $\partial_{t}\gamma^{(1.5)}$ is a symbol with $H^{s-2}\subset
C^{s-3}_{*}$ regularity in $x$, so $\partial_{t}\Gamma_{\alpha}$ is still a
para-differential operator of order $\alpha$. The operator norms of these are
all controlled linearly by $R$, as a consequence of Theorem 2.11.

To summarize, for any real number $\alpha\leq s$, we have

(6.4) $\partial_{t}\Gamma_{\alpha}u=\left(\begin{matrix}&T_{\gamma}\\\
-T_{\gamma}&\end{matrix}\right)\Gamma_{\alpha}u-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}\Gamma_{\alpha}u+\Upsilon_{\alpha}u+\Gamma_{\alpha}\left(\begin{matrix}T_{p}&\\\
&T_{q}\end{matrix}\right)f,$

where $\Upsilon_{\alpha}$ is a para-differential operator of order $\alpha$,
with operator norm controlled linearly by $R$. We now find, using that
$T_{\gamma}$ and $T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}$ are both
approximately anti-self adjoint, the following a priori inequality for a
solution $u$ of (6.3): for any real number $\alpha\leq s$,

$\frac{d}{dt}\|\Gamma_{\alpha}u(t)\|_{L^{2}_{x}}^{2}\lesssim
R\|\Gamma_{\alpha}u(t)\|_{L^{2}_{x}}^{2}+\big{\|}f(\tau);{H^{\alpha+0.5}_{x}\times
H^{\alpha}_{x}}\big{\|}\cdot\|\Gamma_{\alpha}u(t)\|_{L^{2}_{x}}.$

Using the Grönwall inequality and norm equaivalence
$\big{\|}\Gamma_{\alpha}u\big{\|}_{L^{2}_{x}}\simeq\|u\|_{H^{\alpha}_{x}}$,
this implies the _a priori_ energy inequalities for a solution $u$ of (6.3),
provided that it does exist:

$\|u(t)\|_{H^{\alpha}_{x}}\leq
C_{\alpha}e^{C_{\alpha}Rt}\left(\|u(0)\|_{H^{\alpha}_{x}}+\int_{0}^{t}\big{\|}f(\tau);{H^{\alpha+0.5}_{x}\times
H^{\alpha}_{x}}\big{\|}d\tau\right).$

The regularity $\partial_{t}u\in C^{0}_{T}H^{s-1.5}_{x}$ follows from the
equation itself.

We may then apply the standard duality argument for linear hyperbolic systems.
These a priori energy estimates do imply well-posedness of the linear Cauchy
problem (6.3). It is important that $\alpha$ are allowed to be negative. See
for example Section 6.3 in Hörmander’s textbook [Hör97]. The estimates for $u$
are then converted back to estimates for $(\eta,v)$. ∎

###  6.2. Banach Fixed Point Argument

We now formulate a Banach fixed point problem. We shall fix a sufficiently
small $R>0$ to measure the size of the solution. As we shall see shortly, the
major issue in closing a contraction argument is that we need to control some
operator norms under _weaker_ regularity assumptions than $(\zeta,\phi)\in
C_{T}H_{x}^{s+0.5}\times C_{T}H_{x}^{s}$. We thus seek aid from Proposition
2.7 constantly.

The Metric Space and the Map. We will be dealing with the good unknown
$(\zeta,w)$ instead of $(\zeta,\phi)$. We set

$\mathcal{G}:\left(\begin{matrix}\zeta\\\
\phi\end{matrix}\right)\to\left(\begin{matrix}\zeta\\\ w\end{matrix}\right)$

the assignment of $(\zeta,\phi)$ to the corresponding good unknown. We already
know that $\mathcal{G}$ is a diffeomorphism in a neighbourhood of zero in
$C_{T}H^{s+0.5}_{x}\times C_{T}H^{s}_{x}$.

Fix $\bar{\mathfrak{X}}_{R}$ to be the set of all

$(\zeta,w)\in\big{(}C_{T}H^{s+0.5}_{x}\times
C_{T}H^{s}_{x}\big{)}\cap\big{(}C_{T}^{1}H^{s-1}_{x}\times
C_{T}^{1}H^{s-1.5}_{x}\big{)}$

such that $(\zeta,w)$ has norm $\leq R$ in that space, and also
$(\zeta(0),\phi(0))$ has norm $\leq A_{s}R$ in that space, for some $A_{s}\ll
1$ to be specified. We equip $\bar{\mathfrak{X}}_{R}$ with a _weaker_ metric

$\displaystyle
d\big{(}(\zeta_{1},w_{1}),(\zeta_{2},w_{2})\big{)}:=\|\zeta_{1}-\zeta_{2}\|_{C_{T}H^{s-1}_{x}}+\|w_{1}-w_{2}\|_{C_{T}H^{s-1.5}_{x}}.$

By weak compactness of bounded closed convex sets in Hilbert spaces,
$(\bar{\mathfrak{X}}_{R},d)$ is a complete metric space.

Now let $(\zeta,w)\in\bar{\mathfrak{X}}_{R}$ be given, and
$(\zeta,\phi)=\mathcal{G}^{-1}(\zeta,w)$. We define the symbols
$\lambda,h,\mathfrak{b}^{\sharp},\mathfrak{v}^{\sharp}$ as in Section 5, and
the mappings $f_{1},f_{2}$ as in (5.8). Let us consider the _linear_ Cauchy
problem for the unknown $(\eta,v)\in C_{T}H^{s+0.5}_{x}\times C_{T}H^{s}_{x}$:

(6.5) $\partial_{t}\left(\begin{matrix}\eta^{\sharp}\\\
v^{\sharp}\end{matrix}\right)=\left(\begin{matrix}-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}&T_{\lambda}\\\
-T_{h}&-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}\end{matrix}\right)\left(\begin{matrix}\eta^{\sharp}\\\
v^{\sharp}\end{matrix}\right)+\left(\begin{matrix}f_{1}(\zeta,\phi)\\\
f_{2}(\zeta,\phi)\end{matrix}\right).$

Using the operator norm estimates in Subsection 5.3, together with the
quadratic estiamte for $f_{1},f_{2}$ in (5.8), Proposition 6.2 makes sure that
the Cauchy problem for this linear hyperbolic system can be uniquely solved
with given initial data. Thus, (6.5) has a unique solution $(\eta,v)\in
C_{T}H_{x}^{s+0.5}\times C_{T}H^{s}_{x}$ with initial value being
$(\zeta(0),w(0))$, satisfying the energy estimates

(6.6) $\displaystyle\|\eta(t)\|_{H^{s+0.5}_{x}}+\|v(t)\|_{H^{s}_{x}}$
$\displaystyle\leq C_{s}e^{Rt}(A_{s}R+R^{2}t),$
$\displaystyle\|\partial_{t}\eta(t)\|_{H^{s-1}_{x}}+\|\partial_{t}v(t)\|_{H^{s-1.5}_{x}}$
$\displaystyle\leq C_{s}e^{Rt}(A_{s}R+R^{2}t)+C_{s}R^{2}.$

We set the _solution operator_ $\mathcal{S}$ to be the unique solution
$(\eta,v)$ of (6.5) with initial data being $(\zeta(0),\phi(0))$.

The map $\mathscr{F}$ on $\bar{\mathfrak{X}}_{R}$ under consideration is thus
defined by

$\mathscr{F}\left(\begin{matrix}\zeta\\\
w\end{matrix}\right):=\mathcal{S}\mathcal{G}^{-1}\left(\begin{matrix}\zeta\\\
w\end{matrix}\right).$

When $R\simeq 0$ and $T$ is suitably small, if $A_{s}$ is also suitably small,
then for $(\zeta,w)\in\bar{\mathfrak{X}}_{R}$, (6.6) implies

$\displaystyle\big{\|}\mathscr{F}(\zeta,w);{C_{T}H^{s+0.5}_{x}\times
C_{T}H^{s}_{x}}\big{\|}$ $\displaystyle\leq C_{s}e^{C_{s}T}(A_{s}+C_{s}TR)R\ll
R$
$\displaystyle\big{\|}\partial_{t}\mathscr{F}(\zeta,w);{C_{T}H^{s-1}_{x}\times
C_{T}H^{s-1.5}_{x}}\big{\|}$ $\displaystyle\leq
C_{s}e^{C_{s}T}(A_{s}+C_{s}TR)R+C_{s}R^{2}\ll R$

Thus a suitable choice of $A_{s}$ and $T$ makes sure that $\mathscr{F}$ maps
$\bar{\mathfrak{X}}_{R}$ to itself.

Contraction Argument. Let us now consider two different
$(\zeta,w),(\zeta_{1},w_{1})\in\bar{\mathfrak{X}}_{R}$. We use void subscript
or subscript 1 to denote functions or symbols constructed out of
$(\zeta,w),(\zeta_{1},w_{1})\in\bar{\mathfrak{X}}_{R}$ respectively. Write
$(\eta_{j},v_{j})=\mathscr{F}(\zeta_{j},w_{j})$. We would like to prove that
for small $R>0$, there holds

(6.7)
$\|\eta-\eta_{1}\|_{C_{T}H^{s-1}_{x}}+\|v-v_{1}\|_{C_{T}H^{s-1.5}_{x}}\ll
d\big{(}(\zeta,w),(\zeta_{1},w_{1})\big{)},$

which ensures $\mathscr{F}:\bar{\mathfrak{X}}_{R}\to\bar{\mathfrak{X}}_{R}$ is
a contraction.

Write for simplicity $U=\eta-\eta_{1}$, $V=v-v_{1}$. We compute, by the
definition of solution operator, that $U(0)=0$, $V(0)=0$, and

(6.8) $\displaystyle\partial_{t}\left(\begin{matrix}U\\\ V\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}&T_{\lambda}\\\
-T_{h}&-T_{\mathfrak{v}^{\sharp}}\cdot\nabla_{G_{0}}\end{matrix}\right)\left(\begin{matrix}U\\\
V\end{matrix}\right)$
$\displaystyle\quad+\left(\begin{matrix}T_{\mathfrak{v}^{\sharp}-\mathfrak{v}_{1}^{\sharp}}\cdot\nabla_{G_{0}}&-T_{\lambda-\lambda_{1}}\\\
T_{h-h_{1}}&T_{\mathfrak{v}^{\sharp}-\mathfrak{v}_{1}^{\sharp}}\cdot\nabla_{G_{0}}\end{matrix}\right)\left(\begin{matrix}\eta\\\
v\end{matrix}\right)+\left(\begin{matrix}f_{1}(\zeta,w)-f_{1}(\zeta_{1},w_{1})\\\
f_{2}(\zeta,w)-f_{2}(\zeta_{1},w_{1})\end{matrix}\right).$

(6.8) is exactly in the form indicated by Proposition 6.2. We just need to
show that the $H^{s-1}_{x}\times H^{s-1.5}_{x}$ norm of the last two terms of
the right-hand-side in (6.8) is controlled linearly by
$d\big{(}(\zeta,w),(\zeta_{1},w_{1})\big{)}$. But this is exactly the content
of Lemma 6.6, Corollary 6.7 and Lemma 6.8 of [ABZ11], which concerns with
exactly the same quantities. If we want to reproduce the proof, we just
constantly use Proposition 2.10 and Proposition 2.7. For example, the first-
order symbol $h^{(1)}-h_{1}^{(1)}$ contains up to second order derivative of
$\zeta$ and $\zeta_{1}$, so

$\big{\|}h^{(1)}(x,l)-h_{1}^{(1)}(x,l)\big{\|}_{C^{s-4}_{*}}\leq
Cl\|\zeta-\zeta_{1}\|_{H^{s-1}_{x}}.$

By Proposition 2.7, this implies that the para-differential operator
$T_{h^{(1)}-h_{1}^{(1)}}$ is of order 2, and

$\big{\|}T_{h^{(1)}-h_{1}^{(1)}}\eta\big{\|}_{H^{s-1.5}_{x}}\lesssim\|\zeta-\zeta_{1}\|_{H^{s-1}_{x}}\|\eta\|_{H^{s+0.5}_{x}}\lesssim
Rd\big{(}(\zeta,w),(\zeta_{1},w_{1})\big{)}.$

The final step then becomes simple. Applying Proposition 6.2 to (6.8) with
$\alpha=s-1.5$, we find

$\|U\|_{C_{T}H^{s-1}_{x}}+\|V\|_{C_{T}H^{s-1.5}_{x}}\leq
C_{s}e^{C_{s}RT}TRd\big{(}(\zeta,w),(\zeta_{1},w_{1})\big{)}.$

Thus for $T$ suitably small, the mapping $\mathscr{F}$ is a contraction from
$\bar{\mathfrak{X}}_{R}$ to itself, hence has a unique fixed point. The fixed
point of $\mathscr{F}$ is exactly the solution of the para-linearized system
(5.7), thus the original (EQ).

######  Remark 6.1.

In [ABZ11], the existence and uniqueness for solution of the symmetrized
capillary-gravity water waves system is proved as follows: one first prove
energy estimate for a smoothed system with some parameter $\varepsilon$ (e.g.
artificial viscosity), and show that a solution sequence converges to a
genuine solution of the original one as $\varepsilon\to 0$ by a weak
compactness argument, then prove uniqueness by considering the energy estimate
for the difference of two solutions, which is just (6.8). The reason that we
would rather formulate a Banach fixed point theorem is that the latter is more
constructive, and is in fact equivalent to the iterative method used in most
literature on quasi-linear hyperbolic systems, for example Chapter 6 of
[Hör97].

## Acknowledgement

The author would like to thank Professor Carlos Kenig for weekly discussion on
this project, and Professor Gigliola Staffilani for constant support. The
author benefits a lot from discussion with Professor Thomas Alazard, Jean-Marc
Delort, Veronique Fischer, Isabelle Gallagher and David Jerison. Thanks also
goes to the author’s friend, Kai Xu, for comments on representation theory.

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