$\displaystyle+T_{w}\partial_{\alpha}\Pi(X_{\alpha},{\underline{b}})+i\frac{\gamma}{2}\partial_{\alpha}T_{w}X+T_{w}T_{T_{1-\bar{Y}}R_{\alpha}+T_{1-Y}\bar{R}_{\alpha}}X_{\alpha}-i\frac{\gamma}{2}T_{w}T_{T_{1-\bar{Y}}{\mathbf{W}}-T_{1-Y}\bar{{\mathbf{W}}}}X_{\alpha}+G,$
$\displaystyle=$
$\displaystyle\partial_{\alpha}T_{T_{1-\bar{Y}}r_{\alpha}}X+{\mathbf{P}}T_{w_{\alpha}}{\underline{b}}+T_{w}{\mathbf{P}}\left[(1-\bar{Y})R_{\alpha}-i\frac{\gamma}{2}{\mathbf{W}}\right]+i\frac{\gamma}{2}T_{w}{\mathbf{P}}\partial_{\alpha}(\bar{Y}W)$
$\displaystyle+i\frac{\gamma}{2}\partial_{\alpha}T_{w}X+T_{w}T_{T_{1-\bar{Y}}R_{\alpha}}X_{\alpha}-i\frac{\gamma}{2}T_{w}T_{T_{1-\bar{Y}}{\mathbf{W}}-T_{1-Y}\bar{{\mathbf{W}}}}X_{\alpha}+G.$

Here for the last four lines, we use the fact that

$T_{w_{\alpha}}{\mathbf{P}}[(1-\bar{Y})R-i\frac{\gamma}{2}(1-\bar{Y})W]={\mathbf{P}}T_{w_{\alpha}}{\underline{b}}.$

In a similar manner, for other non-vorticity terms, we compute

$\displaystyle-T_{1-\bar{Y}}\partial_{\alpha}T_{r_{\alpha}}X$
$\displaystyle=-\partial_{\alpha}T_{T_{1-\bar{Y}}r_{\alpha}}X+G,$
$\displaystyle-T_{(1-\bar{Y})R_{\alpha}}\partial_{\alpha}T_{w}X$
$\displaystyle=-T_{w}T_{T_{1-\bar{Y}}R_{\alpha}}X_{\alpha}+G.$

For the contributions of corrections with $\gamma$ coefficient, we have

$\displaystyle-\gamma T_{\Im{\mathbf{W}}}\partial_{\alpha}T_{w}X=$
$\displaystyle
i\frac{\gamma}{2}T_{w}T_{T_{1-\bar{Y}}{\mathbf{W}}-T_{1-Y}\bar{{\mathbf{W}}}}X_{\alpha}+G,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}T_{w_{\alpha}}Z=$
$\displaystyle-\frac{\gamma}{2g}T_{\partial_{\alpha}T_{D_{t}}w}Z-\frac{\gamma}{2g}T_{w_{\alpha}}T_{D_{t}}Z+G$
$\displaystyle=$
$\displaystyle\frac{\gamma}{2g}T_{T_{1-\bar{Y}}(r_{\alpha\alpha}+T_{w}R_{\alpha\alpha})}Z-i\frac{\gamma}{2}T_{w_{\alpha}}X+i\frac{\gamma^{2}}{2g}T_{w_{\alpha}}Z+G$
$\displaystyle=$
$\displaystyle\frac{\gamma}{2g}T_{1-\bar{Y}}T_{r_{\alpha\alpha}}Z-i\frac{\gamma}{2}T_{w_{\alpha}}X+i\frac{\gamma^{2}}{2g}T_{w_{\alpha}}Z+G,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}T_{r}X_{\alpha}=$
$\displaystyle-\frac{\gamma}{2g}T_{T_{D_{t}}r}X_{\alpha}-\frac{\gamma}{2g}T_{r}T_{D_{t}}X_{\alpha}+G$
$\displaystyle=$ $\displaystyle
i\frac{\gamma^{2}}{2g}T_{r}X_{\alpha}-i\frac{\gamma}{2}T_{w}X_{\alpha}+\frac{\gamma}{2g}T_{r}T_{1-\bar{Y}}R_{\alpha}+G,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}T_{r_{\alpha}}X=$
$\displaystyle-\frac{\gamma}{2g}T_{\partial_{\alpha}T_{D_{t}}r}X-\frac{\gamma}{2g}T_{r_{\alpha}}T_{D_{t}}X+G$
$\displaystyle=$ $\displaystyle
i\frac{\gamma^{2}}{2g}T_{r_{\alpha}}X-i\frac{\gamma}{2g}T_{T_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha})}X+\frac{\gamma}{2g}T_{r_{\alpha}}T_{1-\bar{Y}}R+G$
$\displaystyle=$ $\displaystyle
i\frac{\gamma^{2}}{2g}T_{r_{\alpha}}X-i\frac{\gamma}{2}T_{w_{\alpha}}X+\frac{\gamma}{2g}T_{r_{\alpha}}T_{1-\bar{Y}}R+G,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}T_{w}R=$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}w}R-\frac{\gamma}{2g}T_{w}T_{D_{t}}R$
$\displaystyle=$
$\displaystyle\frac{\gamma}{2g}T_{T_{1-\bar{Y}}r_{\alpha}+T_{w}R_{\alpha}}R-\frac{\gamma}{2g}T_{w}(iT_{g+{\underline{a}}}Y-i\gamma
R)+G$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2g}T_{1-\bar{Y}}T_{r_{\alpha}}R-i\frac{\gamma}{2g}T_{w}T_{g+{\underline{a}}}Y+i\frac{\gamma^{2}}{2g}T_{w}R+G,$
$\displaystyle-\frac{\gamma}{2g}T_{1-\bar{Y}}\partial_{\alpha}(T_{r_{\alpha}}Z+T_{r}R)=$
$\displaystyle-\frac{\gamma}{2g}T_{1-\bar{Y}}T_{r_{\alpha\alpha}}Z-\frac{\gamma}{g}T_{1-\bar{Y}}T_{r_{\alpha}}R-\frac{\gamma}{2g}T_{r}T_{|1-Y|^{2}}R_{\alpha}+G,$
$\displaystyle i\frac{\gamma}{2}T_{1-\bar{Y}}\partial_{\alpha}T_{w}X=$
$\displaystyle
i\frac{\gamma}{2}T_{w_{\alpha}}X+i\frac{\gamma}{2}T_{w}T_{|1-Y|^{2}}{\mathbf{W}}+G.$

The other $\gamma$ contributions can be absorbed into $G$.

As for the contributions of corrections with $\gamma^{2}$ coefficient, we
compute

$\displaystyle i\frac{\gamma^{2}}{2g}T_{D_{t}}T_{w_{\alpha}}U=$ $\displaystyle
i\frac{\gamma^{2}}{2g}T_{\partial_{\alpha}T_{D_{t}}w}U+i\frac{\gamma^{2}}{2g}T_{w_{\alpha}}T_{D_{t}}U+G$
$\displaystyle=$
$\displaystyle-i\frac{\gamma^{2}}{2g}T_{T_{1-\bar{Y}}r_{\alpha\alpha}}U-i\frac{\gamma^{2}}{2g}T_{w_{\alpha}}T_{1-\bar{Y}}Z+G,$
$\displaystyle
i\frac{\gamma^{2}}{2g}T_{D_{t}}T_{\partial^{-1}_{\alpha}w}X_{\alpha}=$
$\displaystyle
i\frac{\gamma^{2}}{2g}T_{T_{D_{t}}\partial^{-1}_{\alpha}w}X_{\alpha}+i\frac{\gamma^{2}}{2g}T_{\partial^{-1}_{\alpha}w}T_{D_{t}}X_{\alpha}+G$
$\displaystyle=$
$\displaystyle-i\frac{\gamma^{2}}{2g}T_{T_{1-\bar{Y}}r}X_{\alpha}-i\frac{\gamma^{2}}{2g}T_{\partial^{-1}_{\alpha}w}T_{1-\bar{Y}}R_{\alpha}+G,$
$\displaystyle i\frac{\gamma^{2}}{g}T_{D_{t}}T_{w}X=$ $\displaystyle
i\frac{\gamma^{2}}{g}T_{D_{t}w}X+i\frac{\gamma^{2}}{g}T_{w}T_{D_{t}}X+G$
$\displaystyle=$
$\displaystyle-i\frac{\gamma^{2}}{g}T_{T_{1-\bar{Y}}r_{\alpha}}X-i\frac{\gamma^{2}}{g}T_{w}T_{1-\bar{Y}}R+G,$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}T_{D_{t}}T_{r_{\alpha}}Z=$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}T_{\partial_{\alpha}T_{D_{t}}r}Z-\frac{\gamma^{2}}{4g^{2}}T_{r_{\alpha}}T_{D_{t}}Z+G$
$\displaystyle=$ $\displaystyle-\frac{\gamma^{2}}{4g^{2}}T_{-i\gamma
r_{\alpha}+iT_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha})}Z-\frac{\gamma^{2}}{4g^{2}}T_{r_{\alpha}}(igX-i\gamma
Z)+G$ $\displaystyle=$ $\displaystyle
i\frac{\gamma^{3}}{2g^{2}}T_{r_{\alpha}}Z-i\frac{\gamma^{2}}{4g}T_{w_{\alpha}}Z-i\frac{\gamma^{2}}{4g}T_{r_{\alpha}}X+G,$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}T_{D_{t}}T_{r}Z_{\alpha}=$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}T_{T_{D_{t}}r}Z_{\alpha}-\frac{\gamma^{2}}{4g^{2}}T_{r}T_{D_{t}}Z_{\alpha}+G$
$\displaystyle=$ $\displaystyle-\frac{\gamma^{2}}{4g^{2}}T_{-i\gamma
r+iT_{g+{\underline{a}}}(x-T_{w}Y)}R-\frac{\gamma^{2}}{4g^{2}}T_{r}(igT_{1-Y}{\mathbf{W}}-i\gamma
R)+G$ $\displaystyle=$ $\displaystyle
i\frac{\gamma^{3}}{2g^{2}}T_{r}R-i\frac{\gamma^{2}}{4g}T_{w}R-i\frac{\gamma^{2}}{4g}T_{r}T_{1-Y}{\mathbf{W}}+G.$

Here we use (3.12) to change $Z_{\alpha}$ into $R$.

$\displaystyle
i\frac{\gamma^{2}}{2g}T_{1-\bar{Y}}\partial_{\alpha}T_{r_{\alpha}}U=$
$\displaystyle
i\frac{\gamma^{2}}{2g}T_{T_{1-\bar{Y}}r_{\alpha\alpha}}U+i\frac{\gamma^{2}}{2g}T_{r_{\alpha}}X+G,$
$\displaystyle
i\frac{\gamma^{2}}{2g}T_{1-\bar{Y}}\partial_{\alpha}T_{\partial_{\alpha}^{-1}w}R=$
$\displaystyle
i\frac{\gamma^{2}}{2g}T_{1-\bar{Y}}T_{w}R+i\frac{\gamma^{2}}{2g}T_{\partial_{\alpha}^{-1}w}T_{1-\bar{Y}}R_{\alpha}+G,$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{1-\bar{Y}}\partial_{\alpha}T_{w}Z=$
$\displaystyle
i\frac{\gamma^{2}}{4g}T_{w_{\alpha}}T_{1-\bar{Y}}Z+i\frac{\gamma^{2}}{4g}T_{w}R+G,$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{1-\bar{Y}}\partial_{\alpha}T_{r}X=$
$\displaystyle
i\frac{\gamma^{2}}{4g}T_{1-\bar{Y}}T_{r_{\alpha}}X+i\frac{\gamma^{2}}{4g}T_{r}T_{|1-Y|^{2}}{\mathbf{W}}+G.$

The other $\gamma^{2}$ terms have “over differentiated” lowest frequency
variables, and may be absorbed into $G$.

For the contributions of corrections with $\gamma^{3}$ coefficient,

$\displaystyle i\frac{\gamma^{3}}{4g^{2}}T_{D_{t}}T_{r}X=$ $\displaystyle
i\frac{\gamma^{3}}{4g^{2}}T_{T_{D_{t}}r}X+i\frac{\gamma^{3}}{4g^{2}}T_{r}T_{D_{t}}X+G$
$\displaystyle=$ $\displaystyle i\frac{\gamma^{3}}{4g^{2}}T_{-i\gamma
r+iT_{g+{\underline{a}}}(x-T_{w}Y)}X-i\frac{\gamma^{3}}{4g^{2}}T_{r}T_{1-\bar{Y}}R+G$
$\displaystyle=$
$\displaystyle\frac{\gamma^{4}}{4g^{2}}T_{r}X-\frac{\gamma^{3}}{4g}T_{w}X-i\frac{\gamma^{3}}{4g^{2}}T_{r}T_{1-\bar{Y}}R+G,$
$\displaystyle i\frac{\gamma^{3}}{4g^{2}}T_{D_{t}}T_{w}Z=$ $\displaystyle
i\frac{\gamma^{3}}{4g^{2}}T_{D_{t}w}Z+i\frac{\gamma^{3}}{4g^{2}}T_{w}T_{D_{t}}Z+G$
$\displaystyle=$
$\displaystyle-i\frac{\gamma^{3}}{4g^{2}}T_{T_{1-\bar{Y}}(r_{\alpha}+T_{w}R_{\alpha})}Z+i\frac{\gamma^{3}}{4g^{2}}T_{w}(igX-i\gamma
Z)+G$ $\displaystyle=$
$\displaystyle-i\frac{\gamma^{3}}{4g^{2}}T_{r_{\alpha}}Z-\frac{\gamma^{3}}{4g}T_{w}X+\frac{\gamma^{4}}{4g^{2}}T_{w}Z+G,$
$\displaystyle i\frac{\gamma^{3}}{4g^{2}}T_{D_{t}}T_{r_{\alpha}}U=$
$\displaystyle
i\frac{\gamma^{3}}{4g^{2}}T_{T_{D_{t}}r_{\alpha}}U+i\frac{\gamma^{3}}{4g^{2}}T_{r_{\alpha}}T_{D_{t}}U+G$
$\displaystyle=$ $\displaystyle i\frac{\gamma^{3}}{4g^{2}}T_{-i\gamma
r_{\alpha}+iT_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha})}U-i\frac{\gamma^{3}}{4g^{2}}T_{r_{\alpha}}T_{1-\bar{Y}}Z$
$\displaystyle=$
$\displaystyle\frac{\gamma^{4}}{4g^{2}}T_{r_{\alpha}}U-\frac{\gamma^{3}}{4g}T_{w_{\alpha}}U-i\frac{\gamma^{3}}{4g^{2}}T_{r_{\alpha}}T_{1-\bar{Y}}Z,$
$\displaystyle
i\frac{\gamma^{3}}{4g^{2}}T_{D_{t}}T_{\partial_{\alpha}^{-1}w}R=$
$\displaystyle
i\frac{\gamma^{3}}{4g^{2}}T_{T_{D_{t}}\partial_{\alpha}^{-1}w}R+i\frac{\gamma^{3}}{4g^{2}}T_{\partial_{\alpha}^{-1}w}T_{D_{t}}R+G$
$\displaystyle=$
$\displaystyle-i\frac{\gamma^{3}}{4g^{2}}T_{T_{1-\bar{Y}}(r+T_{w}R)}R+i\frac{\gamma^{3}}{4g^{2}}T_{\partial_{\alpha}^{-1}w}(iT_{g+{\underline{a}}}Y-i\gamma
R)+G$ $\displaystyle=$
$\displaystyle-i\frac{\gamma^{3}}{4g^{2}}T_{T_{1-\bar{Y}}r}R-\frac{\gamma^{3}}{4g}T_{\partial_{\alpha}^{-1}w}Y+\frac{\gamma^{4}}{4g^{2}}T_{\partial_{\alpha}^{-1}w}R+G,$
$\displaystyle\frac{\gamma^{3}}{4g}T_{1-\bar{Y}}\partial_{\alpha}T_{w}U=$
$\displaystyle\frac{\gamma^{3}}{4g}T_{1-\bar{Y}}T_{w_{\alpha}}U+\frac{\gamma^{3}}{4g}T_{1-\bar{Y}}T_{w}X+G.$

Here we changed $U_{\alpha}$ to $X$ according to (3.13). The other
$\gamma^{3}$ terms have “over differentiated” lowest frequency variables, and
we put them into $G$.

For the contributions of corrections with $\gamma^{4}$ coefficient,

$\displaystyle\frac{\gamma^{4}}{4g^{2}}T_{D_{t}}T_{w}U=$
$\displaystyle\frac{\gamma^{4}}{4g^{2}}T_{D_{t}w}U+\frac{\gamma^{4}}{4g^{2}}T_{w}T_{D_{t}}U+G$
$\displaystyle=-\frac{\gamma^{4}}{4g^{2}}T_{T_{1-\bar{Y}}(r_{\alpha}+T_{w}R_{\alpha})}U-\frac{\gamma^{4}}{4g^{2}}T_{w}T_{1-\bar{Y}}Z+G$
$\displaystyle=-\frac{\gamma^{4}}{4g^{2}}T_{T_{1-\bar{Y}}r_{\alpha}}U-\frac{\gamma^{4}}{4g^{2}}T_{w}T_{1-\bar{Y}}Z+G,$
$\displaystyle\frac{\gamma^{4}}{4g^{2}}T_{D_{t}}T_{\partial_{\alpha}^{-1}w}X$
$\displaystyle=\frac{\gamma^{4}}{4g^{2}}T_{T_{D_{t}}\partial_{\alpha}^{-1}w}X+\frac{\gamma^{4}}{4g^{2}}T_{\partial_{\alpha}^{-1}w}T_{D_{t}}X+G$
$\displaystyle=-\frac{\gamma^{4}}{4g^{2}}T_{T_{1-\bar{Y}}r}X-\frac{\gamma^{4}}{4g^{2}}T_{\partial_{\alpha}^{-1}w}T_{1-\bar{Y}}R+G.$

The rest of correction terms with $\gamma^{4}$ or $\gamma^{5}$ coefficient are
perturbative.

Summing all of above contributions, we obtain that the total correction is

${\mathbf{P}}T_{w_{\alpha}}{\underline{b}}+T_{w}{\mathbf{P}}\left((1-\bar{Y})R_{\alpha}-i\frac{\gamma}{2}{\mathbf{W}}\right)+G.$

$(2)$ Continuing with the contributions for the frequency balanced corrections
with holomorphic variables in the first equation. The computation is similar
to case $(1)$ in the sense that we simply replace the unbalanced par-Leibniz
error estimate by the corresponding balanced para-Leibniz estimate in Lemma
2.5. As a result, we get the total correction is given by

$\Pi\left(w_{\alpha},{\mathbf{P}}{\underline{b}}\right)+\Pi\left(w,(1-\bar{Y})R_{\alpha}-i\frac{\gamma}{2}{\mathbf{W}}\right)+G.$

$(3)$ Continuing with the frequency balanced corrections with one anti-
holomorphic variable. The computation is similar to case $(2)$.

$\displaystyle-T_{D_{t}}\Pi(w_{\alpha},\bar{X})$
$\displaystyle=-\Pi(\partial_{\alpha}T_{D_{t}}w,\bar{X})-\Pi(w_{\alpha},T_{D_{t}}\bar{X})+G$
$\displaystyle=\Pi(r_{\alpha\alpha},\bar{X})+\Pi(w_{\alpha},T_{1-Y}\bar{R})+G,$
$\displaystyle-T_{1-\bar{Y}}\partial_{\alpha}\Pi(r_{\alpha},\bar{X})$
$\displaystyle=-T_{1-\bar{Y}}\Pi(r_{\alpha\alpha},\bar{X})-\Pi(r_{\alpha},T_{1-\bar{Y}}\bar{X}_{\alpha})+G$
$\displaystyle=-T_{1-\bar{Y}}\Pi(r_{\alpha\alpha},\bar{X})-\Pi(r_{\alpha},\bar{Y})+G.$

For the last term we use (3.11) to replace $T_{1-\bar{Y}}\bar{X}_{\alpha}$ by
$\bar{Y}$. Then we have

$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}\Pi(w_{\alpha},\bar{Z})$
$\displaystyle=-\frac{\gamma}{2g}\Pi(\partial_{\alpha}T_{D_{t}}w,\bar{Z})-\frac{\gamma}{2g}\Pi(w_{\alpha},T_{D_{t}}\bar{Z})+G$
$\displaystyle=\frac{\gamma}{2g}\Pi(r_{\alpha\alpha}+T_{w}R_{\alpha\alpha},\bar{Z})-\frac{\gamma}{2g}\Pi(w_{\alpha},-ig\bar{X}+i\gamma\bar{Z})+G,$
$\displaystyle=\frac{\gamma}{2g}\Pi(r_{\alpha\alpha},\bar{Z})+i\frac{\gamma}{2}\Pi(w_{\alpha},\bar{X})-i\frac{\gamma^{2}}{2g}\Pi(w_{\alpha},\bar{Z})+G,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}\Pi(r_{\alpha},\bar{X})$
$\displaystyle=-\frac{\gamma}{2g}\Pi(\partial_{\alpha}T_{D_{t}}r,\bar{X})-\frac{\gamma}{2g}\Pi(r_{\alpha},T_{D_{t}}\bar{X})+G$
$\displaystyle=-\frac{\gamma}{2g}\Pi(-i\gamma
r_{\alpha}+iT_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha}),\bar{X})+\frac{\gamma}{2g}\Pi(r_{\alpha},T_{1-Y}\bar{R})+G,$
$\displaystyle=i\frac{\gamma^{2}}{2g}\Pi(r_{\alpha},\bar{X})-i\frac{\gamma}{2}\Pi(w_{\alpha},\bar{X})+\frac{\gamma}{2g}\Pi(r_{\alpha},T_{1-Y}\bar{R})+G,$
$\displaystyle-\frac{\gamma}{2g}T_{1-\bar{Y}}\partial_{\alpha}\Pi(r_{\alpha},\bar{Z})$
$\displaystyle=-\frac{\gamma}{2g}T_{1-\bar{Y}}\Pi(r_{\alpha\alpha},\bar{Z})-\frac{\gamma}{2g}\Pi(r_{\alpha},T_{1-\bar{Y}}\bar{R})+G,$
$\displaystyle i\frac{\gamma}{2}T_{1-\bar{Y}}\partial_{\alpha}\Pi(w,\bar{X})$
$\displaystyle=i\frac{\gamma}{2}\Pi(w_{\alpha},T_{1-Y}\bar{X})+i\frac{\gamma}{2}\Pi(w,T_{1-\bar{Y}}\bar{X}_{\alpha})+G.$

For the first term of the last contribution, and the second term of the first
contribution, we can combine them to get

$\Pi\left(w_{\alpha},T_{1-Y}\bar{R}+i\frac{\gamma}{2}T_{1-Y}\bar{X}\right)=\Pi\left(w_{\alpha},\bar{{\mathbf{P}}}\left((1-Y)\bar{R}+i\frac{\gamma}{2}(1-Y)\bar{W}\right)\right)+G=\Pi(w_{\alpha},\bar{{\mathbf{P}}}{\underline{b}})+G,$

which is the other part of the desired balanced truncation
$\Pi(w_{\alpha},{\underline{b}})$ in
${\mathbf{P}}\underline{\mathcal{G}}_{1}$.

For the contributions of $\gamma^{2}$ term, they are

$\displaystyle-i\frac{\gamma^{2}}{2g}T_{D_{t}}\Pi(w_{\alpha},\bar{U})$
$\displaystyle=-i\frac{\gamma^{2}}{2g}\Pi(\partial_{\alpha}T_{D_{t}}w,\bar{U})-i\frac{\gamma^{2}}{2g}\Pi(w_{\alpha},T_{D_{t}}\bar{U})+G$
$\displaystyle=i\frac{\gamma^{2}}{2g}\Pi(r_{\alpha\alpha}+T_{w}R_{\alpha\alpha},\bar{U})+i\frac{\gamma^{2}}{2g}\Pi(w_{\alpha},T_{1-Y}\bar{Z})+G$
$\displaystyle=i\frac{\gamma^{2}}{2g}\Pi(r_{\alpha\alpha},\bar{U})+i\frac{\gamma^{2}}{2g}\Pi(w_{\alpha},T_{1-Y}\bar{Z})+G,$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{D_{t}}\Pi(w,\bar{X})$
$\displaystyle=i\frac{\gamma^{2}}{4g}\Pi(T_{D_{t}}w,\bar{X})+i\frac{\gamma^{2}}{4g}\Pi(w,T_{D_{t}}\bar{X})+G$
$\displaystyle=-i\frac{\gamma^{2}}{4g}\Pi(r_{\alpha}+T_{w}R_{\alpha},\bar{X})-i\frac{\gamma^{2}}{4g}\Pi(w,T_{1-Y}\bar{R})+G$
$\displaystyle=-i\frac{\gamma^{2}}{4g}\Pi(r_{\alpha},\bar{X})-i\frac{\gamma^{2}}{4g}\Pi(w,T_{1-Y}\bar{R})+G,$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}T_{D_{t}}\Pi(r_{\alpha},\bar{Z})$
$\displaystyle=-\frac{\gamma^{2}}{4g^{2}}\Pi(\partial_{\alpha}T_{D_{t}}r,\bar{Z})-\frac{\gamma^{2}}{4g^{2}}\Pi(r_{\alpha},T_{D_{t}}\bar{Z})+G$
$\displaystyle=-\frac{\gamma^{2}}{4g^{2}}\Pi(-i\gamma
r_{\alpha}+iT_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha}),\bar{Z})-\frac{\gamma^{2}}{4g^{2}}\Pi(r_{\alpha},-ig\bar{X}+i\gamma\bar{Z})+G$
$\displaystyle=-i\frac{\gamma^{2}}{4g}\Pi(w_{\alpha},\bar{Z})+i\frac{\gamma^{2}}{4g}\Pi(r_{\alpha},\bar{X})+G,$
$\displaystyle-i\frac{\gamma^{2}}{2g}T_{1-\bar{Y}}\partial_{\alpha}\Pi(r_{\alpha},\bar{U})$
$\displaystyle=-i\frac{\gamma^{2}}{2g}T_{1-\bar{Y}}\Pi(r_{\alpha\alpha},\bar{U})-i\frac{\gamma^{2}}{2g}\Pi(r_{\alpha},T_{1-\bar{Y}}\bar{X})+G,$
$\displaystyle
i\frac{\gamma^{2}}{4g}T_{1-\bar{Y}}\partial_{\alpha}\Pi(w,\bar{Z})$
$\displaystyle=i\frac{\gamma^{2}}{4g}T_{1-\bar{Y}}\Pi(w_{\alpha},\bar{Z})+i\frac{\gamma^{2}}{4g}\Pi(w,T_{1-\bar{Y}}\bar{R})+G.$

The rest of the contributions are given by

$\displaystyle i\frac{\gamma^{3}}{4g^{2}}T_{D_{t}}\Pi(w,\bar{Z})$
$\displaystyle=i\frac{\gamma^{3}}{4g^{2}}\Pi(T_{D_{t}}w,\bar{Z})+i\frac{\gamma^{3}}{4g^{2}}\Pi(w,T_{D_{t}}\bar{Z})+G$
$\displaystyle=-i\frac{\gamma^{3}}{4g^{2}}\Pi(r_{\alpha}+T_{w}R_{\alpha},\bar{Z})+i\frac{\gamma^{3}}{4g^{2}}\Pi(w,-ig\bar{X}+i\gamma\bar{Z})+G$
$\displaystyle=-i\frac{\gamma^{3}}{4g^{2}}\Pi(r_{\alpha},\bar{Z})+\frac{\gamma^{3}}{4g}\Pi(w,\bar{X})-\frac{\gamma^{4}}{4g^{2}}\Pi(w,\bar{Z})+G,$
$\displaystyle-i\frac{\gamma^{3}}{4g^{2}}T_{D_{t}}\Pi(r_{\alpha},\bar{U})$
$\displaystyle=-i\frac{\gamma^{3}}{4g^{2}}\Pi(\partial_{\alpha}T_{D_{t}}r,\bar{U})-i\frac{\gamma^{3}}{4g^{2}}\Pi(r_{\alpha},T_{D_{t}}\bar{U})+G$
$\displaystyle=-i\frac{\gamma^{3}}{4g^{2}}\Pi(-i\gamma
r_{\alpha}+iT_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha}),\bar{U})+i\frac{\gamma^{3}}{4g^{2}}\Pi(r_{\alpha},T_{1-Y}\bar{Z})+G$
$\displaystyle=-\frac{\gamma^{4}}{4g^{2}}\Pi(r_{\alpha},\bar{U})+\frac{\gamma^{3}}{4g}\Pi(w_{\alpha},\bar{U})+i\frac{\gamma^{3}}{4g^{2}}\Pi(r_{\alpha},T_{1-Y}\bar{Z})+G,$
$\displaystyle-\frac{\gamma^{4}}{4g^{2}}T_{D_{t}}\Pi(w,\bar{U})$
$\displaystyle=-\frac{\gamma^{4}}{4g^{2}}\Pi(T_{D_{t}}w,\bar{U})-\frac{\gamma^{4}}{4g^{2}}\Pi(w,T_{D_{t}}\bar{U})+G$
$\displaystyle=\frac{\gamma^{4}}{4g^{2}}\Pi(r_{\alpha},\bar{U})+\frac{\gamma^{4}}{4g^{2}}\Pi(w,T_{1-Y}\bar{Z})+G,$
$\displaystyle-\frac{\gamma^{3}}{4g}T_{1-\bar{Y}}\partial_{\alpha}\Pi(w,\bar{U})$
$\displaystyle=-\frac{\gamma^{3}}{4g}T_{1-\bar{Y}}\Pi(w_{\alpha},\bar{U})-\frac{\gamma^{3}}{4g}\Pi(w,T_{1-\bar{Y}}\bar{X})+G.$

The other contributions have full or over differentiated variables at low
frequency, and can be put into $G$. Collecting all the corrections above, the
total frequency balanced corrections with one anti-holomorphic variable are
given by

$-\Pi(r_{\alpha},\bar{Y})+\Pi(w_{\alpha},\bar{{\mathbf{P}}}{\underline{b}})+\Pi\left(w,i\frac{\gamma}{2}\bar{W}\right)+G.$

Adding the contributions of corrections in cases $(1)$, $(2)$, $(3)$, the sum
is exactly $-{\mathbf{P}}\underline{\mathcal{G}}_{1}+G$.

$(4)$ Next, we compute the second equation in (LABEL:NormalFormGKOne). We
begin with the high-low corrections. Consider first the contributions of non-
vorticity terms,

$\displaystyle-T_{D_{t}}T_{r_{\alpha}}X=$ $\displaystyle-
T_{\partial_{\alpha}T_{D_{t}}r}X-T_{r_{\alpha}}T_{D_{t}}X+K$ $\displaystyle=$
$\displaystyle iT_{\gamma
r_{\alpha}-T_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha})}X+T_{r_{\alpha}}T_{1-\bar{Y}}R+K$
$\displaystyle=$ $\displaystyle i\gamma
T_{r_{\alpha}}X-iT_{g+{\underline{a}}}T_{((1-Y)w)_{\alpha}}X+T_{r_{\alpha}}T_{1-\bar{Y}}R+K,$
$\displaystyle iT_{1-Y}T_{g+{\underline{a}}}\partial_{\alpha}T_{w}X=$
$\displaystyle
iT_{1-Y}T_{g+{\underline{a}}}T_{((1-Y)w)_{\alpha}}W+iT_{1-Y}T_{g+{\underline{a}}}T_{w}T_{1-Y}{\mathbf{W}}+K$
$\displaystyle=$ $\displaystyle
iT_{g+{\underline{a}}}T_{((1-Y)w)_{\alpha}}X+K.$

Then for the contributions of the $\gamma$ terms, they are $-i\gamma
T_{r_{\alpha}}X$ and

$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}T_{r_{\alpha}}Z$
$\displaystyle=-\frac{\gamma}{2g}T_{\partial_{\alpha}T_{D_{t}}r}Z-\frac{\gamma}{2g}T_{r_{\alpha}}T_{D_{t}}Z+K$
$\displaystyle=i\frac{\gamma}{2g}T_{\gamma
r_{\alpha}-T_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha})}Z-\frac{\gamma}{2g}T_{r_{\alpha}}(igX-i\gamma
Z)+K$
$\displaystyle=i\frac{\gamma^{2}}{g}T_{r_{\alpha}}Z-i\frac{\gamma}{2}T_{w_{\alpha}}Z-i\frac{\gamma}{2}T_{r_{\alpha}}X+K,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}T_{r}R$
$\displaystyle=-\frac{\gamma}{2g}T_{D_{t}r}R-\frac{\gamma}{2g}T_{r}T_{D_{t}}R+K$
$\displaystyle=i\frac{\gamma}{2g}T_{\gamma
r-T_{g+{\underline{a}}}(x-T_{w}Y)}R-\frac{\gamma}{2g}T_{r}(igT_{1-Y}{\mathbf{W}}-i\gamma
R)+K$
$\displaystyle=i\frac{\gamma^{2}}{g}T_{r}R-i\frac{\gamma}{2}T_{w}R-i\frac{\gamma}{2}T_{r}T_{1-Y}{\mathbf{W}}+K,$
$\displaystyle i\frac{\gamma}{2}T_{D_{t}}T_{w}X$
$\displaystyle=i\frac{\gamma}{2}T_{D_{t}w}X+i\frac{\gamma}{2}T_{w}T_{D_{t}}X+K$
$\displaystyle=-i\frac{\gamma}{2}T_{T_{1-\bar{Y}}(r_{\alpha}+T_{w}R_{\alpha})}X-i\frac{\gamma}{2}T_{w}T_{1-\bar{Y}}R+K$
$\displaystyle=-i\frac{\gamma}{2}T_{T_{1-\bar{Y}}r_{\alpha}}X-i\frac{\gamma}{2}T_{w}T_{1-\bar{Y}}R+K,$
$\displaystyle
i\frac{\gamma}{2g}T_{1-Y}T_{g+{\underline{a}}}T_{w_{\alpha}}Z=i\frac{\gamma}{2}T_{w_{\alpha}}Z$
$\displaystyle+K,\quad
i\frac{\gamma}{2g}T_{1-Y}T_{g+{\underline{a}}}T_{r}X_{\alpha}=i\frac{\gamma}{2}T_{r}T_{1-Y}{\mathbf{W}}+K,$
$\displaystyle
i\frac{\gamma}{2g}T_{1-Y}T_{g+{\underline{a}}}T_{r_{\alpha}}X=i\frac{\gamma}{2}T_{r_{\alpha}}X$
$\displaystyle+K,\quad
i\frac{\gamma}{2g}T_{1-Y}T_{g+{\underline{a}}}T_{w}R=i\frac{\gamma}{2}T_{w}R+K.$

Next, for the contributions of the $\gamma^{2}$ terms, they are given by

$\displaystyle-i\frac{\gamma^{2}}{2g}T_{r_{\alpha}}Z$
$\displaystyle-i\frac{\gamma^{2}}{2g}T_{r}R-\frac{\gamma^{2}}{2}T_{w}X,$
$\displaystyle i\frac{\gamma^{2}}{2g}T_{D_{t}}T_{r_{\alpha}}U$
$\displaystyle=i\frac{\gamma^{2}}{2g}T_{T_{D_{t}}r_{\alpha}}U+i\frac{\gamma^{2}}{2g}T_{r_{\alpha}}T_{D_{t}}U+K$
$\displaystyle=i\frac{\gamma^{2}}{2g}T_{-i\gamma
r_{\alpha}+iT_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha})}U-i\frac{\gamma^{2}}{2g}T_{r_{\alpha}}T_{1-\bar{Y}}Z+K$
$\displaystyle=\frac{\gamma^{3}}{2g}T_{r_{\alpha}}U-\frac{\gamma^{2}}{2}T_{T_{1-Y}w_{\alpha}}U-i\frac{\gamma^{2}}{2g}T_{r_{\alpha}}T_{1-\bar{Y}}Z+K,$
$\displaystyle i\frac{\gamma^{2}}{2g}T_{D_{t}}T_{\partial_{\alpha}^{-1}w}R$
$\displaystyle=i\frac{\gamma^{2}}{2g}T_{T_{D_{t}}\partial_{\alpha}^{-1}w}R+i\frac{\gamma^{2}}{2g}T_{\partial_{\alpha}^{-1}w}T_{D_{t}}R+K$
$\displaystyle=-i\frac{\gamma^{2}}{2g}T_{T_{1-\bar{Y}}(r+T_{w}R)}R+i\frac{\gamma^{2}}{2g}T_{\partial_{\alpha}^{-1}w}(iT_{g+{\underline{a}}}Y-i\gamma
R)+K$
$\displaystyle=-i\frac{\gamma^{2}}{2g}T_{T_{1-\bar{Y}}r}R-\frac{\gamma^{2}}{2g}T_{\partial_{\alpha}^{-1}w}T_{g+{\underline{a}}}Y+\frac{\gamma^{3}}{2g}T_{\partial_{\alpha}^{-1}w}R+K,$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{D_{t}}T_{w}Z$
$\displaystyle=i\frac{\gamma^{2}}{4g}T_{D_{t}w}Z+i\frac{\gamma^{2}}{4g}T_{w}T_{D_{t}}Z+K$
$\displaystyle=-i\frac{\gamma^{2}}{4g}T_{T_{1-\bar{Y}}(r_{\alpha}+T_{w}R_{\alpha})}Z+i\frac{\gamma^{2}}{4g}T_{w}(igX-i\gamma
Z)+K$
$\displaystyle=-i\frac{\gamma^{2}}{4g}T_{T_{1-\bar{Y}}r_{\alpha}}Z-\frac{\gamma^{2}}{4}T_{w}X+\frac{\gamma^{3}}{4g}T_{w}Z+K,$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{D_{t}}T_{r}X$
$\displaystyle=i\frac{\gamma^{2}}{4g}T_{D_{t}r}X+i\frac{\gamma^{2}}{4g}T_{r}T_{D_{t}}X+K$
$\displaystyle=i\frac{\gamma^{2}}{4g}T_{-i\gamma
r+iT_{g+{\underline{a}}}(x-T_{w}Y)}X-i\frac{\gamma^{2}}{4g}T_{r}T_{1-\bar{Y}}R+K$
$\displaystyle=\frac{\gamma^{3}}{4g}T_{r}X-\frac{\gamma^{2}}{4}T_{w}X-i\frac{\gamma^{2}}{4g}T_{r}T_{1-\bar{Y}}R+K,$
$\displaystyle\frac{\gamma^{2}}{2g}T_{1-Y}T_{g+{\underline{a}}}T_{w_{\alpha}}U=$
$\displaystyle\frac{\gamma^{2}}{2}T_{1-Y}T_{w_{\alpha}}U+K,\quad\frac{\gamma^{2}}{2g}T_{1-Y}T_{g+{\underline{a}}}T_{\partial_{\alpha}^{-1}w}X_{\alpha}=\frac{\gamma^{2}}{2g}T_{g+{\underline{a}}}T_{\partial_{\alpha}^{-1}w}Y+K,$
$\displaystyle\frac{\gamma^{2}}{g}T_{1-Y}T_{g+{\underline{a}}}T_{w}X=$
$\displaystyle\gamma^{2}T_{1-Y}T_{w}X+K,\quad
i\frac{\gamma^{2}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}T_{r_{\alpha}}Z=i\frac{\gamma^{2}}{4g}T_{1-Y}T_{r_{\alpha}}Z+K,$
$\displaystyle
i\frac{\gamma^{2}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}T_{r}Z_{\alpha}=$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{1-Y}T_{r}R+K.$

Continuing with the contributions of the $\gamma^{3}$ corrections, the first
few terms are

$-\frac{\gamma^{3}}{2g}T_{r_{\alpha}}U-\frac{\gamma^{3}}{2g}T_{\partial_{\alpha}^{-1}w}R-\frac{\gamma^{3}}{4g}T_{w}Z-\frac{\gamma^{3}}{4g}T_{r}X.$

The contributions of other $\gamma^{3}$ terms are given by

$\displaystyle\frac{\gamma^{3}}{4g}T_{D_{t}}T_{w}U$
$\displaystyle=\frac{\gamma^{3}}{4g}T_{D_{t}w}U+\frac{\gamma^{3}}{4g}T_{w}T_{D_{t}}U+K$
$\displaystyle=-\frac{\gamma^{3}}{4g}T_{T_{1-\bar{Y}}(r_{\alpha}+T_{w}R_{\alpha})}U-\frac{\gamma^{3}}{4g}T_{w}T_{1-\bar{Y}}Z+K$
$\displaystyle=-\frac{\gamma^{3}}{4g}T_{T_{1-\bar{Y}}r_{\alpha}}U-\frac{\gamma^{3}}{4g}T_{w}T_{1-\bar{Y}}Z+K,$
$\displaystyle\frac{\gamma^{3}}{4g}T_{D_{t}}T_{\partial_{\alpha}^{-1}w}X$
$\displaystyle=\frac{\gamma^{3}}{4g}T_{T_{D_{t}}\partial_{\alpha}^{-1}w}X+\frac{\gamma^{3}}{4g}T_{\partial_{\alpha}^{-1}w}T_{D_{t}}X+K$
$\displaystyle=-\frac{\gamma^{3}}{4g}T_{T_{1-\bar{Y}}(r+T_{w}R)}X-\frac{\gamma^{3}}{4g}T_{\partial_{\alpha}^{-1}w}T_{1-\bar{Y}}R+K$
$\displaystyle=-\frac{\gamma^{3}}{4g}T_{T_{1-\bar{Y}}r}X-\frac{\gamma^{3}}{4g}T_{\partial_{\alpha}^{-1}w}T_{1-\bar{Y}}R+K,$
$\displaystyle\frac{\gamma^{3}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}T_{r}X=$
$\displaystyle\frac{\gamma^{3}}{4g}T_{1-Y}T_{r}X+K,\quad\frac{\gamma^{3}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}T_{w}Z=\frac{\gamma^{3}}{4g}T_{1-Y}T_{w}Z+K,$
$\displaystyle\frac{\gamma^{3}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}T_{r_{\alpha}}U=$
$\displaystyle\frac{\gamma^{3}}{4g}T_{1-Y}T_{r_{\alpha}}U+K,\quad\frac{\gamma^{3}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}T_{\partial_{\alpha}^{-1}w}R=\frac{\gamma^{3}}{4g}T_{1-Y}T_{\partial_{\alpha}^{-1}w}R+K.$

For the contributions of corrections with $\gamma^{4}$ coefficient, they are
given by

$\displaystyle
i\frac{\gamma^{4}}{4g}T_{w}U+i\frac{\gamma^{4}}{4g}T_{\partial_{\alpha}^{-1}w}X,$
$\displaystyle-i\frac{\gamma^{4}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}(T_{w}U+\partial_{\alpha}^{-1}X)$
$\displaystyle=-i\frac{\gamma^{4}}{4g}T_{w}U-i\frac{\gamma^{4}}{4g}T_{\partial_{\alpha}^{-1}w}X+K.$

Adding all the contributions above, we get the total correction

${\mathbf{P}}T_{r_{\alpha}}{\underline{b}}-i\frac{\gamma}{2}T_{1-Y}T_{w}R+T_{(g+{\underline{a}})w}Y+K.$

$(5)$ Continuing with the contributions for the frequency balanced corrections
with holomorphic variables for the second equation of (LABEL:NormalFormGKOne).
The computation is similar to case $(4)$ in the sense that we simply replace
the unbalanced par-Leibniz error estimates by the corresponding balanced para-
Leibniz estimates in Lemma 2.5. Consequently, the total correction is given by

$\Pi\left(r_{\alpha},{\mathbf{P}}(R(1-\bar{Y}))-i\frac{\gamma}{2}{\mathbf{P}}(W(1-\bar{Y}))\right)-i\frac{\gamma}{2}T_{1-Y}\Pi(w,R)+\Pi((g+{\underline{a}})w,Y)+K.$

$(6)$ Finally, we compute the frequency balanced corrections with one anti-
holomorphic variable for the second equation of (LABEL:NormalFormGKOne).

$\displaystyle-T_{D_{t}}\Pi(r_{\alpha},\bar{X})$
$\displaystyle=-\Pi(\partial_{\alpha}T_{D_{t}}r,\bar{X})-\Pi(r_{\alpha},T_{D_{t}}\bar{X})+K$
$\displaystyle=-\Pi(-i\gamma
r_{\alpha}+iT_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha}),\bar{X})+\Pi(r_{\alpha},T_{1-Y}\bar{R})+K$
$\displaystyle=i\gamma\Pi(r_{\alpha},\bar{X})-ig\Pi(T_{1-Y}w_{\alpha},\bar{X})+\Pi(r_{\alpha},T_{1-Y}\bar{R})+K,$
$\displaystyle iT_{1-Y}T_{g+{\underline{a}}}\Pi(w_{\alpha},\bar{X})$
$\displaystyle=ig\Pi(T_{1-Y}w_{\alpha},\bar{X})+K,\quad-i\gamma\Pi(r_{\alpha},\bar{X}),$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}\Pi(r_{\alpha},\bar{Z})$
$\displaystyle=-\frac{\gamma}{2g}\Pi(\partial_{\alpha}T_{D_{t}}r,\bar{Z})-\frac{\gamma}{2g}\Pi(r_{\alpha},T_{D_{t}}\bar{Z})+K$
$\displaystyle=-\frac{\gamma}{2g}\Pi(-i\gamma
r_{\alpha}+iT_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha}),\bar{Z})-\frac{\gamma}{2g}\Pi(r_{\alpha},-ig\bar{X}+i\gamma\bar{Z})+K$
$\displaystyle=-i\frac{\gamma}{2}\Pi(T_{1-Y}w_{\alpha},\bar{Z})+i\frac{\gamma}{2}\Pi(r_{\alpha},\bar{X})+K,$
$\displaystyle i\frac{\gamma}{2}T_{D_{t}}\Pi(w,\bar{X})$
$\displaystyle=i\frac{\gamma}{2}\Pi(T_{D_{t}}w,\bar{X})+i\frac{\gamma}{2}\Pi(w,T_{D_{t}}\bar{X})+K$
$\displaystyle=-i\frac{\gamma}{2}\Pi(T_{1-\bar{Y}}(r_{\alpha}+T_{w}R_{\alpha}),\bar{X})-i\frac{\gamma}{2}\Pi(w,T_{1-Y}\bar{R})+K$
$\displaystyle=-i\frac{\gamma}{2}\Pi(T_{1-\bar{Y}}r_{\alpha},\bar{X})-i\frac{\gamma}{2}T_{1-Y}\Pi(w,\bar{R})+K,$
$\displaystyle
i\frac{\gamma}{2g}T_{1-Y}T_{g+{\underline{a}}}\Pi(w_{\alpha},\bar{Z})$
$\displaystyle=i\frac{\gamma}{2}\Pi(T_{1-Y}w_{\alpha},\bar{Z})+K,$
$\displaystyle
i\frac{\gamma}{2g}T_{1-Y}T_{g+{\underline{a}}}\Pi(r_{\alpha},\bar{X})$
$\displaystyle=i\frac{\gamma}{2}\Pi(r_{\alpha},T_{1-Y}\bar{X})+K.$

The contributions of $\gamma^{2}$ correction terms are given by

$\displaystyle-i\frac{\gamma^{2}}{2g}\Pi(r_{\alpha},\bar{Z})-$ $\displaystyle
i\frac{\gamma^{2}}{2}\Pi(w,\bar{X}),$
$\displaystyle-i\frac{\gamma^{2}}{2g}T_{D_{t}}\Pi(r_{\alpha},\bar{U})$
$\displaystyle=-i\frac{\gamma^{2}}{2g}\Pi(\partial_{\alpha}T_{D_{t}}r,\bar{U})-\frac{\gamma^{2}}{2g}\Pi(r_{\alpha},T_{D_{t}}\bar{U})+K$
$\displaystyle=-i\frac{\gamma^{2}}{2g}\Pi(-i\gamma
r_{\alpha}+iT_{g+{\underline{a}}}(x_{\alpha}-T_{w}Y_{\alpha}),\bar{U})+i\frac{\gamma^{2}}{2g}\Pi(r_{\alpha},T_{1-Y}\bar{Z})+K$
$\displaystyle=-\frac{\gamma^{3}}{2g}\Pi(r_{\alpha},\bar{U})+\frac{\gamma^{2}}{2}\Pi(T_{1-Y}w_{\alpha},\bar{U})+i\frac{\gamma^{2}}{2g}\Pi(r_{\alpha},T_{1-Y}\bar{Z})+K,$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{D_{t}}\Pi(w,\bar{Z})$
$\displaystyle=i\frac{\gamma^{2}}{4g}\Pi(T_{D_{t}}w,\bar{Z})+i\frac{\gamma^{2}}{4g}\Pi(w,T_{D_{t}}\bar{Z})+K$
$\displaystyle=-i\frac{\gamma^{2}}{4g}\Pi(T_{1-\bar{Y}}(r_{\alpha}+T_{w}R_{\alpha}),\bar{Z})+i\frac{\gamma^{2}}{4g}\Pi(w,-ig\bar{X}+i\gamma\bar{Z})+K$
$\displaystyle=-i\frac{\gamma^{2}}{4g}\Pi(T_{1-\bar{Y}}r_{\alpha},\bar{Z})+\frac{\gamma^{2}}{4}\Pi(w,\bar{X})-\frac{\gamma^{3}}{4g}\Pi(w,\bar{Z})+K,$
$\displaystyle-\frac{\gamma^{2}}{2g}T_{1-Y}T_{g+{\underline{a}}}\Pi(w_{\alpha},\bar{U})$
$\displaystyle=-\frac{\gamma^{2}}{2}\Pi(T_{1-Y}w_{\alpha},\bar{U})+K,$
$\displaystyle\frac{\gamma^{2}}{4g}T_{1-Y}T_{g+{\underline{a}}}\Pi(w,\bar{X})$
$\displaystyle=\frac{\gamma^{2}}{4}\Pi(T_{1-Y}w,\bar{X})+K,$ $\displaystyle
i\frac{\gamma^{2}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}\Pi(r_{\alpha},\bar{Z})$
$\displaystyle=i\frac{\gamma^{2}}{4g}\Pi(T_{1-Y}r_{\alpha},\bar{Z})+K.$

Continuing with the contributions of $\gamma^{3}$ and $\gamma^{4}$ terms, they
are

$\displaystyle\frac{\gamma^{3}}{2g}\Pi(r_{\alpha},\bar{U})-$
$\displaystyle\frac{\gamma^{3}}{4g}\Pi(w,\bar{Z})-i\frac{\gamma^{4}}{4g}\Pi(w,\bar{U}),$
$\displaystyle-\frac{\gamma^{3}}{4g}T_{D_{t}}\Pi(w,\bar{U})$
$\displaystyle=-\frac{\gamma^{3}}{4g}\Pi(T_{D_{t}}w,\bar{U})-\frac{\gamma^{3}}{4g}\Pi(w,T_{D_{t}}\bar{U})+K$
$\displaystyle=\frac{\gamma^{3}}{4g}\Pi(T_{1-\bar{Y}}(r_{\alpha}+T_{w}R_{\alpha}),\bar{U})+\frac{\gamma^{3}}{4g}\Pi(w,T_{1-Y}\bar{Z})+K$
$\displaystyle=\frac{\gamma^{3}}{4g}\Pi(T_{1-\bar{Y}}r_{\alpha},\bar{U})+\frac{\gamma^{3}}{4g}\Pi(w,T_{1-Y}\bar{Z})+K,$
$\displaystyle\frac{\gamma^{3}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}\Pi(w,\bar{Z})$
$\displaystyle=\frac{\gamma^{3}}{4g}\Pi(T_{1-Y}w,\bar{Z})+K,$
$\displaystyle-\frac{\gamma^{3}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}\Pi(r_{\alpha},\bar{U})$
$\displaystyle=-\frac{\gamma^{3}}{4g}\Pi(T_{1-Y}r_{\alpha},\bar{U})+K,$
$\displaystyle
i\frac{\gamma^{4}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}\Pi(w,\bar{U})$
$\displaystyle=i\frac{\gamma^{4}}{4g}\Pi(T_{1-Y}w,\bar{U})+K.$

Summing all the contributions above and using the definitions of
${\underline{a}}$ and ${\underline{b}}$, we get that the total correction in
this case is given by

$\Pi\left(r_{\alpha},T_{1-Y}\bar{R}+i\frac{\gamma}{2}T_{1-Y}\bar{X}\right)-i\frac{\gamma}{2}T_{1-Y}\Pi(w,\bar{R})+K=\Pi(r_{\alpha},\bar{{\mathbf{P}}}{\underline{b}})-iT_{1-Y}\Pi(w,\bar{{\mathbf{P}}}{\underline{a}})+K.$

Collecting the contributions of corrections in cases $(4)$, $(5)$, $(6)$
together, the sum is exactly $-{\mathbf{P}}\underline{\mathcal{K}}_{1}+K$. ∎

###  4.5. Normal form analysis for
$(\underline{\mathcal{G}}_{0},\underline{\mathcal{K}}_{0})$

In this subsection, we use the paradifferential normal form corrections to
remove the main part of the source terms
$({\mathbf{P}}\underline{\mathcal{G}}_{0},{\mathbf{P}}\underline{\mathcal{K}}_{0})$
in the linearized equations.

For simplicity regarding the balanced cases, we let $\Pi={\mathbf{P}}\Pi$ to
include the projection ${\mathbf{P}}$. We will then prove the following
result:

######  Proposition 4.13.

Assume that $(w,r)$ solve the paradifferential equations
(LABEL:ParadifferentialLinearEqn), then there exists a linear paradifferential
normal form correction

$(\tilde{w},\tilde{r})=NF(w,r)$

that solves the paradifferential equations

(4.21)
$\left\\{\begin{array}[]{lr}T_{D_{t}}\tilde{w}+T_{1-\bar{Y}}\partial_{\alpha}\tilde{r}+T_{(1-\bar{Y})R_{\alpha}}\tilde{w}+\gamma
T_{\Im{\mathbf{W}}}\tilde{w}=-{\mathbf{P}}\underline{\mathcal{G}}_{0}(w,r)+G&\\\
T_{D_{t}}\tilde{r}+i\gamma\tilde{r}-iT_{1-Y}T_{g+{\underline{a}}}\tilde{w}=-{\mathbf{P}}\underline{\mathcal{K}}_{0}(w,r)+K,&\end{array}\right.$

with the following properties:

  1. (1)

Quadratic correction bound:

$\|NF(w,r)\|_{\dot{\mathcal{H}}^{\frac{1}{4}}}\lesssim_{{\underline{A}}}{\underline{A}^{\sharp}}\left(\|(w,r)\|_{\dot{\mathcal{H}}^{\frac{1}{4}}}+\gamma^{2}\|(w,r)\|_{\dot{\mathcal{H}}^{-\frac{3}{4}}}\right).$

  2. (2)

Secondary correction bound:

$\|NF(w,r)\|_{\dot{\mathcal{H}}^{\frac{1}{4}}}\lesssim_{\underline{A}}{\underline{A}_{\frac{1}{4}}}\left(\|(w,r)\|_{\mathcal{H}^{0}}+\gamma^{2}\|(w,r)\|_{\dot{\mathcal{H}}^{-1}}\right).$

  3. (3)

Cubic error bound:

$\|(G,K)\|_{\dot{\mathcal{H}}^{\frac{1}{4}}}\lesssim_{{\underline{A}^{\sharp}}}\underline{A}_{\frac{1}{4}}{\underline{A}^{\sharp}_{\frac{1}{4}}}\left(\|(w,r)\|_{\dot{\mathcal{H}}^{\frac{1}{4}}}+\gamma^{2}\|(w,r)\|_{\dot{\mathcal{H}}^{-\frac{3}{4}}}\right).$

The idea to construct the paradifferential corrections here is similar to that
for
$({\mathbf{P}}\underline{\mathcal{G}}_{1},{\mathbf{P}}\underline{\mathcal{K}}_{1})$
corrections in Section 4.4. Inspired by the normal form transformation (1.10),
we define the corrections $(\tilde{w},\tilde{r})$ by

$\displaystyle\tilde{w}=$
$\displaystyle-{\mathbf{P}}(\bar{x}{\mathbf{W}})-\frac{\gamma}{2g}{\mathbf{P}}(\bar{r}{\mathbf{W}})-\frac{\gamma}{2g}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)-i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{u}{\mathbf{W}})+i\frac{\gamma^{2}}{4g}{\mathbf{P}}(\bar{x}W)$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R)+i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(\bar{r}W)-i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)-\frac{\gamma^{4}}{4g^{2}}{\mathbf{P}}(\bar{u}W),$
$\displaystyle\tilde{r}=$
$\displaystyle-{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)-\frac{\gamma}{2g}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R)+i\frac{\gamma}{2}{\mathbf{P}}(\bar{x}W)-i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)$
$\displaystyle+i\frac{\gamma^{2}}{4g}{\mathbf{P}}(\bar{r}W)-\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{u}W).$

We remark that this paradifferential correction is roughly the anti-
holomorphic part of the linearization of the normal form transformation (1.10)
at the quadratic level.

These correction bounds are easily verified, and is similar to the estimates
in Section 4.4 using Sobolev embeddings. Lemma 4.7 shows that the main part of
$({\mathbf{P}}\underline{\mathcal{G}}_{0},{\mathbf{P}}\underline{\mathcal{K}}_{0})$
is $(\mathcal{G}_{0,0},\mathcal{K}_{0,0})$. Therefore, it remains to show that
these corrections eliminate $(\mathcal{G}_{0,0},\mathcal{K}_{0,0})$ modulo
perturbative terms.

###### Proof.

We follow the same strategy in Section 4.4, and put all the balanced cubic and
higher order terms into $(G,K)$. We will consider first the contributions of
corrections for the first equation of (LABEL:NormalFormGKZero), then for the
second equation of (LABEL:NormalFormGKZero). For each case, we again compute
according to the power of the vorticity $\gamma$. When computing the para-
material derivatives of paraproducts, we apply the para-Leibniz rule Lemma 2.5
to distribute para-material derivatives to each variable. For the expressions
of material derivatives and para-material derivatives of each variable, we use
the leading terms plus errors measured in either $BMO$ or $L^{4}$ based spaces
in corresponding Lemmas in Section 3.2 and Section 4.3.

$(1)$ We first consider the first equation of (LABEL:NormalFormGKZero). We
begin by computing the contributions of corrections of non-vorticity and
$\gamma$ terms.

$\displaystyle-T_{D_{t}}{\mathbf{P}}(\bar{x}{\mathbf{W}})$
$\displaystyle=-{\mathbf{P}}(D_{t}\bar{x}{\mathbf{W}})-{\mathbf{P}}(\bar{x}T_{D_{t}}{\mathbf{W}})+G$
$\displaystyle={\mathbf{P}}(T_{1-Y}(T_{1-\bar{Y}}\bar{r}_{\alpha}+T_{\bar{x}}\bar{R}_{\alpha}){\mathbf{W}})+{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha})+G$
$\displaystyle={\mathbf{P}}(T_{1-\bar{Y}}\bar{r}_{\alpha}T_{1-Y}{\mathbf{W}})+{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha})+G$
$\displaystyle={\mathbf{P}}(T_{1-\bar{Y}}\bar{r}_{\alpha}T_{1+{\mathbf{W}}}Y)+T_{1-\bar{Y}}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R_{\alpha})+G.$

Here we use (2.16) to move para-coefficients $T_{1-Y}$, $T_{1-\bar{Y}}$. We
also use (3.11) to change $T_{1-Y}{\mathbf{W}}$ to $T_{1+{\mathbf{W}}}Y$.

$\displaystyle-
T_{1-\bar{Y}}\partial_{\alpha}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)=$
$\displaystyle-
T_{1-\bar{Y}}{\mathbf{P}}(\bar{x}_{\alpha}T_{1+{\mathbf{W}}}R)-T_{1-\bar{Y}}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R_{\alpha})+G,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}{\mathbf{P}}(\bar{r}{\mathbf{W}})=$
$\displaystyle-\frac{\gamma}{2g}{\mathbf{P}}(D_{t}\bar{r}{\mathbf{W}})-\frac{\gamma}{2g}{\mathbf{P}}(\bar{r}T_{D_{t}}{\mathbf{W}})+G$
$\displaystyle=$
$\displaystyle-\frac{\gamma}{2g}{\mathbf{P}}((i\gamma\bar{r}-iT_{g+{\underline{a}}}(\bar{x}-T_{\bar{w}}\bar{Y})){\mathbf{W}})$
$\displaystyle+\frac{\gamma}{2g}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha})+G$
$\displaystyle=$
$\displaystyle-i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{r}{\mathbf{W}})+i\frac{\gamma}{2}{\mathbf{P}}(\bar{x}{\mathbf{W}})+\frac{\gamma}{2g}T_{1-\bar{Y}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R_{\alpha})+G,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)=$
$\displaystyle-\frac{\gamma}{2g}{\mathbf{P}}(D_{t}\bar{x}T_{1+{\mathbf{W}}}R)-\frac{\gamma}{2g}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}T_{D_{t}}R)-\frac{\gamma}{2g}{\mathbf{P}}(\bar{x}T_{D_{t}{\mathbf{W}}}R)+G$
$\displaystyle=$
$\displaystyle\frac{\gamma}{2g}{\mathbf{P}}(T_{1-Y}(T_{1-\bar{Y}}\bar{r}_{\alpha}+T_{\bar{x}}\bar{R}_{\alpha})T_{1+{\mathbf{W}}}R)$
$\displaystyle-i\frac{\gamma}{2g}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}(T_{g+{\underline{a}}}Y-\gamma
R))+\frac{\gamma}{2g}{\mathbf{P}}(\bar{x}T_{T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha}}R)+G$
$\displaystyle=$
$\displaystyle\frac{\gamma}{2g}T_{1-\bar{Y}}{\mathbf{P}}(\bar{r}_{\alpha}R)-i\frac{\gamma}{2}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}Y)+i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)+G,$
$\displaystyle-\frac{\gamma}{2g}T_{1-\bar{Y}}\partial_{\alpha}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R)=$
$\displaystyle-\frac{\gamma}{2g}T_{1-\bar{Y}}{\mathbf{P}}(\bar{r}_{\alpha}R)-\frac{\gamma}{2g}T_{1-\bar{Y}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R_{\alpha})+G,$
$\displaystyle
i\frac{\gamma}{2}T_{1-\bar{Y}}\partial_{\alpha}{\mathbf{P}}(\bar{x}W)=$
$\displaystyle
i\frac{\gamma}{2}T_{1-\bar{Y}}{\mathbf{P}}(\bar{x}_{\alpha}W)+i\frac{\gamma}{2}T_{1-\bar{Y}}{\mathbf{P}}(\bar{x}{\mathbf{W}})+G.$

Continuing with the contributions of corrections of $\gamma^{2}$ terms, we
compute

$\displaystyle-i\frac{\gamma^{2}}{2g}T_{D_{t}}{\mathbf{P}}(\bar{u}{\mathbf{W}})=$
$\displaystyle-i\frac{\gamma^{2}}{2g}{\mathbf{P}}(T_{D_{t}}\bar{u}{\mathbf{W}})-i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{u}T_{D_{t}}{\mathbf{W}})+G$
$\displaystyle=$ $\displaystyle
i\frac{\gamma^{2}}{2g}{\mathbf{P}}(T_{1-Y}(T_{1-\bar{Y}}\bar{r}+T_{\bar{x}}\bar{R}){\mathbf{W}})+i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha})+G$
$\displaystyle=$ $\displaystyle
i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{r}{\mathbf{W}})+i\frac{\gamma^{2}}{2g}T_{1-\bar{Y}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R_{\alpha})+G,$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{D_{t}}{\mathbf{P}}(\bar{x}W)=$
$\displaystyle
i\frac{\gamma^{2}}{4g}{\mathbf{P}}(D_{t}\bar{x}W)+i\frac{\gamma^{2}}{4g}{\mathbf{P}}(\bar{x}T_{D_{t}}W)+G$
$\displaystyle=$
$\displaystyle-i\frac{\gamma^{2}}{4g}{\mathbf{P}}(T_{1-Y}(T_{1-\bar{Y}}\bar{r}_{\alpha}+T_{\bar{x}}\bar{R}_{\alpha})W)-i\frac{\gamma^{2}}{4g}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R)+G$
$\displaystyle=$
$\displaystyle-i\frac{\gamma^{2}}{4g}{\mathbf{P}}(T_{1-\bar{Y}}\bar{r}_{\alpha}T_{1-Y}W)-i\frac{\gamma^{2}}{4g}T_{1-\bar{Y}}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)+G,$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}T_{D_{t}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R)=$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}{\mathbf{P}}(D_{t}\bar{r}T_{1+{\mathbf{W}}}R)-\frac{\gamma^{2}}{4g^{2}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}T_{D_{t}}R)$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}{\mathbf{P}}(\bar{r}T_{D_{t}{\mathbf{W}}}R)+G$
$\displaystyle=$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}{\mathbf{P}}((i\gamma\bar{r}-iT_{g+{\underline{a}}}(\bar{x}-T_{\bar{w}}\bar{Y}))T_{1+{\mathbf{W}}}R)$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}(iT_{g+{\underline{a}}}Y-i\gamma
R))+\frac{\gamma^{2}}{4g^{2}}{\mathbf{P}}(\bar{r}T_{T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha}}R)+G$
$\displaystyle=$ $\displaystyle
i\frac{\gamma^{2}}{4g}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)-\frac{\gamma^{2}}{4g}i{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}Y)+G,$
$\displaystyle-i\frac{\gamma^{2}}{2g}T_{1-\bar{Y}}\partial_{\alpha}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)=$
$\displaystyle-i\frac{\gamma^{2}}{2g}T_{1-\bar{Y}}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)-i\frac{\gamma^{2}}{2g}T_{1-\bar{Y}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R_{\alpha})+G,$
$\displaystyle
i\frac{\gamma^{2}}{4g}T_{1-\bar{Y}}\partial_{\alpha}{\mathbf{P}}(\bar{r}W)=$
$\displaystyle
i\frac{\gamma^{2}}{4g}T_{1-\bar{Y}}{\mathbf{P}}(\bar{r}_{\alpha}W)+i\frac{\gamma^{2}}{4g}T_{1-\bar{Y}}{\mathbf{P}}(\bar{r}{\mathbf{W}})+G.$

For the contributions of corrections of $\gamma^{3}$ and $\gamma^{4}$ terms,
we compute

$\displaystyle i\frac{\gamma^{3}}{4g^{2}}T_{D_{t}}{\mathbf{P}}(\bar{r}W)=$
$\displaystyle
i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(D_{t}\bar{r}W)+i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(\bar{r}T_{D_{t}}W)+G$
$\displaystyle=$ $\displaystyle
i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}((i\gamma\bar{r}-iT_{g+{\underline{a}}}(\bar{x}-T_{\bar{w}}\bar{Y}))W)-i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R)+G$
$\displaystyle=$
$\displaystyle-\frac{\gamma^{4}}{4g^{2}}{\mathbf{P}}(\bar{r}W)+\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{x}W)-i\frac{\gamma^{3}}{4g^{2}}T_{1-\bar{Y}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R)+G,$
$\displaystyle-i\frac{\gamma^{3}}{4g^{2}}T_{D_{t}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)=$
$\displaystyle-i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(T_{D_{t}}\bar{u}T_{1+{\mathbf{W}}}R)-i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}T_{D_{t}}R)$
$\displaystyle-i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(\bar{u}T_{D_{t}{\mathbf{W}}}R)+G$
$\displaystyle=$ $\displaystyle
i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(T_{1-Y}(T_{1-\bar{Y}}\bar{r}+T_{\bar{x}}\bar{R})T_{1+{\mathbf{W}}}R)$
$\displaystyle-i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}(iT_{g+{\underline{a}}}Y-i\gamma
R))+i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(\bar{u}T_{T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha}}R)+G$
$\displaystyle=$ $\displaystyle
i\frac{\gamma^{3}}{4g^{2}}{\mathbf{P}}(T_{1-\bar{Y}}\bar{r}R)+\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}Y)-\frac{\gamma^{4}}{4g^{2}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)+G,$
$\displaystyle-\frac{\gamma^{3}}{4g}T_{1-\bar{Y}}\partial_{\alpha}{\mathbf{P}}(\bar{u}W)=$
$\displaystyle-\frac{\gamma^{3}}{4g}{\mathbf{P}}(T_{1-\bar{Y}}\bar{x}W)-\frac{\gamma^{3}}{4g}{\mathbf{P}}(T_{1-\bar{Y}}\bar{u}{\mathbf{W}})+G,$
$\displaystyle-\frac{\gamma^{4}}{4g^{2}}T_{D_{t}}{\mathbf{P}}(\bar{u}W)=$
$\displaystyle-\frac{\gamma^{4}}{4g^{2}}{\mathbf{P}}(T_{D_{t}}\bar{u}W)-\frac{\gamma^{4}}{4g^{2}}{\mathbf{P}}(\bar{u}T_{D_{t}}W)+G$
$\displaystyle=$
$\displaystyle\frac{\gamma^{4}}{4g^{2}}{\mathbf{P}}(T_{1-Y}(T_{1-\bar{Y}}\bar{r}+T_{\bar{x}}\bar{R})W)+\frac{\gamma^{4}}{4g^{2}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R)+G$
$\displaystyle=$
$\displaystyle\frac{\gamma^{4}}{4g^{2}}{\mathbf{P}}(\bar{r}W)+\frac{\gamma^{4}}{4g^{2}}T_{1-\bar{Y}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)+G.$

The contributions of other corrections are perturbative and can be absorbed
into $G$. Collecting all the contributions of the corrections together, we
find that the total contributions for the first equation is exactly
$-\mathcal{G}_{0,0}+G$.

$(2)$ Then we consider the second equation of (LABEL:NormalFormGKZero). We
begin by computing the contributions of corrections of non-vorticity and
$\gamma$ terms.

$\displaystyle-T_{D_{t}}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)=$
$\displaystyle-{\mathbf{P}}(D_{t}\bar{x}T_{1+{\mathbf{W}}}R)-{\mathbf{P}}(\bar{x}T_{D_{t}{\mathbf{W}}}R)-{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}T_{D_{t}}R)+K$
$\displaystyle=$
$\displaystyle{\mathbf{P}}(T_{1-Y}(T_{1-\bar{Y}}\bar{r}_{\alpha}+T_{\bar{x}}\bar{R}_{\alpha})T_{1+{\mathbf{W}}}R)+{\mathbf{P}}(\bar{x}T_{T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha}}R)$
$\displaystyle-i{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}(T_{g+{\underline{a}}}Y-\gamma
R))+K$ $\displaystyle=$
$\displaystyle{\mathbf{P}}(T_{1-\bar{Y}}\bar{r}_{\alpha}R)-iT_{g+{\underline{a}}}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}Y)+i\gamma{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)+K,$
$\displaystyle
iT_{1-Y}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{x}{\mathbf{W}})=$
$\displaystyle
iT_{g+{\underline{a}}}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}Y)+K,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R)=$
$\displaystyle-\frac{\gamma}{2g}{\mathbf{P}}(D_{t}\bar{r}T_{1+{\mathbf{W}}}R)-\frac{\gamma}{2g}{\mathbf{P}}(\bar{r}T_{D_{t}{\mathbf{W}}}R)-\frac{\gamma}{2g}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}T_{D_{t}}R)+K$
$\displaystyle=$
$\displaystyle-i\frac{\gamma}{2g}{\mathbf{P}}((\gamma\bar{r}-T_{g+{\underline{a}}}(\bar{x}-T_{\bar{w}}\bar{Y}))T_{1+{\mathbf{W}}}R)+\frac{\gamma}{2g}{\mathbf{P}}(\bar{r}T_{T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha}}R)$
$\displaystyle-i\frac{\gamma}{2g}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}(T_{g+{\underline{a}}}Y-\gamma
R))+K$ $\displaystyle=$ $\displaystyle
i\frac{\gamma}{2}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)-i\frac{\gamma}{2g}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}Y)+K,$
$\displaystyle i\frac{\gamma}{2}T_{D_{t}}{\mathbf{P}}(\bar{x}W)=$
$\displaystyle
i\frac{\gamma}{2}{\mathbf{P}}(D_{t}\bar{x}W)+i\frac{\gamma}{2}{\mathbf{P}}(\bar{x}T_{D_{t}}W)+K$
$\displaystyle=$
$\displaystyle-i\frac{\gamma}{2}{\mathbf{P}}(T_{1-Y}(T_{1-\bar{Y}}\bar{r}_{\alpha}+T_{\bar{x}}\bar{R}_{\alpha})W)-i\frac{\gamma}{2}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R)+K$
$\displaystyle=$
$\displaystyle-i\frac{\gamma}{2}{\mathbf{P}}(T_{1-\bar{Y}}\bar{r}_{\alpha}W)-i\frac{\gamma}{2}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)+K,$
$\displaystyle
i\frac{\gamma}{2g}T_{1-Y}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{r}{\mathbf{W}})=$
$\displaystyle
i\frac{\gamma}{2g}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{r}T_{1-Y}{\mathbf{W}})+K=i\frac{\gamma}{2g}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}Y)+K,$
$\displaystyle
i\frac{\gamma}{2g}T_{1-Y}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)=$
$\displaystyle
i\frac{\gamma}{2}{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R)+K,\quad\text{ and
}-i\gamma{\mathbf{P}}(\bar{x}T_{1+{\mathbf{W}}}R).$

Continuing with the contributions of corrections of $\gamma^{2}$ terms, we
compute

$\displaystyle-i\frac{\gamma^{2}}{2g}T_{D_{t}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)=$
$\displaystyle-i\frac{\gamma^{2}}{2g}{\mathbf{P}}(T_{D_{t}}\bar{u}T_{1+{\mathbf{W}}}R)-i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{u}T_{D_{t}{\mathbf{W}}}R)$
$\displaystyle-i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}T_{D_{t}}R)+K$
$\displaystyle=$ $\displaystyle
i\frac{\gamma^{2}}{2g}{\mathbf{P}}(T_{1-Y}(T_{1-\bar{Y}}\bar{r}+T_{\bar{x}}\bar{R})T_{1+{\mathbf{W}}}R)+i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{u}T_{T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha}}R)$
$\displaystyle+\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}(T_{g+{\underline{a}}}Y-\gamma
R))+K$ $\displaystyle=$ $\displaystyle
i\frac{\gamma^{2}}{2g}{\mathbf{P}}(T_{1-\bar{Y}}\bar{r}R)+\frac{\gamma^{2}}{2g}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}Y)-\frac{\gamma^{3}}{2g}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)+K,$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{D_{t}}{\mathbf{P}}(\bar{r}W)=$
$\displaystyle
i\frac{\gamma^{2}}{4g}{\mathbf{P}}(D_{t}\bar{r}W)+i\frac{\gamma^{2}}{4g}{\mathbf{P}}(\bar{r}T_{D_{t}}W)+K$
$\displaystyle=$
$\displaystyle-\frac{\gamma^{2}}{4g}{\mathbf{P}}((\gamma\bar{r}-T_{g+{\underline{a}}}(\bar{x}-T_{\bar{w}}\bar{Y}))W)-i\frac{\gamma^{2}}{4g}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R)+K$
$\displaystyle=$
$\displaystyle-\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{r}W)+\frac{\gamma^{2}}{4}{\mathbf{P}}(\bar{x}W)-i\frac{\gamma^{2}}{4g}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R)+K,$
$\displaystyle-\frac{\gamma^{2}}{2g}T_{1-Y}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{u}{\mathbf{W}})=$
$\displaystyle-\frac{\gamma^{2}}{2g}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}Y)+K,$
$\displaystyle\frac{\gamma^{2}}{4g}T_{1-Y}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{x}W)=$
$\displaystyle\frac{\gamma^{2}}{4}{\mathbf{P}}(\bar{x}W)+K,$ $\displaystyle
i\frac{\gamma^{2}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R)=$
$\displaystyle
i\frac{\gamma^{2}}{4g}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R)+K,$
$\displaystyle-i\frac{\gamma^{2}}{2g}{\mathbf{P}}(\bar{r}T_{1+{\mathbf{W}}}R)-$
$\displaystyle\frac{\gamma^{2}}{2}{\mathbf{P}}(\bar{x}W).$

Finally, for the contributions of corrections of $\gamma^{3}$ and $\gamma^{4}$
terms, we get

$\displaystyle-\frac{\gamma^{3}}{4g}T_{D_{t}}{\mathbf{P}}(\bar{u}W)=$
$\displaystyle-\frac{\gamma^{3}}{4g}{\mathbf{P}}(T_{D_{t}}\bar{u}W)-\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{u}T_{D_{t}}W)+K$
$\displaystyle=$
$\displaystyle\frac{\gamma^{3}}{4g}{\mathbf{P}}(T_{1-Y}(T_{1-\bar{Y}}\bar{r}+T_{\bar{x}}\bar{R})W)+\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R)+K$
$\displaystyle=$
$\displaystyle\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{r}W)+\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)+K,$
$\displaystyle\frac{\gamma^{3}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{r}W)=$
$\displaystyle\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{r}W)+K,$
$\displaystyle-\frac{\gamma^{3}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)=$
$\displaystyle-\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)+K,$
$\displaystyle
i\frac{\gamma^{4}}{4g^{2}}T_{1-Y}T_{g+{\underline{a}}}{\mathbf{P}}(\bar{u}W)=$
$\displaystyle i\frac{\gamma^{4}}{4g}{\mathbf{P}}(\bar{u}W)+K,$
$\displaystyle\frac{\gamma^{3}}{2g}{\mathbf{P}}(\bar{u}T_{1+{\mathbf{W}}}R)-$
$\displaystyle\frac{\gamma^{3}}{4g}{\mathbf{P}}(\bar{r}W)-i\frac{\gamma^{4}}{4g}{\mathbf{P}}(\bar{u}W).$

Putting together all the contributions of the corrections, we get that the
total contributions for the second equation is exactly $-\mathcal{K}_{0,0}+K$.
∎

##  5\. Energy estimates for the full system

In this section, we prove energy estimates for system (1.7), as in the
statement of Theorem 1.3.

As $({\mathbf{W}},R)$ solve the linearized system (LABEL:linearizedeqn), the
results in the previous section show that the full system is well-posed for
initial data in $\dot{\mathcal{H}}^{\frac{1}{4}}$. However, the estimates for
the linearized system no longer hold for other Sobolev index
$s\neq\frac{1}{4}$. We consider in this section for general $s\geq 0$.

###  5.1. Reduction to paradifferential linearized $({\mathbf{W}},R)$ system

The system (LABEL:linearizedeqn) follows from linearizing (1.6) and
diagonalizing to switch to good variables. In this section, we instead
consider direct linearization from (1.7). Let the linearized variables around
a solution $({\mathbf{W}},R)$ of the water waves system (1.7) be
$(\hat{w},\hat{r})$. We begin by computing the linearizations of the auxiliary
functions:

$\displaystyle\delta
a=i[\bar{{\mathbf{P}}}(\bar{{\hat{r}}}R_{\alpha})+\bar{{\mathbf{P}}}(\bar{R}{\hat{r}}_{\alpha})-{\mathbf{P}}({\hat{r}}\bar{R}_{\alpha})-{\mathbf{P}}(R\bar{{\hat{r}}}_{\alpha})],$
$\displaystyle\delta{\underline{b}}=\delta b-i\frac{\gamma}{2}\delta
b_{1},\quad\delta
b=2\Re{\mathbf{P}}[(1-\bar{Y}){\hat{r}}-(1-\bar{Y})^{2}R\bar{{\hat{w}}}],$
$\displaystyle\delta
b_{1}=2i\Im{\mathbf{P}}[(1-\bar{Y})\partial^{-1}_{\alpha}{\hat{w}}-(1-\bar{Y})^{2}W\bar{{\hat{w}}}],$
$\displaystyle\delta
N=2\Re{\mathbf{P}}[\partial^{-1}_{\alpha}{\hat{w}}\bar{R}_{\alpha}+W\bar{{\hat{r}}}_{\alpha}-\bar{{\hat{w}}}R-\bar{W}_{\alpha}{\hat{r}}],$
$\displaystyle\delta
M=2\Re{\mathbf{P}}[{\hat{r}}\bar{Y}_{\alpha}-2R(1-\bar{Y})\bar{{\hat{w}}}\bar{Y}_{\alpha}+R(1-\bar{Y})^{2}\bar{{\hat{w}}}_{\alpha}-\bar{{\hat{r}}}_{\alpha}Y-\bar{R}_{\alpha}(1-Y)^{2}{\hat{w}}],$
$\displaystyle\delta
M_{1}=2i\Im{\mathbf{P}}\partial_{\alpha}[\partial^{-1}_{\alpha}{\hat{w}}\bar{Y}+W(1-\bar{Y})^{2}\bar{{\hat{w}}}],\quad\delta\underline{M}=\delta
M-i\frac{\gamma}{2}\delta M_{1}.$

Then the linearized system can be written as

$\left\\{\begin{aligned}
D_{t}{\hat{w}}&+(1-\bar{Y})(1+W_{\alpha}){\hat{r}}_{\alpha}+(1-\bar{Y})R_{\alpha}{\hat{w}}=(1+W_{\alpha})\delta\underline{M}+\underline{M}{\hat{w}}-W_{\alpha\alpha}\delta{\underline{b}}\\\
&+(1-\bar{Y})^{2}(1+W_{\alpha})R_{\alpha}\bar{{\hat{w}}}+i\gamma
W_{\alpha}{\hat{w}}-i\frac{\gamma}{2}\bar{W}_{\alpha}{\hat{w}}-i\frac{\gamma}{2}W_{\alpha}\bar{{\hat{w}}},\\\
D_{t}{\hat{r}}&+i\gamma{\hat{r}}-i(g+a)(1-Y)^{2}{\hat{w}}=-R_{\alpha}\delta{\underline{b}}-i(1-Y)\delta
a+i\frac{\gamma}{2}(1-Y)^{2}{\hat{w}}(R+\bar{R})\\\
&+i\frac{\gamma}{2}W_{\alpha}(1-Y)({\hat{r}}+\bar{{\hat{r}}})+i\frac{\gamma}{2}(1-Y)\delta
N-i\frac{\gamma}{2}N(1-Y)^{2}{\hat{w}}.\end{aligned}\right.$

Next, we rewrite the linearized system in the paradifferential framework. We
consider the cases where ${\hat{w}}$ and ${\hat{r}}$ terms are the highest
frequencies for the paraproduct types contributions. After applying the
projection ${\mathbf{P}}$, we can eliminate all
$(\bar{{\hat{w}}},\bar{{\hat{r}}})$ terms, as well as all
$({\hat{w}},{\hat{r}})$ terms inside the anti-holomorphic projection
$\bar{{\mathbf{P}}}$. Using (1.8) to simplify, we obtain the following linear
paradifferential equations:

$\left\\{\begin{array}[]{lr}T_{D_{t}}{\hat{w}}+T_{{\underline{b}}_{\alpha}}{\hat{w}}+\partial_{\alpha}T_{1-\bar{Y}}T_{1+W_{\alpha}}{\hat{r}}-i\frac{\gamma}{2}\partial_{\alpha}T_{1+W_{\alpha}}T_{1-\bar{Y}}\partial^{-1}_{\alpha}{\hat{w}}+i\frac{\gamma}{2}{\hat{w}}=0&\\\
T_{D_{t}}{\hat{r}}+T_{{\underline{b}}_{\alpha}}{\hat{r}}+i\gamma{\hat{r}}-iT_{(1-Y)^{2}}T_{g+{\underline{a}}}{\hat{w}}+T_{\underline{M}}{\hat{r}}+i\frac{\gamma}{2}T_{b_{\alpha}+M}\partial^{-1}_{\alpha}{\hat{w}}+\gamma
T_{\Im W_{\alpha}Y}{\hat{r}}=0.&\end{array}\right.$

We claim that the differentiated system (1.7) can be recast in the above
paradifferential type equations with source terms. We let $(G,K)$ be the
favorable balanced cubic or higher source terms that satisfy

(5.1)
$\|(G,K)\|_{\dot{\mathcal{H}}^{s}}\lesssim_{\underline{A}}\underline{A}^{2}_{\frac{1}{4}}\left(\|({\mathbf{W}},R)\|_{\dot{\mathcal{H}}^{s}}+\gamma\|({\mathbf{W}},R)\|_{\dot{\mathcal{H}}^{s-\frac{1}{2}}}\right).$

The rest of the source terms are either quadratic in $({\mathbf{W}},R)$ or
unbalanced cubic terms. These unfavorable source terms will be later removed
by paradifferential normal form corrections. This reduction result is stated
as following:

######  Lemma 5.1.

The differentiated system (1.7) can be rewritten as paradifferential equations
for the variable $({\hat{w}},{\hat{r}})=({\mathbf{W}},R)$, of the form

(5.2)
$\left\\{\begin{array}[]{lr}T_{D_{t}}{\hat{w}}+T_{{\underline{b}}_{\alpha}}{\hat{w}}+\partial_{\alpha}T_{1-\bar{Y}}T_{1+W_{\alpha}}{\hat{r}}-i\frac{\gamma}{2}\partial_{\alpha}T_{1+W_{\alpha}}T_{1-\bar{Y}}\partial^{-1}_{\alpha}{\hat{w}}+i\frac{\gamma}{2}{\hat{w}}=\mathcal{G}({\mathbf{W}},R)+G,&\\\
T_{D_{t}}{\hat{r}}+i\gamma{\hat{r}}+T_{{\underline{b}}_{\alpha}}{\hat{r}}-iT_{(1-Y)^{2}}T_{g+{\underline{a}}}{\hat{w}}-i\frac{\gamma}{2}T_{b_{\alpha}}\partial^{-1}_{\alpha}{\hat{w}}=\mathcal{K}({\mathbf{W}},R)+K,&\end{array}\right.$

where $(G,K)$ are balanced cubic source terms that satisfy (5.1), and

$\left\\{\begin{aligned}
\mathcal{G}({\mathbf{W}},R)=&-\partial_{\alpha}[\Pi({\mathbf{W}},T_{1-\bar{Y}}R)+\Pi({\mathbf{W}},T_{1-Y}\bar{R})-\Pi(\bar{Y},T_{1+{\mathbf{W}}}R)]\\\
&+i\frac{\gamma}{2}\partial_{\alpha}[\Pi({\mathbf{W}},T_{1-\bar{Y}}W)-\Pi({\mathbf{W}},T_{1-Y}\bar{W})-\Pi(X,\bar{Y})],\\\
\mathcal{K}({\mathbf{W}},R)=&-T_{1-\bar{Y}}\Pi(R_{\alpha},R)-T_{1-Y}\Pi(R_{\alpha},\bar{R})-T_{1-Y}\Pi(\bar{R}_{\alpha},R)\\\
&+i\frac{\gamma}{2}\Pi(R_{\alpha},T_{1-\bar{Y}}W-T_{1-Y}\bar{W})-iT_{1-Y}T_{g+{\underline{a}}}\Pi(Y,{\mathbf{W}})\\\
&+i\frac{\gamma}{2}T_{1-Y}\Pi(W,\bar{R}_{\alpha})-i\frac{\gamma}{2}T_{1-Y}\Pi(\bar{{\mathbf{W}}},R)+i\frac{\gamma}{2}\Pi(R+\bar{R},Y).\end{aligned}\right.$

Again we include the projection ${\mathbf{P}}$ in $\Pi$ implicitly as in the
previous section.

###### Proof.

Recall that the differentiated system (1.7) can be rewritten as

$\left\\{\begin{array}[]{lr}D_{t}{\mathbf{W}}+(1+\mathbf{W})(1-\bar{Y})R_{\alpha}=(1+\mathbf{W})\underline{M}+i\dfrac{\gamma}{2}\mathbf{W}(\mathbf{W}-\bar{\mathbf{W}})&\\\
D_{t}R+i\gamma
R=i[g-(g+{\underline{a}})(1-Y)]+i\frac{\gamma}{2}(R+\bar{R}).&\end{array}\right.$

Here the system is written in an algebraic way for convenience. We add the
projection ${\mathbf{P}}$ to above system to eliminate the anti-holomorphic
parts. For our computation below, the balanced cubic terms are put in $(G,K)$
without further specification.

  1. (1)

For the first term in the first equation, we expand ${\underline{b}}$ to write

$\displaystyle{\mathbf{P}}D_{t}{\mathbf{W}}$
$\displaystyle=T_{D_{t}}{\mathbf{W}}+T_{{\mathbf{W}}_{\alpha}}{\mathbf{P}}{\underline{b}}+\Pi({\mathbf{W}}_{\alpha},{\underline{b}})$
$\displaystyle=T_{D_{t}}{\mathbf{W}}+T_{{\mathbf{W}}_{\alpha}}T_{1-\bar{Y}}R-i\frac{\gamma}{2}T_{{\mathbf{W}}_{\alpha}}T_{1-\bar{Y}}W$
$\displaystyle+\Pi({\mathbf{W}}_{\alpha},T_{1-\bar{Y}}R+T_{1-Y}\bar{R})-i\frac{\gamma}{2}\Pi({\mathbf{W}}_{\alpha},T_{1-\bar{Y}}W-T_{1-Y}\bar{W})+G.$

For the second term in the first equation, we expand using paraproducts

$\displaystyle{\mathbf{P}}[(1+{\mathbf{W}})(1-\bar{Y})R_{\alpha}]$
$\displaystyle=T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha}+T_{1-\bar{Y}}T_{R_{\alpha}}{\mathbf{W}}$
$\displaystyle-
T_{1+{\mathbf{W}}}\Pi(\bar{Y},R_{\alpha})+T_{1-\bar{Y}}\Pi({\mathbf{W}},R_{\alpha})+G.$

For the first source term in the first equation, we use (1.8) and (3.7) to
write

$\displaystyle(1+{\mathbf{W}})\underline{M}$
$\displaystyle=T_{1+{\mathbf{W}}}\underline{M}+G=T_{1+{\mathbf{W}}}{\mathbf{P}}\left[R\bar{Y}_{\alpha}-\bar{R}_{\alpha}Y-i\frac{\gamma}{2}(W\bar{Y})_{\alpha}\right]+G$
$\displaystyle=T_{1+{\mathbf{W}}}\left(T_{\bar{Y}_{\alpha}}R-T_{\bar{R}_{\alpha}}Y-i\frac{\gamma}{2}T_{\bar{Y}_{\alpha}}W-i\frac{\gamma}{2}T_{\bar{Y}}{\mathbf{W}}\right)$
$\displaystyle+T_{1+{\mathbf{W}}}\left[\Pi(\bar{Y}_{\alpha},R)-\Pi(\bar{R}_{\alpha},Y)-i\frac{\gamma}{2}\partial_{\alpha}\Pi(W,\bar{Y})\right]+G.$

For the last term in the first equation,

$i\dfrac{\gamma}{2}\mathbf{W}(\mathbf{W}-\bar{\mathbf{W}})=iT_{\frac{\gamma}{2}({\mathbf{W}}-\bar{{\mathbf{W}}})}{\mathbf{W}}+i\frac{\gamma}{2}T_{\mathbf{W}}{\mathbf{W}}+i\frac{\gamma}{2}\Pi({\mathbf{W}},{\mathbf{W}}-\bar{{\mathbf{W}}}).$

Combining all these terms, we use (3.11) to get

$\displaystyle
T_{1-\bar{Y}}T_{R_{\alpha}}{\mathbf{W}}+T_{1+{\mathbf{W}}}T_{\bar{R}_{\alpha}}Y-iT_{\frac{\gamma}{2}({\mathbf{W}}-\bar{{\mathbf{W}}})}{\mathbf{W}}=T_{{\underline{b}}_{\alpha}}+G,$
$\displaystyle
T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha}-T_{1+{\mathbf{W}}}T_{\bar{Y}_{\alpha}}R+T_{{\mathbf{W}}_{\alpha}}T_{1-\bar{Y}}R=\partial_{\alpha}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R,$
$\displaystyle
i\frac{\gamma}{2}(-T_{{\mathbf{W}}_{\alpha}}T_{1-\bar{Y}}W+T_{1+{\mathbf{W}}_{\alpha}}T_{\bar{Y}_{\alpha}}W+T_{\bar{Y}}{\mathbf{W}}-T_{{\mathbf{W}}}{\mathbf{W}})=-i\frac{\gamma}{2}\partial_{\alpha}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}W+i\frac{\gamma}{2}W.$

Hence, we obtain the paradifferential ${\mathbf{W}}$ equation.

  2. (2)

For the second $R$ equation, we expand in the similar way to write

$\displaystyle{\mathbf{P}}D_{t}R$
$\displaystyle=T_{D_{t}}R+T_{R_{\alpha}}{\mathbf{P}}{\underline{b}}+\Pi(R_{\alpha},{\underline{b}})$
$\displaystyle=T_{D_{t}}R+T_{R_{\alpha}}T_{1-\bar{Y}}R-i\frac{\gamma}{2}T_{R_{\alpha}}T_{1-\bar{Y}}W$
$\displaystyle+\Pi(R_{\alpha},T_{1-\bar{Y}}R+T_{1-Y}\bar{R})-i\frac{\gamma}{2}\Pi(R_{\alpha},T_{1-\bar{Y}}W-T_{1-Y}\bar{W})+K.$

For the first term on the right hand side of the second equation,

$\displaystyle i{\mathbf{P}}[g-(g+{\underline{a}})(1-Y)]$
$\displaystyle=iT_{g+{\underline{a}}}Y-iT_{1-Y}{\mathbf{P}}{\underline{a}}+i\Pi({\underline{a}},Y)$
$\displaystyle=iT_{g+{\underline{a}}}Y-T_{1-Y}T_{\bar{R}_{\alpha}}R-T_{1-Y}\Pi(\bar{R}_{\alpha},R)$
$\displaystyle-i\frac{\gamma}{2}T_{1-Y}R+i\frac{\gamma}{2}T_{1-Y}T_{\bar{R}_{\alpha}}W-i\frac{\gamma}{2}T_{1-Y}T_{\bar{{\mathbf{W}}}}R$
$\displaystyle+i\frac{\gamma}{2}T_{1-Y}\Pi(W,\bar{R}_{\alpha})-i\frac{\gamma}{2}T_{1-Y}\Pi(\bar{{\mathbf{W}}},R)+i\frac{\gamma}{2}\Pi(R+\bar{R},Y)+K,$

where again using $(4)$ of Lemma 3.9, we get

$iT_{g+{\underline{a}}}Y=iT_{(1-Y)^{2}}T_{g+{\underline{a}}}{\mathbf{W}}-iT_{1-Y}T_{g+{\underline{a}}}\Pi(Y,{\mathbf{W}})+K.$

Similar to the case of the first equation, we combine using (1.8) and (3.7),

$\displaystyle-
T_{R_{\alpha}}T_{1-\bar{Y}}R-T_{1-Y}T_{\bar{R}_{\alpha}}R-i\frac{\gamma}{2}T_{1-Y}R-i\frac{\gamma}{2}T_{1-Y}T_{\bar{{\mathbf{W}}}}R+i\frac{\gamma}{2}R$
$\displaystyle=$ $\displaystyle-
T_{R_{\alpha}(1-\bar{Y})+\bar{R}_{\alpha}(1-Y)}R-i\frac{\gamma}{2}T_{1-Y}R+i\frac{\gamma}{2}T_{1-Y}T_{1+{\mathbf{W}}}R-i\frac{\gamma}{2}T_{1-Y}T_{\bar{{\mathbf{W}}}}R+K$
$\displaystyle=$ $\displaystyle-
T_{R_{\alpha}(1-\bar{Y})+\bar{R}_{\alpha}(1-Y)}R+i\frac{\gamma}{2}T_{{\mathbf{W}}-\bar{{\mathbf{W}}}}R+K=-T_{{\underline{b}}_{\alpha}}R+K,$
$\displaystyle-i\frac{\gamma}{2}T_{R_{\alpha}}T_{1-\bar{Y}}W-i\frac{\gamma}{2}T_{1-Y}T_{\bar{R}_{\alpha}}W=-i\frac{\gamma}{2}T_{{\underline{b}}_{\alpha}}W+K.$

Hence, we obtain the paradifferential $R$ equation.

∎

The rest of Section 5 is devoted into the study of paradifferential system
(5.2).

###  5.2. Well-posedness for the paradifferential flow

In this subsection, we first consider the linear part of the paradifferential
equations (5.2), namely

(5.3)
$\left\\{\begin{array}[]{lr}T_{D_{t}}{\hat{w}}+T_{{\underline{b}}_{\alpha}}{\hat{w}}+\partial_{\alpha}T_{1-\bar{Y}}T_{1+W_{\alpha}}{\hat{r}}-i\frac{\gamma}{2}\partial_{\alpha}T_{1+W_{\alpha}}T_{1-\bar{Y}}\partial^{-1}_{\alpha}{\hat{w}}+i\frac{\gamma}{2}{\hat{w}}=0&\\\
T_{D_{t}}{\hat{r}}+i\gamma{\hat{r}}+T_{{\underline{b}}_{\alpha}}{\hat{r}}-iT_{(1-Y)^{2}}T_{g+{\underline{a}}}{\hat{w}}-i\frac{\gamma}{2}T_{b_{\alpha}}\partial^{-1}_{\alpha}{\hat{w}}=0,&\end{array}\right.$

Although it is possible to perform the analysis similar as previouly done in
Section 4, a more convenient way is to reduce (5.3) to the paradifferential
equations (LABEL:SourceParadifferential). The result is as follows:

######  Proposition 5.2.

Suppose that $({\hat{w}},{\hat{r}})$ solve (5.3), then there exists a bounded
linear transformation which is independent of the Sobolev index $s$, such that
it turns $({\hat{w}},{\hat{r}})$ into $(w,r)$ that solve
(LABEL:SourceParadifferential), and it satisfies the following properties: for
any $s\in\mathbb{R}$, we have

  1. (1)

Invertibility:

$\|(w_{\alpha},r_{\alpha})-({\hat{w}},{\hat{r}})\|_{\dot{\mathcal{H}}^{s}}\lesssim_{\underline{A}}{\underline{A}}\|(w,r)\|_{\dot{\mathcal{H}}^{s}}.$

  2. (2)

Perturbative source term:

$\|(G,K)\|_{\dot{\mathcal{H}}^{s}}\lesssim_{A}\underline{A}^{2}_{\frac{1}{4}}\left(\|(w,r)\|_{\dot{\mathcal{H}}^{s}}+\gamma\|(w,r)\|_{\dot{\mathcal{H}}^{s-\frac{1}{2}}}\right).$

As a consequence, Proposition 4.3 holds with (5.3) in place of
(LABEL:ParadifferentialFlow).

###### Proof.

When computing the linearized system, the linearization of $R$ is given by

$\delta R=\frac{r_{\alpha}+R_{\alpha}w}{1+{\mathbf{W}}}.$

This suggests that the connection between $(w,r)$ and $({\hat{w}},{\hat{r}})$
is given by the relation

$({\hat{w}},{\hat{r}})=\left(w_{\alpha},\frac{r_{\alpha}+R_{\alpha}w}{1+{\mathbf{W}}}\right).$

Given a solution $({\hat{w}},{\hat{r}})$ to (5.3), we therefore define $(w,r)$
at the paradifferential level:

(5.4)
$(w,r):=(\partial^{-1}_{\alpha}{\hat{w}},\partial^{-1}_{\alpha}T_{1+{\mathbf{W}}}{\hat{r}}-\partial^{-1}_{\alpha}T_{R_{\alpha}}\partial^{-1}_{\alpha}{\hat{w}}).$

Clearly, this definition of $(w,r)$ satisfies the invertibility property. It
remains to show that $(w,r)$ satisfy (LABEL:SourceParadifferential) with
perturbative source terms.

We plug in the relation (5.4) into (LABEL:ParadifferentialFlow) and compute
the corresponding source terms. Here, the source terms are acceptable if they
satisfy

$\|G\|_{\dot{H}^{s}}+\|K\|_{\dot{H}^{s+\frac{1}{2}}}\lesssim_{\underline{A}}{\underline{A}}^{2}_{\frac{1}{4}}\|(w,r)\|_{\dot{\mathcal{H}}^{s+1}}.$

For the first equation,

$\displaystyle\partial_{\alpha}(T_{D_{t}}w+T_{1-\bar{Y}}\partial_{\alpha}r+T_{(1-\bar{Y})R_{\alpha}}w+\gamma
T_{\Im{\mathbf{W}}}w)$ $\displaystyle=$ $\displaystyle
T_{D_{t}}{\hat{w}}+T_{{\underline{b}}_{\alpha}}{\hat{w}}+\partial_{\alpha}T_{1-\bar{Y}}T_{1+W_{\alpha}}{\hat{r}}+\gamma\partial_{\alpha}T_{\Im
W_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}}$ $\displaystyle=$ $\displaystyle
T_{D_{t}}{\hat{w}}+T_{{\underline{b}}_{\alpha}}{\hat{w}}+\partial_{\alpha}T_{1-\bar{Y}}T_{1+W_{\alpha}}{\hat{r}}-i\frac{\gamma}{2}\partial_{\alpha}T_{1+W_{\alpha}}T_{1-\bar{Y}}\partial^{-1}{\hat{w}}+i\frac{\gamma}{2}{\hat{w}}$
$\displaystyle+\left(\gamma\partial_{\alpha}T_{\Im
W_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}}+i\frac{\gamma}{2}\partial_{\alpha}T_{1+W_{\alpha}}T_{1-\bar{Y}}\partial^{-1}{\hat{w}}-i\frac{\gamma}{2}{\hat{w}}\right)$
$\displaystyle=$
$\displaystyle\left(T_{D_{t}}{\hat{w}}+T_{{\underline{b}}_{\alpha}}{\hat{w}}+\partial_{\alpha}T_{1-\bar{Y}}T_{1+W_{\alpha}}{\hat{r}}-i\frac{\gamma}{2}\partial_{\alpha}T_{1+W_{\alpha}}T_{1-\bar{Y}}\partial^{-1}{\hat{w}}+i\frac{\gamma}{2}{\hat{w}}\right)+G$
$\displaystyle=$ $\displaystyle G.$

For the second equation, we use Lemma 2.5 to distribute para-material
derivatives, and use para-commutators (2.11), para-products (2.12) to write

$\displaystyle\partial_{\alpha}(T_{D_{t}}r+i\gamma
r-iT_{1-Y}T_{g+{\underline{a}}}w)$ $\displaystyle=$ $\displaystyle
T_{D_{t}}(T_{1+{\mathbf{W}}}{\hat{r}}-T_{R_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}})+T_{{\underline{b}}_{\alpha}}(T_{1+{\mathbf{W}}}{\hat{r}}-T_{R_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}})$
$\displaystyle+i\gamma(T_{1+{\mathbf{W}}}{\hat{r}}-T_{R_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}})-i\partial_{\alpha}T_{1-Y}T_{g+{\underline{a}}}w$
$\displaystyle=$ $\displaystyle
T_{D_{t}{\mathbf{W}}}{\hat{r}}+T_{1+{\mathbf{W}}}T_{D_{t}}{\hat{r}}-T_{R_{\alpha}}T_{D_{t}}\partial_{\alpha}^{-1}{\hat{w}}-T_{\partial_{\alpha}D_{t}R}w+T_{1+{\mathbf{W}}}T_{{\underline{b}}_{\alpha}}{\hat{r}}$
$\displaystyle+i\gamma T_{1+{\mathbf{W}}}{\hat{r}}-i\gamma
T_{R_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}}+iT_{Y_{\alpha}}T_{g+{\underline{a}}}w-iT_{1-Y}T_{g+{\underline{a}}}w_{\alpha}-iT_{1-Y}T_{{\underline{a}}_{\alpha}}w+K$
$\displaystyle=$ $\displaystyle
T_{1+{\mathbf{W}}}(T_{D_{t}}+T_{{\underline{b}}_{\alpha}}){\hat{r}}-T_{R_{\alpha}}\partial^{-1}_{\alpha}(T_{D_{t}}+T_{{\underline{b}}_{\alpha}}){\hat{w}}+T_{D_{t}{\mathbf{W}}}{\hat{r}}-T_{\partial_{\alpha}D_{t}R}w$
$\displaystyle+i\gamma T_{1+{\mathbf{W}}}{\hat{r}}-i\gamma
T_{R_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}}+iT_{Y_{\alpha}}T_{g+{\underline{a}}}w-iT_{1-Y}T_{g+{\underline{a}}}w_{\alpha}-iT_{1-Y}T_{{\underline{a}}_{\alpha}}w+K$
$\displaystyle=$ $\displaystyle
iT_{1+{\mathbf{W}}}T_{(g+{\underline{a}})(1-Y)^{2}}{\hat{w}}-i\gamma
T_{1+{\mathbf{W}}}{\hat{r}}+i\frac{\gamma}{2}T_{1+{\mathbf{W}}}T_{b_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}}+T_{R_{\alpha}}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}{\hat{r}}$
$\displaystyle-i\frac{\gamma}{2}T_{R_{\alpha}}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}\partial_{\alpha}^{-1}{\hat{w}}+i\frac{\gamma}{2}T_{R_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}}-T_{(1+{\mathbf{W}})(1-\bar{Y})R_{\alpha}}{\hat{r}}+i\gamma
T_{R_{\alpha}}w+iT_{\partial_{\alpha}(g+{\underline{a}})(1-Y)}w$
$\displaystyle-i\frac{\gamma}{2}T_{R_{\alpha}+\bar{R}_{\alpha}}w+i\gamma
T_{1+{\mathbf{W}}}{\hat{r}}-i\gamma
T_{R_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}}-iT_{\partial_{\alpha}(g+{\underline{a}})(1-Y)}w-iT_{1-Y}T_{g+{\underline{a}}}w_{\alpha}+K$
$\displaystyle=$ $\displaystyle
i\frac{\gamma}{2}T_{1+{\mathbf{W}}}T_{b_{\alpha}}\partial_{\alpha}^{-1}{\hat{w}}-i\frac{\gamma}{2}T_{R_{\alpha}}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}\partial_{\alpha}^{-1}{\hat{w}}-i\frac{\gamma}{2}T_{\bar{R}_{\alpha}}w+K$
$\displaystyle=$ $\displaystyle K.$

Here we have harmlessly replaced $T_{D_{t}}{\mathbf{W}}$ and $T_{D_{t}}R$ by
$D_{t}{\mathbf{W}}$ and $D_{t}R$.

Finally, the $\dot{\mathcal{H}}^{s+1}$ well-posedness of
(LABEL:ParadifferentialFlow) proved in Proposition 4.3 implies the
$\dot{\mathcal{H}}^{s}$ well-posedness of (5.3). ∎

###  5.3. The paradifferential normal form transformation

Having settled the local well-posedness of the linear system (5.3), we now
consider the right hand side source terms of (5.2). Although the source terms
$(\mathcal{G},\mathcal{K})$ are not directly perturbative, we are able to use
a paradifferential normal form transformation to eliminate them modulo
perturbative terms. Precisely, we can construct paradifferential normal form
variables $({\mathbf{W}}_{NF},R_{NF})$ that satisfy the following result:

######  Proposition 5.3.

Suppose that $({\mathbf{W}},R)$ solve the system (1.7), then there exist
paradifferential normal form variables $({\mathbf{W}}_{NF},R_{NF})$ that
satisfy the following system

$\left\\{\begin{array}[]{lr}T_{D_{t}}{\mathbf{W}}_{NF}+T_{{\underline{b}}_{\alpha}}{\mathbf{W}}_{NF}+\partial_{\alpha}T_{1-\bar{Y}}T_{1+W_{\alpha}}R_{NF}-i\frac{\gamma}{2}\partial_{\alpha}T_{1+W_{\alpha}}T_{1-\bar{Y}}\partial^{-1}_{\alpha}{\mathbf{W}}_{NF}+i\frac{\gamma}{2}{\mathbf{W}}_{NF}=\tilde{\mathcal{G}}&\\\
T_{D_{t}}R_{NF}+i\gamma
R_{NF}+T_{{\underline{b}}_{\alpha}}R_{NF}-iT_{(1-Y)^{2}}T_{g+{\underline{a}}}{\mathbf{W}}_{NF}-i\frac{\gamma}{2}T_{b_{\alpha}}\partial^{-1}_{\alpha}{\mathbf{W}}_{NF}=\tilde{\mathcal{K}},&\end{array}\right.$

such that for any $s\geq 0$, we have

  1. (1)

Invertibility:

$\|({\mathbf{W}}_{NF},R_{NF})-({\mathbf{W}},R)\|_{\dot{\mathcal{H}}^{s}}\lesssim_{\underline{A}}{\underline{A}}\left(\|({\mathbf{W}},R)\|_{\dot{\mathcal{H}}^{s}}+\gamma^{2}\|({\mathbf{W}},R)\|_{\dot{\mathcal{H}}^{s-1}}\right).$

  2. (2)

Perturbative source terms:

$\|(\tilde{\mathcal{G}},\tilde{\mathcal{K}})\|_{\dot{\mathcal{H}}^{s}}\lesssim_{\underline{A}}{\underline{A}}^{2}_{\frac{1}{4}}\left(\|({\mathbf{W}},R)\|_{\dot{\mathcal{H}}^{s}}+\gamma^{2}\|({\mathbf{W}},R)\|_{\dot{\mathcal{H}}^{s-1}}\right).$

###### Proof.

We choose the paradifferential normal form variables
$({\mathbf{W}}_{NF},R_{NF})$ to be $({\mathbf{W}}_{NF},R_{NF})$
$:=({\mathbf{W}}+\tilde{{\mathbf{W}}},R+\tilde{R})$, where the
paradifferential corrections $(\tilde{{\mathbf{W}}},\tilde{R})$ are given by

$\displaystyle\tilde{{\mathbf{W}}}=$
$\displaystyle-\partial_{\alpha}\Pi({\mathbf{W}},2\Re
X)-\frac{\gamma}{2g}\partial_{\alpha}\Pi({\mathbf{W}},2\Re
Z)-\frac{\gamma}{2g}\partial_{\alpha}\Pi(R,2\Re
X)-\frac{\gamma^{2}}{2g}\partial_{\alpha}\Pi({\mathbf{W}},2\Im U)$
$\displaystyle+i\frac{\gamma^{2}}{4g}\partial_{\alpha}\Pi(X,X)+i\frac{\gamma^{2}}{4g}\partial_{\alpha}\Pi(X,2\Re
X)-\frac{\gamma^{2}}{4g^{2}}\partial_{\alpha}\Pi(R,2\Re
Z)+i\frac{\gamma^{3}}{4g^{2}}\partial_{\alpha}\Pi(X,2\Re Z)$
$\displaystyle-\frac{\gamma^{3}}{4g^{2}}\partial_{\alpha}\Pi(R,2\Im
U)+i\frac{\gamma^{4}}{2g^{2}}\partial_{\alpha}\Pi(X,2\Im U),$
$\displaystyle\tilde{R}=$ $\displaystyle-\Pi(R_{\alpha},2\Re
X)-\Pi(T_{1-\bar{Y}}\bar{{\mathbf{W}}},R)-\frac{\gamma}{2g}\partial_{\alpha}\Pi(R,2\Re
Z)+i\frac{\gamma}{2}\partial_{\alpha}\Pi(X,2\Re X)$
$\displaystyle-i\frac{\gamma}{4}\partial_{\alpha}\Pi(X,X)-\frac{\gamma^{2}}{2g}\partial_{\alpha}\Pi(R,2\Im
U)+i\frac{\gamma^{2}}{4g}\partial_{\alpha}\Pi(X,2\Re
Z)+i\frac{\gamma^{3}}{4g}\partial_{\alpha}\Pi(X,2\Im U).$

We remark that the quadratic part of this paradifferential correction is
nothing but roughly the balanced part of the derivative of the normal form
transformation (1.10), after switching to the good variables
$({\mathbf{W}},R)$.

Clearly, $(\tilde{{\mathbf{W}}},\tilde{R})$ satisfy the invertibility property
using direct computation, it only suffices to check that the source terms
$(\tilde{\mathcal{G}},\tilde{\mathcal{K}})$ are perturbative.

We insert these corrections into the system (5.3) and compute the
corresponding source terms. For both equations, the terms having
$T_{{\underline{b}}_{\alpha}}$ are perturbative. For convenience of the
computation below, we recall in Section 3.2, the leading terms of para-
material derivatives.

$\displaystyle
T_{D_{t}}{\mathbf{W}}+T_{1+{\mathbf{W}}}T_{1-\bar{Y}}R_{\alpha}=E_{1},\quad$
$\displaystyle\|E_{1}\|_{BMO}\lesssim_{\underline{A}}{\underline{A}}^{2}_{\frac{1}{4}},$
$\displaystyle T_{D_{t}}R-iT_{g+{\underline{a}}}Y+i\gamma R=E_{2},\quad$
$\displaystyle\|E_{2}\|_{BMO^{\frac{1}{2}}}\lesssim_{\underline{A}}{\underline{A}}^{2}_{\frac{1}{4}},$
$\displaystyle T_{D_{t}}X+T_{1-\bar{Y}}R=E_{3},\quad$
$\displaystyle\||D|E_{3}\|_{BMO}+\gamma^{2}\|E_{3}\|_{BMO}\lesssim_{{\underline{A}^{\sharp}}}\underline{A}^{2}_{\frac{1}{4}},$
$\displaystyle T_{D_{t}}Z-igX+i\gamma Z=E_{4},\quad$
$\displaystyle\gamma^{2}\||D|^{\frac{1}{2}}E_{4}\|_{BMO}+\gamma^{3}\|E_{4}\|_{BMO}\lesssim_{{\underline{A}^{\sharp}}}\underline{A}^{2}_{\frac{1}{4}},$
$\displaystyle T_{D_{t}}U+T_{1-\bar{Y}}Z=E_{5},\quad$
$\displaystyle\gamma^{2}\||D|E_{5}\|_{BMO}\lesssim_{{\underline{A}^{\sharp}}}\underline{A}^{2}_{\frac{1}{4}}.$

We will use Lemma 2.5 to distribute para-material derivatives. Let $(G,K)$ be
the perturbative source terms that satisfy the perturbative source terms bound
in the proposition. In the following, we will put the perturbative source
terms into $(G,K)$ for simplicity.

We first compute the source term of the first equation of (5.3). For the first
term of the first equation,

$\displaystyle-T_{D_{t}}\partial_{\alpha}\Pi({\mathbf{W}},2\Re X)$
$\displaystyle=\partial_{\alpha}\left(\Pi(T_{(1+{\mathbf{W}})(1-\bar{Y})}R_{\alpha},2\Re
X)+\Pi({\mathbf{W}},T_{1-\bar{Y}}R+T_{1-Y}\bar{R})\right)+G,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}\partial_{\alpha}\Pi({\mathbf{W}},2\Re
Z)$
$\displaystyle=\frac{\gamma}{2g}\partial_{\alpha}\left(\Pi(T_{(1+{\mathbf{W}})(1-\bar{Y})}R_{\alpha},2\Re
Z)+\Pi({\mathbf{W}},2g\Im X-2\gamma\Im Z)\right)+G,$
$\displaystyle-\frac{\gamma}{2g}T_{D_{t}}\partial_{\alpha}\Pi(R,2\Re X)$
$\displaystyle=\frac{\gamma}{2g}\partial_{\alpha}\left(\Pi(-iT_{g+{\underline{a}}}Y+i\gamma
R,2\Re X)+\Pi(R,T_{1-\bar{Y}}R+T_{1-Y}\bar{R})\right)+G,$
$\displaystyle-\frac{\gamma^{2}}{2g}T_{D_{t}}\partial_{\alpha}\Pi({\mathbf{W}},2\Im
U)$
$\displaystyle=\frac{\gamma^{2}}{2g}\partial_{\alpha}\left(\Pi(T_{(1+{\mathbf{W}})(1-\bar{Y})}R_{\alpha},2\Im
U)+\Pi({\mathbf{W}},2\Im T_{1-\bar{Y}}Z)\right)+G,$ $\displaystyle
i\frac{\gamma^{2}}{4g}T_{D_{t}}\partial_{\alpha}\Pi(X,X)$
$\displaystyle=-i\frac{\gamma^{2}}{2g}\partial_{\alpha}\Pi(T_{1-\bar{Y}}R,X)+G,$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{D_{t}}\partial_{\alpha}\Pi(X,2\Re X)$
$\displaystyle=-i\frac{\gamma^{2}}{4g}\partial_{\alpha}\left(\Pi(T_{1-\bar{Y}}R,2\Re
X)+\Pi(X,T_{1-\bar{Y}}R+T_{1-Y}\bar{R})\right)+G,$
$\displaystyle-\frac{\gamma^{2}}{4g^{2}}T_{D_{t}}\partial_{\alpha}\Pi(R,2\Re
Z)$
$\displaystyle=\frac{\gamma^{2}}{4g^{2}}\partial_{\alpha}\left(\Pi(-iT_{g+{\underline{a}}}Y+i\gamma
R,2\Re Z)+\Pi(R,2g\Im X-2\gamma\Im Z)\right)+G,$ $\displaystyle
i\frac{\gamma^{3}}{4g^{2}}T_{D_{t}}\partial_{\alpha}\Pi(X,2\Re Z)$
$\displaystyle=-i\frac{\gamma^{3}}{4g^{2}}\partial_{\alpha}\left(\Pi(T_{1-\bar{Y}}R,2\Re
Z)+\Pi(X,2g\Im X-2\gamma\Im Z)\right)+G,$
$\displaystyle-\frac{\gamma^{3}}{4g^{2}}T_{D_{t}}\partial_{\alpha}\Pi(R,2\Im
U)$
$\displaystyle=\frac{\gamma^{3}}{4g^{2}}\partial_{\alpha}\left(\Pi(-iT_{g+{\underline{a}}}Y+i\gamma
R,2\Im U)+\Pi(R,2\Im T_{1-\bar{Y}}Z)\right)+G,$ $\displaystyle
i\frac{\gamma^{4}}{2g^{2}}T_{D_{t}}\partial_{\alpha}\Pi(X,2\Im U)$
$\displaystyle=-i\frac{\gamma^{4}}{2g^{2}}\partial_{\alpha}\left(\Pi(T_{1-\bar{Y}}R,2\Im
U)+\Pi(X,2\Im T_{1-\bar{Y}}Z)\right)+G.$

For the third term of the first equation, we apply the para-associativity
(2.14) and the relations (3.11), (3.12), (3.13),

$\displaystyle-\partial_{\alpha}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}\Pi(R_{\alpha},2\Re
X)$
$\displaystyle=-\partial_{\alpha}\Pi(T_{(1-\bar{Y})(1+{\mathbf{W}})}R_{\alpha},2\Re
X)+G,$
$\displaystyle-\partial_{\alpha}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}\Pi(T_{1-\bar{Y}}\bar{{\mathbf{W}}},R)$
$\displaystyle=-\partial_{\alpha}\Pi(\bar{Y},T_{1+{\mathbf{W}}}R)+G,$
$\displaystyle-\frac{\gamma}{2g}\partial_{\alpha}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}\partial_{\alpha}\Pi(R,2\Re
Z)$
$\displaystyle=-\frac{\gamma}{2g}\partial_{\alpha}\left(\Pi(T_{(1-\bar{Y})(1+{\mathbf{W}})}R_{\alpha},2\Re
Z)+\Pi(R,2\Re T_{1-\bar{Y}}R)\right)+G,$ $\displaystyle
i\frac{\gamma}{2}\partial_{\alpha}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}\partial_{\alpha}\Pi(X,2\Re
X)$ $\displaystyle=i\frac{\gamma}{2}\partial_{\alpha}\left(\Pi(Y,2\Re
X)+\Pi(X,2\Re{\mathbf{W}})\right)+G,$
$\displaystyle-i\frac{\gamma}{4}\partial_{\alpha}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}\partial_{\alpha}\Pi(X,X)$
$\displaystyle=-i\frac{\gamma}{2}\partial_{\alpha}\Pi({\mathbf{W}},T_{1-\bar{Y}}W)+G,$
$\displaystyle-\frac{\gamma^{2}}{2g}\partial_{\alpha}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}\partial_{\alpha}\Pi(R,2\Im
U)$
$\displaystyle=-\frac{\gamma^{2}}{2g}\partial_{\alpha}\left(\Pi(T_{(1+{\mathbf{W}})(1-\bar{Y})}R_{\alpha},2\Im
U)+\Pi(T_{1-\bar{Y}}R,2\Im X)\right)+G,$ $\displaystyle
i\frac{\gamma^{2}}{4g}\partial_{\alpha}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}\partial_{\alpha}\Pi(X,2\Re
Z)$
$\displaystyle=i\frac{\gamma^{2}}{4g}\partial_{\alpha}\left(\Pi({\mathbf{W}},2\Re
Z)+\Pi(X,T_{1-\bar{Y}}R+T_{1-Y}\bar{R})\right)+G,$ $\displaystyle
i\frac{\gamma^{3}}{4g}\partial_{\alpha}T_{1-\bar{Y}}T_{1+{\mathbf{W}}}\partial_{\alpha}\Pi(X,2\Im
U)$
$\displaystyle=i\frac{\gamma^{3}}{4g}\partial_{\alpha}\left(\Pi({\mathbf{W}},2\Im
U)+\Pi(X,2\Im X)\right)+G.$

Here, when the derivatives fall on the paradifferential coefficients
$T_{1+{\mathbf{W}}}$ or $T_{1-\bar{Y}}$, these terms are perturbative and may
go to $G$.

For the last two terms of the first equation, they are

$-i\frac{\gamma}{2}\partial_{\alpha}T_{1+{\mathbf{W}}}T_{1-\bar{Y}}\partial^{-1}_{\alpha}\tilde{{\mathbf{W}}}+i\frac{\gamma}{2}\tilde{{\mathbf{W}}}=-i\frac{\gamma}{2}T_{{\mathbf{W}}-\bar{Y}-{\mathbf{W}}\bar{Y}}\tilde{{\mathbf{W}}}-i\frac{\gamma}{2}T_{{\mathbf{W}}_{\alpha}}T_{1-\bar{Y}}\tilde{{\mathbf{W}}}+i\frac{\gamma}{2}T_{1+{\mathbf{W}}}T_{\bar{Y}_{\alpha}}\tilde{{\mathbf{W}}}.$

Since the paradifferential correction $\tilde{{\mathbf{W}}}$ satisfies

$\|\tilde{{\mathbf{W}}}\|_{\dot{\mathcal{H}}^{s}}\lesssim_{\underline{A}}{\underline{A}}\left(\|({\mathbf{W}},R)\|_{\dot{\mathcal{H}}^{s}}+\gamma^{2}\|({\mathbf{W}},R)\|_{\dot{\mathcal{H}}^{s-1}}\right),$

the sum of the last two terms can be absorbed into $G$.

Gathering all terms for the first equation, we see that the contribution of
paradifferential corrections cancel the source term $\tilde{\mathcal{G}}$
modulo perturbative terms.

Next, we perform the computation for the contribution of the paradifferential
corrections in the second equation of (5.3). For the first term of the second
equation,

$\displaystyle-T_{D_{t}}\Pi(R_{\alpha},2\Re X)$
$\displaystyle=-i\Pi(T_{g+{\underline{a}}}Y_{\alpha}-\gamma R_{\alpha},2\Re
X)+\Pi(R_{\alpha},T_{1-\bar{Y}}R+T_{1-Y}\bar{R})+K,$ $\displaystyle-
T_{D_{t}}\Pi(T_{1-\bar{Y}}\bar{{\mathbf{W}}},R)$
$\displaystyle=T_{1-Y}\Pi(\bar{R}_{\alpha},R)-i\Pi(T_{1-\bar{Y}}\bar{{\mathbf{W}}},T_{g+{\underline{a}}}Y-\gamma
R)+K,$ $\displaystyle-\frac{\gamma}{2g}T_{D_{t}}\partial_{\alpha}\Pi(R,2\Re
Z)$
$\displaystyle=-\frac{\gamma}{2g}\partial_{\alpha}\left(\Pi(iT_{g+{\underline{a}}}Y-i\gamma
R,2\Re Z)+\Pi(R,-2g\Im X+2\gamma\Im Z)\right)+K,$ $\displaystyle
i\frac{\gamma}{2}T_{D_{t}}\partial_{\alpha}\Pi(X,\bar{X})$
$\displaystyle=-i\frac{\gamma}{2}\partial_{\alpha}\left(\Pi(T_{1-\bar{Y}}R,\bar{X})+\Pi(T_{1-Y}\bar{R},X)\right)+K,$
$\displaystyle-i\frac{\gamma}{4}T_{D_{t}}\partial_{\alpha}\Pi(X,X)$
$\displaystyle=i\frac{\gamma}{2}\partial_{\alpha}\Pi(T_{1-\bar{Y}}R,X)+K,$
$\displaystyle-\frac{\gamma^{2}}{2g}T_{D_{t}}\partial_{\alpha}\Pi(R,2\Im U)$
$\displaystyle=\frac{\gamma^{2}}{2g}\partial_{\alpha}\left(\Pi(i\gamma
R-iT_{g+{\underline{a}}}Y,2\Im U)+\Pi(R,2\Im T_{1-\bar{Y}}Z)\right)+K,$
$\displaystyle i\frac{\gamma^{2}}{4g}T_{D_{t}}\partial_{\alpha}\Pi(X,2\Re Z)$
$\displaystyle=-i\frac{\gamma^{2}}{4g}\partial_{\alpha}\left(\Pi(T_{1-\bar{Y}}R,2\Re
Z)+\Pi(X,2g\Im X-2\gamma\Im Z)\right)+K,$ $\displaystyle
i\frac{\gamma^{3}}{4g}T_{D_{t}}\partial_{\alpha}\Pi(X,2\Im U)$
$\displaystyle=-i\frac{\gamma^{3}}{4g}\partial_{\alpha}\left(\Pi(T_{1-\bar{Y}}R,2\Im
U)+\Pi(X,2\Im T_{1-\bar{Y}}Z)\right)+K.$

The second term of the second equation is given by

$\displaystyle-i\gamma\Pi(R_{\alpha},2\Re
X)-i\gamma\Pi(T_{1-\bar{Y}}\bar{{\mathbf{W}}},R)-i\frac{\gamma^{2}}{2g}\partial_{\alpha}\Pi(R,2\Re
Z)-\frac{\gamma^{2}}{2}\partial_{\alpha}\Pi(X,2\Re X)$
$\displaystyle+\frac{\gamma^{2}}{4}\partial_{\alpha}\Pi(X,X)-i\frac{\gamma^{3}}{2g}\partial_{\alpha}\Pi(R,2\Im
U)-\frac{\gamma^{3}}{4g}\partial_{\alpha}\Pi(X,2\Re
Z)-\frac{\gamma^{4}}{4g}\partial_{\alpha}\Pi(X,2\Im U).$

For the fourth term of the second equation, the contribution of
$T_{\underline{a}}$ is perturbative, and may be absorbed into $K$. We apply
the para-associativity (2.14) to write,

$\displaystyle
iT_{(1-Y)^{2}}T_{g+{\underline{a}}}\partial_{\alpha}\Pi({\mathbf{W}},2\Re X)$
$\displaystyle=i\Pi(T_{g+{\underline{a}}}Y_{\alpha},2\Re X)$
$\displaystyle+iT_{g+{\underline{a}}}\left(T_{1-Y}\Pi({\mathbf{W}},Y)+\Pi(Y,\bar{X}_{\alpha})\right)+K,$
$\displaystyle
i\frac{\gamma}{2g}T_{(1-Y)^{2}}T_{g+{\underline{a}}}\partial_{\alpha}\Pi({\mathbf{W}},2\Re
Z)$
$\displaystyle=\frac{\gamma}{2g}\partial_{\alpha}\Pi(iT_{g+{\underline{a}}}Y,2\Re
Z)+K,$ $\displaystyle
i\frac{\gamma}{2g}T_{(1-Y)^{2}}T_{g+{\underline{a}}}\partial_{\alpha}\Pi(R,2\Re
X)$ $\displaystyle=i\frac{\gamma}{2}\partial_{\alpha}\Pi(T_{1-\bar{Y}}R,2\Re
X)+K,$ $\displaystyle
i\frac{\gamma^{2}}{2g}T_{(1-Y)^{2}}T_{g+{\underline{a}}}\partial_{\alpha}\Pi({\mathbf{W}},2\Im
U)$
$\displaystyle=\frac{\gamma^{2}}{2g}\partial_{\alpha}\Pi(iT_{g+{\underline{a}}}Y,2\Im
U)+K,$
$\displaystyle\frac{\gamma^{2}}{4g}T_{(1-Y)^{2}}T_{g+{\underline{a}}}\partial_{\alpha}\Pi(X,X)$
$\displaystyle=\frac{\gamma^{2}}{4}\partial_{\alpha}\Pi(X,X)+K,$
$\displaystyle\frac{\gamma^{2}}{4g}T_{(1-Y)^{2}}T_{g+{\underline{a}}}\partial_{\alpha}\Pi(X,2\Re
X)$ $\displaystyle=\frac{\gamma^{2}}{4}\partial_{\alpha}(X,2\Re X)+K,$
$\displaystyle
i\frac{\gamma^{2}}{4g^{2}}T_{(1-Y)^{2}}T_{g+{\underline{a}}}\partial_{\alpha}\Pi(R,2\Re
Z)$ $\displaystyle=i\frac{\gamma^{2}}{4g}\partial_{\alpha}\Pi(R,2\Re Z)+K,$
$\displaystyle\frac{\gamma^{3}}{4g^{2}}T_{(1-Y)^{2}}T_{g+{\underline{a}}}\partial_{\alpha}\Pi(X,2\Re
Z)$ $\displaystyle=\frac{\gamma^{3}}{4g}\partial_{\alpha}\Pi(X,2\Re Z)+K,$
$\displaystyle
i\frac{\gamma^{3}}{4g^{2}}T_{(1-Y)^{2}}T_{g+{\underline{a}}}\partial_{\alpha}\Pi(R,2\Im
U)$ $\displaystyle=i\frac{\gamma^{3}}{4g}\partial_{\alpha}\Pi(R,2\Im U)+K,$
$\displaystyle\frac{\gamma^{4}}{2g^{2}}T_{(1-Y)^{2}}T_{g+{\underline{a}}}\partial_{\alpha}\Pi(X,2\Im
U)$ $\displaystyle=\frac{\gamma^{4}}{2g}\partial_{\alpha}\Pi(X,2\Im U)+K.$

The last term of the left hand side of the second equation
$-i\frac{\gamma}{2}T_{b_{\alpha}}\partial^{-1}_{\alpha}\tilde{{\mathbf{W}}}$
is perturbative and may absorbed into $K$. After cancellations, the source
term of the paradifferential corrections of the second equation equals
$-\tilde{\mathcal{K}}$ modulo perturbative $K$ terms. Therefore, we construct
the paradifferential normal form variables $({\mathbf{W}}_{NF},R_{NF})$ that
satisfy the desired properties. ∎

Having switched into the paradifferential normal form variables
$({\mathbf{W}}_{NF},R_{NF})$, the differentiated system (1.7) is reduced to
the paradifferential equations (5.3) with perturbative source terms. Theorem
1.3 then follows directly from the last part of Proposition 5.2.

##  6\. The proof of local well-posedness

In this section, we prove the main result of this article, namely the low
regularity well-posedness Theorem 1.4. The result from [17] asserts that the
local well-posedness holds for more regular data. We will first establish the
$\mathcal{H}^{n}$ bounds for regular solutions. Then, we use those regular
solutions as starting points to construct rough solutions. Precisely, we
obtain regular solutions by truncating the rough initial data in frequency, so
that we get a continuous family of solutions, thereby estimating only a
solution for linearized equations at each step. Finally, we prove the
continuous dependence on the initial data in $\mathcal{H}^{\frac{3}{4}}$.

Note that we can make use of the space time scaling (1.3). By choosing the
scaling parameter $\lambda$ small enough, we can make
${\underline{A}^{\sharp}}\ll 1$, at the price to turn the vorticity $\gamma$
into $\lambda\gamma$. The argument for the proof below is similar to the proof
in [4]. For simplicity, we only give an outline of the approach here. Please
check Section 7 of [4] for detailed proof such as the use of frequency
envelops.

###### Outline of proof of Theorem 1.4.

The proof of the theorem is divided into the following four steps:

$(1)$ $\mathcal{H}^{n}$ bounds for regular solutions. Suppose for integer
$n\geq 1$ we have an $\mathcal{H}^{n}$ solution $({\mathbf{W}},R)$ that
satisfies the the initial condition

$\|({\mathbf{W}}_{0},R_{0})\|_{\dot{\mathcal{H}}^{\frac{3}{4}}}+\gamma^{2}\|({\mathbf{W}}_{0},R_{0})\|_{\dot{\mathcal{H}}^{-\frac{1}{4}}}\leq\mathcal{M}_{0}\ll
1.$

In order to show that there exists a time $T=T(\mathcal{M}_{0})$ such that the
solution exists in $C([0,T];\mathcal{H}^{n})$ and satisfies the bounds

(6.1)
$\displaystyle\|({\mathbf{W}},R)\|_{L^{\infty}(0,T;\dot{\mathcal{H}}^{\frac{3}{4}})}+\gamma^{2}\|({\mathbf{W}},R)\|_{L^{\infty}(0,T;\dot{\mathcal{H}}^{-\frac{1}{4}})}<\mathcal{M}(\mathcal{M}_{0}),$
(6.2) $\displaystyle\|({\mathbf{W}},R)\|_{L^{\infty}(0,T;\mathcal{H}^{n})}\leq
C(\mathcal{M}_{0})\|({\mathbf{W}}_{0},R_{0})\|_{\mathcal{H}^{n}},$

we make the bootstrap assumption

$\|({\mathbf{W}},R)\|_{L^{\infty}(0,T;\dot{\mathcal{H}}^{\frac{3}{4}})}+\gamma^{2}\|({\mathbf{W}},R)\|_{L^{\infty}(0,T;\dot{\mathcal{H}}^{-\frac{1}{4}})}<2\mathcal{M}.$

Using the bootstrap assumption and Sobolev embedding, we are able to bound the
control parameters

${\underline{A}_{\frac{1}{4}}}\lesssim\mathcal{M},\quad{\underline{A}^{\sharp}_{\frac{1}{4}}}\lesssim\mathcal{M}.$

Using the energy estimate with $s=\frac{3}{4}$ in Theorem 1.3, we have

$\|({\mathbf{W}},R)(t)\|_{\dot{\mathcal{H}}^{\frac{3}{4}}}+\gamma^{2}\|({\mathbf{W}},R)(t)\|_{\dot{\mathcal{H}}^{-\frac{1}{4}}}\lesssim
e^{Ct}\left(\|({\mathbf{W}}_{0},R_{0})\|_{\dot{\mathcal{H}}^{\frac{3}{4}}}+\gamma^{2}\|({\mathbf{W}}_{0},R_{0})\|_{\dot{\mathcal{H}}^{-\frac{1}{4}}}\right)\lesssim
e^{Ct}\mathcal{M}_{0}.$

By choosing large $\mathcal{M}$ and suitable $T$, the bound (6.1) holds for
$t\in[0,T]$.

Applying Theorem 1.3 and Gronwall’s inequality for each integer $k$ between
$1$ and $n$, we have

$\|({\mathbf{W}},R)(t)\|_{\dot{\mathcal{H}}^{k}}+\gamma^{2}\|({\mathbf{W}},R)(t)\|_{\dot{\mathcal{H}}^{k-1}}\lesssim
e^{Ct}\left(\|({\mathbf{W}},R)(t)\|_{\dot{\mathcal{H}}^{k}}+\gamma^{2}\|({\mathbf{W}},R)(t)\|_{\dot{\mathcal{H}}^{k-1}}\right).$

Summing above inequalities for $k$ from $1$ to $n$, we get (6.2). Therefore we
have the $\mathcal{H}^{n}$ bounds for regular solutions up to time $T$.

$(2)$ Construction of rough solutions,
$({\mathbf{W}},R)\in\mathcal{H}^{\frac{3}{4}}$. We regularize the initial
data, $(W_{<k}(0),Q_{<k}(0))$ and $({\mathbf{W}}_{<k}(0),R_{<k}(0))$ by
truncating at frequency $2^{-k}$. We are able to establish the bound of
regularized initial data using frequency envelops. The corresponding solutions
will be regular, with a uniform lifespan bound. Here $k$ can be viewed as a
continuous parameter rather than a discrete parameter. Then

$(w^{k},r^{k})=(\partial_{k}W_{<k},\partial_{k}Q_{<k}-R_{<k}\partial_{k}W_{<k})$

solve the corresponding linearized equations around
$({\mathbf{W}}_{<k},R_{<k})$. For the high frequency part of the regularized
solutions, we apply the energy estimates for the full equation Theorem 1.3.
Next, using Theorem 4.2 for the linearized variables $(w^{k},r^{k})$, one can
establish the difference bound
$({\mathbf{W}}_{<k+1}-{\mathbf{W}}_{<k},R_{<k+1}-R_{<k})$ in
$\mathcal{H}^{\frac{3}{4}}$. Summing up with respect to $k$, it follows that
the sequence $({\mathbf{W}}_{<k},R_{<k})$ converges to a solution
$({\mathbf{W}},R)$ with uniform $\mathcal{H}^{\frac{3}{4}}$ bound in time
interval $[0,T]$.

$(3)$ Continuous dependence on the data for rough solutions. Consider an
arbitrary sequence $({\mathbf{W}}_{j},R_{j})(0)$ that converges to
$({\mathbf{W}}_{0},R_{0})$ in $\mathcal{H}^{\frac{3}{4}}$ topology. Using
again the frequency truncation, we get the approximate solutions
$({\mathbf{W}}_{j}^{k},R_{j}^{k})$, respectively $({\mathbf{W}}^{k},R^{k})$.
Due to the continuous dependence for the regular solutions which is proved in
Theorem 1.1 of [15], we have for each $k$

$({\mathbf{W}}^{k}_{j},R^{k}_{j})-({\mathbf{W}}^{k},R^{k})\rightarrow
0\quad\text{in }\mathcal{H}^{n},n\geq 1.$

On the other hand, letting $k$ go to infinity, we have for the initial data

$({\mathbf{W}}^{k}_{j},R^{k}_{j})(0)-({\mathbf{W}}_{j},R_{j})(0)\rightarrow
0\quad\text{in }\mathcal{H}^{\frac{3}{4}},\quad\text{uniformly in }j.$

Using the frequency envelope analysis, we further get the uniform convergence
for the solution:

$({\mathbf{W}}^{k}_{j},R^{k}_{j})-({\mathbf{W}}_{j},R_{j})\rightarrow
0\quad\text{in }\mathcal{H}^{\frac{3}{4}},\quad\text{uniformly in }j.$

We can again let $k$ go to infinity to conclude that

$({\mathbf{W}}_{j},R_{j})-({\mathbf{W}},R)\rightarrow 0\quad\text{in
}\mathcal{H}^{\frac{3}{4}},$

which shows the continuous dependence on the data in
$\mathcal{H}^{\frac{3}{4}}$.

$(4)$ Continuation of solutions. Here we show that the solution can be
continued for as long as $\max\\{{\underline{A}^{\sharp}},{\underline{A}}\\}$
remains small and
${\underline{A}_{\frac{1}{4}}}(\gamma^{\frac{1}{4}}+{\underline{A}_{\frac{1}{4}}})\in
L^{1}_{t}$. This is a direct consequence of the energy estimates Theorem 1.3.
∎

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