M_3M_\sigma\|U^\delta\|^2_{H^\sigma}+M_3\|U\|_{H^{\sigma+1}}\|U^\delta\|_{H^\sigma}\|U^\delta\|_{H^{3/2}},
\end{split}$$ and $$\begin{split}
|I^\delta_P+IV_P^\delta|&=2\Big|\sum_{j=1}^2\int_{\mathbb{R}^2}R_jPX^\delta\cdot\Lambda\big[P(|\nabla|\Lambda^{-1}X'\cdot R_jY^\delta+|\nabla|\Lambda^{-1}X^\delta\cdot R_jY)-|\nabla|\Lambda^{-1}X'\cdot R_jPY^\delta\big]\,dx\Big|\\
&\lesssim M_3\|U^\delta\|_{H^\sigma}^2+M_\sigma\|U^\delta\|_{H^\sigma}\|U^\delta\|_{H^3}+\sum_{j=1}^2\Big|\int_{\mathbb{R}^2}R_jPX^\delta\cdot\Lambda\big[P|\nabla|\Lambda^{-1}X'\cdot R_jY^\delta\big]\,dx\Big|\\
&+\sum_{j=1}^2\Big|\int_{\mathbb{R}^2}R_jPX^\delta\cdot\Lambda\big[P|\nabla|\Lambda^{-1}X^\delta\cdot R_jY\big]\,dx\Big|+\sum_{j=1}^2\Big|\int_{\mathbb{R}^2}R_jPX^\delta\cdot\Lambda\big[|\nabla|\Lambda^{-1}X^\delta\cdot PR_jY\big]\,dx\Big|.
\end{split}$$ Using also the estimate $$\|\Lambda(f\cdot g)-\Lambda f\cdot g\|_{L^2}\lesssim \|f\|_{L^2}\|g\|_{H^3},$$ see the proof of Lemma \[tech1.5\] (iii), it follows that $$\sum_{j=1}^2\Big|\int_{\mathbb{R}^2}R_jPX^\delta\cdot\Lambda\big[|\nabla|\Lambda^{-1}X^\delta\cdot PR_jY\big]\,dx\Big|\lesssim \|U^\delta\|_{H^\sigma}\|U^\delta\|_{H^{3/2}}\|U\|_{H^{\sigma+1}}+M_\sigma\|U^\delta\|_{H^\sigma}\|U^\delta\|_{H^3}.$$ In addition, for $\widetilde{Y}\in\{Y,Y^\delta\}$, $$\begin{split}
\sum_{j=1}^2\Big|\int_{\mathbb{R}^2}&R_jPX^\delta\cdot\Lambda\big[P|\nabla|\Lambda^{-1}X^\delta\cdot R_j\widetilde{Y}\big]\,dx\Big|\lesssim M_3\|U^\delta\|_{H^\sigma}^2+\sum_{j=1}^2\Big|\int_{\mathbb{R}^2}R_jPX^\delta\cdot P|\nabla|X^\delta\cdot R_j\widetilde{Y}\,dx\Big|\\
&\lesssim M_3\|U^\delta\|_{H^\sigma}^2+\sum_{j,m=1}^2\Big|\int_{\mathbb{R}^2}\partial_mR_jPX^\delta\cdot PR_mX^\delta\cdot R_j\widetilde{Y}\,dx\Big|\lesssim M_3\|U^\delta\|_{H^\sigma}^2.
\end{split}$$ Finally $$\begin{split}
\sum_{j=1}^2\Big|\int_{\mathbb{R}^2}R_jPX^\delta\cdot\Lambda\big[P|\nabla|\Lambda^{-1}X\cdot R_jY^\delta\big]\,dx\Big|\lesssim \|U^\delta\|_{H^\sigma}\|U^\delta\|_{H^{3/2}}\|U\|_{H^{\sigma+1}},
\end{split}$$ and it follows that $$|I^\delta_P+IV_P^\delta|\lesssim M_\sigma\|U^\delta\|_{H^\sigma}^2+\|U^\delta\|_{H^\sigma}\|U^\delta\|_{H^{3/2}}\|U\|_{H^{\sigma+1}}.$$ The inequality follows from these estimates.

To prove we notice that the previous estimates can be improved if $P=\mathrm{Id}$ and $F=F'=0$. Indeed, similar arguments as before show that $$\begin{split}
&|VI^\delta_{\mathrm{Id}}|=0,\qquad |III^\delta_{\mathrm{Id}}|\lesssim M_3^2\|U^\delta\|^2_{L^2},\qquad |II^\delta_{\mathrm{Id}}|\lesssim M_3\|U^\delta\|_{L^2}^2,\\
&|V^\delta_{\mathrm{Id}}|\lesssim M_3^2\|U^\delta\|_{L^2}^2,\qquad |I^\delta_{\mathrm{Id}}+IV^\delta_{\mathrm{Id}}|\lesssim M_3\|U^\delta\|_{L^2}^2,
\end{split}$$ and the desired inequality follows.

We can now complete the proof of the proposition, using the definitions, Lemma \[energylem1\], Lemma \[enerdif\], and the Bona–Smith argument [@BoSm].

Assume $M\in[3,40]\cap\mathbb{Z}$ is fixed and $U_0\in H^M$ is a given data satisfying $$\|U_0\|_{H^3}\leq \delta_0:=\delta_1,$$ where $\delta_1$ is the small constant in Lemma \[energylem1\]. Using Lemma \[energylem1\], for any $k\in\mathbb{Z}_+$ and $\varepsilon>0$ we construct a solution $U_k^\varepsilon\in C([0,1]:H^{K+2})$ of the equation , with initial condition $U_k^\varepsilon(0)=P_{(-\infty,k]}U_0$. It follows from and that $$\label{fgh3}
\sup_{t\in[0,1]}\|U_k^\varepsilon(t)\|_{H^3}\lesssim \|U_0\|_{H^3},\qquad \sup_{t\in[0,1]}\|U_k^\varepsilon(t)\|_{H^{M+d}}\lesssim 2^{dk}\|U_0\|_{H^M},\qquad d\in\{0,1,2\}.$$

We fix $k$ and apply now Lemma \[enerdif\] to the solutions $U:=U_k^\varepsilon$ and $U':=U_k^{\varepsilon'}$, 