$U$.

\[MainThm2\] Assume $N=30$, $N_0=20$, and $\delta=1/100$. There is $\overline{\varepsilon}$ sufficiently small such that if $$\|U_0\|_{H^N}+\|U_0\|_{Y^{N_0}}\leq\overline{\varepsilon}$$ then there is a unique global solution $U\in C([0,\infty):H^N)$ of the initial-value problem $$\label{invp}
\begin{cases}
&(\partial_t+i\Lambda)U=\frac{i}{4}\sum_{j=1}^2\Lambda R_j\big[|\nabla|\Lambda^{-1}(U+\overline{U})\cdot R_j(U-\overline{U})\big]+\frac{i}{8}\sum_{j=1}^2|\nabla|\big[R_j(U-\overline{U})\cdot R_j(U-\overline{U})\big],\\
&U(0)=U_0.
\end{cases}$$ In addition, $U\in X^N_T$ for any $T\geq 1$ and $$\label{invp2}
 \sup_{t\in[0,\infty)}\big[(1+t)^{-\delta}\|e^{it\Lambda}U(t)\|_{H^N}+\|e^{it\Lambda}U(t)\|_{Y^{N_0}}\big]\lesssim \|U_0\|_{H^N}+\|U_0\|_{Y^{N_0}}.$$

It is easy to see that Theorem \[MainThm2\] implies Theorem \[MainThm\], since $$n-1=\Lambda^{-1}|\nabla|(\Re U),\qquad v_1=R_1(\Im U),\qquad v_2=R_2(\Im U).$$ On the other hand, Theorem \[MainThm2\] is a consequence of Proposition \[LocExLemma\] and Proposition \[Norm\] below. We start with the local existence theory:

\[LocExLemma\] (i) There exists $\delta_0>0$ such that for any $U_0\in H^3$ satisfying $$\label{SmallNess}
\Vert U_0\Vert_{H^3}\le \delta_0,$$ there exists a unique solution $U\in C([0,1]:H^3)$ of such that $U(0)=U_0$ and $$\sup_{t\in[0,1]}\|U(t)\|_{H^3}\lesssim \Vert U_0\Vert_{H^3}.$$

\(ii) Assume in addition that $U_0\in H^M$, $M\in[3,40]\cap\mathbb{Z}$. Then $U\in C([0,1]:H^M)$ and for any $0\le s\le t\le 1$, $$\label{EGrowth}
E_M(t)\le E_M(s)+C\int_s^t\|U(t')\|_{Z'}\cdot E_M(t^\prime)dt^\prime$$ where the $M$-th order energy, defined in below, satisfies $E_M(t)\simeq \Vert U(t)\Vert_{H^M}^2$ uniformly in time and $$\label{Z'norm}
 \|f\|_{Z'}:=\sup_{k\in\mathbb{Z}}(2^{k/2}+2^{2k})\|P_k f\|_{L^\infty}.$$

\(iii) With $N_0$ as in Theorem \[MainThm2\], assume $T\geq 1$ and $U\in C([0,T]:H^{N_0})$ is a solution of with the property that $U_0\in Y^{N_0}$. Then $$\begin{split}
&\sup_{t\in[0,T]}\|e^{it\Lambda}U(t)\|_{Y^{N_0}}\leq C\big(\|U_0\|_{Y^{N_0}},\sup_{t\in[0,T]}\|U(t)\|_{H^{N_0}},T\big),\\
&\lim_{t'\to t}\|e^{it'\Lambda}U(t')-e^{it\Lambda}U(t)\|_{Y^{N_0}}=0,\qquad\text{ for any }t\in[0,T].
\end{split}$$

The main conclusion in the 