irrotational, since $\omega=\hbox{curl}(v)$ satisfies a well-known transport equation. Hence we may assume that $v=\nabla h$ for some velocity potential $h$. We also set $n=1+\rho$ and becomes $$\label{EP2}
\begin{split}
\partial _{t}\rho+\Delta h+\partial_\al(\rho\partial_\al h)&=0,\\
\partial _{t}h+(1/2)(\partial_\al h\partial_\al h)+a\rho&=b\phi.\\
\Delta \phi&=\rho.
\end{split}$$

Let $|\nabla|$ denote the operator on $\mathbb{R}^2$ defined by the Fourier multiplier $\xi\mapsto |\xi|$, and let $g:=|\nabla|^{-1}\rho$. In terms of $g,h$ the system becomes $$\label{EP3}
\begin{split}
&\partial_tg-|\nabla|h+|\nabla|^{-1}\partial_j(|\nabla|g\cdot\partial_j h)=0,\\
&\partial_th+(a|\nabla|+b|\nabla|^{-1})g+(1/2)(\partial_j h\partial_j h)=0.
\end{split}$$ Letting $$U:=\sqrt{a\vert\nabla\vert^2+b} g+i|\nabla|h,$$ we derive the equation $$\label{norm1}
(\partial_t+i\Lambda)U=\frac{i}{4}\sum_{j=1}^2R_j\Lambda\big[|\nabla|\Lambda^{-1}(U+\overline{U})\cdot R_j(U-\overline{U})\big]+\frac{i}{8}\sum_{j=1}^2|\nabla|\big[R_j(U-\overline{U})\cdot R_j(U-\overline{U})\big],$$ where $$\Lambda:=\sqrt{a|\nabla|^2+b},\qquad R_j:=\partial_j/|\nabla|.$$

We fix $\varphi:\mathbb{R}\to[0,1]$ an even smooth function supported in $[-8/5,8/5]$ and equal to $1$ in $[-5/4,5/4]$. Let $$\varphi_k(x):=\varphi(|x|/2^k)-\varphi(|x|/2^{k-1})\text{ for any }k\in\mathbb{Z},\,x\in\mathbb{R}^2,\qquad \varphi_I:=\sum_{m\in I\cap\mathbb{Z}}\varphi_m\text{ for any }I\subseteq\mathbb{R}.$$ Let $P_k$, $k\in\mathbb{Z}$, denote the operator on $\mathbb{R}^2$ defined by the Fourier multiplier $\xi\to \varphi_k(\xi)$. Similarly, for any $I\subseteq \mathbb{R}$ let $P_I$ denote the operator on $\mathbb{R}^2$ defined by the Fourier multiplier $\xi\to \varphi_I(\xi)$. For $T\geq 1$ and integers $1\leq N_0<N$ we define $$\label{sec4}
\begin{split}
X^N_T(\mathbb{R}^2\times[0,T]):=\{&f\in C([0,T]:H^N(\mathbb{R}^2)):e^{it\Lambda}f\in C([0,T]:Y^{N_0}(\mathbb{R}^2))\\
&\text{ and }\|f\|_{X^N_T}:=\sup_{t\in[0,T]}(1+t)^{-\delta}\|f(t)\|_{H^N}+\sup_{t\in[0,T]}\|e^{it\Lambda}f(t)\|_{Y^{N_0}}<\infty\},
\end{split}$$ where $$\label{sec5}
\begin{split}
&Y^{N_0}(\mathbb{R}^2):=\{\phi\in L^2(\mathbb{R}^2):\|\phi\|_{Y^{N_0}}:=\|\phi\|_{H^{N_0}}+\|\phi\|_Z<\infty\},\\
&\|\phi\|_Z:=\sup_{k\in\mathbb{Z}}(2^{k/10}+2^{10k})\Big[\|P_k\phi\|_{L^2}+\sum_{j\in\mathbb{Z}_+}2^j\|\varphi_j(x)\cdot P_k\phi(x)\|_{L^2}\Big].
\end{split}$$

The spaces $X^N_T$ are our main spaces, and we use them to control our “smooth” solutions. First we control the high energy norm $H^N$ which is allowed to grow 