electrically neutral: $$\int_{\mathbb{R}^2}(n(x,0)-1)dx=0,$$ which is then conserved by the flow. This condition is also necessary in order to ensure finiteness of the physical (conserved) energy $$\label{Energy}
E=\frac{1}{2}\int_{\mathbb{R}^2}\left[n\vert v\vert^2+a (n-1)^2+b\vert\nabla\phi\vert^2\right]dx,$$ hence we find it an acceptable assumption.

The irrotationality assumption is also propagated by the flow. It removes a component in the system that is only transported and does not obey nice decay estimates.

Theorem \[MainThm\] follows from the more precise Theorem \[MainThm2\] below. In order to attack this problem, we restate it as a quasilinear dispersive equation. The main difficulties then come from the slow decay of the norms, the quadratic power of the nonlinearity and some nonlocal terms in the nonlinearity (Riesz transforms) that prevent good localization of the small frequencies and convenient use of the only almost integrable norm ($L^\infty$).

The first two problems are classical in the study of quasilinear dispersive systems and several methods have been developed to address these difficulties, including normal form transformations [@Sha] and commuting vector fields [@Kla]. More recently refinements and new developments from Gustafson-Nakanishi-Tsai [@GNT2; @GNT] and Germain-Masmoudi-Shatah [@GerMas; @GerMasSha; @GerMasSha2] have led to progress in dealing with various physical problems. Our analysis is in the framework of this general scheme, 