Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D

This paper is concerned with the Cauchy problem of $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness Sobolev space $H^s({\mathbb{R}}^2)$ for $s > -1/4$, and these are optimal up to endpoint. utilize nonlinear version classical Loomis-Whitney inequality develop an almost orthogonal decomposition set resonant frequencies. As a corollary, we obtain global $L^2({\mathbb{R}}^2)$.