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

中文翻译：

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二维Zakharov-Kuznetsov方程Cauchy问题的整体适定性

本文涉及二维Zakharov-Kuznetsov方程的柯西问题。我们证明了双线性估计，这暗示了Sobolev空间中时间适定性的局部性$H^s（{\mathbb{[R}}^2）$ 对于 $s>-\mathrm{1个}/4$，这是最佳的端点。我们利用经典Loomis-Whitney不等式的非线性形式，并开发了谐振频率集的几乎正交分解。作为推论，我们获得了${\mathrm{大号}}^2（{\mathbb{[R}}^2）$。