Local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in the fully periodic case with initial data in Sobolev spaces $H^s$, $s>1$, is proved. Frequency dependent time localization is utilized to control the derivative nonlinearity. The new ingredient to improve on previous results is a nonlinear Loomis-Whitney-type inequality.

中文翻译：

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周期 Zakharov-Kuznetsov 方程的 Loomis-Whitney 型不等式和低正则适定性

证明了二维Zakharov-Kuznetsov方程在Sobolev空间$H^s$,$s>1$中初始数据完全周期情况下的局部适定性。利用频率相关的时间定位来控制导数非线性。改进先前结果的新要素是非线性 Loomis-Whitney 型不等式。