In this paper, we first establish some new dyadic bilinear estimates of KP-II equation. Then following some ideas in [7], we consider the Cauchy problem of the 2-D KP-II equation in negative Sobolev space with respect to y direction

$$\begin{aligned} \partial _t u + \partial _{xxx}u+\partial _{x}^{-1}(\partial _{yy} )u +\partial _x (u^2) =0, \ (x,y,t) \in {\mathbb {R}}^3. \end{aligned}$$

It follows that the 2D-KP-II is locally well-posed in space \(H^{(s_0,s_1)}\) with \(s_0=0, s_1\ge -41/360\) ( or \(s_0\ge -1/3\) and \(s_1\ge -1/360\)) although \(s_1\ge -41/360\) is not critical case.

中文翻译：

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负 Sobolev 空间中 Kadomtsev-Petviashvili-II 相对于 y 方向的适定性

在本文中，我们首先建立了一些新的KP-II方程的二进双线性估计。然后按照[7]中的一些想法，我们考虑负 Sobolev 空间中关于 y 方向的二维 KP-II 方程的柯西问题

$$\begin{对齐} \partial _t u + \partial _{xxx}u+\partial _{x}^{-1}(\partial _{yy} )u +\partial _x (u^2) =0 , \ (x,y,t) \in {\mathbb {R}}^3。\end{对齐}$$

由此可见，2D-KP-II 在空间\(H^{(s_0,s_1)}\)中局部适定，其中\(s_0=0, s_1\ge -41/360\)（或\(s_0 \ge -1/3\)和\(s_1\ge -1/360\) ) 尽管\(s_1\ge -41/360\)不是关键情况。