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Well-Posedness of the Kadomtsev–Petviashvili-II in the Negative Sobolev Space with Respect to y Direction

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Abstract

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.

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Acknowledgements

The author would like to thank Professor Gustavo Ponce for his useful arguments. The author would like to thank the referees’ useful comments.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, People’s Republic of China

    Zhaohui Huo

  2. Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Zhaohui Huo

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Huo, Z. Well-Posedness of the Kadomtsev–Petviashvili-II in the Negative Sobolev Space with Respect to y Direction. J Geom Anal 34, 84 (2024). https://doi.org/10.1007/s12220-023-01533-1

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