Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces

Wei Yan; Yimin Zhang; Yongsheng Li; Jinqiao Duan

doi:10.3934/dcds.2021097

Article Contents

2021, Volume 41, Issue 12: 5825-5849. Doi: 10.3934/dcds.2021097

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Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces

1.
School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China
2.
Center for Mathematical Sciences, Wuhan University of Technology, Wuhan, Hubei 430070, China
3.
School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China
4.
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

^* Corresponding author: Yimin Zhang
^* Corresponding author: Yimin Zhang

Received: January 31, 2021

Revised: April 30, 2021

Early access: July 2021

Published: December 2021

This work is supported by the NSFC under grant numbers 11771127, 11571118 and 11471330. The first author is also supported by the education department of Henan province under the grant number 21A110014. The third author is also supported by the Fundamental Research Funds for the Central Universities of China under the grant number 2017ZD094

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Abstract

We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation

$ \begin{align*} \partial_{x}\left(u_{t}-\beta\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-\gamma u = 0 \end{align*} $

in the anisotropic Sobolev spaces $ H^{s_{1},s_{2}}(\mathbb{R}^{2}) $. When $ \beta <0 $ and $ \gamma >0, $ we prove that the Cauchy problem is locally well-posed in $ H^{s_{1}, s_{2}}(\mathbb{R}^{2}) $ with $ s_{1}>-\frac{1}{2} $ and $ s_{2}\geq 0 $. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang(Transactions of the American Mathematical Society, 364(2012), 3395–3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in $ H^{s_{1},0}(\mathbb{R}^{2}) $ with $ s_{1}<-\frac{1}{2} $ in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not $ C^{3} $. When $ \beta <0,\gamma >0, $ by using the $ U^{p} $ and $ V^{p} $ spaces, we prove that the Cauchy problem is locally well-posed in $ H^{-\frac{1}{2},0}(\mathbb{R}^{2}) $.

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Mathematics Subject Classification: Primary: 35Q53;Secondary: 35B30.

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