Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space

Nobu Kishimoto; Minjie Shan; Yoshio Tsutsumi

doi:10.3934/dcds.2020078

Article Contents

2020, Volume 40, Issue 3: 1283-1307. Doi: 10.3934/dcds.2020078

This issue Previous Article Hausdorff dimension of a class of three-interval exchange maps Next Article Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space

1.
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
2.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
3.
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

Received: September 2018

Revised: August 2019

Published: December 2019

The first author N.K is partially supported by JSPS KAKENHI Grant-in-Aid for Young Researchers (B) (16K17626). The second author M.S is supported by China Postdoctoral Science Foundation grant 2019M650872. The third author Y.T is partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) (17H02853) and Grant-in-Aid for Exploratory Research (16K13770)

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Abstract

The global well-posedness for the KP-Ⅱ equation is established in the anisotropic Sobolev space $ H^{s, 0} $ for $ s>-\frac{3}{8} $. Even though conservation laws are invalid in the Sobolev space with negative index, we explore the asymptotic behavior of the solution by the aid of the $ I $-method in which Colliander, Keel, Staffilani, Takaoka, and Tao introduced a series of modified energy terms. Moreover, a-priori estimate of the solution leads to the existence of global attractor for the weakly damped, forced KP-Ⅱ equation in the weak topology of the Sobolev space when $ s>-\frac{1}{8} $.

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Mathematics Subject Classification: Primary: 35Q55, 35A01, 35B41; Secondary: 35B45.

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