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

Abstract

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