Global well-posedness of the Cauchy problem for a fifth-order KP-I equation in anisotropic Sobolev spaces

TL;DR

This paper proves the local and global well-posedness of a fifth-order KP-I equation in anisotropic Sobolev spaces, extending previous results and broadening the understanding of its mathematical properties.

Contribution

It establishes new well-posedness results for the fifth-order KP-I equation in specific Sobolev spaces, improving upon earlier research in the field.

Findings

Local well-posedness in $H^{s_1, s_2}$ with $s_1 > -9/8$, $s_2 ext{ non-negative}$

Global well-posedness in $H^{s_1, 0}$ with $s_1 > -4/7$

Significant extension of previous mathematical results on KP-I equations

Abstract

In this paper, we consider the Cauchy problem for the fifth-order KP-I equation \begin{align*} u_t + \partial_x^5u+\partial_x^{-1}\partial_y^2u + \frac{1}{2}\partial_x(u^2)=0. \end{align*} Firstly, we establish the local well-posedness of the problem in the anisotropic Sobolev spaces $H^{s_1, s_2}(\mathbb{R}^2)$ with $s_1>-\frac{9}{8}$ and $s_2\geq 0$. Secondly, we establish the global well-posedness of the problem in $H^{s_1,0}(\mathbb{R}^2)$ with $s_1>-\frac{4}{7}$. Our result improves considerably the results of Saut and Tzvetkov (J. Math.\ Pures Appl.\ 79(2000), 307--338.) and Li and Xiao (J. Math.\ Pures Appl.\ 90(2008), 338--352.) and Guo, Huo and Fang (J. Diff.\ Eqns.\ 263 (2017), 5696--5726).