Global well-posedness of the generalized KP-II equation in anisotropic Sobolev spaces

TL;DR

This paper establishes local and global well-posedness results for the generalized KP-II equation in anisotropic Sobolev spaces, extending previous work and covering a broader range of the parameter .

Contribution

It proves local and global well-posedness of the generalized KP-II equation in anisotropic Sobolev spaces for  and , improving prior results for certain  values.

Findings

Local well-posedness in H^{s_1,s_2} for s_1>rac{1}{4}-rac{3}{8}\u0014, s_20.

Global well-posedness in H^{s_1,0} for -rac{(3-4)^2}{28}<s_10.

Extension of well-posedness results to broader parameter ranges  .

Abstract

In this paper, we consider the Cauchy problem for the generalized KP-II equation \begin{eqnarray*} u_{t}-|D_{x}|^{\alpha}u_{x}+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,\alpha\geq4. \end{eqnarray*} The goal of this paper is two-fold. Firstly, we prove that the problem is locally well-posed in anisotropic Sobolev spaces H^{s_{1},\>s_{2}}(\R^{2}) with s_{1}>\frac{1}{4}-\frac{3}{8}\alpha, s_{2}\geq 0 and \alpha\geq4. Secondly, we prove that the problem is globally well-posed in anisotropic Sobolev spaces H^{s_{1},\>0}(\R^{2}) with -\frac{(3\alpha-4)^{2}}{28\alpha}<s_{1}\leq0. and \alpha\geq4. Thus, our global well-posedness result improves the global well-posedness result of Hadac (Transaction of the American Mathematical Society, 360(2008), 6555-6572.) when 4\leq \alpha\leq6.