On the Cauchy-problem for generalized Kadomtsev-Petviashvili-II equations

TL;DR

This paper proves local well-posedness for the generalized Kadomtsev-Petviashvili-II equation in near-critical Sobolev spaces using advanced harmonic analysis techniques.

Contribution

It establishes local well-posedness results for a broad class of generalized KP-II equations in anisotropic Sobolev spaces, extending previous work.

Findings

Proves local well-posedness in almost critical spaces.

Utilizes bilinear Strichartz estimates and multilinear interpolation.

Combines smoothing and maximal function estimates effectively.

Abstract

The Cauchy-problem for the generalized Kadomtsev-Petviashvili-II equation $$u_t + u_{xxx} + \partial_x^{-1}u_{yy}= (u^l)_x, \quad l \ge 3,$$ is shown to be locally well-posed in almost critical anisotropic Sobolev spaces. The proof combines local smoothing and maximal function estimates as well as bilinear refinements of Strichartz type inequalities via multilinear interpolation in $X_{s,b}$-spaces.