Bilinear space-time estimates for linearised KP-type equations on the three-dimensional torus with applications

TL;DR

This paper establishes bilinear space-time estimates for linearised KP-type equations on a 3D torus, leading to new well-posedness results for the periodic KP-II equation in anisotropic Sobolev spaces.

Contribution

It introduces bilinear estimates in Bourgain spaces for linearised KP equations on the 3D torus, enabling new well-posedness results for the periodic KP-II equation.

Findings

Proves bilinear estimates in Bourgain spaces for KP-type equations.

Establishes local and global well-posedness for periodic KP-II in specific Sobolev spaces.

Shows the KP-II problem is well-posed for data with minimal regularity in anisotropic Sobolev spaces.

Abstract

A bilinear estimate in terms of Bourgain spaces associated with a linearised Kadomtsev-Petviashvili-type equation on the three-dimensional torus is shown. As a consequence, time localized linear and bilinear space time estimates for this equation are obtained. Applications to the local and global well-posedness of dispersion generalised KP-II equations are discussed. Especially it is proved that the periodic boundary value problem for the original KP-II equation is locally well-posed for data in the anisotropic Sobolev spaces $H^s_xH^{\e}_y(\T^3)$, if $s \ge \frac12$ and $\e > 0$.