On KP-II type equations on cylinders

TL;DR

This paper investigates the generalized KP-II equations on cylindrical domains, establishing bilinear estimates and proving local well-posedness using Bourgain space techniques, advancing understanding of dispersive PDEs on such geometries.

Contribution

It introduces bilinear Strichartz estimates independent of dispersion, and applies them to prove local well-posedness for generalized KP-II equations on cylinders.

Findings

Bilinear Strichartz estimates depend only on domain dimension.

Bilinear estimates for nonlinear terms are established.

Local well-posedness of the Cauchy problem is proved.

Abstract

In this article we study the generalized dispersion version of the Kadomtsev-Petviashvili II equation, on $\T \times \R$ and $\T \times \R^2$. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the domain but not on the dispersion. Their analogues in terms of Bourgain spaces are then used as the main tool for the proof of bilinear estimates of the nonlinear terms of the equation and consequently of local well-posedness for the Cauchy problem.