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

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 x s H y ε ( T 3 ) , if s ⩾ 1 2 and ε > 0 .