Well-posedness and scattering for the KP-II equation in a critical space

The Cauchy problem for the Kadomtsev–Petviashvili-II equation (ut+uxxx+uux)x+uyy=0 is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space is derived. Additionally, it is proved that for arbitrarily large initial data the Cauchy problem is locally well-posed in the homogeneous space and in the inhomogeneous space , respectively.