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

Abstract

The Cauchy problem for the Kadomtsev–Petviashvili-II equation $(u_{t} + u_{xxx} + uu_{x})_{x} + u_{yy} = 0$ is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space H\limits^{˙}^{−\frac{1}{2},0}(\mathbb{R}^{2}) is derived. Additionally, it is proved that for arbitrarily large initial data the Cauchy problem is locally well-posed in the homogeneous space H\limits^{˙}^{−\frac{1}{2},0}(\mathbb{R}^{2}) and in the inhomogeneous space $H^{−\frac{1}{2},0}(\mathbb{R}^{2})$, respectively.