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

Abstract

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

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