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

TL;DR

This paper establishes global well-posedness and scattering results for the KP-II equation in a critical Sobolev space, and also proves local well-posedness for large initial data in related spaces.

Contribution

It introduces new well-posedness results for the KP-II equation in a critical space, extending understanding of the equation's behavior for large and small initial data.

Findings

Global well-posedness and scattering for small data in rac{-1/2,0}{ ext{Sobolev space}}.

Local well-posedness for large data in the same space.

Well-posedness results in both homogeneous and inhomogeneous Sobolev spaces.

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 \dot H^{-1/2,0}(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^{-1/2,0}(R^2) and in the inhomogeneous space H^{-1/2,0}(R^2), respectively.