Well-posedness for the Kadomtsev-Petviashvili II equation

Abstract

We study the well-posedness for the Cauchy problem of the KP II equation. We prove the local well-posedness in the anisotropic Sobolev spaces $H_{x,y}^{-1/4+\epsilon,0}$ and in the anisotropic homogeneous Sobolev spaces $H_{x,y}^{-1/2+4\epsilon,0}\cap\dot{H}_{x,y}^{-1/2+\epsilon,0}$. The first result is an improvement of the result in $L^2$ obtained by J. Bourgain [2].