Well-posedness for the Kadomtsev-Petviashvili II equation.(English)Zbl 0994.35108

Summary: 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 $$H_{x,y}^{-1/4 +\varepsilon,0}$$ and in the anisotropic homogeneous Sobolev spaces $$H_{x,y}^{-1/2+4 \varepsilon,0} \cap H_{x,y}^{-1/2+ \varepsilon,0}$$. The first result is an improvement of the result in $$L^2$$ obtained by J. Bourgain [Geom. Funct. Anal. 3, Mo. 4, 315-341 (1993; Zbl 0787.35086)].

MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 35B35 Stability in context of PDEs 35B60 Continuation and prolongation of solutions to PDEs

Zbl 0787.35086