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

Takaoka, Hideo

Adv. Differ. Equ. 5, No. 10-12, 1421-1443 (2000).

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

Keywords:

Cauchy problem; local well-posedness; anisotropic homogeneous Sobolev spaces

Citations:

Zbl 0787.35086