## Well-posedness and scattering for the KP-II equation in a critical space.(English)Zbl 1169.35372

Summary: 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 $$\dot H ^{-\frac 1 2,0}(\mathbb R^2)$$ 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 $$\dot H ^{-\frac 1 2,0}(\mathbb R^2)$$, respectively.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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