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

  • Martin Hadac

    Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, Beringstraße 1, 53115 Bonn, Germany
  • Sebastian Herr

    Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, Beringstraße 1, 53115 Bonn, Germany
  • Herbert Koch

    Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, Beringstraße 1, 53115 Bonn, Germany

Abstract

The Cauchy problem for the Kadomtsev–Petviashvili-II equation (ut+uxxx+uux)x+uyy=0(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 H\limits^{˙}^{−\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 H\limits^{˙}^{−\frac{1}{2},0}(\mathbb{R}^{2}) and in the inhomogeneous space H12,0(R2)H^{−\frac{1}{2},0}(\mathbb{R}^{2}), respectively.

Cite this article

Martin Hadac, Sebastian Herr, Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, pp. 917–941

DOI 10.1016/J.ANIHPC.2008.04.002