Scattering theory below energy space for two-dimensional nonlinear Schrödinger equation

Abstract

The purpose of this paper is to illustrate the I-method by studying low-regularity solutions of the nonlinear Schrödinger equation in two space dimensions. By applying this method, together with the interaction Morawetz estimate, (see [J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Commun. Pure Appl. Math.62 (2009) 920–968; F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. École Norm. Sup.42 (2009) 261–290]), we establish global well-posedness and scattering for low-regularity solutions of the equation iut+ Δu = λ1|u|p1u + λ2|u|p2u under certain assumptions on parameters. This is the first result of this type for an equation which is not scale-invariant. In the first step, we establish global well-posedness and scattering for low regularity solutions of the equation iut+ Δu = |u|pu, for a suitable range of the exponent p extending the result of Colliander, Grillakis and Tzirakis [Commun. Pure Appl. Math.62 (2009) 920–968].