Scattering theory below energy space for two dimensional nonlinear Schrödinger equation

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The purpose of this paper is to illustrate the I-method by studying low-regularity solutions of the nonlinear Schr[o]dinger 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. Ecole Norm. Sup. 42(2009)261-290]), establish global well-posedness and scattering for low-regularity solutions of the equation iuₜ + Δu = λ₁|u|ᵖ¹ u + λ₂|u|ᵖ² u 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|ᵖ u, for a suitable range of the exponent p extending the result of Colliander, Grillakis and Tzirakis [Commun. Pure Appl. Math. 62(2009)920-968].