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 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 $i{u}_t+\Delta u={\lambda }_1|u{|}^{p_1}u+{\lambda }_2|u{|}^{p_2}u$ 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 $i{u}_t+\Delta u=|u{|}^{p}u$, for a suitable range of the exponent $p$ extending the result of Colliander, Grillakis and Tzirakis [Commun. Pure Appl. Math. 62(2009)920-968].

30pages

References in corpus (1)

• Global well-posedness for Schrödinger equations with derivative