# American Institute of Mathematical Sciences

May  2008, 7(3): 467-489. doi: 10.3934/cpaa.2008.7.467

## Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $R$

 1 University of Toronto, Toronto, Ontario, M5S 2E4 2 Mathematics Department, University of California, Evans Hall, Berkeley, CA 94720 3 Institute for Advanced Study and M.S.R.I., United States 4 Academy of Mathematics and System Sciences, Chinese Academy of Sciences and M.S.R.I., China

Received  May 2007 Revised  October 2007 Published  February 2008

We consider the Cauchy problem for a family of semilinear defocusing Schrödinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz estimate giving a priori $L_{t,x}^8$ spacetime control on solutions.
Citation: J. Colliander, Justin Holmer, Monica Visan, Xiaoyi Zhang. Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $R$. Communications on Pure & Applied Analysis, 2008, 7 (3) : 467-489. doi: 10.3934/cpaa.2008.7.467
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