Global well-posedness of critical nonlinear Schrödinger equations below $L^2$

Yonggeun Cho; Gyeongha Hwang; Tohru Ozawa

doi:10.3934/dcds.2013.33.1389

Article Contents

2013, Volume 33, Issue 4: 1389-1405. Doi: 10.3934/dcds.2013.33.1389

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Global well-posedness of critical nonlinear Schrödinger equations below $L^2$

1.
Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756
2.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747
3.
Department of Applied Physics, Waseda University, Tokyo 169-8555

Received: April 30, 2011

Revised: July 31, 2012

Published: March 2013

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Abstract

The global well-posedness on the Cauchy problem of nonlinear Schrödinger equations (NLS) is studied for a class of critical nonlinearity below $L^2$ in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index $s_c$ is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 42B37.

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