On ill-posedness for the one-dimensional periodic cubic Schrodinger equation

Details On ill-posedness for the one-dimensional periodic cubic Schrodinger equation On ill-posedness for the one-dimensional periodic cubic Schrodinger equation

2008-06-27

arxiv.org/abs/0806.4538v2

Mathematics

Analysis of PDEs

To appear in Mathematical Research Letters

Scientific paper

We prove the ill-posedness in $ H^s(\T) $, $s<0$, of the periodic cubic Schr\"odinger equation in the sense that the flow-map is not continuous from $H^s(\T) $ into itself for any fixed $ t\neq 0 $. This result is slightly stronger than the one obtained by Christ-Colliander-Tao where the discontinuity of the solution map is established. Moreover our proof is different and clarifies the ill-posedness phenomena. Our approach relies on a new result on the behavior of the associated flow-map with respect to the weak topology of $ L^2(\T) $.

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