Mathematical Research Letters

Volume 16 (2009)

Number 1

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

Pages: 111 – 120

DOI: http://dx.doi.org/10.4310/MRL.2009.v16.n1.a11

Author

Luc Molinet (Université Paris 13)

Abstract

We prove the ill-posedness in $ H^s(\T) $, $s<0$, of the periodic cubic Schrödinger 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 in \cite{CCT2} 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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