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

• Autores: Luc Molinet
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 16, Nº 1, 2009, págs. 111-120
• Idioma: inglés
• DOI: 10.4310/mrl.2009.v16.n1.a11
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• Resumen

  • 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 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) $.