Global well-posedness in L^2 for the periodic Benjamin-Ono equation

Details Global well-posedness in L^2 for the periodic Benjamin-Ono equation Global well-posedness in L^2 for the periodic Benjamin-Ono equation

2006-01-10

arxiv.org/abs/math/0601217v2

Mathematics

Analysis of PDEs

47 pages

Scientific paper

We prove that the Benjamin-Ono equation is globally well-posed in $ H^s(\T) $ for $ s\ge 0 $. Moreover we show that the associated flow-map is Lipschitz on every bounded set of $ {\dot H}^s(\T) $, $s\ge 0$, and even real-analytic in this space for small times. This result is sharp in the sense that the flow-map (if it can be defined and coincides with the standard flow-map on $ H^\infty(\T) $) cannot be of class $ C^{1+\alpha} $, $\alpha>0 $, from $ {\dot H}^s(\T) $ into $ {\dot H}^s(\T) $ as soon as $ s< 0 $.

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