A remark on ill-posedness issues for KdV and mKdV equations

Luc Molinet (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

In this brief article we give an elementary proof of the following alternative for the Cauchy problem associated with the KdV and the mKdV equations: Either there exists no $ T>0 $ such that the solution map $ u_0\mapsto u $ associated with mKdV is continuous at the origin from $ H^s(\R) $, $s<0 $, into $ {\mathcal D}'(]0,T[\times \R) $ or there exists no $ T>0 $ and no $ R>0 $ such that the solution map associated with KdV is continuous from the ball $ B(0,R) $ of $H^s(\R) $, $s<-1 $, into $ {\mathcal D}'(]0,T[\times \R) $.

Mots clés: KdV equation; mKdV equation; Cauchy problem; ill-posedness

Domaines: Equations aux dérivées partielles [math.AP]

Luc Molinet. A remark on ill-posedness issues for KdV and mKdV equations. 2010. ⟨hal-00473966v1⟩