Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case

Luc Molinet (1) , Stéphane Vento (2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 LAGA - Laboratoire Analyse, Géométrie et Applications

We complete the known results on the local Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in $ H^{-1}(\R) $ with a solution-map that is analytic from $H^{-1}(\R) $ to $C([0,T];H^{-1}(\R))$ whereas it is ill-posed in $ H^s(\R) $, as soon as $ s<-1 $, in the sense that the flow-map $u_0\mapsto u(t) $ cannot be continuous from $ H^s(\R) $ to even ${\cal D}'(\R) $ at any fixed $ t>0 $ small enough. As far as we know, this is the first result of this type for a dispersive-dissipative equation. The framework we develop here should be very useful to prove similar results for other dispersive-dissipative models

Mots clés: Cauchy problem; well-posedness; ill-posedness; KdV-Burgers equation

Domaines: Equations aux dérivées partielles [math.AP]

Luc Molinet, Stéphane Vento. Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case. 2009. ⟨hal-00436652v1⟩