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

Molinet, Luc¹  ; Vento, Stéphane^2, 3, 4

¹ Laboratoire de Mathématiques et Physique Théorique Université François Rabelais Tours Fédération Denis Poisson-CNRS Parc Grandmont, 37200 Tours, France
² L.A.G.A., Institut Galilée Université Paris 13 93430 Villetaneuse, France
³ Laboratoire de Mathématiques et Physique Théorique Université François Rabelais Tours Fédération Denis Poisson-CNRS Parc Grandmont, 37200 Tours, France [e-mail]
⁴ L.A.G.A., Institut Galilée Université Paris 13 93430 Villetaneuse, France [e-mail]

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 531-560

Résumé

We complete the known results on the Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in $H^{-1}\left(ℝ\right)$ with a solution-map that is analytic from $H^{-1}\left(ℝ\right)$ to $C([0,T];H^{-1}\left(ℝ\right))$ whereas it is ill-posed in $H^s\left(ℝ\right)$, as soon as $s<-1$, in the sense that the flow-map $u_0\mapsto u\left(t\right)$ cannot be continuous from $H^s\left(ℝ\right)$ to even ${𝒟}^{\text{'}}\left(ℝ\right)$ 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 useful to prove similar results for other dispersive-dissipative models.

Détail

Publié le : 2018-06-21

Classification : 35E15, 35M11, 35Q53, 35Q60