Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 531-560.

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 () with a solution-map that is analytic from H -1 () to C([0,T];H -1 ()) whereas it is ill-posed in H s (), as soon as s<-1, in the sense that the flow-map u 0 u(t) cannot be continuous from H s () to even 𝒟 ' () 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.

Publié le :
Classification : 35E15, 35M11, 35Q53, 35Q60
Molinet, Luc 1 ; 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 Luc.Molinet@lmpt.univ-tours.fr
4 L.A.G.A., Institut Galilée Université Paris 13 93430 Villetaneuse, France vento@math.univ-paris13.fr
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     title = {Sharp ill-posedness and well-posedness results for the {KdV-Burgers} equation: the real line case},
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Molinet, Luc; Vento, Stéphane. Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 531-560. http://archive.numdam.org/item/ASNSP_2011_5_10_3_531_0/

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