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, Serie 5, Volume 10 (2011) no. 3, p. 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.

Published online : 2018-06-21
Classification:  35E15,  35M11,  35Q53,  35Q60
@article{ASNSP_2011_5_10_3_531_0,
     author = {Molinet, Luc and Vento, St\'ephane},
     title = {Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {3},
     year = {2011},
     pages = {531-560},
     zbl = {1238.35136},
     mrnumber = {2905378},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_3_531_0}
}
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, Serie 5, Volume 10 (2011) no. 3, pp. 531-560. http://www.numdam.org/item/ASNSP_2011_5_10_3_531_0/

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