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 ${H}^{-1}\left(ℝ\right)$ with a solution-map that is analytic from ${H}^{-1}\left(ℝ\right)$ to $C\left(\left[0,T\right];{H}^{-1}\left(ℝ\right)\right)$ whereas it is ill-posed in ${H}^{s}\left(ℝ\right)$, as soon as $s<-1$, in the sense that the flow-map ${u}_{0}↦u\left(t\right)$ 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.

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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