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 with a solution-map that is analytic from to whereas it is ill-posed in , as soon as , in the sense that the flow-map cannot be continuous from to even at any fixed 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.
@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}, pages = {531--560}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {3}, year = {2011}, mrnumber = {2905378}, zbl = {1238.35136}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2011_5_10_3_531_0/} }
TY - JOUR AU - Molinet, Luc AU - Vento, Stéphane TI - Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2011 SP - 531 EP - 560 VL - 10 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2011_5_10_3_531_0/ LA - en ID - ASNSP_2011_5_10_3_531_0 ER -
%0 Journal Article %A Molinet, Luc %A Vento, Stéphane %T Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2011 %P 531-560 %V 10 %N 3 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2011_5_10_3_531_0/ %G en %F 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, Série 5, Tome 10 (2011) no. 3, pp. 531-560. http://www.numdam.org/item/ASNSP_2011_5_10_3_531_0/
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