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

2009/11/27 by Molinet, Luc, Vento, Stéphane
#35M11 #35Q 53 #35Q60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.0911.5256

Abstract

We complete the known results on the local Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in H^-1(\R) with a solution-map that is analytic from H^-1(\R) to C([0,T];H^-1(\R)) whereas it is ill-posed in H^s(\R) , as soon as s0 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 very useful to prove similar results for other dispersive-dissipative models

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