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

TL;DR

This paper establishes sharp well-posedness and ill-posedness results for the KdV-Burgers equation on the real line, showing well-posedness in $H^{-1}$ and ill-posedness below that Sobolev space, a novel finding for dispersive-dissipative equations.

Contribution

It provides the first sharp well-posedness and ill-posedness results for the KdV-Burgers equation, highlighting the critical Sobolev space at $H^{-1}$ and developing a framework applicable to similar models.

Findings

Well-posedness in $H^{-1}( )$ with an analytic solution map.

Ill-posedness for $H^s( )$ when $s < -1$, with discontinuous flow map.

First such results for a dispersive-dissipative PDE.

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 $ s<-1 $, in the sense that the flow-map $u_0\mapsto u(t) $ cannot be continuous from $ H^s(\R) $ to even ${\cal D}'(\R) $ 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 very useful to prove similar results for other dispersive-dissipative models