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

TL;DR

This paper establishes the well-posedness of the KdV-Burgers equation in the Sobolev space H^{-1} on the torus, while demonstrating ill-posedness in spaces with lower regularity, highlighting the limits of dissipation effects.

Contribution

It provides a precise threshold for well-posedness and ill-posedness of the KdV-Burgers equation in periodic Sobolev spaces, extending understanding of its regularity properties.

Findings

Well-posed in H^{-1} with analytic solution map

Ill-posed in H^s for s < -1, with discontinuous flow map

Dissipation does not improve the critical index beyond certain limits

Abstract

We prove that the KdV-Burgers is globally well-posed in $ H^{-1}(\T) $ with a solution-map that is analytic from $H^{-1}(\T) $ to $C([0,T];H^{-1}(\T))$ whereas it is ill-posed in $ H^s(\T) $, as soon as $ s<-1 $, in the sense that the flow-map $u_0\mapsto u(t) $ cannot be continuous from $ H^s(\T) $ to even ${\cal D}'(\T) $ at any fixed $ t>0 $ small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dissipation part of the KdV-Burgers equation allows to lower the $ C^\infty $ critical index with respect to the KdV equation, it does not permit to improve the $ C^0$ critical index .