Sharp ill-posedness and well-posedness results for dissipative KdV equations on the real line

TL;DR

This paper investigates the well-posedness and ill-posedness of a generalized dissipative KdV equation on the real line, establishing sharp results in Sobolev spaces using advanced functional analysis techniques.

Contribution

It provides the first sharp global well-posedness and ill-posedness results for the dissipative KdV equations in specific Sobolev spaces, extending previous frameworks.

Findings

Sharp global well-posedness in $H^{-p/2}$

Sharp ill-posedness in $H^s$ for $s < -p/2$

Use of Besov-Bourgain spaces for bilinear estimates

Abstract

This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation \begin{equation*} \left\{\begin{array}{l} \partial_tu+\partial_x^3u+L_pu+u\partial_xu=0, u(0,\,x)=u_0(x). \end{array} \right. \end{equation*} where $L_p$ is a dissipative multiplicator operator. Using Besov-Bourgain Spaces, we establish a bilinear estimate and following the framework developed in Molinet, L. & Vento, S. (2011) we prove sharp global well-posedness in the Sobolev spaces $H^{-p/2}(I\!\!R)$ and sharp ill-posedness in $H^s(I\!\!R)$ when $s<-p/2$ with $p \geq 2$.