Global well-posedness for dissipative Korteweg-de Vries equations

TL;DR

This paper establishes the global well-posedness of dissipative KdV equations in certain Sobolev spaces by deriving optimal bilinear estimates in Bourgain's spaces, covering a range of dissipation parameters.

Contribution

It provides the first optimal bilinear estimates for dissipative KdV equations, leading to improved well-posedness results across different dissipation regimes.

Findings

Global well-posedness for $s > -3/4$ when $0<\alpha\leq 1/2$

Global well-posedness for $s > -3/(5-2\alpha)$ when $\alpha > 1/2$

Optimal bilinear estimates in Bourgain's type spaces

Abstract

This paper is devoted to the well-posedness for dissipative KdV equations $u_t+u_{xxx}+|D_x|^{2\alpha}u+uu_x=0$, $0<\alpha\leq 1$. An optimal bilinear estimate is obtained in Bourgain's type spaces, which provides global well-posedness in $H^s(\R)$, $s>-3/4$ for $\alpha\leq1/2$ and $s>-3/(5-2\alpha)$ for $\alpha>1/2$.