Global well-posedness for dissipative Korteweg-de Vries equations

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 $\alpha>1/2$.