Global well-posedness for dissipative Korteweg-de Vries equations

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