  We study the Cauchy problem for the dissipative Benjamin-Ono equations
$u_t+\H u_{xx}+|D|^\alpha u+uu_x=0$ with $0\leq\alpha\leq 2$. When
$0\leq\alpha< 1$, we show the ill-posedness in $H^s(\R)$, $s\in\R$, in the
sense that the flow map $u_0\mapsto u$ (if it exists) fails to be $\C^2$ at the
origin. For $1<\alpha\leq 2$, we prove the global well-posedness in $H^s(\R)$,
$s>-\alpha/4$. It turns out that this index is optimal.
