  In this paper, we study the Cauchy problem for the Benjamin-Ono-Burgers
equation $\partial_t u-\epsilon \partial_x^2 u+\mathcal{H}\partial_x^2u+u
u_x=0$, where $\mathcal{H}$ denotes the Hilbert transform. We obtain that it is
uniformly locally well-posed for small data in the refined Sobolev space
$\widetilde{H}^\sigma(\mathbb{R})$($\sigma\geq 0$), whose low-frequency part is
scaling critical and high-frequency part is equal to Sobolev space
$H^\sigma$($\sigma\geq 0$). Furthermore, we also obtain its inviscid limit
behavior in $\widetilde{H}^\sigma(\mathbb{R})$($\sigma\geq 0$).
