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 $\widetilde{ H}^\sigma(\mathbb{ R})$($\sigma\geq 0$).
Categories: math.AP