Uniform local well-posedness and inviscid limit for the Benjamin-Ono-Burgers equation

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 ${\tilde{H}}^{\sigma }(\mathbb{R})$($\sigma \ge 0$), whose low-frequency part is scaling critical and high-frequency part is equal to Sobolev space $H^{\sigma }$($\sigma \ge 0$). Furthermore, we also obtain its inviscid limit behavior in ${\tilde{H}}^{\sigma }(\mathbb{R})$($\sigma \ge 0$).