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

In this paper, we study the Cauchy problem for Benjamin-Ono-Burgers equation $${\partial _t}u - \epsilon \partial _x^2u + {\cal H}\partial u{u_x} = 0$$ , where $${\cal H}$$ denotes Hilbert transform operator. We obtain that it is uniformly locally well-posed small data in refined Sobolev space $${\tilde H^\sigma }(\mathbb{R})\,\,(\sigma \geqslant 0)$$ which a subspace of L2(ℝ). It worth noting low-frequency part }(\mathbb{R})$$ scaling critical, and thus necessary. The high-frequency equal to Hσ (ℝ) (σ ⩾ 0) reduces Furthermore, also its inviscid limit behavior 0).