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

Abstract

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