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

## TL;DR

This paper establishes uniform local well-posedness and inviscid limit results for the Benjamin-Ono-Burgers equation in a refined Sobolev space, advancing understanding of its behavior for small initial data.

## Contribution

It proves uniform local well-posedness and inviscid limit behavior for the Benjamin-Ono-Burgers equation in a novel refined Sobolev space, handling low and high-frequency components.

## Key findings

- Uniform local well-posedness for small data in the refined Sobolev space
- Inviscid limit behavior in the same function space
- Handling of low-frequency scaling criticality

## Abstract

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 $\widetilde{H}^\sigma(\mathbb{R})$($\sigma\geq 0$).