Global well-posedness and inviscid limit for the generalized Benjamin–Ono–Burgers equation

ABSTRACT

This paper deals with the Cauchy problem for the generalized Benjamin–Ono–Burgers equation ${\mathrm{\partial }}_{t}u+\mathcal{H}{\mathrm{\partial }}_x^{2}u-\nu u_{xx}+{\mathrm{\partial }}_x\left(u^{k+1}/\right(k+1\left)\right)=0$, $k⩾4$, where $\mathcal{H}$ denotes Hilbert transform. We obtain its global well-posedness results in Besov Spaces if $k⩾4$ and the initial data in ${\dot{B}}_{2,1}^{s_k}$ are sufficiently small, where $s_k:=1/2-1/k$ corresponds to the critical scaling regularity index. Furthermore, we prove its global well-posedness and inviscid limit behavior in Sobolev spaces.

KEYWORDS:

• Generalized Benjamin–Ono–Burgers equation
• Cauchy problem
• inviscid limit behavior

2010 MSC:

• 35Q53
• 35Q55
• 35A01

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

L. Han is supported in part by the Fundamental Research Funds for the Central Universities [grant number 2018MS054], M. Chen is partially supported by China Postdoctoral Science Foundation [grant number 2019M650019].