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 ${\partial }_{t}u-\epsilon {\partial }_x^{2}u+\mathcal{H}{\partial }_x^{2}u+u{u}_x=0$, 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 ${\tilde{H}}^{\sigma }(\mathbb{R})$($\sigma \ge 0$).

References in corpus (2)

• Nonlinear wave interactions for the Benjamin-Ono equation
• Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations