Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations

Abstract

We study the Cauchy problem for the dissipative Benjamin-Ono equations $u_t+{\stackrel{˝}{u}}_{xx}+|D{|}^{\alpha }u+u{u}_x=0$ with $0\le \alpha \le 2$. When $0\le \alpha <1$, we show the ill-posedness in $H^s(\mathbb{R})$, $s\in \mathbb{R}$, in the sense that the flow map $u_0\mapsto u$ (if it exists) fails to be $\C^2$ at the origin. For $1<\alpha \le 2$, we prove the global well-posedness in $H^s(\mathbb{R})$, $s>-\alpha /4$. It turns out that this index is optimal.

References in corpus (1)

• Nonlinear wave interactions for the Benjamin-Ono equation