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

Abstract

We study the Cauchy problem for the dissipative Benjamin--Ono equations $u_{t} + \mathcal{H} u_{xx} + \lvert D\rvert^{\alpha} u + uu_{x} = 0$ with $0 \leq \alpha \leq 2$. When $0 \leq \alpha < 1$, 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 $\mathcal{C}^{2}$ at the origin. For $1 < \alpha \leq 2$, we prove the global well-posedness in $H^{s}(\mathbb{R})$, $s > -\alpha/4$. It turns out that this index is optimal.