Well-posedness for Fractional Growth-Dissipative Benjamin-Ono Equations

This paper is devoted to study the Cauchy problem for the fractional dissipative BO equations $u_t+\mathcal{H}u_{xx}-(D_x^{\alpha}-D_x^{\beta})u+uu_x=0$, $0< \alpha < \beta$. When $1<\beta <2$, we prove GWP in $H^s(\mathbb{R})$, $s>-\beta/4$. For $\beta\geq 2$, we show GWP in $H^s(\mathbb{R})$, $s>\max\{3/2-\beta , \, -\beta/2\}$. We establish that our results are sharp in the sense that the flow map $u_0\mapsto u$ fails to be $C^2$ in $H^s(\mathbb{R})$, for $s<-\beta/2$, and it fails to be $C^3$ in $H^s(\mathbb{R})$ when $s<\min\{3/2-\beta , \, -\beta/4\}$. When $0< \beta<1$, we show ill-posedness in $H^s(\mathbb{R})$, $s\in \mathbb{R}$. Finally, if $\beta >3/2$, we prove GWP in $H^s(\mathbb{T})$, $s>\max\{3/2-\beta , \, -\beta/2\}$, and we deduce lack of $C^2$ regularity in $H^s(\mathbb{T})$ when $s<-\beta/2$, in particular we get sharp results when $\beta \geq 3$.