Well-posedness for Fractional Growth-Dissipative Benjamin-Ono Equations
This paper is devoted to study the Cauchy problem for the fractional dissipative BO equations $u_t+\mathcal{H}u_{xx}-(D_x^α-D_x^β)u+uu_x=0$, $0< α< β$. When $1<β<2$, we prove GWP in $H^s(\mathbb{R})$, $s>-β/4$. For $β\geq 2$, we show GWP in $H^s(\mathbb{R})$, $s>\max\{3/2-β, \, -β/2\}$. 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 $s<-β/2$, and it fails to be $C^3$ in $H^s(\mathbb{R})$ when $s<\min\{3/2-β, \, -β/4\}$. When $0< β<1$, we show ill-posedness in $H^s(\mathbb{R})$, $s\in \mathbb{R}$. Finally, if $β>3/2$, we prove GWP in $H^s(\mathbb{T})$, $s>\max\{3/2-β, \, -β/2\}$, and we deduce lack of $C^2$ regularity in $H^s(\mathbb{T})$ when $s<-β/2$, in particular we get sharp results when $β\geq 3$.