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

Abstract

This paper is devoted to study the Cauchy problem for the fractional dissipative BO equations $u_t+\mathcal{H}{u}_{xx}-(D_x^{\alpha }-D_x^{\beta })u+u{u}_x=0$, $0<\alpha <\beta$. When $1<\beta <2$, we prove GWP in $H^s(\mathbb{R})$, $s>-\beta /4$. For $\beta \ge 2$, 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 $\beta \ge 3$.

30 pages