Well-posedness for equations of Benjamin-Ono type

Abstract

The Cauchy problem $u_t - |D|^{\alpha}u_x + uu_x=0$ in $(-T,T) \times \mathbb{R}$, $u(0)=u_0$, is studied for $1< \alpha <2$. Using suitable spaces of Bourgain type, local well-posedness for initial data $u_0 \in H^s(\mathbb{R}) \cap \dot{H}^{-\omega}(\mathbb{R})$ for any $s > -\tfrac{3}{4}(\alpha-1)$, $\omega:=1/\alpha-1/2$ is shown. This includes existence, uniqueness, persistence, and analytic dependence on the initial data. These results are sharp with respect to the low frequency condition in the sense that if $\omega<1/\alpha-1/2$, then the flow map is not $C^2$ due to the counterexamples previously known. By using a conservation law, these results are extended to global well-posedness in $H^s(\mathbb{R}) \cap \dot{H}^{-\omega}(\mathbb{R})$ for $s \geq 0$, $\omega=1/\alpha-1/2$, and real valued initial data.