Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity

Abstract

We establish the local well-posedness of the generalized Benjamin-Ono equation ${\partial }_{t}u+\mathcal{H}{\partial }_x^{2}u\pm u^k{\partial }_{x}u=0$ in $H^s(\mathbb{R})$, $s>1/2-1/k$ for $k\ge 12$ and without smallness assumption on the initial data. The condition $s>1/2-1/k$ is known to be sharp since the solution map $u_0\mapsto u$ is not of class ${\mathcal{C}}^{k+1}$ on $H^s(\mathbb{R})$ for $s<1/2-1/k$. On the other hand, in the particular case of the cubic Benjamin-Ono equation, we prove the ill-posedness in $H^s(\mathbb{R})$, $s<1/3$.