Differential and Integral Equations

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

Stéphane Vento

Abstract

We establish the local well posedness of the generalized Benjamin-Ono equation $\partial_tu+\mathcal{H}\partial_x^2u\pm u^k\partial_xu=0$ in $H^s({\mathbb{R}})$, $s > 1/2-1/k$ for $k\geq 12$ 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 $H^s({\mathbb{R}})$, $s < 1/3$.

Article information

Source
Differential Integral Equations, Volume 22, Number 5/6 (2009), 425-446.

Dates
First available in Project Euclid: 20 December 2012