Well-posedness for the fifth-order KdV equation in the energy space

We prove that the initial value problem (IVP) associated to the fifth order KdV equation {equation} \label{05KdV} \partial_tu-\alpha\partial^5_x u=c_1\partial_xu\partial_x^2u+c_2\partial_x(u\partial_x^2u)+c_3\partial_x(u^3), {equation} where $x\in \mathbb{R}$, $t\in \mathbb{R}$, $u=u(x,t)$ is a real-valued function and $\alpha ,c_1,c_2,c_3$ are real constants with $\alpha \mathrm{\ne }0$, is locally well-posed in $H^s(\mathbb{R})$ for $s\ge 2$. In the Hamiltonian case (\textit i.e. when $c_1=c_2$), the IVP associated to \eqref{05KdV} is then globally well-posed in the energy space $H^2(\mathbb{R})$.