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

Abstract:

We prove that the initial value problem (IVP) associated to the fifth-order KdV equation \begin{equation*} \tag {0.1} \partial _tu-\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), \end{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 $\alpha \neq 0$, is locally well-posed in $H^s(\mathbb R)$ for $s \ge 2$. In the Hamiltonian case (i.e. when $c_1=c_2$), the IVP associated to (0.1) is then globally well-posed in the energy space $H^2(\mathbb R)$.