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 (0.1) ∂ t u − ∂ x 5 u = c 1 ∂ x u ∂ x 2 u + c 2 ∂ x ( u ∂ x 2 u ) + c 3 ∂ x ( u 3 ) , \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 ∈ R x \in \mathbb R , t ∈ R t \in \mathbb R , u = u ( x , t ) u=u(x,t) is a real-valued function and α ,   c 1 ,   c 2 ,   c 3 \alpha , \ c_1, \ c_2, \ c_3 are real constants with α ≠ 0 \alpha eq 0 , is locally well-posed in H s ( R ) H^s(\mathbb R) for s ≥ 2 s \ge 2 . In the Hamiltonian case ( i.e. when c 1 = c 2 c_1=c_2 ), the IVP associated to (0.1) is then globally well-posed in the energy space H 2 ( R ) H^2(\mathbb R) .