  We consider the Cauchy problem of the fifth-order equation arising from the
Korteweg-de Vries (KdV) hierarchy u_t + u_{xxxxx} + c_1u_{x} u_{xx} + c_2u
u_{x} = 0 x,t \in \R
  We prove a priori bound of solutions for H^s(\R) with s >= 5/4 and the local
well-posedness for s >= 2. The method is a short time X^{s,b} space, which is
first developed by Ionescu-Kenig-Tataru in the context of the KP-I equation. In
addition, we use a weight on localized X^{s,b} structures to reduce the
contribution of high-low frequency interaction where the low frequency has
large modulation. As an immediate result from a conservation law, we have the
fifth-order equation in the KdV hierarchy is globally well-posed in the energy
space H^2.

Analysis of PDEs