Almost Sharp Global Well-Posedness for a class of Dissipative and   Dispersive Equations on R in Low Regularity Sobolev Spaces
  In this paper we obtain global well-posedness in low order Sobolev spaces of higher order KdV type equations with dissipation. The result is optimal in the sense that the flow-map is not twice continuously differentiable in rougher spaces. The solution is shown to be smooth for positive times. 
Mikael Signahl
math.AP