Well-posedness of the Cauchy problem of a water wave equation with low regularity initial data

In this paper we investigate local well-posedness of the IVP of a class of fifth-order nonlinear dispersive equations including the equation of Benney modeling the interaction of short and long water waves. By introducing the modified Sobolev spaces H ( s , ω ) ( R ) and the modified Bourgain space X s , ω , b ( R 2 ) , and making bilinear estimates in the space X s , ω , b ( R 2 ) , we prove that the IVP of this class of equations is locally well-posed in H ( s , 1 4 ) ( R ) for any s > − 1 4 .