The well posedness of the dissipative Korteweg-de Vries equations with low regularity data

We study the Cauchy problem of a dissipative version of the KdV equation with rough initial data. By working in a Bourgain type space we prove the local and global well posedness results for Sobolev spaces of negative order, and the order number is lower than the well known value −34. In some sense this paper is intended to show how the Bourgain type space is applicable to the study of semilinear equations with a linear part which contain both dissipative and dispersive terms.