Bilinear estimates with applications to the generalized Benjamin-Ono-Burgers equations

Abstract

In this paper, we deal with the well-posedness issues of the generalized Benjamin-Ono-Burgers (gBOB) equations which are interpolated between the ordinary BOB equation and the KdV-Burgers equation with respect to the dispersive terms. We solve the initial-value problem (IVP) with data below $H ^{-1 /2}$, where $s = -1/2$ is the threshold for the well posedness of the Burgers equation. Our proof is based on the method by L. Molinet and F. Ribaud, which is analogous to that developed by J. Bourgain and C.E. Kenig, G. Ponce, and L. Vega. Interestingly, it is known that such a method cannot be applied to the Benjamin-Ono equation with initial data in $H ^s(\mathbb R)$, $s \in \mathbb R$.