We consider a family of dispersion generalized Benjamin-Ono equations (dgBO)
which are critical with respect to the L2 norm and interpolate between the
critical modified (BO) equation and the critical generalized Korteweg-de Vries
equation (gKdV). First, we prove local well-posedness in the energy space for
these equations, extending results by Kenig, Ponce and Vega concerning the
(gKdV) equations. Second, we address the blow up problem in the spirit of works
of Martel and Merle on the critical (gKdV) equation, by studying rigidity
properties of the (dgBO) flow in a neighborhood of solitons. We prove that when
the model is close to critical (gKdV), solutions of negative energy close to
solitons blow up in finite or infinite time in the energy space. The blow up
proof requires in particular extensions to (dgBO) of monotonicity results for
localized versions of L2 norms by pseudo-differential operator tools.Comment: Submitte
