  In this paper we prove a global regularity result for a quadratic quasilinear
model associated to the water waves system with surface tension and no gravity
in dimension two (the capillary waves system). The model we consider here
retains most of the difficulties of the full capillary water waves system,
including the delicate time-resonance structure and modified scattering. It is
slightly simpler, however, at the technical level and our goal here is to
present our method in this simplified situation. The full system is the subject
of a forthcoming paper.
