The lifespan of small data solutions in two dimensional capillary water   waves
  This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions. 
Mihaela Ifrim and Daniel Tataru
math.AP