  We consider the 2D gravity water waves equation on an infinite domain. We
prove a local wellposedness result which allows interfaces with corners and
cusps as initial data and which is such that the time of existence of solutions
is uniform even as the gravity parameter $g \to 0$. For $g>0$, we prove an
improved blow up criterion for these singular solutions and we also prove an
existence result for $g = 0$. Moreover the energy estimate used to prove this
result is scaling invariant.
  As an application of this energy estimate, we then consider the water wave
equation with no gravity where the fluid domain is homeomorphic to the disc. We
prove a local wellposedness result which allows for interfaces with angled
crests and cusps as initial data and then by a rigidity argument, we show that
there exists initial interfaces with angled crests for which the energy blows
up in finite time, thereby proving the optimality of this local wellposedness
result. For smooth initial data, this local wellposedness result gives a longer
time of existence as compared to previous results when the initial velocity is
small and we also improve upon the blow up criterion.
