  A class of semi-bounded solutions of the two-dimensional incompressible Euler
equations satisfying either periodic or Dirichlet boundary conditions is
examined. For smooth initial data, new blowup criteria in terms of the initial
concavity profile is presented and the effects that the boundary conditions
have on the global regularity of solutions is discussed. In particular, by
deriving a formula for a general solution along Lagrangian trajectories, we
describe how periodicity can prevent blow-up. This is as opposed to Dirichlet
boundary conditions which, as we will show, allow for the formation of
singularities in finite time. Lastly, regularity of solutions arising from
non-smooth initial data is briefly discussed.
