  We consider the Euler system set on a bounded convex planar domain, endowed
with impermeability boundary conditions. This system is a model for the
barotropic mode of the Primitive Equations on a rectangular domain. We show the
existence of weak solutions with L^p vorticity for 4/3<= p <= 2, extending and
enriching a previous result of Taylor.
  In the physically interesting case of a rectangular domain, a similar result
holds for all 2<p<\infty as well. Moreover, we show the uniqueness of solutions
with bounded initial vorticity. The main tool is a new BMO regularity estimate
for the Dirichlet problem on domains with corners.
