  We consider rotational initial data for the two-dimensional incompressible
Euler equations on an annulus. Using the convex integration framework, we show
that there exist infinitely many admissible weak solutions (i.e. such with
non-increasing energy) for such initial data. As a consequence, on bounded
domains there exist admissible weak solutions which are not dissipative in the
sense of P.-L. Lions, as opposed to the case without physical boundaries.
Moreover we show that admissible solutions are dissipative provided they are
H\"{o}lder continuous near the boundary of the domain.
