  We examine the two-dimensional Euler equations including the local energy
(in)equality as a differential inclusion and show that the associated
relaxation essentially reduces to the known relaxation for the Euler equations
considered without local energy (im)balance. Concerning bounded solutions we
provide a sufficient criterion for a globally dissipative subsolution to induce
infinitely many globally dissipative solutions having the same initial data,
pressure and dissipation measure as the subsolution. The criterion can easily
be verified in the case of a flat vortex sheet giving rise to the
Kelvin-Helmholtz instability. As another application we show that there exists
initial data, for which associated globally dissipative solutions realize every
dissipation measure from an open set in
$\mathcal{C}^0(\mathbb{T}^2\times[0,T])$. In fact the set of such initial data
is dense in the space of solenoidal $L^2(\mathbb{T}^2;\mathbb{R}^2)$ vector
fields.
