  This note addresses the question of energy conservation for the 2D Euler
system with an $L^p$-control on vorticity. We provide a direct argument, based
on a mollification in physical space, to show that the energy of a weak
solution is conserved if $\omega = \nabla \times u \in L^{\frac32}$. An example
of a 2D field in the class $\omega \in L^{\frac32 - \epsilon}$ for any
$\epsilon>0$, and $u\in B^{1/3}_{3,\infty}$ (Onsager critical space) is
constructed with non-vanishing energy flux. This demonstrates sharpness of the
kinematic argument. Finally we prove that any solution to the Euler equation
produced via a vanishing viscosity limit from Navier-Stokes, with $\omega \in
L^p$, for $p>1$, conserves energy. This is an Onsager-supercritical condition
under which the energy is still conserved, pointing to a new mechanism of
energy balance restoration.
