  In these notes we discuss the conservation of the energy for weak solutions
of the two-dimensional incompressible Euler equations. Weak solutions with
vorticity in $L^\infty_t L^p_x$ with $p\geq 3/2$ are always conservative, while
for less integrable vorticity the conservation of the energy may depend on the
approximation method used to construct the solution. Here we prove that the
canonical approximations introduced by DiPerna and Majda provide conservative
solutions when the initial vorticity is in the class $L(\log L)^\alpha$ with
$\alpha>1/2$.
