  We consider energy conservation in a two-dimensional incompressible and
inviscid flow through weak solutions of the filtered-Euler equations, which
describe a regularized Euler flow based on a spatial filtering. We show that
the energy dissipation rate for the filtered weak solution with vorticity in
$L^p$, $p > 3/2$ converges to zero in the limit of the filter parameter.
Although the energy defined in the whole space is not finite in general, we
formally extract a time-dependent part, which is well-defined for filtered
solutions, from the energy and define the energy dissipation rate as its
time-derivative. Moreover, the limit of the filtered weak solution is a weak
solution of the Euler equations and it satisfies a local energy balance in the
sense of distributions. For the case of $p = 3/2$, we find the same result as
$p > 3/2$ by assuming Onsager's critical condition for the family of the
filtered solutions.
