  In this paper we give elementary proofs of energy conservation for weak
solutions to the Euler and Navier-Stokes equations in the class of H\"older
continuous functions, relaxing some of the assumptions on the time variable
(both integrability and regularity at initial time) and presenting them in a
unified way. Then, in the final section we prove (for the Navier-Stokes
equations) a result of energy conservation in presence of a solid boundary and
with Dirichlet boundary conditions. This result seems the first one -- in the
viscous case -- with H\"older type assumptions, but without additional
assumptions on the pressure.
