Energy conservation for weak solutions of incompressible fluid equations: the Hölder case and connections with Onsager's conjecture

In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Hölder 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 Hölder type assumptions, but without additional assumptions on the pressure.