Energy conservation or anomalous dissipation for incompressible fluids

We provide an overview and propose elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes
equations in the class of Holder continuous functions. Our focus is on exploring the interplay between space and time regularity.

Additionally, we delve into the potential extension of these results to the Navier-Stokes equations in the presence of a solid boundary. Specifically, we consider the case of Dirichlet boundary conditions and our approach avoids any additional assumptions on the kinematic pressure.