Energy conservation for weak solutions of incompressible Newtonian fluid equations n H"older spaces with Dirichlet boundary conditions in the half-space

We investigate sufficient H\"older continuity conditions on Leray--Hopf (weak) solutions to the in unsteady Navier--Stokes equations in three dimensions guaranteeing energy conservation. Our focus is on the half-space case with homogeneous Dirichlet boundary conditions. This problem is more technically challenging, if compared to the Cauchy or periodic cases, and has not been previously addressed. At present are known a few sub-optimal results obtained through Morrey embedding results based on conditions for the gradient of the velocity in Sobolev spaces. Moreover, the results in this paper are obtained without any additional assumption neither on the pressure nor the flux of the velocity, near to the boundary.