Remarks on the Energy Equality for the 3D Navier-Stokes Equations

In a recent paper, jointly with L. C. Berselli, we study the problem of energy conservation for solutions of the initial boundary value problem associated with the 3D Navier-Stokes equations with Dirichlet boundary conditions. While the energy equality is satisfied for strong solutions, the dissipation phenomenon is expected to be connected with the roughness of the solutions. A natural question is, then, which regularity is needed for a weak solution in order to conserve the energy. The importance of this issue was brought out in evidence by Onsager’s work.