On the weak solutions to the navier-Stokes equations: a possible gap related to the energy equality
Paolo Maremonti
TL;DR
The paper investigates whether Leray–Hopf weak solutions of the 3D Navier–Stokes equations satisfy energy equality. By constructing suitable weak solutions as limits of mollified problems and establishing strong gradient convergence for the mollified sequence, it proves a dichotomy: either energy equality holds almost everywhere in time, or a quantified energy gap arises, described via a finite family of time intervals and a limiting procedure as a parameter tends to 1. A refined 'special energy equality' is derived, which reduces to full energy balance when a certain integer indicator vanishes; otherwise, the gap is precisely captured by endpoint contributions from these intervals. These results connect the energy balance to regularity and provide a framework to assess potential non-uniqueness and dissipation phenomena in weak solutions.
Abstract
It is well known that a Leray-Hopf weak solution enjoys an energy inequality. Here, we investigate the energy equality related to a suitable weak solution to the Navier-Stokes initial boundary value problem. The term suitable is meant in the sense that for our goals we achieve a weak solution whose existence is based as limit of solutions to the mollified Navier-Stokes system. In the case of a weak regularity of the solution, our results justify the possible gap for the energy equality in terms of "kinetic energy". However, if there is a sufficient regularity, e.g., like the continuity of the L2-norm of the weak solution, then the energy equality holds.