On the weak solutions to the navier-Stokes equations: a possible gap related to the energy equality

TL;DR

This paper investigates the conditions under which weak solutions to the Navier-Stokes equations satisfy the energy equality, highlighting a potential gap in the case of weak regularity and confirming equality under higher regularity.

Contribution

It introduces a notion of suitable weak solutions as limits of mollified solutions and clarifies when the energy equality holds or may fail.

Findings

Energy inequality always holds for Leray-Hopf solutions.

Energy equality may fail for weakly regular solutions.

Higher regularity ensures the energy equality holds.

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.