Notes on Navier–Stokes Regularity and Energy Conservation

This paper first proves the regularity criteria of the weak solution for the 3D Navier–Stokes equations involving the positive part of the intermediate eigenvalue of the strain matrix in Besov space. And then, it shows two energy conservation criterion of very weak solution via control on integrability of strain tensor. To our best knowledge, for three‐dimensional flow, it is the first result of energy conservation criterion by control on the integrability of strain tensor in the framework.