Energy equality for the Navier-Stokes equations in weak-in-time Onsager spaces

TL;DR

This paper proves energy equality for weak solutions of the 3D Navier-Stokes equations in a broad class of function spaces with weak-in-time regularity, advancing understanding of Onsager's conjecture.

Contribution

It establishes energy equality under weaker conditions than previous results, using optimal space regularity and weak-in-time assumptions.

Findings

Energy equality holds for weak solutions in new function spaces.

Conditions are weaker than all prior classical results.

Heuristics suggest the results are potentially sharp.

Abstract

Onsager's conjecture for the 3D Navier-Stokes equations concerns the validity of energy equality of weak solutions with regards to their smoothness. In this note we establish energy equality for weak solutions in a large class of function spaces. These conditions are weak-in-time with optimal space regularity and therefore weaker than all previous classical results. Heuristics using intermittency argument suggests the possible sharpness of our results.