Remarks on the “Onsager Singularity Theorem” for Leray–Hopf Weak Solutions: The Hölder Continuous Case

Abstract

In this paper, we first present an overview of the results related to energy conservation in spaces of Hölder-continuous functions for weak solutions to the Euler and Navier–Stokes equations. We then consider families of weak solutions to the Navier–Stokes equations with Hölder-continuous velocities with norms uniformly bound in terms of viscosity. We finally provide the proofs of our original results that extend the range of allowed exponents for inviscid limits producing solutions to the Euler equations satisfying the energy equality, and improve the so-called “Onsager singularity” theorem.