Onsager's Conjecture for the Incompressible Euler Equations in the H\"{o}log Spaces $C^{0,\alpha}_\lambda(\bar{\Omega})$

TL;DR

This paper extends Onsager's conjecture results for the incompressible Euler equations to new Hölog spaces, establishing energy equality for weak solutions with specific regularity, including the critical case.

Contribution

It introduces a novel class of Hölog spaces for Onsager's conjecture, proving energy equality for weak solutions with minimal regularity, including the critical exponent case.

Findings

Energy equality holds for solutions in $C^{0,rac{1}{3}}_ ext{log}$ spaces.

Extension of previous results to new Hölog functional spaces.

Proofs follow established methods, adapted to the new space setting.

Abstract

In this note we extend a 2018 result of Bardos and Titi \cite{BT} to a new class of functional spaces $C^{0,\alpha}_\lambda(\bar{\Omega})$. It is shown that weak solutions $\,u\,$ satisfy the energy equality provided that $u\in L^3((0,T);C^{0,\alpha}_\lambda(\bar{\Omega}))$ with $\alpha\geq\frac{1}{3}$ and $\lambda>0$. The result is new for $\,\alpha = \,\frac{1}{3}\,.$ Actually, a quite stronger result holds. For convenience we start by a similar extension of a 1994 result of Constantin, E, and Titi, \cite{CET}, in the space periodic case. The proofs follow step by step those of the above authors. For the readers convenience, and completeness, proofs are presented in a quite complete form.