Onsager’s Conjecture for the Incompressible Euler Equations in the Hölog Spaces $C^{0,\alpha }_{\lambda }(\bar{\Omega })$

Abstract

In this note we extend a 2018 result of Bardos and Titi (Arch Ration Mech Anal 228(1):197–207, 2018) 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 $\alpha \ge \frac{1}{3}$ 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 and Titi (Commun Math Phys 165:207–209, 1994), 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.