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 [1] to a new class of functional spaces C 0,α λ (Ω). It is shown that weak solutions u satisfy the energy equality provided that u ∈ L 3 ((0, T ); C 0,α λ (Ω)) with α ≥ 1 3 and λ > 0. The result is new for α = 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,[7], 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.Mathematics Subject Classification: 35Q31, 76B03.

Remarks on anomalous dissipation for passive scalars

“…In this derivation, both the (averaged) spectrum dissipation rate χ and ¯ are assumed independent of κ and ν, respectively. By Onsager's conjecture [45], which has been rigorously established now (we refer to [46][47][48] and references therein), the flow u can only be Anomalous scalar dissipation for fluid velocities with Hölder regularity was recently investigated in [24]. The authors gave sufficient conditions for anomalous dissipation in terms of the mixing rates of the advecting flow.…”

“…In this derivation, both the (averaged) spectrum dissipation rate χfalse¯ and ϵfalse¯ are assumed independent of κ and ν, respectively. By Onsager’s conjecture [45], which has been rigorously established now (we refer to [46–48] and references therein), the flow u can only be Hölder continuous of exponent less than 1/3 for anomalous energy dissipation to occur. The Obukhov–Corrsin theory also predicts the scaling of structure functions, if the fluid flow obeys Kolmogorov’s scaling of fully developed isotropic, homogeneous turbulence.…”