Onsager's Conjecture for Subgrid Scale $\alpha$-Models of Turbulence

TL;DR

This paper proves an Onsager-type conjecture for various subgrid scale $\alpha$-models of turbulence, identifying the specific regularity thresholds needed for these models to conserve energy-like quantities, extending Onsager's classical result.

Contribution

It establishes energy conservation criteria for several $\alpha$-models of turbulence, revealing different regularity exponents than the classical $1/3$ for Euler equations.

Findings

Different H"older exponents ensure energy conservation in $\alpha$-models.

Models are smoother than Euler, allowing lower regularity thresholds.

Results contrast with the universal $1/3$ Onsager exponent for conservation laws.

Abstract

The first half of Onsager's conjecture states that the Euler equations of an ideal incompressible fluid conserve energy if $u (\cdot ,t) \in C^{0, \theta} (\mathbb{T}^3)$ with $\theta > \frac{1}{3}$. In this paper, we prove an analogue of Onsager's conjecture for several subgrid scale $\alpha$-models of turbulence. In particular we find the required H\"older regularity of the solutions that ensures the conservation of energy-like quantities (either the $H^1 (\mathbb{T}^3)$ or $L^2 (\mathbb{T}^3)$ norms) for these models. We establish such results for the Leray-$\alpha$ model, the Euler-$\alpha$ equations (also known as the inviscid Camassa-Holm equations or Lagrangian averaged Euler equations), the modified Leray-$\alpha$ model, the Clark-$\alpha$ model and finally the magnetohydrodynamic Leray-$\alpha$ model. In a sense, all these models are inviscid regularisations of the Euler equations; and formally converge to the Euler equations as the regularisation length scale $\alpha \rightarrow 0^+$. Different H\"older exponents, smaller than $1/3$, are found for the regularity of solutions of these models (they are also formulated in terms of Besov and Sobolev spaces) that guarantee the conservation of the corresponding energy-like quantity. This is expected due to the smoother nonlinearity compared to the Euler equations. These results form a contrast to the universality of the $1/3$ Onsager exponent found for general systems of conservation laws by (Gwiazda et al., 2018; Bardos et al., 2019).