Onsager's Conjecture for Subgrid Scale $alpha$-Models of Turbulence
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abstract
The first half of Onsager's conjecture states that the Euler equations of an ideal incompressible fluid conserve energy if $u (cdot ,t) in C^{0, heta} (mathbb{T}^3)$ with $ heta > 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 $alpha ightarrow 0^+$. 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).
