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

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 Hölder 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 $\alpha \to 0^{+}$. Different Hölder 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).

arXiv comment38 pages

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Cited by in corpus (3)

• Four-thirds law of energy and magnetic helicity in electron and Hall magnetohydrodynamic fluids
• On Energy Conservation for the Hydrostatic Euler Equations: An Onsager Conjecture
• A probabilistic approach to drift estimation from stochastic data