Onsager’s conjecture for subgrid scale [formula omitted]-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(⋅,t)∈C0,θ(T3) with θ>13. In this paper, we prove an analogue of Onsager’s conjecture for several subgrid scale α-models of turbulence. In particular we find the required Hölder regularity of the solutions that ensures the conservation of energy-like quantities (either the H1(T3) or L2(T3) norms) for these models.We establish such results for the Leray-α model, the Euler-α equations (also known as the inviscid Camassa–Holm equations or Lagrangian averaged Euler equations), the modified Leray-α model, the Clark-α model and finally the magnetohydrodynamic Leray-α 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 α→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).