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Onsager's conjecture for subgrid scale α-models of turbulence

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Boutros, DW 
Titi, ES 

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 H"older 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 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 Bardos et al. 2019.

Description

Keywords

Onsager?s conjecture, Energy conservation, Subgrid scale turbulence models

Journal Title

Physica D: Nonlinear Phenomena

Conference Name

Journal ISSN

0167-2789
1872-8022

Volume Title

Publisher

Elsevier BV
Sponsorship
Engineering and Physical Sciences Research Council (EP/K032208/1)
Engineering and Physical Sciences Research Council (EP/R014604/1)