Analytical study of certain magnetohydrodynamic-α models

Abstract

In this paper we present an analytical study of a subgrid scale turbulence model of the threedimensional magnetohydrodynamic (MHD) equations, inspired by the Navier-Stokes-α (also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes-α model. Specifically, we show the global well-posedness and regularity of solutions of a certain MHD-α model (which is a particular case of the Lagrangian averaged magnetohydrodynamic-α model without enhancing the dissipation for the magnetic field). We also introduce other subgrid scale turbulence models, inspired by the Leray-α and the modified Leray-α models of turbulence. Finally, we discuss the relation of the MHD-α model to the MHD equations by proving a convergence theorem, that is, as the length scale α tends to zero, a subsequence of solutions of the MHD-α equations converges to a certain solution (a Leray-Hopf solution) of the three-dimensional MHD equations.

Global existence for two regularized MHD models in three space-dimension

“…In this case, Linshiz-Titi [12] have shown a global existence result in a threedimensional periodic box when ν > 0 and µ > 0, while Fan-Ozawa [8] have achieved the same result in the whole space R 2 for both (ν = 1, µ = 0) and (ν = 0, µ = 1).…”

“…As pointed out in [12] (see also the suggested bibliography), at the moment, there is no possibility to compute the turbulent behavior of fluids neither analytically nor via direct numerical simulation (this task is prohibitively expensive and disputable as well due to sensitivity of perturbation errors in the initial data). Hence, one can try to focus only on certain statistical features of the physical phenomenon through the employment of suitable models.…”

“…Linshiz-Titi [12] have suggested several models. For instance, filtering only the velocity field, one can consider the following model:…”

Global well‐posedness for two modified‐Leray‐α‐MHD models with partial viscous terms

“…The existence and uniqueness of weak solutions have been proved in [1]. When B ≡ 0, the above system has been studied in [2].…”

Existence and Convergence of an MHD Approximate Deconvolution Model

“…In the recent years, the topic has attracted the interest of many researchers and, especially for the study of existence, uniqueness, regularity, and estimates on the number of degrees of freedom, we recall [6,7,[10][11][12][13]16].…”