Analysis of a General Family of Regularized Navier-Stokes and MHD Models

TL;DR

This paper provides a unified analysis of a broad family of regularized Navier-Stokes and MHD models, establishing fundamental mathematical properties and extending previous results to a larger class of models.

Contribution

It introduces a comprehensive framework that encompasses many existing models, offering new unified existence, regularity, and stability results, along with attractor and determining operator analyses.

Findings

Established existence and regularity for the entire model family

Proved uniqueness and stability under certain conditions

Derived estimates for the global attractor's dimension

Abstract

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. We give a unified analysis of the entire three-parameter family using only abstract mapping properties of the principle dissipation and smoothing operators, and then use specific parameterizations to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish results for singular perturbations, including the inviscid and alpha limits. Next we show existence of a global attractor for the general model, and give estimates for its dimension. We finish by establishing some results on determining operators for subfamilies of dissipative and non-dissipative models. In addition to establishing a number of results for all models in this general family, the framework recovers most of the previous results on existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operators for well-known members of this family.