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

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 > 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 NavierStokes-α model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-αlike 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 of models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the α → 0 limit in α models. Next we show existence of a global attractor for the general model, and then give estimates for the dimension of the global attractor. We finish by establishing some results on determining operators for the two distinct subfamilies of dissipative and non-dissipative models. In addition to establishing a number of new technical results for all models in this general family, the framework we develop can recover most of the existing existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operator results for the well-known specific members of this family of regularized Navier-Stokes and MHD models. Analyzing the more abstract generalized model allows for a simpler analysis that helps clarify the core common features of the various specific models and results. To make the paper reasonably self-contained, we include supporting material on spaces involving time, Sobolev spaces, and Gronwall-type inequalities. Date: March 11, 2009.