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We consider a general family of regularized Navier-Stokes andMagnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannianmanifolds with or without boundary, with n greater than or equal to 2. Thisfamily captures most of the specific regularized models that have been proposedand analyzed in the literature, including the Navier-Stokes equations, theNavier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alphamodel, the Simplified Bardina model, the Navier-Stokes-Voight model, theNavier-Stokes-alpha-like models, and certain MHD models, in addition torepresenting a larger 3-parameter family of models not previously analyzed. Wegive a unified analysis of the entire three-parameter family using onlyabstract mapping properties of the principle dissipation and smoothingoperators, and then use specific parameterizations to obtain the sharpestresults. We first establish existence and regularity results, and underappropriate assumptions show uniqueness and stability. We then establishresults for singular perturbations, including the inviscid and alpha limits.Next we show existence of a global attractor for the general model, and giveestimates for its dimension. We finish by establishing some results ondetermining operators for subfamilies of dissipative and non-dissipativemodels. In addition to establishing a number of results for all models in thisgeneral family, the framework recovers most of the previous results onexistence, regularity, uniqueness, stability, attractor existence anddimension, and determining operators for well-known members of this family.

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