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 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.Comment: 37 pages; references added, minor typos corrected, minor changes to revise for publicatio

Should NICE guidelines be universally accepted for the evaluation of stable coronary disease? A debate

The 2016 National Institute of Health and Care Excellence clinical guideline for the assessment and diagnosis of chest pain positions coronary computed tomography angiography as the first test for all stable chest pain patients without confirmed coronary artery disease and discards the previous emphasis on calculation of pre-test likelihood recommended in their 2012 edition of the guidelines. On the other hand, the American College of Cardiology Foundation/American Heart Association and the European Society of Cardiology guidelines continue to present the stress testing functional modalities as the tests of choice. The aim of this review is to present, in the form of a debate, the pros and cons of these paradigm changing recommendations, with an emphasis on literature review and projection of future needs, with conclusions to be drawn by the reader.Cardiolog