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

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 ≥ 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-α model, the Leray-α model, the modified Leray-α model, the simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-α-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. This family of models has become particularly important in the development of mathematical and computational models of turbulence. 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 and the number of degrees of freedom in terms of a generalized Grashof number. We then establish some results on determining operators for the two distinct subfamilies of dissipative and non-dissipative models. We finish by deriving some new length-scale estimates in terms of the Reynolds number, which allows for recasting the Grashof number-based results into analogous statements involving the Reynolds number. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, and determining operators for the well-known specific members of this family of regularized Navier-Stokes and MHD models, the framework we develop also makes possible a number of new results for all models in the general family, including some new results for several of the well-studied models. Analyzing the more abstract generalized model allows for a simpler analysis that helps bring out the core common structure of the various regularized Navier-Stokes and magnetohydrodynamics models, and also helps clarify the common features of many of the existing and new results. To make the paper reasonably self-contained, we include supporting material on spaces involving time, Sobolev spaces, and Grönwall-type inequalities.

Singular Limits of Voigt Models in Fluid Dynamics

“…Unlike the 3D Navier-Stokes equations, existence and uniqueness of globally-defined smooth solutions can be rigorously proven for the 3D NSV equations, as well as the fact that the latter recovers the NSE in the limit as a certain length scale α goes to zero [26,33]. This property was already noted for other important regularized models such as the Navier-Stokes α-model and the Leray α-model [2,50].…”

“…We emphasize that all the results proven in this article are also valid in a more general setting, when Ω is a compact Riemannian manifold with or without boundary and in the presence of other boundary conditions. We refer the reader to [21,26] for more details. System (2.1)-(2.3) can be rewritten as an abstract evolution equation of the form…”

“…In recent years, many regularized equations have been proposed for the purpose of direct numerical simulations of turbulent incompressible flows modeled by the Navier-Stokes (NSE) equations [26] (for models of two-phase incompressible fluid flows, we refer the reader to [21] for further discussion and results). One such regularized model is the Navier-Stokes-Voigt (NSV) equations introduced by Oskolkov [41] as a model for the motion of a linear, viscoelastic, incompressible fluid.…”

Mathematical Results for Some α Models of Turbulence with Critical and Subcritical Regularizations

“…Existence and uniqueness of solutions of other modifications of the Navier-Stokes equations have been studied by Ladyzhenskaya [12] Lions [16], Málek et al [17]. Our task is to find the critical relation between the regularizations θ 1 and θ 2 (see We note that the α family considered here is a particular case of the general study in [10] where the results do not recover the critical case 2θ 1 + θ 2 = 1 2 . The Leray-α model with critical regularization is studied in [3].…”

“…Fractionnal order Laplace operator has been used in another α models of turbulence in [18,3,10]. Existence and uniqueness of solutions of other modifications of the Navier-Stokes equations have been studied by Ladyzhenskaya [12] Lions [16], Málek et al [17].…”