We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of non-CMC weak solutions using a combination of priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions and fixed-point techniques for the Hamiltonian constraint, Riesz-Schauder theory for the momentum constraint, together with a topological fixed-point argument for the coupled system. Although we present general existence results for non-CMC weak solutions when the rescaled background metric is in any of the three Yamabe classes, an important new feature of the results we present for the positive Yamabe class is the absence of the near-CMC assumption, if the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small, and if the energy density of matter is not identically zero. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant, giving what is apparently the first existence results for non-CMC solutions without the near-CMC assumption. Using a coupled topological fixed-point argument that avoids near-CMC conditions, we establish existence of coupled non-CMC weak solutions with (positive) conformal factor ϕ ∈ W s,p , where p ∈ (1,∞) and s(p) ∈ (1 + 3/p,∞). In the CMC case, the regularity can be reduced to p ∈ (1,∞) and s(p) ∈ (3/p, ∞) ∩ [1,∞). In the case of s = 2, we reproduce the CMC existence results of Choquet-Bruhat [10], and in the case p = 2, we reproduce the CMC existence results of Maxwell [33], but with a proof that goes through the same analysis framework that we use to obtain the non-CMC results. The non-CMC results on closed manifolds here extend the 1996 non-CMC result of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed in the case of the positive Yamabe class; (2) regularity is extended down to the maximum allowed by the background metric and the matter; and (3) the result holds for all three Yamabe classes. This last extension was also accomplished recently by Allen, Clausen and Isenberg, although their result is restricted to the near-CMC case and to smoother background metrics and data.

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.