A Note On Determining Projections for
Non-Homogeneous Incompressible Fluids

In the following, we consider a viscous incompressible fluid in $\Omega\subset\mathbb{R}^{d}$, where $\Omega$ is an open bounded domain with Lipschitz continuous boundary, and where $d=2$ or $d=3$. Given the kinematic viscosity $\nu>0$, and the vector volume force function $f(x,t)$ for each $x\in\Omega$ and $t\in(0,\infty)$, the governing Navier-Stokes equations (NSE) for the fluid velocity vector $u=u(x,t)$ and the scalar pressure field $p=p(x,t)$ are given by:

One is also provided with initial conditions $u(0)=u_{0}$, as well as boundary conditions on $\partial\Omega\times(0,\infty)$. Our goal in this article is to examine some questions about a concept known as determining degrees of freedom in the flow described by (1.1)–(1.2). While the classical setting involves the assumption of a constant bulk viscosity $\nu>0$, our particular interest here is in the case of non-constant viscosity, representing either a fluid with viscosity $\nu(t)$ that changes over time (such as an oil that loses viscosity as it degrades), or a fluid with viscosity $\nu(x)$ varying spatially (as in the case of two-phase or multi-phase fluid models), or both, represented by a viscosity $\nu(x,t)$ that changes over space and time. We will assume that $\nu$ is everywhere positive as a function of time and/or space, and will also assume that it a priori satisfies some uniform pointwise upper and lower bounds, based on some underlying physical considerations. Although we do not consider dependence of the viscosity on fluid velocity in this work, we note that there has long been an active numerical simulation community that studies this case, and there is now also growing interest in the analysis of the Navier-Stokes equations with variable viscosity; cf. [20, 19, 7, 6, 18, 8].

The notion of determining modes for the NSE was first introduced in [10] as an attempt to identify and estimate the number of degrees of freedom in turbulent flows; a thorough discussion of the role of determining sets in turbulence theory can be found in [5]. This core idea later led to the study of Inertial Manifolds [11]. Estimates of the number of determining modes under various assumptions have been developed sine the early 1980’s; some examples include [9, 26]. The notion of determining nodes and related concepts were introduced in [12, 13], followed by determining volumes in [14, 24], and various estimates of their number in different modeling scenarios can be found in e.g. [25, 26]. A unified framework for modes, nodes, and volumes was presented in [2, 3], including the relationship to Inertial Manifolds. In [21, 23], we extended the results of [2, 3] to the more general setting of weak solutions lying in a suitably defined divergence-free solution space $V$ (see §2 below). In particular, we showed that if a projection operator $R_{N}\colon V\to V_{N}\subset L^{2}(\Omega)$ into a subset $V_{N}$ with $N=\dim(V_{N})<\infty$, satisfies an approximation inequality for $\gamma>0$ of the form,

then the operator $R_{N}$ is a determining projection for the system (1.1)–(1.2) in the sense of Definition 1.1 below, provided $N$ is large enough. Furthermore, in [21, 23], we also derived explicit bounds on the dimension $N$ which guarantees that $R_{N}$ is determining, and we gave explicit constructions of determining projections for both smooth and weak solutions using “rough” finite element quasi-interpolants. Our more recent article [22] generalized these results further, to a more general family of regularized NSE and MHD models that includes (1.1)–(1.2) as a special case. This area of research has continued to generate substantial activity; a survey through 2009 appears in our earlier article [22], and much more recent related activity includes [16, 34, 15] and the references therein, among many other related works that are too numerous to list here.

Bounds on the number of determining modes, nodes, and volumes are usually phrased in terms of a generalized Grashof number, defined for the two-dimensional NSE as:

where $\lambda_{1}$ is the smallest eigenvalue of the Stokes operator and $c_{\rho}=1/\sqrt{\lambda_{1}}$ is the related (best) Poincaré constant. Here, $F=\limsup_{t\rightarrow\infty}(\int_{\Omega}|f(x,t)|^{2})^{1/2}$ if $f\in L^{2}(\Omega)$ for almost every $t$, or $F=\limsup_{t\rightarrow\infty}\sqrt{\lambda_{1}}\|f\|_{H^{-1}(\Omega)}$ if $f\in H^{-1}(\Omega)$ for almost every $t$. Due to the failure of the Sobolev embedding $H^{1}\hookrightarrow C^{0}$ in dimensions 2 and 3, determining node analysis, which was based on point-wise interpolants of the velocity, was limited to $H^{2}$-regular solutions, although it was understood that determining modes and volume elements made sense under weaker conditions. To construct a general analysis framework for the case of weak (e.g., $H^{1}$-regular solutions) solutions to (1.1)–(1.2), in [23] we introduced the notions of determining projections and determining functionals, which we now define. (The standard spaces $H$, $V$, and $V^{\prime}$ for (1.1)–(1.2) are reviewed below in §2.)

The analysis of whether $R_{N}$ or $\{l_{i}\}_{i=1}^{N}$ are determining can be reduced to an analysis of the approximation properties of $R_{N}$. Note that in this construction, the basis $\{\phi_{i}\}_{i=1}^{N}$ need not span a subspace of the solution space $V$, so that the functions $\phi_{i}$ need not, for example, be divergence-free. Note that Definition 1.1 encompasses each of the notions of determining modes, nodes, and volumes by making particular choices for the sets of functions $\{\phi_{i}\}_{i=1}^{N}$ and $\{l_{i}\}_{i=1}^{N}$.

We briefly review some background material and notation, following the approach taken in our earlier articles [21, 23, 22], which in turn followed the notational conventions used in [4, 29, 32, 33]. To keep the discussion in this section as clear and concise as possible, we have placed some technical results that are repeatedly used throughout the paper in Appendix A.

Let $\Omega\subset\mathbb{R}^{d}$ denote an open bounded set. The embedding and other standard results we will need to rely on are known to hold for example if the domain $\Omega$ has a locally Lipschitz boundary, denoted as $\Omega\in\mathcal{C}^{0,1}$ (cf. [1]). For example, open bounded convex sets $\Omega\subset\mathbb{R}^{d}$ satisfy $\Omega\in\mathcal{C}^{0,1}$ (Corollary 1.2.2.3 in [17]), so that convex polyhedral domains are in $\mathcal{C}^{0,1}$. Let $H^{k}(\Omega)$ denote the usual Sobolev spaces $W^{k,2}(\Omega)$. Employing multi-index notation, the distributional partial derivative of order $|\alpha|$ is denoted $D^{\alpha}$, so that the (integer-order) norms and semi-norms in $H^{k}(\Omega)$ may be denoted

where $|\Omega|$ represents the measure of $\Omega$. Fractional order Sobolev spaces and norms may be defined for example through Fourier transform and extension theorems, or through interpolation. A fundamentally important subspace is the $k=1$ case of

for which the Poincaré Inequality holds. (See Lemma A.2 in Appendix A.) The spaces above extend naturally (cf. [32]) to product spaces of vector functions $u=(u_{1},u_{2},\ldots,u_{d})$, which are denoted with the same letters but in bold-face; for example, $\mathbf{H}^{k}_{0}(\Omega)=\left(H^{k}_{0}(\Omega)\right)^{d}$. The inner-products and norms in these product spaces are extended in the natural Euclidean way; the convention here will be to subscript these extended vector norms the same as the scalar case.

Define now the space $\mathcal{V}$ of divergence free $\mathbf{C}^{\infty}$ vectors with compact support as

Two subspaces of $\mathbf{L}^{2}(\Omega)$ and $\mathbf{H}^{1}_{0}(\Omega)$ are fundamental to the study of the NSE:

We use a fairly standard notation (cf. [32]) for inner-products and norms in $H$ and $V$ is:

Thanks to the Poincaré inequality, the $H^{1}$-semi-inner-product $[u,v]_{H^{1}(\Omega)}$ is an inner-product on $V$, and the $H^{1}$-semi-norm $|u|_{H^{1}(\Omega)}$ is a norm on $V$.

The NSE (1.1)–(1.2) with homogeneous Dirichlet (no-slip) boundary conditions are equivalent (cf. [32]) to the functional differential equation:

where the Stokes operator $A$ and bilinear map $B$ are defined as

where the operator $P$ is the Leray orthogonal projector, $P\colon H_{0}^{1}\to V$ and $P\colon L^{2}\to H$, respectively. Weak formulations of the NSE will use the bilinear Dirichlet form $a(\cdot,\cdot)$ and trilinear form $b(\cdot,\cdot,\cdot)$ as:

Again, thanks to the Poincaré inequality, the form $a(\cdot,\cdot)$ is actually an inner-product on V, and as noted above, the induced semi-norm $|\cdot|_{H^{1}(\Omega)}=a(\cdot,\cdot)^{1/2}$ is in fact a norm on V, equivalent to the $H^{1}$-norm. A priori bounds and various symmetries can be derived for the trilinear form $b(\cdot,\cdot,\cdot)$; the results of this type that we will need are collected together in Appendix A.

A general weak formulation of the NSE (1.1)–(1.2) can be written as (cf. [32, 33, 4]):

Here, the space $C_{w}([0,T];H)$ is the subspace of $L^{\infty}([0,T];H)$ of weakly continuous functions, and $<\cdot,\cdot>$ denotes the duality pairing between $V$ and $V^{\prime}$, where $H$ is the Riesz-identified pivot space in the Gelfand triple $V\subset H=H^{\prime}\subset V^{\prime}$. Note that since the Stokes operator can be uniquely extended to $A\colon V\to V^{\prime}$, and since it can be shown that $B\colon V\times V\to V^{\prime}$ (cf. [4, 33] for both results), the functional form (2.3) still makes sense for weak solutions, and the total operator represents a mapping $V\to V^{\prime}$.

Consider now two forcing functions $f,g\in L^{2}([0,\infty];V^{\prime})$ and corresponding weak solutions $u$ and $v$ to (2.3) in either the two- or three-dimensional case. Subtracting equations (2.3) for $u$ and $v$ yields an equation for the difference function $w=u-v$, namely

Since the residual of (2.6) lies in the dual space $V^{\prime}$, for almost every $t$, we can consider duality pairing of (2.6) with a function in $V$, and in particular with $w\in V$, which yields

Using the notation (2.1)–(2.2) going forward, it can be shown (cf. [32, 33]) that

in the sense of distributions. Lemma A.4 in Appendix A establishes the symmetry relation $b(w,u,w)=b(u,u,w)-b(v,v,w),\forall u,v,w\in V$, so the function $w=u-v$ must satisfy

Equation (2.8) will be the starting point for our analysis of determining projections below. In the introduction, we highlighted an approximation property (1.3) that we will assume that a determining projection satisfies, and we will give an explicit example of such a projection below from [21, 23]. A useful consequence of property (1.3) that was noted in [23] is the following.

Finally, we note key tools for establishing a number of a priori estimates in Sections 4 and 5 will be both classical and generalized Gronwall inequalities (cf. [9, 24]) which we include in Appendix A.

We now give an overview of the general framework for constructing determining projections for the NSE for both two and three spatial dimensions, represented by Theorem 3.1 below. To make use of the framework to establish bounds on the number of determining degrees of freedom for weak solutions, one must assume, or establish, a single a priori bound for solutions to the equations, (inequality (3.2) below) and also provide a projection operator that satisfies a single approximation inequality (inequality (2.9) above).

Our earlier results in [21, 23] are included as particular instances of this framework, and we include in Appendix B the well-known a priori bounds for constant viscosity in the $d=2$ and $d=3$ cases that were used in [21, 23, 22]. The remainder of the article then turns to the necessary a priori bounds for non-constant viscosity.

In this section, we develop the a priori estimates needed to apply the determining projection framework from §3 (Theorem 3.1) to the case of a time-varying viscosity function $\nu=\nu(t)$. The first is an $L^{2}$ estimate that is used to prove the second and third estimates, following the strategy for the case of constant viscosity (see Appendix B). The second estimate is what is needed for use with with Theorem 3.1 in different contexts. The time-varying viscosity is assumed to satisfy $\nu\in L^{1}(0,T)$ and obey the a priori pointwise bounds:

where

The first estimate gives a bound on the $L^{2}$-norm of a weak solution to (1.1)–(1.2).

The following estimate gives the a bound on the time-averaged $H^{1}$-semi-norm of a weak solution to (1.1)–(1.2).