Introduction

In this chapter we show how the dimension of the global attractor ℕ can be estimated for the Navier-Stokes equations. The approach is an extension of that developed in Chapter 4 for ordinary differential equations where it was shown that if N-dimensional volume elements in the system phase space contract to zero, then the attractor dimension dL(ℕ) must be bounded by N. For partial differential equations the technical chore remains the same; namely, to derive estimates on the spectrum of the linearized evolution operator, linearized around solutions on the attractor, and to perform this operation in some function space instead of an a priori finite dimensional phase space. As we saw in Chapter 4 in the context of the Lorenz equations, this requires some knowledge of the location of the attractor, i.e., a priori estimates on the solutions. This approach is pursued in section 9.2 which deals with the 2d Navier-Stokes equations. It was shown in Chapter 7 that a global attractor si exists in this case, and we have good control of the solutions on the attractor. It turns out that the result for periodic boundary conditions is quite sharp, within logarithms of both the conventional heuristic estimate for the number of degrees of freedom in a 2d turbulent flow and rigorous lower bounds.

The 3d Navier-Stokes equations on a periodic domain are the concern of section 9.3. The lack of a regularity proof for this case results in some uncertainty concerning the very existence of a compact attractor. To achieve any formal estimate of the attractor dimension it is necessary to assume that H1 remains bounded for all t.

Introduction

The Navier-Stokes equations of fluid dynamics are a formulation of Newton's laws of motion for a continuous distribution of matter in the fluid state, characterized by an inability to support shear stresses. We will restrict our attention to the incompressible Navier-Stokes equations for a single component Newtonian fluid. Although they may be derived systematically from the microscopic description in terms of a Boltzmann equation, albeit with some additional fundamental assumptions, in this chapter we present a heuristic derivation designed to illustrate the elements of the physics contained in the equations.

Euler's equations for an incompressible fluid

First we consider an ideal inviscid fluid. The dependent variables in the so-called Eulerian description of fluid mechanics are the fluid density ρ(x, t), the velocity vector field u(x, t), and the pressure field ρ(x, t). Here x ∈ Rd is the spatial coordinate in a d-dimensional region of space (d typically takes values 2 or 3, with a default value of 3 in this chapter). An infinitesimal element of the fluid of volume δ V located at position x at time t has mass δm = ρ(x,t)δV and is moving with velocity u(x,t) and momentum δmu(x,t). The normal force directed into the infinitesimal volume across a face of area nda centered at x, where n is the outward directed unit vector normal to the face, is —np(x, t)δa. The pressure is the magnitude of the force per unit area, or normal stress, imposed on elements of the fluid from neighboring elements. These definitions are illustrated in Figure 1.1.

This book is not meant to be a review or a reference work, nor did we write it as a research monograph. It is not a text on fluid mechanics, and it is not an analysis course book. Rather, our goal is to outline one specific challenge that faces the next generation of applied mathematicians and mathematical physicists. The problem, which we believe is not widely appreciated in these communities, is that it is not at all certain whether one of the fundamental models of classical mechanics, of wide utility in engineering applications, is actually self-consistent.

The suspect model is embodied in the Navier-Stokes equations of incompressible fluid dynamics. These equations are nothing more than a continuum formulation of Newton's laws of motion for material “trying to get out of its own way.” They are a set of nonlinear partial differential equations which are thought to describe fluid motions for gases and liquids, from laminar to turbulent flows, on scales ranging from below a millimeter to astronomical lengths. Only for the simplest examples are they exactly soluble, though, usually corresponding to laminar flows. In many important applications, including turbulence, they must be modified and matched, truncated and closed, or otherwise approximated analytically or numerically in order to extract any predictions. On its own this is not a fundamental barrier, for a good approximation can sometimes be of equal or greater utility than a complicated exact result.

The issue is that it has never been shown that the Navier-Stokes equations, in three spatial dimensions, possess smooth solutions starting from arbitrary initial conditions, even very smooth, physically reasonable initial conditions. It is possible that the equations produce solutions which exhibit finite-time singularities.

Introduction

In this chapter we explore the problem of establishing the existence and uniqueness of solutions of the Navier-Stokes equations. The existence issue touches on the question of the self-consistency of the physical model embodied in the Navier-Stokes equations; if no solutions exist, then the theory is empty. The question of uniqueness relates to the predictive power of the model. In classical mechanics, uniqueness of the solutions of the equations of motion is the cornerstone of classical determinism. A breakdown of uniqueness signals the introduction of other effects, effects which are not contained in the dynamical equations, into the system's evolution. For the incompressible Navier-Stokes equations these are not trivial questions either mathematically or physically.

In the next section we review the standard theory of existence and uniqueness of solutions of ordinary differential equations (ODEs), stressing the importance of either a global Lipschitz condition, or a local Lipschitz condition along with a priori bounds, for global and/or local existence and for uniqueness. The subsequent section is concerned with constructing Galerkin approximations and the so-called “weak” solutions of the Navier-Stokes partial differential equations (PDEs). Existence questions are necessarily more involved for PDEs due to the fact that there may be a selection of function spaces available in which to look for solutions, each of which typically admits a variety of topologies, and hence a variety of notions of convergence. Without entering into the details of the functional analysis (which is not the aim of this book and for which a number of complete, authoritative texts already exist) we set out to explain the notion of weak solutions and to focus on the essential ingredients used to prove their existence.

Introduction

Turbulent motion in fluids is a familiar phenomenon from our everyday experience, but it is nevertheless an extremely difficult thing to define quantitatively. For the most part, the best that can be done to define turbulence is to list some of its characteristics: It is unsteady chaotic flow, apparently random, with fluid motions distributed over a relatively wide range of length and time scales. The complicated spatio-temporal structure of turbulent velocity fields renders their analytical description impossible, and the large number of degrees of freedom and the wide range of scales in turbulent flows result in difficult problems for numerical analysis, taxing both the speed and memory capacities of present day computers.

Statistical turbulence theory and the closure problem

Because of the effectively random behavior of turbulent flows, it is natural to attempt a statistical formulation. This is the classical approach to turbulence theory. The idea is to decompose a turbulent velocity vector field into a mean and a fluctuating part in an attempt to extract the relevant mean physical quantities. The “mean” in this approach may be a time average – appropriate for a steady configuration which, although fluctuating at all times, has well behaved time averaged characteristics – or an ensemble average where the average is over initial conditions in some class. To illustrate this decomposition we will consider only a steady state turbulent flow, assuming that well defined time averages of all quantities exist.

All the solutions considered so far have had very regular streamlines or particle paths but a glance at the weather or a smoke plume shows that spatial and temporal complexity arises in many common situations. In this chapter we will look at some situations where, in spite of this complexity, a mathematical model can be made of at least some aspects of the physical problem. We begin with the problem of flow in a porous medium where the spatial complexity can be averaged to give a smoothly varying macroscopic model.

Flow in a porous medium

Surprisingly, the Hele-Shaw model of Chapter 4 can be used as a model for the geometrically complicated problem of flow of a viscous fluid through a porous medium. Such a model is relevant in many practical situations such as oil recovery, hydrology, soil mechanics, filter design, and fluidized beds as well as numerous other phenomena in the earth sciences. The most basic model assumes that, besides there being an obvious microscopic flow scale defined by the ‘pore’ size and illustrated in figure 5.1, there is a much larger macroscopic scale over which the problem is to be studied; this macroscopic scale may be the overall dimensions of the industrial device or the oil field. We can make progress by working on an intermediate scale which is small compared to the macroscopic scale yet still contains enough pores for an averaged velocity u and pressure p to be defined.

In this chapter we consider relatively low Reynolds number flow of a thin film. Such a film may exist between two rigid walls, as in a bearing, or in a droplet, e.g. paint, spreading under gravity on a rigid surface. In either case the geometry of the problem allows us to simplify equation (3.2) in a way that is similar to the technique used to derive boundary layer theory in Chapter 2. The differences are that the order of magnitude of the width of the thin layer is dictated by the data of the problem and, since the layer is confined geometrically, there is no need to match with an outer flow.

Lubrication theory for slider bearings

The simple observation that a sheet of paper can slide across a smooth floor shows that a thin layer of fluid can support a relatively large normal load while offering very little resistance to tangential motion. More important mechanical examples occur in the lubrication of machinery and this motivates the study of slider bearings. A slider bearing consists of a thin layer of viscous fluid confined between nearly parallel walls that are in relative tangential motion.

A two-dimensional bearing is shown in figure 4.1 in which the plane y = 0 moves with constant velocity U in the x-direction and the top of the bearing (the slider) is fixed.