Viscous Fluids and the Navier—Stokes Equations

Abstract

In this chapter we apply, step-by-step, the following functional-analytical results:

• (i) Leray-Schauder principle (Theorem 6.A).

• (ii) Main theorem about pseudomonotone operators (Theorem 27.A).

• (iii) Main theorem about first-order evolution equations (Theorem 23.A).

• (iv) Implicit function theorem (Theorem 4.B).

• (v) Main theorem of analytic bifurcation theory (Theorem 8.A).

For nonlinear equations, such as the Navier—Stokes equations, it is known that a regular solution for the nonstationary problem need not exist for all times t ≥ 0. At some finite time the solution may go to infinity or loose its regularity. Even if the solution exists for all t ≥ 0, it need not converge towards the solution of the stationary problem as t → + ∞, when the boundary conditions and the forces converge towards a stationary situation.

Olga Aleksandrovna Ladyženskaja (1970)

For many years now, the Navier—Stokes equations have attracted the attention of engineers and mathematicians. The reason lies in the great number of interesting and difficult problems which are connected with them and which lead to important applications. Many of these problems are still unsolved. Beginning with the work of Jean Leray, during the 1930s, a number of deeper results were obtained for individual solutions of these equations. But from a physical point of view, the study of only individual solutions is not always justified. For large Reynolds numbers, i.e., roughly speaking, for large velocities or small viscosities, the flow becomes turbulent. Therefore, it is reasonable to look for a statistical description, analogously to the kinetic gas theory.

Mark losifovič Višik and Andrei Vladimirovič Fursikov (1980)

The distinguishing feature of a turbulent flow is that its velocity field appears to be random and varies unpredictably. The flow does, however, satisfy the Navier—Stokes differential equations, which are not random. This contrast is the source of much of what is interesting in turbulence theory….

If the Reynolds number Re is small, then the Navier—Stokes equations have a unique solution which can be observed in nature. When Re is large, even when a solution can be obtained, it is not observed in nature. The reason lies in the fact that the solutions are unstable; very small perturbations, too small to be measured by the experimenter, can be amplified and induce large changes in the flow. Note that this fact makes uniqueness theorems rather meaningless for a person trying to describe physical reality, since the possible uniqueness rests on an assumed uniqueness of the data, which cannot be assured in any meaningful sense….

The main problem of turbulence theory is to isolate statistical properties of solutions at the Navier—Stokes equations which are independent of the precise statistical properties of random data, if this is possible, and then use the knowledge thus acquired to construct reasonable predictive procedures for specific problems. One should keep in mind that a practical person is usually interested only in mean properties of a small number of functionals of the flow (e.g., lift and drag in the case of a flow past a wing), and these could conceivably be obtained even when the details of the flow are unknown….

Alexandre Joel Chorin (1975)

The recent improvement of our understanding of the nature of turbulence has three different roots. The first is the injection of new mathematical ideas from the theory of dynamical systems (strange attractors). The second is the availability of powerful computers which permit, among other things, experimental mathematics on dynamical systems and numerical simulation of hydrodynamic equations. The third is the improvement of experimental techniques, in particular, Doppler measurements of velocities by use of a laser beam, and then numerical Fourier analysis of the time series obtained….

Extensive computer studies of low-dimensional dynamical systems have shown that sensitive dependence on initial data is quite common, but mostly appears in systems for which we have no good mathematical theory.

David Ruelle(1983)