In this preliminary chapter we first outline some questions which will be treated in this book. Then we derive the Navier-Stokes equations. Though the derivation will not be used later, it is of interest to understand the underlying logical and physical assumptions, because the mathematical theory of the equations is not complete. There is no existence proof except for small time intervals. Thus it has been questioned whether the N-S equations really describe general flows. If one changes the stress tensor such that diffusion increases when the velocities become large, then existence can be shown. This change of the equations does not seem to be justified physically, however. For example, certain similarity laws valid for the Navier-Stokes equations are well-established experimentally, but the modified equations do not allow the corresponding similarity transformations. Possibly a lack of mathematical ingenuity is the reason for the missing existence proof, and the N-S equations are physically correct.

The N-S equations form a quasilinear differential system, and much of our understanding of such systems is gained through the study of linearized equations. These will, in general, have variable coefficients. By freezing the coefficients in such a problem, one obtains systems with constant coefficients. It is much easier to analyse the latter, as will be later shown in Chapter 2. However, the relation between variable-coefficient and constant-coefficient equations is not trivial. The fundamental ideas of linearization and localization are discussed in Section 1.3.