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.