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.