One of the most difficult problems in mathematics and physics is to
find an accurate, practical description of turbulent flows. Turbulence
is ubiquitous in nature, occurring in very diverse physical settings,
such as aerodynamics, geophysics, weather and climate modeling, ocean
and atmospheric flows, star formation, blood flow in the heart, and
many others. This problem is not only untenable by current
mathematical tools, but direct numerical simulation of detailed
turbulent flows has proven to be computationally prohibitive, even
using the most powerful state-of-the-art computers. A major piece of
the puzzle of understanding these phenomena is widely believed to lie
in a system of nonlinear PDEs known as the Navier-Stokes equations,
which are the subject of one of the seven $1,000,000 Clay Millennium
Prize problems. I will discuss give an introduction to the
Navier-Stokes equations and discuss their relationship to turbulence
and the Millennium problem.