We've all heard of the
Navier-Stokes equations: they are the fundamental equations that describe any kind of fluid flow and are, in a sense, Newton's second law written specifically for fluids. The real charm of these equations is that they describe an
enormous number of physical phenomena, right from relatively simple flows in pipes to massive air currents in the atmosphere.
These equations also have another, more notorious, claim to fame : nobody has ever proven that solutions to these set of equations exist in 3 dimensions ( the real world ). Even if we assume that 3-d solutions exist, nobody has proven till now that the solution will be physically meaningful ( that they do not contain 'spikes' or 'jumps' ). Here we are, with a set of equations that we
know describe a large set of physical phenomena. We know the physical phenomena exist, but we haven't yet
proven in theory, that our equations make sense!
The Clay Mathematics Institute includes the proof of existence and smoothness of the Navier-Stokes equations as one of its
seven Millennium problems.
Terrence Tao - the Australian child prodigy who became a full professor at UCLA at the ripe age of 24 - has written an very detailed analysis of the problem from a purely mathematical point of view. Take a look:
Why global regularity for Navier-Stokes equation is hard
