Millennium Problem Status: The Navier-Stokes Equations

If you have trouble paying the mortgage, or want to treat yourself, in addition to trying your luck in the lottery, you can always take paper and pencil and try to solve one of the seven Millennium Prize Problems proposed by the Clay Mathematics Institute, each endowed with a million-dollar prize. Bad news, one of them has already been solved by the Russian Gregory Perelman, but he refused to collect the million dollars. One of the seven problems is the resolution of the Navier-Stokes equations. As is known, the Navier-Stokes equations governs the movement of fluids and is of paramount importance not only in engineering but in many other fields such as medicine.

The Navier-Stokes equations are a second-order, non-linear, and strongly coupled partial differential equations. The non-linearities of the equations make it extremely difficult to solve analytically, having only found some partial solutions for very simple problems. The way to tackle these equations is through numerical methods, either by finite element or finite difference methods. The problem is that, like any approximation, numerical methods do not provide exact solutions, but only close to reality, so that they may not fully capture elements of it. It may happen that the results of a simulation, as time increases enough, bear no resemblance to reality.

Turbulence is origin of all pains, and it is caused by the viscosity of the fluid. Although the problem has not yet been solved, there is some consensus in the scientific community that the solutions to the Navier-Stokes equations do not necessarily have to be unique or smooth (differentiable at least once), resulting in phenomena of breakage or explosion (vortices) even when starting from a completely laminar initial state and with limited kinetic energy.