For the month of October, Calculus Humor will dedicate Monday Math Mysteries to some of the infamous "Millennium Prize problems" that still have yet to be solved! For you physics buffs out there, this first one involves the Navier-Stokes equations, which are hugely important in the field of fluid mechanics. Enjoy!

The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, one of the pillars of fluid mechanics. These equations describe the motion of fluids in space. Solutions of the Navier–Stokes equations remain one of the greatest unsolved problems in physics, despite its immense importance in science and engineering. Though there have been found to be many applications of the Navier-Stokes Equations, theoretically there has been only small steps made in actually finding exact solutions. For example, for the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proven that smooth solutions always exist, or that if they do exist they have bounded kinetic energy.

Using multivariable calculus, linear algebra, fluid dynamics, and perhaps some good programming, can you make any headway into the problem? Remember, there's a $1,000,000 prize in store for the winner.

Background:Background:

The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, one of the pillars of fluid mechanics. These equations describe the motion of fluids in space. Solutions of the Navier–Stokes equations remain one of the greatest unsolved problems in physics, despite its immense importance in science and engineering. Though there have been found to be many applications of the Navier-Stokes Equations, theoretically there has been only small steps made in actually finding exact solutions. For example, for the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proven that smooth solutions always exist, or that if they do exist they have bounded kinetic energy.

**Your Task:**

The Clay Mathematics Institute has offered a US $1,000,000 cash prize to the first person who can provide a clear solution to the problem, which is formally stated below:**Prove or give a counter-example of the following statement:***In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.*Using multivariable calculus, linear algebra, fluid dynamics, and perhaps some good programming, can you make any headway into the problem? Remember, there's a $1,000,000 prize in store for the winner.