posted at 06:50
Author Name: Andy Kiersz
Navier-Stokes Millennium Prize Problems
The Navier-Stokes equations are a set of differential equations that describe how the speed of a fluid's flow will change based on forces coming from within the fluid like pressure and viscosity and on any external forces, like gravity, acting on the fluid. There are many natural situations, like the fluid flows modeled by the Navier-Stokes equations, in which it is easy to mathematically describe the rate at which some quantity is changing, but in which a description of the quantity itself is not immediately apparent. Solving a differential equation means starting with the equation, which describes how the quantity being modeled changes across time and space, and some initial conditions for the state of the system at the beginning of the simulation, and coming up with a formula for the quantity at any time and place. For some more complicated differential equations, like the Navier-Stokes equations, mathematicians don't know how to find solutions generally. If the Navier-Stokes equations are going to be solved either some previously unnoticed quantity that can control small-scale chaotic behavior needs to be discovered, or a radically new mathematical approach to solving differential equations needs to be invented. Tao's "Machine" is designed to increasingly quickly make smaller and smaller copies of itself, until the time between reproductions is essentially zero, causing the kind of breakdown of the system that, if shown for the actual Navier-Stokes equations, would provide the negative answer to the problem, showing that in some cases the equations are unsolvable. His "Averaged" version of the equations has many similarities to the actual equations. Most importantly, the "Averaged" equations have basically the same large-scale energy constraint as the actual equations.

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