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The Navier-Stokes equations capture in a few succinct terms one of the most ubiquitous features of the physical world: the flow of fluids. The equations, which date to the 1820's, are today used to model everything from ocean currents to turbulence in the wake of an airplane to the flow of blood in the heart.
While physicists consider the equations to be as reliable as a hammer, mathematicians eye them warily. To a mathematician, it means little that the equations appear to work. They want proof that the equations are unfailing: that no matter the fluid, and no matter how far into the future you forecast its flow, the mathematics of the equations will still hold. Such a guarantee has proved elusive. The first person (or team) to prove that the Navier-Stokes equations will always work — or to provide an example where they don’t — stands to win one of seven Millennium Prize Problems endowed by the Clay Mathematics Institute, along with the associated $1 million reward and many mathematicians have developed many ways of trying to solve the problem. (Im going to add the whole problem explained version the button below: )
To see how the equations can break down, first imagine the flow of an ocean current. Within it there may be a multitude of crosscurrents, with some parts moving in one direction at one speed and other areas moving in other directions at other speeds. These crosscurrents interact with one another in a continually evolving interplay of friction and water pressure that determines how the fluid flows.
Mathematicians model that interplay using a map that tells you the direction and magnitude of the current at every position in the fluid. This map, which is called a vector field, is a snapshot of the internal dynamics of a fluid. The Navier-Stokes equations take that snapshot and play it forward, telling you exactly what the vector field will look like at every subsequent moment in time.
A map of the wind functions much like a vector field: At every point, the wind has a particular direction and magnitude.
They describe fluid flows as reliably as Newton’s equations predict the future positions of the planets; physicists employ them all the time, and they’ve consistently matched experimental results.
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