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The Navier-Stokes equations and smoothness

The Navier-Stokes equations are a set of equations used to describe the motion of viscous fluids. They have been in routine use since the mid 1880s and are now applied to fields such as hydrodynamics, aerodynamics, weather modeling, and most recently video-game programming. (For precise details of the equations see Wikipedia)

Despite their everyday use in research fields, the equations are not fully understood - for example, solutions of the Navier-Stokes equations often featureTurbulenceplugin-autotooltip__plain plugin-autotooltip_bigTurbulence

Due to lack of understanding of precise underlying mechanisms, turbulent flow in liquids, gases and powders (etc) can't be exactly described.

This can be very significant when attempting to predict the behaviour of complex natural phenomena - weather systems for example. Or when making predictions about dynamic forces and frictions in turbulence around man-made tech such as aircraft, ships, turbines etc. etc..
, which is critically important to many real-world systems (and which itself is poorly understood)

In addition, for 3-D equations, it has never been proved or disproved that smooth solutions (i.e. turbulence free) always exist.

This is officially titled 'The Navier–Stokes existence and smoothness problem' and, since 2000, the Clay Mathematics Institute is offering a $1m prize for its solution.

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.

(ref.)

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