The Navier-Stokes equations are differential equations in four variables: pressure (p) and the three components of velocity. They describe the flow of water, air and other fluids.
The first 3 equations, shown above, describe the time derivative of the velocity. In this notation, there is an implicit sum (Einstein convention) over indices that appear twice. The fourth equation says that the divergence of the spacial flow of the velocity is zero. This messes things up, because although the first 3 equations are local, meaning one can compute a solution at each grid point without needing to know the solution everywhere else, the fourth is not.
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