Who comes up “unsolvable” problems like the Navier Stokes problem, and why is solving such unsolvable equations so important?
In: Mathematics
Sometimes when trying to determine how something moves, it’s easier to describe how motion changes over time than it is to describe the complete trajectory of how something moves.
This is the realm of differential equations.
How a leaf moves around in the wind is a good example. Who knows where it will end up. But if we know the direction of the wind at a bunch of points, then we can determine what direction the leaf will go for a short distance, and then determine what the direction of the wind is at the new location, and repeat this process.
The Navier Stokes Equation is one such scenario, in this case dealing with fluid flow. It takes into account the momentum of a fluid, the energy and pressure of the fluid, as well as the viscosity of the fluid, in all three directions, as well as the fact that fluid is incompressible. All this information can be used to determine what fluid will do in the next split second, which will provide information that can be used again to find the next split second, and so on. This gives approximate solutions that are usually sufficient for engineering purposes.
The key takeaway is that the answers we get from doing this apply to the specific conditions present (Energy, momentum, pressure, etc), and cannot be solved “generally”
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