Why are people interested in solving the Navier-Stokes equations if people can find an good approximate solution? Why are people interested in solving the Navier-Stokes equations if people can find a good approximate solution?
Also especially when people have supercomputers?
 A: The issue is that it's actually exceptionally hard to find a good approximation, even with supercomputers. Furthermore, any solution you compute with sufficient accuracy is typically only relevant for the precise initial conditions you specified. In reality, parameters vary and it is more interesting to solve for broader problems that incorporate this variability.
In addition, most problems where the Navier-Stokes equations apply include other multi-physical phenomena. For example: aeroelastic problems with compressible flow; combustion chemistry in flow fields; magnetohydrodynamics of confined plasmas.
In such a case, solving the Navier-Stokes equations gets you part of a solution. There's a whole lot more that goes on. If we could establish solvability of N-S, we could develop methods that were more efficient in solving these problems. In some cases, we do just this. For example, Large Eddy Simulation uses known properties of the N-S equations to compute fluid profiles. It does this by essentially filtering the N-S equations; to paint with a broad brush, filtering reduces the computational complexity and makes the equations easier to solve with sufficient accuracy.
