The Navier-Stokes equation is one of the 7 "millennium problems" announced by Clay Mathematics Institute (CMI). A million-dollar reward (each) will be awarded for an analytical solution to each problem.
There is no analytical solution to the Navier-Stokes equations in their most general form. [Until 2023]
The Navier-Stokes equations are nonlinear because the terms in the equations do not have a simple linear relationship with each other. This means that the equations cannot be solved using traditional linear techniques, and more advanced methods must be used instead.