What type equation Navier-Stokes is: Elliptic, parabolic, or hyperbolic? Should it give always the same answer no matter what is the initial condition? How these statements could be proved?
The question assumes that there is a strict delineation between equations, but there isn't. On paper, of course, the Navier-Stokes equations have a parabolic character because there is a non-zero diffusion term. But, in reality, we say that equations are "hyperbolic" when we mean that they are advection dominated, and "parabolic" when they are diffusion dominated, and the Navier-Stokes equations can be either depending on whether your Reynolds number is large or small.
I have tried to make this point in lecture #26 at http://www.math.tamu.edu/~bangerth/videos.html in more detail.