Can somebody explain: do singularities / blow-ups in solutions have any relation to physical phenomena in fluid dynamics or are they purely artifacts of how the N-S equations may not accurately describe what actually happens in the physical world?
i always found it fascinating how existence and uniqueness of solutions for the basic types of PDEs (Laplace, wave, heat...) follows from boundary conditions of just the right type intuition tells us, i.e. either value or derivative for Laplace (corresponding to fixing voltage or charge on the conductors), both value and derivative for wave (corresponding to initial position and velocity of the parts of the string, as we'd expect from classical mechanics), and also something about the solutions for the heat equation being unstable for negative times (which totally makes sense when you think of "diffusion" -- can't unmix it).