I was discussing this with my friend the other night. I mentioned that proving existence of smooth solutions for the Navier-Stokes equations would win a $1,000,000 prize from the Clay Mathematics Institute, and also that turbulence is considered one of the unsolved problems of physics; as a matter of fact, more than one physicist has been quoted as stating something along the lines of expecting turbulence to be a trickier problem to solve than the mysteries of QED or relativity. My friend believes this is BS, stating that he can visualize the solution physically but is not mathematically skilled enough to state it in mathematical terms. For the record he claims that he was a child prodigy who had designed state-of-the-art avionics system at a very young age. I want to know what exactly about turbulence is considered so elusive? I have self-studied ODEs (and to maybe a lesser extent, PDEs) to a point of great proficiency (and I invite readers to test it), and though lately I have mostly been self-studying real analysis, abstract algebra, and quantum mechanics (particularly as relates to chemistry, I've studied the elements a lot as well for the purpose of an ambitious art project of mine), basically just whatever captures my fancy and obsessive drive during that day or week or so (though I have been trying to stick with certain topics at least long enough to gain some knowledge before jumping to another one so I won't be stuck at the start when I get back to it later) and I am about ready to dive in to the study of... I believe this is what you call continuum mechanics? I was very much displeased by the introductory classical mechanics course I aced at university and so I am very much wanting for some classical mechanics problems to tackle that are actually at my mathematical level. From what I matter, fluids in such problems are modeled as a continuum, as though there are no individual particles bouncing around within. But isn't that model flawed in the case of turbulence? I mean, given initial conditions in a continuum sense, one cannot guess, for example, the exact shape that smoke will spread and curl into after a given time such that the flow ceases to be laminar? Would one not have to know the state of the individual particles to be able to determine that? The motion of the particles, given a body of fluid at a given velocity and pressure and temperature and other relevant conditions, presented with a vacuum or another fluid at different conditions, well the rate at which it diffuses for instance can be modeled stochastically based on the average velocity of the particles, although, if you view the particles closely enough, you will find that there will randomly be great holes in the distribution on the boundary between the two fluids (as an analogy from an unrelated field think van der Waals forces which happen because of moments when the electron distribution is dense enough to attract nonpolar molecules together) and at such a moment a burst of Fluid A will not diffuse into Fluid B but pass as a wisp of Fluid A conditions surrounded by fluid with Fluid B conditions (although diffusion continues to happen through this wisp, and more wisps may trail out of it) So couldn't one model where and when such a wisp is most likely to pass through, how large it is likely to be, and when, taking into account the homogenization of the fluids due to diffusion, more wisps will trail out? The shape of the wisps would be based statistically given the motion of the particles in the wisping fluid, and well, more insight might be gained by using such a model? Please tell me the equations were originally derived taking the many particle model into account... Pardon if this is all incomprehensible or just plain wrong; I do not know all the proper terminology so I am just describing how it happens in my head, not sure about my friend's head though. Thank you for reading, and I hope to gain some insight from others who might know more than I do.