We explore how the Navier–Stokes equations extend Newton’s laws to viscous fluids, why nonlinear convective terms spawn turbulence, and the stubborn 3D math question at their core: do smooth solutions always exist or can they blow up? From airplanes and weather to blood flow, these equations underpin real-world physics and engineering—yet a million-dollar prize hinges on proving existence and smoothness (or finding a counterexample). A rigorous, accessible dive into one of mathematics’ most enduring mysteries.

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