Navier-Stokes Equations

The Navier-Stokes equations are partial differential equations that describe the motion of fluid substances, such as water or air. They apply Newton's second law of motion to fluid dynamics, relating the fluid's velocity, pressure, density, and temperature. In their most common form, they describe the motion of a viscous fluid and include terms for inertia, pressure gradients, viscous forces, and external body forces like gravity. For viscous, incompressible fluids, the change in velocity field determines the flow and pressure distribution, and complex turbulence can arise depending on the conditions. Due to the nonlinear nature of fluid motion, small differences in initial conditions can lead to vastly different outcomes, making solutions very difficult to obtain. These equations are used to analyze phenomena such as airflow around wings, weather and ocean currents, blood flow, and fluid dynamics in pipes. The problem of proving whether smooth solutions exist for the 3D incompressible Navier-Stokes equations is one of the Millennium Prize Problems and remains unsolved for the general case.