The Navier-Stokes equations describe the motion of fluid substances such as liquids and gases. They are fundamental in understanding how fluids behave under various forces and conditions, making them essential in fields like engineering, meteorology, and oceanography. These equations are a set of nonlinear partial differential equations that express the conservation of momentum and mass in fluid flow. Solving the Navier-Stokes equations can be complex, and they are known for their role in the unsolved problem of existence and smoothness in three dimensions, which is a major topic in mathematical research.