 Navier-Stokes existence and smoothness

One of the Millennium Prize Problems.

See also: existence and uniqueness of solutions of partial differential equations.

The Navier–Stokes existence and smoothness problem is a major unsolved problem in mathematics that deals with the mathematical framework of fluid dynamics. Specifically, it pertains to the behavior of solutions to the Navier–Stokes equations, which describe the motion of viscous fluid substances. The Navier–Stokes equations are a set of nonlinear partial differential equations that describe how the velocity field of a fluid evolves over time under various forces.