Millennium Prize 1
Navier–Stokes Existence and Smoothness
The motion of a viscous incompressible fluid is described by the Navier–Stokes equations, first written down by Claude-Louis Navier in 1822 and given their modern form by George Gabriel Stokes. Whether smooth solutions to these equations can always be continued for all time (or whether they can spontaneously develop a singularity at some finite time) is one of the deepest open problems in mathematics, and one of the seven Clay Millennium Prize Problems, carrying a 1,000,000$ prize for a solution.