The Navier–Stokes existence and smoothness problem asks whether smooth, globally defined solutions always exist for the three-dimensional Navier–Stokes equations, which describe the motion of fluids. Despite their immense practical use in science and engineering, turbulence remains one of the greatest unsolved problems in physics, and mathematicians have neither proven that smooth solutions always exist nor found a counter-example. In May 2000, the Clay Mathematics Institute designated this as one of its seven Millennium Prize Problems, offering a $1,000,000 prize for a solution. The specific challenge is to prove or provide a counter-example to the statement that, given an initial velocity field in three dimensions, there exists a smooth, globally defined vector velocity and scalar pressure field that solve the equations. These equations model the motion of viscous Newtonian fluids, such as liquids and non-rarefied gases, based on Newton's second law.