The
Navier–Stokes existence and smoothness problem concerns the
mathematical properties of solutions to the
Navier–Stokes equations, one of the pillars of
fluid mechanics (such as with
turbulence). These equations describe the motion of a fluid (that is, a liquid or a gas) in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include
turbulence, which remains one of the greatest
unsolved problems in physics, despite its immense importance in science and engineering.