Navier–Stokes existence and smoothness
Navier–Stokes existence and smoothness
!Flow visualization of a turbulent jet, made by [laser-induced fluorescence. The jet exhibits a wide range of length scales, an important characteristic of turbulent flows.](//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Falsecolorimageofthefarfieldofasubmergedturbulentjet.jpg/250px-Falsecolorimageofthefarfieldofasubmergedturbulentjet.jpg)
Navier–Stokes existence and smoothness
The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid 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.
Navier–Stokes existence and smoothness
Even more basic (and seemingly intuitive) properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have neither proved that smooth solutions always exist, nor found any counter-examples. This is called the Navier–Stokes existence and smoothness problem.
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