<Tr> <Td> <Ul> <Li> </Li> <Li> </Li> <Li> </Li> </Ul> </Td> </Tr> <Ul> <Li> </Li> <Li> </Li> <Li> </Li> </Ul> <P> 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 . </P> <P> Even much more basic 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 not yet proved that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass . This is called the Navier--Stokes existence and smoothness problem . </P>

Existence and smoothness of the navier stokes equation
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