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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.
The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing.
What does Navier-Stokes equation describe?
Navier-Stokes equation, in fluid mechanics, a partial differential equation that describes the flow of incompressible fluids. The equation is a generalization of the equation devised by Swiss mathematician Leonhard Euler in the 18th century to describe the flow of incompressible and frictionless fluids.
What are the assumptions of the Navier-Stokes equations?
The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum, in other words is not made up of discrete particles but rather a continuous substance.
The Navier-Stokes equations represent the conservation of momentum, while the continuity equation represents the conservation of mass.
Unsolved problems have been used to indicate a rare mathematical talent in fiction. The Navier-Stokes problem features in The Mathematician’s Shiva (2014), a book about a prestigious, deceased, fictional woman mathematician named Rachela Karnokovitch taking the proof to her grave in protest of academia.
Do Navier-Stokes solutions always have bounded energy?
Even 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.
Are there any solutions to the Navier-Stokes equations in two dimensions?
The Navier–Stokes problem in two dimensions was solved by the 1960s: there exist smooth and globally defined solutions. is sufficiently small then the statement is true: there are smooth and globally defined solutions to the Navier–Stokes equations.
Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of solenoidal (” divergence -free”) functions. For this flow of a homogeneous medium, density and viscosity are constants.