In fluid mechanics, the Navier-Stokes equations are nonlinear partial differential equations that describe the flow of fluids. For example: they govern the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena.
Solution of flow equations by numerical methods is called computational fluid dynamics. There is hope that some problems of this equation can be solved with the help of solution method for flows of any macrostructure.
It is a famous open question whether smooth initial conditions always lead to smooth solutions for all times; a $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute for the answer to this question.