The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluids. These equations govern everything in fluid dynamics from water and lava, to currents and the movement of an avalanche.
The Navier-Stokes consist of two main components: the conservation of mass equation and the conservation of momentum equation. The conservation of mass equation represents the principle of mass conservation and states that the rate of change of density within a fluid element is equal to the negative divergence of the velocity field. In simpler terms, a change in density is proportional to a fluid expanding or contracting.
The conservation of momentum equations is essentially Newton’s second law, F=ma. It couples the effects of pressure, viscous forces, and external forces.
Essentially Navier Stokes is a one to one correspondence with Newtons Laws for fluids.
Despite its complexity, the Navier-Stokes equations have been extensively studied and are crucial for understanding various phenomena in fluid dynamics. Yet one question remains unsolved.
One of the most significant unsolved problems in mathematics and physics is the existence and smoothness of solutions to the Navier-Stokes equations. The smoothness of a function means that if a small change in the input occurs, a small change in the output must occur as well. Right now, no such proof exists. That’s the reason why we can’t predict weather for more than a week or so. Our knowledge on the smoothness of Navier-Stokes isn’t studied well enough.
The Clay Mathematics Institute has listed the Navier-Stokes existence and smoothness problem as one of the seven Millennium Prize Problems, offering a prize of one million dollars for its solution.
