The Euler equations describe the motion of an inviscid flow (that is, an ideal fluid that has no viscosity). They were derived by Euler more than 250 years ago and the simply express the balance laws for momentum and mass in differential form. If the fluid is assumed to be incompressible, they take the following form
uit + div (u ui) + pi = 0
div u = 0
where u= (u1, ... , un) is the velocity of the fluid and p the pressure.
The existence of solutions to this system of partial differential equations in 3 space dimension is still an unsolved question. The regularity of the solutions for the corresponding equations for a viscous fluid (the Navier-Stokes equations) is one of the celebrated millennium problems. However, there are several other important questions related to these equations and which are linked to fundamental questions in fluid dynamics.