The three-dimensional Euler equations: Where do we stand?

The three-dimensional Euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists. While much has been discovered, the nature of solutions is still largely a mystery. This paper surveys some of the issues, such as singularity formation, that have cost so much effort in the last 25 years. In this light we review the Beale–Kato–Majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted. A different line of endeavour focuses on work concerning the pressure Hessian and how it may be used and modelled. The Euler equations are finally discussed in terms of their membership of a class of general Lagrangian evolution equations. Using Hamilton’s quaternions, these are reformulated in an elegant manner to describe the motion and rotation of fluid particles.