Asymptotics of violent surface motion

The very fact that the motion of a free boundary is violent can be of considerable value in analysing the mathematical models of its evolution. For example the smallness of the angle between the undisturbed free surface of a liquid and an impacting smooth solid almost allows the problem to be formulated as a classical 'mixed boundary value' problem. This paper gives an informal review of several such configurations and indicates where the violence of the motion enables asymptotic analysis to give new insight, both qualitatively and quantitatively. However, many of these situations are in practice irreversible and we will explain why this can cause severe difficulties with the mathematics.