We use the method of characteristics to prove the short-time existence
of smooth solutions of the unsteady inviscid Prandtl equations, and present
a simple explicit solution that forms a singularity in finite time. We give
numerical and asymptotic solutions which indicate that this singularity
persists for nonmonotone solutions of the viscous Prandtl equations. We
also solve the linearization of the inviscid Prandtl equation about shear
flow. We show that the resulting problem is weakly, but not strongly,
well-posed, and that it has an unstable continuous spectrum when the shear
flow has a critical point, in contrast with the behavior of the linearized
Euler equations.