Singularity Formation and Instability in the Unsteady Inviscid and Viscous Prandtl Equations

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 the viscous Prandtl equations. We also solve the linearization of the inviscid Prandtl equations about a 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.