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Description

We discuss the singularity formation and instability in the unsteady inviscid and viscous Prandtl equations. We present a simple explicit solution of the inviscid Prandtl equations that forms a singularity in finite time. The inclusion of viscosity does not apear to prevent this singularity formation. We have also explicitly solved the linearization of the inviscid Prandtl equations about a shear flow. The resulting equations are weakly, but not strongly, well-posed, and have a continuous spectrum which is unstable when the unperturbed shear flow profile has a critical point. Finally, we consider briefly the effect of viscosity on this instability. From our numerical solution of the viscous equations, we see a similar instability to the inviscid one.